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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Several kinds of oscillations in forced Liénard equations

Joël Blot1*, Souhila Boudjema2 and Philippe Cieutat3

Author Affiliations

1 Laboratoire SAMM EA 4543, Université Paris 1 Panthéon-Sorbonne, centre P.M.F., 90 rue de Tolbiac, Paris cedex 13, 75634, France

2 Département de Mathématiques, Faculté des Sciences, Université de Skikda, Skikda, Algérie

3 Laboratoire de Mathématiques de Versailles, UMR-CNRS 8100, Université Versailles-Saint-Quentin-en-Yvelines, 45 avenue des États-Unis, Versailles cedex, 78035, France

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Boundary Value Problems 2013, 2013:66  doi:10.1186/1687-2770-2013-66

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/66


Received:5 November 2012
Accepted:8 March 2013
Published:29 March 2013

© 2013 Blot et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Near an equilibrium we study the existence of asymptotically a.p. (almost periodic), asymptotically a.a. (almost automorphic), pseudo a.p., pseudo a.a., weighed pseudo a.p. and weighed pseudo a.a. solutions of Liénard differential equations in the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M1">View MathML</a>, where the forcing term possesses a similar nature, and where p is a parameter in a Banach space. We use a perturbation method around an equilibrium. We also study two special cases of the previous family of equations that are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M3">View MathML</a>.

MSC: 34C27, 34C99, 47J07.

Keywords:
asymptotically almost periodic functions; asymptotically almost automorphic functions; pseudo almost periodic functions; pseudo almost automorphic functions; weighted pseudo almost periodic functions; weighted pseudo almost automorphic functions; Liénard equations

1 Introduction

We consider the following family of forced Liénard equations:

where P is a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M6">View MathML</a> are two functions for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M8">View MathML</a> is a function.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M9">View MathML</a> is almost periodic in the Bohr sense (respectively almost automorphic), in [1], Theorem 3.1 (respectively Theorem 3.2), we have proven the existence of an almost periodic (respectively almost automorphic) solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M10">View MathML</a> near an equilibrium, by using the perturbation method in the setting of Nonlinear Functional Analysis.

In the present paper, we extend this result to the frameworks of asymptotically almost periodic, asymptotically almost automorphic, pseudo almost periodic, pseudo almost automorphic, weighted pseudo almost periodic and weighted pseudo almost automorphic functions.

We also consider two special cases of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>), which are

On the existence of such solutions for the two previous cases, we obtain similar results as the one obtained on (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>).

Martínez-Amores and Torres in [2], then Campos and Torres in [3] described the dynamics of equation (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M14">View MathML</a>) in the periodic case, namely the forcing term e is periodic. Then Cieutat extended these results to the almost periodic case in [4], Ait Dads, Cieutat and Lhachimi did to the pseudo almost periodic case in [5], and Cieutat, Fatajou and N’Guérékata did to the almost automorphic case in [6].

For almost periodic solutions of second-order differential systems, results on the differentiable dependence were established in [7] by Blot, Cieutat and Mawhin.

Our approach of the problem is to transform it into a nonlinear equation with parameters in Banach function spaces, and to apply the implicit function theorem of the differential calculus in Banach spaces. To realize our aim, we use the Nemytskii operators (also called superposition operators) and state some properties on these operators. Then we establish, for a linear differential equation in a Banach space, a result on the existence and uniqueness of the solutions described above. We also extend the well-know result on the almost periodicity of the derivative of an almost periodic function to the weighted pseudo almost periodic and weighted pseudo almost automorphic cases.

Now we give a brief description of the contents of the paper. In Section 2, we fix our notation and we recall some definitions. In Section 3, we establish some results on the differentiation of weighted pseudo almost periodic functions and weighted pseudo almost automorphic functions, then on a linear differential equation and on the Nemytskii operators. In Section 4, we state the main theorem (Theorem 4.1) and we give the proof of this theorem. In Section 5, we establish two corollaries (Corollary 5.1 and Corollary 5.2) of the two special cases of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>).

2 Notation

When X and Y are Banach spaces, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M16">View MathML</a> stands for the space of all bounded linear operators from X into Y, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M17">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M18">View MathML</a>, respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M19">View MathML</a>) stands for the space of continuous (respectively Fréchet continuously differentiable, respectively twice Fréchet continuously differentiable) functions from X into Y.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M20">View MathML</a> stands for the space of bounded continuous functions from ℝ into X. We also define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M22">View MathML</a>. Endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M23">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M24">View MathML</a>, respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M25">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M26">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M27">View MathML</a>, respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M28">View MathML</a>) is a Banach space.

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M29">View MathML</a> is called an almost periodic function (in the Bohr sense) when it satisfies the following criterion (due to Bochner): <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M30">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M20">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M32">View MathML</a> denotes the space of almost periodic functions from ℝ into X. Endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M32">View MathML</a> is a Banach space which is invariant by translation [8], that is to mean that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M35">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M36">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M37">View MathML</a>.

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M29">View MathML</a> is called an almost automorphic function (in the Bochner sense) when, for all real sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M39">View MathML</a>, there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M40">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M39">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M43">View MathML</a> exists in X, and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M45">View MathML</a> exists. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M46">View MathML</a> denotes the space of almost automorphic functions from ℝ into X. Endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M46">View MathML</a> is a Banach space which is invariant by translation [9].

We also consider the following other function spaces which one can find in [10]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M49">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M50">View MathML</a> the space of asymptotically almost periodic functions [8].

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M51">View MathML</a> the space of asymptotically almost automorphic functions [9].

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M52">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M53">View MathML</a> the space of pseudo almost periodic functions [11,12].

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M54">View MathML</a> the space of pseudo almost automorphic functions [13,14].

Endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M59">View MathML</a> are Banach spaces which are invariant by translation (cf. respectively [8,9,11] and [14]).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M60">View MathML</a> be the set of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M61">View MathML</a> which are positive and locally Lebesgue-integrable over ℝ. For a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M62">View MathML</a> and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M63">View MathML</a>, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M64">View MathML</a>.

We define the following spaces:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M65">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M66">View MathML</a>, where (H) is the following condition due to [15]:

(H) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M36">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M68">View MathML</a> and a bounded interval I such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M69">View MathML</a> a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M70">View MathML</a>,

which is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M72">View MathML</a> (cf. Remark 3.4 in [15] or Remark 3.1 in [16]).

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M73">View MathML</a>, we consider the following spaces:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M74">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M75">View MathML</a> the space of weighted pseudo almost periodic functions [17,18].

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M76">View MathML</a> the space of weighted pseudo almost automorphic functions [19].

Let μ be a positive measure on ( is a Lebesgue σ-field on ℝ) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M80">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M81">View MathML</a>. A function u is called μ-pseudo almost periodic (respectively μ-pseudo almost automorphic) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M82">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M83">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M84">View MathML</a>) and ϕ is a function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M85">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M86">View MathML</a> is the measure of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M87">View MathML</a>. The set of such functions is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M88">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M89">View MathML</a>) cf.[15] (respectively [16]). When the measure μ satisfies the following condition:

(C) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M36">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M91">View MathML</a> and a bounded interval I such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M92">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M93">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M94">View MathML</a>,

then the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M88">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M89">View MathML</a>) is a Banach space which is invariant by translation, cf. Corollary 2.31 and Theorem 3.3 in [15] (respectively Theorem 4.9 and Theorem 3.5 in [16]).

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M97">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M98">View MathML</a>) is a special case of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M88">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M89">View MathML</a>) in the following sense: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M101">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M102">View MathML</a>), where the measure μ is absolutely continuous with respect to the Lebesgue measure and its Radon-Nikodym derivative is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M103">View MathML</a>. The function ρ satisfies hypothesis (H) if and only if the measure μ satisfies condition (C), cf. Remark 3.4 in [15] (respectively Remark 3.1 in [16]). Consequently, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M104">View MathML</a>, the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M97">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M98">View MathML</a> are Banach spaces which are invariant by translation.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107">View MathML</a> denotes one the following spaces: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M112">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M113">View MathML</a> endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M114">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M115">View MathML</a>, and when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107">View MathML</a> is a Banach space which is invariant by translation. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M118">View MathML</a> denotes the space of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M119">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M120">View MathML</a>. Endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M122">View MathML</a> is a Banach space. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M123">View MathML</a> denotes the space of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M124">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M125">View MathML</a>. Endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M123">View MathML</a> is a Banach space.

3 Preliminary results

For the proof of the main result, we need the following lemmas.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M129">View MathML</a>. If the derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M130">View MathML</a>is uniformly continuous, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M131">View MathML</a>, thus<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M132">View MathML</a>.

Remark 3.2 Lemma 3.1 is well know for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M46">View MathML</a>. In the scalar case, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M58">View MathML</a>, this result is proved in [12], Corollary 5.6, p.59.

Proof Consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M136">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M137">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M138">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107">View MathML</a> is a translation invariant vectorial space, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M140">View MathML</a>. The equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M141">View MathML</a> shows that the uniform continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M142">View MathML</a> implies that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M143">View MathML</a> with values in the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107">View MathML</a> converges uniformly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M142">View MathML</a> on ℝ. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M146">View MathML</a>, and from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M118">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M148">View MathML</a>. □

Lemma 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M150">View MathML</a>. If the spectrum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M151">View MathML</a>ofAdoes not intersect the imaginary axis, then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M152">View MathML</a>, there exists a unique solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107">View MathML</a>of the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M154">View MathML</a>

(3.1)

Moreover, the solutionuis in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M118">View MathML</a>.

Proof Applying Theorem 4.1, p.81 in [20] (or Theorem 4 in [21]), Equation (3.1) admits a unique bounded solution on ℝ which is given by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M156">View MathML</a>

(3.2)

where G is the principal Green function for Equation (3.1). The Green function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M157">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M158">View MathML</a>, and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M159">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M160">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M161">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M162">View MathML</a>.

Now we prove that the bounded solution u defined by (3.2) belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M164">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M165">View MathML</a>), this result is a straightforward consequence of Theorem 3.8 in [15] (respectively Theorem 3.9 in [16]) and when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M166">View MathML</a>, this result is proved in [21], Proposition 3. Then we deduce the result for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M167">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M168">View MathML</a>).

Note that the case of the pseudo almost periodic (respectively pseudo almost automorphic) functions is a special case of the weighted pseudo almost periodic (respectively weighted pseudo almost automorphic) functions by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M169">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M42">View MathML</a>; remark that the associated measure is exactly the Lebesgue measure.

And so it suffices to prove the cases of weighted pseudo almost periodic functions and of weighted pseudo almost automorphic functions.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M171">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M172">View MathML</a>), this result is a straightforward consequence of Theorem 3.8 in [15] (respectively Theorem 3.9 in [16]). Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M173">View MathML</a>, and from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107">View MathML</a>, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M175">View MathML</a>. Since u satisfies Equation (3.1), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M131">View MathML</a>, and from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M118">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M132">View MathML</a>. □

Lemma 3.4LetXandYbe two finite-dimensional Banach spaces, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M179">View MathML</a>be a continuous mapping. Then the Nemytskii operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M180">View MathML</a>, defined by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M181">View MathML</a>, is continuous.

Remark 3.5 Contrary to the asymptotically almost periodic case and, in particular, for the almost periodic case, when the dimension of the Banach spaces X and Y is infinite, Lemma 3.4 does not hold for the pseudo almost periodic case, and thus for the weighted pseudo almost periodic case, without additional assumptions. This is due to the fact that the range of a pseudo almost periodic function is only bounded, but not relatively compact, contrary to the asymptotically almost periodic case. This last observation still holds when the word almost periodic is replaced by almost automorphic.

Proof When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M183">View MathML</a>, replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M184">View MathML</a> by ℝ, this result is a variation of Theorem 8.4 in [22].

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M186">View MathML</a>, replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M184">View MathML</a> by ℝ, the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M188">View MathML</a> is a variation of Theorem 2.15 in [9]. Moreover, using Lemma 1 in [23], we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M189">View MathML</a> is continuous, and so its restriction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M57">View MathML</a> is also continuous.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M192">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M193">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M194">View MathML</a>), this result is a straightforward consequence of Theorem 4.1 (respectively Theorem 4.2) in [24].

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M196">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M198">View MathML</a>), using Corollary 4.12 in [15] (respectively Corollary 5.10 in [16]), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M199">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M200">View MathML</a>). Moreover, using Lemma 1 in [23], we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M189">View MathML</a> is continuous, and so its restriction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M97">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M203">View MathML</a>) is also continuous. □

Lemma 3.6LetXandYbe two finite-dimensional Banach spaces, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M179">View MathML</a>be a continuously Fréchet-differentiable mapping. Then the Nemytskii operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M180">View MathML</a>is continuously Fréchet-differentiable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M107">View MathML</a>, and we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M207">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M208">View MathML</a>.

Proof When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M183">View MathML</a>, replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M184">View MathML</a> by ℝ, this result is a variation of Theorem 8.5 in [22].

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M186">View MathML</a>, using Lemma 1 in [23], we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M214">View MathML</a> is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a> and that we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M216">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M217">View MathML</a>. Now, using Theorem 2.15 in [9], we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M218">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M219">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M220">View MathML</a>. And so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M221">View MathML</a> and the announced formula for its Fréchet-differential is proven.

As in the proof of Lemma 3.3, the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M58">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M59">View MathML</a>) is a corollary of the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M97">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M98">View MathML</a>). And so it suffices to prove the cases of the weighted pseudo almost periodic functions and of the weighted pseudo almost automorphic functions.

To prove the result in the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M196">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M228">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M229">View MathML</a>), note that, using Lemma 1 in [23], we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M189">View MathML</a> is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a> and that we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M232">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M217">View MathML</a>. Now, using Corollary 4.12 in [15] (respectively Corollary 5.10 in [16]), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M234">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M235">View MathML</a>) and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M236">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M237">View MathML</a>) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M238">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M98">View MathML</a>).

Consequently, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M240">View MathML</a> (respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M241">View MathML</a>) and the announced formula for its Fréchet-differential is proven. □

Lemma 3.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M243">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M244">View MathML</a>. Ifxis a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M247">View MathML</a>andxsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M249">View MathML</a>.

Proof Lemma 3.2 in [4] asserts that if x is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M251">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M252">View MathML</a>, therefore the derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M253">View MathML</a> is uniformly continuous, and by help of Lemma 3.1, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M254">View MathML</a>. By using Lemma 3.4, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M255">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M256">View MathML</a> are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>. Applying again Lemma 3.4 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M258">View MathML</a> and using the continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M259">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M260">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M261">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M262">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M263">View MathML</a>, and from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M249">View MathML</a>. □

4 The main result

First we announce the main result of the paper.

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116">View MathML</a>. Under the following assumptions:

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M267">View MathML</a>,

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M268">View MathML</a>,

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M269">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M270">View MathML</a>,

(A4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M271">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M272">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M273">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M274">View MathML</a>,

there exist a neighborhoodof 0 in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264">View MathML</a>, a neighborhoodof 0 inPand a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>-mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M279">View MathML</a>fromintowhich satisfies the following conditions:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M282">View MathML</a>,

(ii) for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M284">View MathML</a>is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>,

(iii) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M287">View MathML</a>is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M283">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M291">View MathML</a>.

To prove Theorem 4.1, we define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M292">View MathML</a> by setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M293">View MathML</a>

(4.1)

when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M295">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M243">View MathML</a>. By using Lemma 3.7, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M294">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M298">View MathML</a> if and only if x is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>.

Under (A2) and (A3), note that 0 is a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M301">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264">View MathML</a>, and so the following equality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M303">View MathML</a>

(4.2)

Lemma 4.2Under (A1)-(A3), the operator Φ is well defined and it is of class<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M305">View MathML</a>. Moreover, the partial differential of Φ with respect to the first variable, at the point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M306">View MathML</a>, is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M307">View MathML</a>

when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M308">View MathML</a>.

Proof First we introduce linear operators: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M309">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M310">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M311">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M313">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M314">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M315">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M316">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M317">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M318">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M319">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M294">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M321">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M322">View MathML</a>, these linear operators are continuous; and consequently, the following assertion holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M323">View MathML</a>

Now we define the Nemytskii operators build on the functions f and g: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M324">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M325">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M326">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M327">View MathML</a>. By using Lemma 3.6, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M328">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M329">View MathML</a> are of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M331">View MathML</a> assimilated to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M332">View MathML</a>, and Lemma 3.6 provides formulas for the differentials of these Nemytskii operators:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M333">View MathML</a>

(4.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M334">View MathML</a>

(4.4)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M335">View MathML</a>.

We can assimilate a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M295">View MathML</a> to the constant function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M337">View MathML</a> that belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M338">View MathML</a>, which permits us to look at P as a closed vector subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M338">View MathML</a>. Then we can consider the following restrictions of the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M328">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M329">View MathML</a>: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M342">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M343">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M344">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M345">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M346">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M295">View MathML</a>.

Since the restriction of a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>-mapping to a Banach subspace is also a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>-mapping, a straightforward consequence of the continuous differentiability of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M328">View MathML</a> and of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M329">View MathML</a> is that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M352">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M353">View MathML</a> are of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>, and the consequences of (4.3) and (4.4) are the following formulas:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M355">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M335">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M295">View MathML</a>.

Now we consider the following operators: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M358">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M359">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M360">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M361">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M362">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M363">View MathML</a>. We consider the Nemystkii operator build on B, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M364">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M365">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M366">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M367">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M368">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M369">View MathML</a> are linear continuous, they are of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>. Since B is bilinear continuous, it is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>; and consequently, using Lemma 3.6, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M372">View MathML</a> is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M374">View MathML</a>. Denoting by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M375">View MathML</a> the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M376">View MathML</a>, ε is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a> after (A3), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M378">View MathML</a> is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a> as a composition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>-mappings. And so we can assert that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M368">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M382">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M372">View MathML</a> and C are of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>.

Now we note that the following equality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M385">View MathML</a>

(4.5)

Since all the mappings which are present in the previous formula are of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>, using the usual rules of the differential calculus in Banach spaces, we obtain that Φ is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M308">View MathML</a>, by using the classical formulas of the differential calculus in Banach spaces and (4.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M389">View MathML</a>

which implies, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M42">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M391">View MathML</a>

which is the announced formula. □

Lemma 4.3Under (A1)-(A4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M392">View MathML</a>is a bijection from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264">View MathML</a>onto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M395">View MathML</a>. We want to prove that there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M308">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M397">View MathML</a>. Using the formula provided by Lemma 4.2, this equation is equivalent to saying that y is a solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264">View MathML</a> of the following second-order linear differential equation (which is a Duffing equation):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M399">View MathML</a>

(4.6)

Rewriting this second-order equation in the form of a first-order system, we obtain the following equivalent differential system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M400">View MathML</a>

(4.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M401">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M402">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M403">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116">View MathML</a> and with condition (A4), the assumptions of Lemma 3.3 are fulfilled, and we can assert that there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M405">View MathML</a>, which is a solution of (4.7). Therefore the first coordinate of X, denoted by y, is the unique solution of (4.6) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264">View MathML</a> since y and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M407">View MathML</a>, and then y is the unique element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264">View MathML</a> which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M409">View MathML</a>. □

Proof of Theorem 4.1 By using (4.2), Lemma 4.2 and Lemma 4.3, we can use the implicit function theorem ([25], p.61) that permits us to say that there exist a neighborhood of 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264">View MathML</a>, a neighborhood of 0 in P and a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>-mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M279">View MathML</a> from into , which satisfies the following conditions:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M282">View MathML</a>, that is, the condition (i) of Theorem 4.1.

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M418">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M283">View MathML</a>, that ensures that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M284">View MathML</a> is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M283">View MathML</a>, that is, the conclusion (ii) of Theorem 4.1.

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M424">View MathML</a> that implies the conclusion (iii) of Theorem 4.1.

And so Theorem 4.1 is proven. □

5 Special cases

We consider the equation

which is a special case of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>) by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M427">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M428">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M429">View MathML</a> defined as the identity operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M430">View MathML</a>.

On the existence solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M14">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M432">View MathML</a>, we establish the following result.

Corollary 5.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116">View MathML</a>. Under the following assumptions:

(A5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M434">View MathML</a>,

(A6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M435">View MathML</a>,

(A7) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M436">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M437">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M438">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M439">View MathML</a>,

there exist a neighborhoodof 0 in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>, a neighborhoodof 0 in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264">View MathML</a>and a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>-mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M445">View MathML</a>frominto, which satisfies the following conditions:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M282">View MathML</a>,

(ii) for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M449">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M450">View MathML</a>is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M14">View MathML</a>) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M432">View MathML</a>,

(iii) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M287">View MathML</a>is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M14">View MathML</a>) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M449">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>, then we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M457">View MathML</a>.

The second special case of the equation (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M11">View MathML</a>) is

when q belongs to a Banach space Q, and by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M460">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M461">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M462">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M463">View MathML</a>. On the existence solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M464">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M432">View MathML</a>, we establish the following result.

Corollary 5.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M116">View MathML</a>. Under the following assumptions:

(A8) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M467">View MathML</a>,

(A9) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M468">View MathML</a>,

(A10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M469">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M470">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M471">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M472">View MathML</a>,

there exist a neighborhood<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M473">View MathML</a>of 0 in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>, a neighborhood<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M475">View MathML</a>of 0 in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M264">View MathML</a>, a neighborhood<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M477">View MathML</a>of 0 inQand a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M215">View MathML</a>-mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M479">View MathML</a>from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M480">View MathML</a>into<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M475">View MathML</a>which satisfies the following conditions:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M482">View MathML</a>,

(ii) for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M483">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>and for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M485">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M486">View MathML</a>is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M464">View MathML</a>) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>,

(iii) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M489">View MathML</a>is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M464">View MathML</a>) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M483">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M246">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M485">View MathML</a>, then we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M494">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

The authors are very pleased to contribute to this special issue in honor of Jean Mawhin, an international expert in the field of nonlinear analysis and differential equations, whose opinion is ever very important and useful to us.

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