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

Open Access Research

Periodic solutions to the Liénard type equations with phase attractive singularities

Robert Hakl1* and Manuel Zamora2

Author Affiliations

1 Institute of Mathematics, Academy of Sciences of the Czech Republic, Žižkova 22, Brno, 616 62, Czech Republic

2 Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, Campus de Fuentenueva s/n, Granada, 18071, Spain

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Boundary Value Problems 2013, 2013:47  doi:10.1186/1687-2770-2013-47


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/47


Received:10 December 2012
Accepted:18 February 2013
Published:6 March 2013

© 2013 Hakl and Zamora; 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

Sufficient conditions are established guaranteeing the existence of a positive ω-periodic solution to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M2">View MathML</a> are continuous functions with possible singularities at zero and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M3">View MathML</a> is a Carathéodory function. The results obtained are rewritten for the equation of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M4">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M6">View MathML</a>, δ are non-negative constants, c, μ, ν, γ are real numbers, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M7">View MathML</a>. The last equation also covers the so-called Rayleigh-Plesset equation, frequently used in fluid mechanics to model the bubble dynamics in liquid. In the paper, the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M8">View MathML</a>, i.e., the case which covers the attractive singularity of the function g, is studied. The results obtained assure that there exists a positive ω-periodic solution to the above-mentioned equation if the power μ or ν is sufficiently large.

MSC: 34C25, 34B16, 34B18, 76N15.

Keywords:
Rayleigh-Plesset equation; singular equation; periodic solution; upper and lower function

1 Introduction

The topic of singular boundary value problems has been of substantial and rapidly growing interest for many scientists and engineers. The importance of such investigation is emphasized by the fact that numerical simulations of solutions to such problems usually break down near singular points.

On the other hand, problems of this type arise frequently in applied science. Namely, in fluid mechanics, since 1917 the physicists have used the Rayleigh equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M9">View MathML</a>

to model the bubble dynamics in liquid, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M10">View MathML</a> is the ratio of the bubble at the time t, ρ is the liquid density, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M11">View MathML</a> is the pressure in the liquid at a large distance from the bubble, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M12">View MathML</a> is the pressure in the liquid at the bubble boundary. In 1949, Plesset proposed to use a more exact equation involving the surface-tension constant S and the coefficient of the liquid viscosity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M13">View MathML</a>, which was finally improved by adding a term with polytropic coefficient k in 1977, nowadays known as a Rayleigh-Plesset equation (see [1])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M14">View MathML</a>

The transformation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M15">View MathML</a> in the previous equation leads to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M16">View MathML</a>

Consequently, the class of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M17">View MathML</a>

(1.1)

with non-negative constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M6">View MathML</a>, δ, real numbers c, μ, ν, γ, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M20">View MathML</a>, plays an important role in fluid mechanics. Therefore, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M21">View MathML</a>

(1.2)

subjected to the periodic conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M22">View MathML</a>

(1.3)

is investigated in the presented paper. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M23">View MathML</a> are continuous, having possible singularities at zero, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M3">View MathML</a> is a Carathéodory function, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M25">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M27">View MathML</a> is continuous for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>, and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M29">View MathML</a>, there exists a non-negative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M30">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M31">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M33">View MathML</a>. By a solution to (1.2), (1.3) we understand a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M34">View MathML</a> which is positive, absolutely continuous together with its first derivative, satisfies (1.2) almost everywhere on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M35">View MathML</a>, and verifies (1.3). In spite of the fact that the problem (1.2), (1.3) was investigated by many mathematicians (see, e.g., [2-27]), most of the mentioned works deal with the repulsive case and/or when f has no singularity. However, the physical model, covered by equation (1.1), justifies considering the types of equations with a singular friction-like term.

A particular case of (1.1) is the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M36">View MathML</a>

(1.4)

studied by Lazer and Solimini. Their results were published in 1987 (see [11]) and they proved, among others, that (1.4), (1.3) has at least one solution if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37">View MathML</a>, provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M38">View MathML</a> is bounded. Recently, we have proved (see [28]) that this result cannot be extended to the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M38">View MathML</a> is a general integrable (and so unbounded) function unless some additional conditions are introduced. In particular, (1.4), (1.3) is solvable for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M40">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M42">View MathML</a>; and, moreover, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M43">View MathML</a>, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M20">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37">View MathML</a> such that (1.4), (1.3) has no solution. At this point, we would like to emphasize the important fact that the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M42">View MathML</a> can be weakened if (1.4) is generalized to equation (1.1), see Remark 2.2 below.

The structure of the paper is as follows. After the introduction and basic notation, we recall the definition of lower and upper functions to the problem (1.2), (1.3), and we formulate the classical theorem on the existence of a solution to (1.2), (1.3) in the case when there exists a couple of well-ordered lower and upper functions. In Section 2, we establish our main results and their consequences. Sections 3 and 4 are devoted to auxiliary propositions and proofs of the main results, respectively.

For convenience, we finish the introduction with a list of notations which are used throughout the paper:

ℕ is the set of all natural numbers, ℝ is the set of all real numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M49">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M50">View MathML</a> is the Banach space of continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M34">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M52">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M53">View MathML</a>, resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M54">View MathML</a>, is the set of continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M55">View MathML</a>, resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M56">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M57">View MathML</a> is the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M56">View MathML</a> which are continuous together with their first derivative.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M59">View MathML</a> is a set of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M34">View MathML</a> such that u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M61">View MathML</a> are absolutely continuous.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M62">View MathML</a> is the Banach space of the Lebesgue integrable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M63">View MathML</a> endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M64">View MathML</a>

For a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M65">View MathML</a>, its mean value is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M66">View MathML</a>

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M67">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M68">View MathML</a>

The following definitions of lower and upper functions are suitable for us. For more general definitions, one can see, e.g., [[12], Definition 8.2].

Definition 1.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M69">View MathML</a> is called a lower function to the problem (1.2), (1.3) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M70">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M72">View MathML</a>

Definition 1.2 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M73">View MathML</a> is called an upper function to the problem (1.2), (1.3) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M74">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M76">View MathML</a>

The following theorem is well known in the theory of differential equations (see, e.g., [[12], Theorem 8.12]).

Theorem 1.1Letαandβbe lower and upper functions to the problem (1.2), (1.3) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M77">View MathML</a>

(1.5)

Then there exists a solutionuto the problem (1.2), (1.3) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M78">View MathML</a>

2 Main results

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M79">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M80">View MathML</a>be non-decreasing functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M81">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M83">View MathML</a>

(2.1)

and let there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M84">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M85">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M86">View MathML</a>

(2.3)

Let, moreover, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M87">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M88">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M89">View MathML</a>

(2.5)

and let either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M90">View MathML</a>

(2.6)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M91">View MathML</a>

(2.7)

Furthermore, let us suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92">View MathML</a>fulfills at least one of the following conditions:

(a) there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M93">View MathML</a>of positive numbers such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M94">View MathML</a>

(2.8)

and there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M96">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M97">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M98">View MathML</a>

(2.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M99">View MathML</a>

(2.10)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M102">View MathML</a>

(2.11)

(b) the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M103">View MathML</a>is non-increasing and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M104">View MathML</a>

(2.12)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>andσis given by (2.11).

Besides, let us suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M107">View MathML</a>fulfills at least one of the following conditions:

(c) there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M108">View MathML</a>of positive numbers such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M109">View MathML</a>

and there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M110">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M111">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M112">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M113">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>;

(d) the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M115">View MathML</a>is non-increasing and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M116">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M113">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>.

Then there exists at least one solution to the problem (1.2), (1.3).

Remark 2.1 Note that there exists a suitable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M119">View MathML</a> such that (2.10) holds, e.g., if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M120">View MathML</a>

For equation (1.1), from Theorem 2.1 we get the following assertion.

Corollary 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M125">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M126">View MathML</a>

If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M127">View MathML</a>

then (1.1), (1.3) has at least one solution.

Remark 2.2 In [28], it is proved, among others, that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M128">View MathML</a>

(2.13)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37">View MathML</a>, has a positive ω-periodic solution if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M42">View MathML</a>. Moreover, there is also an example introduced showing that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M43">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M40">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37">View MathML</a> such that (2.13), (1.3) has no positive solution.

Corollary 2.1 says that if a friction-like term or sub-linear term are added to (2.13), the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M42">View MathML</a> can be weakened. For example,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M136">View MathML</a>

has a positive solution satisfying (1.3) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M137">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M138">View MathML</a>, provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37">View MathML</a>. Also, the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M140">View MathML</a>

subjected to the boundary conditions (1.3) is solvable for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M137">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M142">View MathML</a>, provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37">View MathML</a>.

Example 2.1 As it was mentioned in the introduction, the particular case of (1.1) is the so-called Rayleigh-Plesset equation frequently used in fluid mechanics. This equation has the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M144">View MathML</a>

(2.14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M40">View MathML</a>, c, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M6">View MathML</a> are positive constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M148">View MathML</a> (see [9,10]).

The results dealing with the existence of positive ω-periodic solutions of (2.14) were established in [10] provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M38">View MathML</a> is bounded from above (see [[10], Theorems 4.4, 4.6, 4.7]). However, Corollary 2.1 says that in the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M150">View MathML</a>, the problem (2.14), (1.3) is solvable if one of the following items is fulfilled:

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M151">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M37">View MathML</a>;

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M153">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M154">View MathML</a>;

3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M155">View MathML</a>.

Thus, Corollary 2.1 assures that the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M38">View MathML</a> is not necessary.

Corollary 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M160">View MathML</a>. Let, moreover, either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M161">View MathML</a>or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M162">View MathML</a>

(2.15)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M163">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M165">View MathML</a>

Then the problem (1.1), (1.3) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M166">View MathML</a>has at least one solution.

Remark 2.3 According to [29] and Theorem 1.1, it can be easily verified that the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M167">View MathML</a>

(2.16)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M137">View MathML</a>, has a positive solution if and only if the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M170">View MathML</a> holds (see notation in [29]).

Indeed, according to [[29], Definition 1.1], the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M170">View MathML</a> implies the existence of a positive solution v to the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M172">View MathML</a>

Therefore there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M173">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M174">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M175">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>. By setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M177">View MathML</a>

one can easily realize that α and β are, respectively, lower and upper functions to (2.16) satisfying (1.5). Now, the existence of a positive solution to (2.16) follows from Theorem 1.1.

On the other hand, the existence of a positive solution to (2.16) implies the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M170">View MathML</a> (see [[29], Theorem 2.1]).

However, one of the optimal effective conditions guaranteeing such an inclusion is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M179">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M180">View MathML</a>

(see [[29], Corollary 2.5]). Therefore, the condition (2.15) is natural in a certain sense.

When the right-hand side of equation (1.3) does not depend on u, i.e., when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M181">View MathML</a>, then (1.3) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M182">View MathML</a>

(2.17)

From Theorem 2.1, for equation (2.17) we get the following assertion.

Corollary 2.3Let there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M185">View MathML</a>

and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M186">View MathML</a>

Let, moreover, either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M187">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M188">View MathML</a>

Then there exists at least one solution to the problem (2.17), (1.3).

In the following result, the assumptions do not depend on the friction-like term. On the other hand, a certain smallness of oscillation of the primitive to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M38">View MathML</a> is supposed. Clearly, Theorems 2.1 and 2.2 are independent.

Theorem 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M79">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M80">View MathML</a>be non-decreasing functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M81">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M193">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M194">View MathML</a>

(2.18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M195">View MathML</a>

(2.19)

Let, moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M196">View MathML</a>

(2.20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M197">View MathML</a>

(2.21)

and let there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M198">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M199">View MathML</a>

(2.22)

Besides, let us suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M107">View MathML</a>fulfills at least one of the conditions (c) or (d) of Theorem 2.1. Then there exists at least one solution to the problem (1.2), (1.3).

In the particular case, when equation (1.2) has the form (1.1), the following assertion immediately follows from Theorem 2.2.

Corollary 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M123">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M193">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M203">View MathML</a>

Let, moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M204">View MathML</a>

Then the problem (1.1), (1.3) has at least one solution.

Remark 2.4 The consequence of Theorem 2.2 for the problem (2.17), (1.3) coincides with the result obtained in [[10], Theorem 3.6].

3 Auxiliary propositions

In what follows, we will show the existence of a solution to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M205">View MathML</a>

(3.1)

satisfying the boundary conditions (1.3). Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M206">View MathML</a> is a non-decreasing function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M40">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M208">View MathML</a>. Together with (3.1), for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M209">View MathML</a>, consider the auxiliary equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M210">View MathML</a>

(3.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M211">View MathML</a>

(3.3)

Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M212">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M213">View MathML</a>

(3.5)

The following three results can be found in [10].

Lemma 3.1 (see [[10], Corollary 2.17])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M215">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M216">View MathML</a>

(3.6)

Let, moreover, there exist a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M93">View MathML</a>of positive numbers such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M218">View MathML</a>

(3.7)

and let there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M219">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M97">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M221">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M222">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>. Then there exists an upper functionβto the problem (3.1), (1.3) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M224">View MathML</a>

(3.8)

Lemma 3.2 (see [[10], Corollary 2.18])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M215">View MathML</a>be such that (3.6) holds. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M227">View MathML</a>is a non-increasing function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M228">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M222">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>, then there exists an upper functionβto the problem (3.1), (1.3) satisfying (3.8).

Lemma 3.3 (see [[10], Corollary 2.11])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M231">View MathML</a>be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M232">View MathML</a>

Then there exists a lower functionαto the problem (2.17), (1.3) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M233">View MathML</a>

(3.9)

Lemma 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184">View MathML</a>be such that (2.2) holds. Let, moreover, there exist a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M93">View MathML</a>of positive numbers such that (3.7) is fulfilled, and let there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M219">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M97">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M239">View MathML</a>

(3.10)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>. Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M242">View MathML</a>and an upper functionβto the problems (3.2), (1.3) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243">View MathML</a>satisfying (3.8).

Proof

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M244">View MathML</a>

(3.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M245">View MathML</a>

(3.12)

Then, obviously, in view of (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M246">View MathML</a>

(3.13)

and, consequently, on account of (2.2), (3.5), (3.10), and (3.13), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M242">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M248">View MathML</a>

(3.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M249">View MathML</a>

(3.15)

Therefore, according to Lemma 3.1, there exists an upper function β to (3.2), (1.3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M250">View MathML</a> satisfying (3.8). Obviously, in view of (3.4) and the non-negativity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92">View MathML</a>, it follows that β is also an upper function to (3.2), (1.3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243">View MathML</a>. □

Lemma 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184">View MathML</a>be such that (2.2) holds. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M227">View MathML</a>is a non-increasing function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M256">View MathML</a>

(3.16)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>, then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M242">View MathML</a>and an upper functionβto the problems (3.2), (1.3) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243">View MathML</a>satisfying (3.8).

Proof Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M262">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M263">View MathML</a> by (3.11) and (3.12). Then, obviously, in view of (3.3), we have that (3.13) holds and, consequently, on account of (2.2), (3.5), (3.13), and (3.16), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M242">View MathML</a> such that (3.14) is valid and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M265">View MathML</a>

(3.17)

Therefore, according to Lemma 3.2, there exists an upper function β to (3.2), (1.3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M250">View MathML</a> satisfying (3.8). Obviously, in view of (3.4) and the non-negativity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92">View MathML</a>, it follows that β is also an upper function to (3.2), (1.3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243">View MathML</a>. □

Lemma 3.6Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M269">View MathML</a>

(3.18)

and let either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M270">View MathML</a>

(3.19)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M271">View MathML</a>

(3.20)

Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M272">View MathML</a>, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M273">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M209">View MathML</a>and any positive solutionuof (3.2), (1.3) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M275">View MathML</a>

(3.21)

we have the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M276">View MathML</a>

(3.22)

Proof Assume that (3.20) is fulfilled. Let u be a positive solution to (3.2), (1.3) satisfying (3.21). Then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M277">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M278">View MathML</a>

(3.23)

Define the operator ϑ of ω-periodic prolongation by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M279">View MathML</a>

(3.24)

Then, obviously, from (3.2) and (1.3) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M280">View MathML</a>

(3.25)

The integration of (3.25) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M281">View MathML</a> to t, on account of (3.23), yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M282">View MathML</a>

(3.26)

From (3.21), (3.23), and (3.24) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M283">View MathML</a>

(3.27)

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M284">View MathML</a>

(3.28)

According to (3.18), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M285">View MathML</a>

(3.29)

Thus, using (3.3), (3.20), (3.21), and (3.27)-(3.29) in (3.26), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M286">View MathML</a>

(3.30)

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M287">View MathML</a>

Then, on account of (3.24) and (3.30), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M288">View MathML</a>

(3.31)

On the other hand, the integration of (3.25) from t to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M289">View MathML</a>, with respect to (3.23), results in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M290">View MathML</a>

(3.32)

Now, using (3.3), (3.20), (3.21), and (3.27)-(3.29) in (3.32), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M291">View MathML</a>

(3.33)

Therefore, in view of (3.24), from (3.33) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M292">View MathML</a>

(3.34)

Consequently, (3.31) and (3.34) result in (3.22).

Now, suppose that (3.19) is fulfilled. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M293">View MathML</a>

(3.35)

Then, according to (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M294">View MathML</a>

(3.36)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M295">View MathML</a>

Analogously to the above-proved, using (3.19) instead of (3.20), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M296">View MathML</a>

(3.37)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M297">View MathML</a>

Thus, (3.35) and (3.37) yield (3.22). □

Lemma 3.7Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M298">View MathML</a>

(3.38)

and let either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M299">View MathML</a>

(3.39)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M300">View MathML</a>

(3.40)

Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M272">View MathML</a>, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M302">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M209">View MathML</a>and any positive solutionuof (3.2), (1.3) satisfying (3.21), we have the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M304">View MathML</a>

(3.41)

Proof Let u be a positive solution to (3.2), (1.3) satisfying (3.21). Thus, the integration of (3.2) from 0 to ω, in view of (1.3) and (3.4), yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M305">View MathML</a>

(3.42)

On the other hand, (3.38) implies the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M306">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M307">View MathML</a>

(3.43)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M308">View MathML</a> be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M309">View MathML</a>

(3.44)

Obviously, either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M310">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M311">View MathML</a>

(3.45)

Obviously, it is sufficient to show the estimate (3.41) is valid just in the case when (3.45) is fulfilled. Let, therefore, (3.45) hold.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M312">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>, then applying (3.43) in (3.42) we obtain a contradiction. Thus, there exist points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M314">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M315">View MathML</a>

(3.46)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M316">View MathML</a>

(3.47)

where ϑ is the operator defined by (3.24). Obviously, (3.25) holds.

Assume that (3.39) holds. Then, according to Lemma 3.6, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M273">View MathML</a> such that (3.22) holds. The integration of (3.25) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M318">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M319">View MathML</a>, in view of (3.4), (3.21), (3.22), (3.39), (3.43), (3.44), and (3.46), results in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M320">View MathML</a>

(3.48)

Note that in view of (3.46), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M321">View MathML</a>. Consequently, from (3.48) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M322">View MathML</a>

(3.49)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M323">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M324">View MathML</a> does not depend on k. Therefore, if we apply (3.39) in (3.49), it can be easily seen, with respect to (3.44), that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M302">View MathML</a> such that (3.41) holds.

If (3.40) holds, we integrate (3.25) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M326">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M327">View MathML</a> and apply similar steps as above, just using (3.47) instead of (3.46). Finally, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M328">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M329">View MathML</a>

Therefore, also in this case, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M302">View MathML</a> such that (3.41) holds. □

Lemma 3.8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184">View MathML</a>be such that (2.2) holds. Let, moreover, (3.38) be fulfilled, and let either (3.39) or (3.40) be valid. Let, in addition, there exist a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M93">View MathML</a>of positive numbers such that (3.7) holds, and let there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M219">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M97">View MathML</a>such that (3.10) is fulfilled, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>. Then there exists a positive solutionuto (3.1), (1.3).

Proof According to Lemma 3.4, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M242">View MathML</a> and an upper function β to the problems (3.2), (1.3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243">View MathML</a> satisfying (3.8). On the other hand, in view of (3.4) and (3.38), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M340">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M342">View MathML</a>

Thus, if we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M343">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>, according to Theorem 1.1, there exists a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M345">View MathML</a> to (3.2), (1.3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M243">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M347">View MathML</a>

(3.50)

Moreover, according to Lemmas 3.6 and 3.7, in view of (3.50), there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M272">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M273">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M302">View MathML</a>, not depending on k, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M351">View MathML</a>

(3.51)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M352">View MathML</a>

(3.52)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M353">View MathML</a>

(3.53)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M354">View MathML</a>

Therefore, according to the Arzelà-Ascoli theorem, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M355">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M356">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M357">View MathML</a>

(3.54)

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M345">View MathML</a> are solutions to (3.2), (1.3), in view of (3.3), (3.52), and (3.54), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M359">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M360">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M361">View MathML</a> is a positive solution to (3.1), (1.3). □

The following assertion can be proved analogously to Lemma 3.8, just Lemma 3.5 is used instead of Lemma 3.4.

Lemma 3.9Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184">View MathML</a>be such that (2.2) holds. Let, moreover, (3.38) be fulfilled, and let either (3.39) or (3.40) be valid. Let, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M227">View MathML</a>be a non-increasing function and let (3.16) be fulfilled, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>. Then there exists a positive solutionuto (3.1), (1.3).

Lemma 3.10Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M79">View MathML</a>be non-decreasing, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184">View MathML</a>be such that (2.2) holds. Let, moreover, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M87">View MathML</a>such that (2.4) and (2.5) are valid, and let either (2.6) or (2.7) be fulfilled. Let, in addition, there exist a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M93">View MathML</a>of positive numbers such that (2.8) holds and let there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M96">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M97">View MathML</a>such that (2.9) and (2.10) are fulfilled, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>andσis given by (2.11). Then there exists a lower functionαto the problem (3.1), (1.3).

Proof Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92">View MathML</a> is a positive function, from (2.4) and (2.11) we obtain that σ is a positive increasing function. Therefore, there exists an inverse function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M378">View MathML</a> to σ which is also increasing. Moreover, in view of (2.4) and (2.11), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M379">View MathML</a>

(3.55)

Consider the auxiliary equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M380">View MathML</a>

(3.56)

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M381">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M382">View MathML</a>. Then from (2.2) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M383">View MathML</a>

(3.57)

and, in view of (3.55), from (2.5) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M384">View MathML</a>

(3.58)

Furthermore, the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M385">View MathML</a> in (2.6), resp (2.7), with respect to (2.11) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M386">View MathML</a>

(3.59)

resp.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M387">View MathML</a>

(3.60)

Moreover, put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M388">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M389">View MathML</a>. Then from (2.8), in view of (3.55), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M390">View MathML</a>

(3.61)

Finally, (2.10) results in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M391">View MathML</a>

and so, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92">View MathML</a> is a non-decreasing function, from (2.9) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M393">View MathML</a>

(3.62)

Therefore, applying Lemma 3.8, according to (3.57)-(3.62), there exists a positive solution u to the problem (3.56), (1.3).

Now, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M394">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>, i.e., in view of (2.11),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M396">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M69">View MathML</a> is a positive function and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M398">View MathML</a>

Thus, it can be easily seen that α is a lower function to the problem (3.1), (1.3). □

Analogously to the proof of Lemma 3.10, one can prove the following assertion applying Lemma 3.9 instead of Lemma 3.8.

Lemma 3.11Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M79">View MathML</a>be non-decreasing, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M82">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M184">View MathML</a>be such that (2.2) holds. Let, moreover, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M87">View MathML</a>such that (2.4) and (2.5) are valid, and let either (2.6) or (2.7) be fulfilled. Let, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M103">View MathML</a>be a non-increasing function and let (2.12) be fulfilled, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M100">View MathML</a>for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>andσis given by (2.11). Then there exists a lower functionαto the problem (3.1), (1.3).

4 Proofs of the main results

Proof of Theorem 2.1 According to Lemmas 3.1, 3.2, 3.10, and 3.11, the conditions of the theorem guarantee a well-ordered couple of lower and upper functions, therefore the result is a direct consequence of Theorem 1.1. □

Proof of Corollary 2.1 It follows from Theorem 2.1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M406">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M407">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M408">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M409">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M410">View MathML</a>

Then items (a) and (c) of Theorem 2.1 are fulfilled. □

Proof of Corollary 2.2 It follows from Theorem 2.1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M406">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M412">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M413">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M414">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M415">View MathML</a>. Then items (b) and (d) of Theorem 2.1 are fulfilled. □

Proof of Corollary 2.3 It immediately follows from Theorem 2.1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M406">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M417">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M418">View MathML</a>). □

Proof of Theorem 2.2

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M419">View MathML</a>

(4.1)

Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M92">View MathML</a> is a positive function, from (2.20) and (4.1) we obtain that σ is an increasing function. Therefore, there exists an inverse function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M378">View MathML</a> to σ which is also increasing.

Consider the auxiliary equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M422">View MathML</a>

(4.2)

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M381">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M382">View MathML</a>. Then from (2.20) and (2.21), in view of (4.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M425">View MathML</a>

Therefore, according to Lemma 3.3, there exists a lower function w to the problem (4.2), (1.3) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M426">View MathML</a>

(4.3)

Now, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M427">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M28">View MathML</a>, i.e., in view of (4.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M429">View MathML</a>

Obviously, with respect to (4.3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M430">View MathML</a> is a positive function satisfying (3.9), and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M431">View MathML</a>

Thus, on account of (2.18), (3.9), and (4.2), it can be easily seen that α is a lower function to the problem (1.2), (1.3).

The existence of an upper function β to (1.2), (1.3) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M432">View MathML</a>

(4.4)

follows from (2.19) and Lemma 3.1, resp. 3.2.

Obviously, in view of (3.9) and (4.4), we have that (1.5) holds. Thus the theorem follows from Theorem 1.1. □

Proof of Corollary 2.4 It follows from Theorem 2.2 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M406">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M407">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/47/mathml/M435">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

RH and MZ obtained the results in a joint research. Both authors read and approved the final manuscript.

Acknowledgements

The first author was supported by RVO: 67985840; the second author was supported by Ministerio de Educación y Ciencia, Spain, project MTM2011-23652.

References

  1. Plesset, MS, Prosperetti, A: Bubble dynamics and cavitation. Annu. Rev. Fluid Mech.. 9, 145–185 (1977). Publisher Full Text OpenURL

  2. Habets, P, Sanchez, L: Periodic solutions of some Liénard equations with singularities. Proc. Am. Math. Soc.. 109, 1135–1144 (1990). Publisher Full Text OpenURL

  3. Bonheure, D, Fabry, C, Smets, D: Periodic solutions of forced isochronous oscillators at resonance. Discrete Contin. Dyn. Syst.. 8(4), 907–930 (2002)

  4. Bonheure, D, De Coster, C: Forced singular oscillators and the method of upper and lower solutions. Topol. Methods Nonlinear Anal.. 22, 297–317 (2003)

  5. Martins, RF: Existence of periodic solutions for second-order differential equations with singularities and the strong force condition. J. Math. Anal. Appl.. 317, 1–13 (2006). Publisher Full Text OpenURL

  6. Mawhin, J: Topological degree and boundary value problems for nonlinear differential equations. In: Furi M, Zecca P (eds.) Topological Methods for Ordinary Differential Equations, pp. 74–142. Springer, Berlin (1993)

  7. Omari, P, Ye, W: Necessary and sufficient conditions for the existence of periodic solutions of second-order ordinary differential equations with singular nonlinearities. Differ. Integral Equ.. 8, 1843–1858 (1995)

  8. Yuan, R, Zhang, Z: Existence of positive periodic solutions for the Liénard differential equations with weakly repulsive singularity. Acta Appl. Math.. 111, 171–178 (2010). Publisher Full Text OpenURL

  9. Hakl, R, Torres, P, Zamora, M: Periodic solutions of singular second order differential equations: the repulsive case. Topol. Methods Nonlinear Anal.. 39, 199–220 (2012)

  10. Hakl, R, Torres, P, Zamora, M: Periodic solutions of singular second order differential equations: upper and lower functions. Nonlinear Anal.. 74, 7078–7093 (2011). Publisher Full Text OpenURL

  11. Lazer, AC, Solimini, S: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc.. 99, 109–114 (1987). Publisher Full Text OpenURL

  12. Rachůnková, I, Staněk, S, Tvrdý, M: Solvability of Nonlinear Singular Problems for Ordinary Differential Equations, Hindawi Publishing Corporation, New York (2008)

  13. Zhang, M: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl.. 203, 254–269 (1996). Publisher Full Text OpenURL

  14. Chu, J, Li, M: Positive periodic solutions of Hill’s equation with singular nonlinear perturbations. Nonlinear Anal.. 69, 276–286 (2008). Publisher Full Text OpenURL

  15. Chu, J, Nieto, JJ: Recent existence results for second-order singular periodic differential equations. Bound. Value Probl.. 2009, Article ID 540863 (2009)

  16. Chu, J, Torres, PJ: Applications of Schauder’s fixed point theorem to singular differential equations. Bull. Lond. Math. Soc.. 39, 653–660 (2007). Publisher Full Text OpenURL

  17. Chu, J, Torres, PJ, Zhang, M: Periodic solutions of second order non-autonomous singular dynamical systems. J. Differ. Equ.. 239, 196–212 (2007). Publisher Full Text OpenURL

  18. Franco, D, Torres, PJ: Periodic solutions of singular systems without the strong force condition. Proc. Am. Math. Soc.. 136, 1229–1236 (2008)

  19. Franco, D, Webb, JRL: Collisionless orbits of singular and nonsingular dynamical systems. Discrete Contin. Dyn. Syst.. 15, 747–757 (2006)

  20. Hakl, R, Torres, P: On periodic solutions of second-order differential equations with attractive-repulsive singularities. J. Differ. Equ.. 248, 111–126 (2010). PubMed Abstract | Publisher Full Text | PubMed Central Full Text OpenURL

  21. Rachůnková, I, Tvrdý, M, Vrkoč, I: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differ. Equ.. 176, 445–469 (2001). Publisher Full Text OpenURL

  22. Torres, PJ: Existence of one-signed periodic solutions of some second order differential equations via a Kranoselskii fixed point theorem. J. Differ. Equ.. 190, 643–662 (2003). Publisher Full Text OpenURL

  23. Torres, PJ: Non-collision periodic solutions of forced dynamical systems with weak singularities. Discrete Contin. Dyn. Syst.. 11, 693–698 (2004)

  24. Torres, PJ: Weak singularities may help periodic solutions to exists. J. Differ. Equ.. 232, 277–284 (2007). Publisher Full Text OpenURL

  25. Torres, PJ: Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity. Proc. R. Soc. Edinb., Sect. A, Math.. 137, 195–201 (2007)

  26. Yan, P, Zhang, M: Higher order nonresonance for differential equations with singularities. Math. Methods Appl. Sci.. 26, 1067–1074 (2003). Publisher Full Text OpenURL

  27. Zhang, M: Periodic solutions of equations of Emarkov-Pinney type. Adv. Nonlinear Stud.. 6, 57–67 (2006)

  28. Hakl, R, Zamora, M: On the open problems connected to the results of Lazer and Solimini. Proc. R. Soc. Edinb., Sect. A, Math. (to appear) http://www.math.cas.cz/fichier/preprints/IM_20120709132356_18.pdf

  29. Hakl, R, Torres, P: Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign. Appl. Math. Comput.. 217, 7599–7611 (2011). Publisher Full Text OpenURL