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Extremal mild solutions for impulsive fractional evolution equations with nonlocal initial conditions

Jia Mu

Author Affiliations

School of Mathematics and Computer Science Institute, Northwest University for Nationalities, Lanzhou, Gansu, People’s Republic of China

Boundary Value Problems 2012, 2012:71  doi:10.1186/1687-2770-2012-71


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/71


Received:11 November 2011
Accepted:20 February 2012
Published:5 July 2012

© 2012 Mu; 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

In this article, the theory of positive semigroup of operators and the monotone iterative technique are extended for the impulsive fractional evolution equations with nonlocal initial conditions. The existence results of extremal mild solutions are obtained. As an application that illustrates the abstract results, an example is given.

Keywords:
impulsive fractional evolution equations; nonlocal initial conditions; extremal mild solutions; monotone iterative technique

1 Introduction

In this article, we use the monotone iterative technique to investigate the existence of extremal mild solutions of the impulsive fractional evolution equation with nonlocal initial conditions in an ordered Banach space X

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M2">View MathML</a> is the Caputo fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M4">View MathML</a> is a linear closed densely defined operator, −A is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M10">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M11">View MathML</a> is continuous (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M12">View MathML</a> will be defined in Section 2), the impulsive function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M13">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M14">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M16">View MathML</a> represent the right and left limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M17">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M18">View MathML</a>, respectively.

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary real or complex order. The subject is as old as differential calculus, and goes back to the time when Leibnitz and Newton invented differential calculus. Fractional derivatives have been extensively applied in many fields which have been seen an overwhelming growth in the last three decades. Examples abound: models admitting backgrounds of heat transfer, viscoelasticity, electrical circuits, electro-chemistry, economics, polymer physics, and even biology are always concerned with fractional derivative [1-6]. Fractional evolution equations have attracted many researchers in recent years, for example, see [7-14]. A strong motivation for investigating the problem (1.1) comes form physics. For example, fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M19">View MathML</a>, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M20">View MathML</a>

(1.2)

we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M21">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M22">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M23">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M24">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M27">View MathML</a> are the fractional derivatives of order α, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M29">View MathML</a>, respectively.

The existence results to evolution equations with nonlocal conditions in Banach space was studied first by Byszewski [15,16]. Deng [17] indicated that, using the nonlocal condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M30">View MathML</a> to describe for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give better result than using the usual local Cauchy problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M31">View MathML</a>. For example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M32">View MathML</a> can be given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M33">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M34">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M35">View MathML</a>) are given constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M36">View MathML</a>. On the other hand, the differential equations involving impulsive effects appear as a natural description of observed evolution phenomena introduction of the basic theory of impulsive differential equations, we refer the reader to [18] and the references therein. The study of impulsive evolution equations with nonlocal initial conditions has attracted a great deal of attention in fractional dynamics and its theory has been treated in several works [12-14]. They use the contraction mapping principle, the Krasnoselskii fixed point theorem and the Schaefer fixed point theorem.

To the authors’ knowledge, there are no studies on the existence of solutions for the impulsive fractional evolution equations with nonlocal initial conditions by using the monotone iterative technique in the presence of lower and upper solutions. Nevertheless, the monotone iterative technique concerning upper and lower solutions is a powerful tool to solve the differential equations with various kinds of boundary conditions, see [19-21]. This technique is that, for the considered problem, starting from a pair ordered lower and upper, one constructs two monotone sequences such that them uniformly converge to the extremal solutions between the lower and upper solutions. In this article, based on Mu [8], we obtained the existence of extremal mild solutions of the problem (1.1) by using the monotone iterative technique.

In following section, we introduce some preliminaries which are used throughout this article. In Section 3, by combining the theory of positive semigroup of linear operators and the monotone iterative technique coupled with the method of upper and lower solutions, we construct two groups of monotone iterative sequences, and then prove these sequences monotonically converge to the maximal and minimal mild solutions of the problem (1.1), respectively, under some monotone conditions and noncompactness measure conditions of f, g, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M37">View MathML</a>. In Section 4, in order to illustrate our results, an impulsive fractional partial differential equation with nonlocal initial condition is also considered.

2 Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this article.

Definition 2.1[22]

The fractional integral of order α with the lower limit zero for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M38">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M39">View MathML</a>

(2.1)

provided the right side is pointwise defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M40">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M41">View MathML</a> is the gamma function.

Definition 2.2[22]

The Riemann-Liouville derivative of order α with the lower limit zero for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M38">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M43">View MathML</a>

(2.2)

Definition 2.3[22]

The Caputo fractional derivative of order α for a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M38">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M45">View MathML</a>

(2.3)

Remark 2.4

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M46">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M47">View MathML</a>

(2.4)

(ii) The Caputo derivative of a constant is equal to zero.

(iii) If f is an abstract function with values in X, then the integrals and derivatives which appear in Definitions 2.1-2.3 are taken in Bochner’s sense.

Let X be an ordered Banach space with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M48">View MathML</a> and partial order ≤, whose positive cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M49">View MathML</a> (θ is the zero element of X) is normal with normal constant N. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M50">View MathML</a> be the Banach space of all continuous X-value functions on interval I with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M51">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M52">View MathML</a>. Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M53">View MathML</a> is an ordered Banach space with norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M54">View MathML</a> and the partial order ≤ reduced by the positive cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M55">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M56">View MathML</a> is also normal with the same normal constant N. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M58">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M59">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M61">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M62">View MathML</a>, denote the ordered interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M63">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M53">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M65">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60">View MathML</a>) in X. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M67">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M68">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M69">View MathML</a> we denote the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M70">View MathML</a> with the graph norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M71">View MathML</a>. An abstract function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M72">View MathML</a> is called a solution of (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M17">View MathML</a> satisfies all the equalities of (1.1). We note that −A is the infinitesimal generator of a uniformly bounded analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>). This means there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M76">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M77">View MathML</a>

(2.5)

Definition 2.5 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M78">View MathML</a> and satisfies inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M79">View MathML</a>

(2.6)

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80">View MathML</a> is called a lower solution of problem (1.1); if all inequalities of (2.6) are inverse, we call it an upper solution of the problem (1.1).

Lemma 2.6[7]

Ifhsatisfies a uniform H<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M81">View MathML</a>lder condition, with exponent<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M82">View MathML</a>, then the unique solution of the linear initial value problem (LIVP) for the fractional evolution equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M83">View MathML</a>

(2.7)

is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M84">View MathML</a>

(2.8)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M85">View MathML</a>

(2.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M86">View MathML</a>is a probability density function defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M87">View MathML</a>.

Remark 2.7[9,10]

(2.10)
(2.11)

Remark 2.8[10]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M93">View MathML</a>.

Definition 2.9 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M94">View MathML</a>, by the mild solution of IVP (2.7), we mean that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M95">View MathML</a> satisfying the integral Equation (2.8).

Form Definition 2.9, we can easily obtain the following result.

Lemma 2.10For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M98">View MathML</a>, the LIVP for the linear impulsive fractional evolution equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M99">View MathML</a>

(2.12)

has the unique mild solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M100">View MathML</a>given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M101">View MathML</a>

(2.13)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M102">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M103">View MathML</a>are given by (2.9).

Remark 2.11 We note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M103">View MathML</a> do not possess the semigroup properties. The mild solution of (2.12) can be expressed only by using piecewise functions.

Definition 2.12 A <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M106">View MathML</a>-semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M107">View MathML</a> is called a positive semigroup, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M108">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>.

Definition 2.13 A bounded linear operator K on X is called to be positive, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M111">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M109">View MathML</a>.

Remark 2.14 By (2.9) and Remark 2.8, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M103">View MathML</a> are positive, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M107">View MathML</a> is a positive semigroup.

Remark 2.15 From Remark 2.14, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>) is a positive semigroup generated by −A, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M119">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M98">View MathML</a>, then the mild solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M100">View MathML</a> of (2.12) satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M123">View MathML</a>. For the applications of positive operators semigroup, one can refer to [23-25].

Now, we recall some properties of the measure of noncompactness will be used later. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M124">View MathML</a> denotes the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [26]. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M125">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M127">View MathML</a>. If B is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M50">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M129">View MathML</a> is bounded in X, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M130">View MathML</a>. If E is a precompact set in X, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M131">View MathML</a>.

Lemma 2.16[27]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M132">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M133">View MathML</a>) be a bounded and countable set. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M134">View MathML</a>is Lebesgue integral onI, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M135">View MathML</a>

(2.14)

In order to prove our results, we also need a generalized Gronwall inequality for fractional differential equation.

Lemma 2.17[28]

Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M137">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M138">View MathML</a>is a nonnegative function locally integrable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M139">View MathML</a> (some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M140">View MathML</a>), and suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M17">View MathML</a>is nonnegative and locally integrable on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M139">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M143">View MathML</a>

(2.15)

on this interval; then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M144">View MathML</a>

(2.16)

3 Main results

Theorem 3.1LetXbe an ordered Banach space, whose positive conePis normal with normal constantN. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>) is positive, the Cauchy problem (1.1) has a lower solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M147">View MathML</a>and an upper solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M148">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M149">View MathML</a>, and the following conditions are satisfied:

(H1) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M150">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M151">View MathML</a>

(3.1)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M153">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M154">View MathML</a>is increasing inxfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M155">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M32">View MathML</a>is decreasing inufor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M157">View MathML</a>.

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M158">View MathML</a>is increasing inxfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M155">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M160">View MathML</a>.

(H4) There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M161">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M162">View MathML</a>

(3.2)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60">View MathML</a>, and increasing or decreasing monotonic sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M164">View MathML</a>.

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M165">View MathML</a>is precompact inX, for any increasing or decreasing monotonic sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M166">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M167">View MathML</a>.

Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169">View MathML</a>, which can be obtained by a monotone iterative procedure starting from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169">View MathML</a>, respectively.

Proof It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M172">View MathML</a> generates an positive analytic semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M173">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M175">View MathML</a>. By Remark 2.14, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M176">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M178">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>) are positive. By (2.5) and Remark 2.8, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M180">View MathML</a>

(3.3)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M184">View MathML</a>. We define a mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M185">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M186">View MathML</a>

(3.4)

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M185">View MathML</a> is continuous. By Lemma 2.10, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M188">View MathML</a> is a mild solution of problem (1.1) if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M189">View MathML</a>

(3.5)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M190">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M191">View MathML</a>, from the positivity of operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M178">View MathML</a>, (H1), (H2), and (H3), we have inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M194">View MathML</a>

(3.6)

Now, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M196">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M197">View MathML</a>. By Definition 2.5, Lemma 2.10, the positivity of operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M178">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M200">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M201">View MathML</a>

(3.7)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M202">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M203">View MathML</a>

(3.8)

Continuing such a process interval by interval to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M204">View MathML</a>, by (3.4), we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M195">View MathML</a>. Similarly, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M196">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M188">View MathML</a>, in view of (3.6), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M208">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M209">View MathML</a> is a continuous increasing monotonic operator. We can now define the sequences

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M210">View MathML</a>

(3.9)

and it follows from (3.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M211">View MathML</a>

(3.10)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M133">View MathML</a> . By (3.10) and the normality of the positive cone P, then B and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M215">View MathML</a> are bounded. It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M216">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M217">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M219">View MathML</a>

(3.11)

From (H4), (H5), (3.3), (3.4), (3.9), (3.11), Lemma 2.16 and the positivity of operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M178">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M200">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M222">View MathML</a>

(3.12)

By (3.12) and Lemma 2.17, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M223">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M224">View MathML</a>. In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M225">View MathML</a>. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M226">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M227">View MathML</a> are precompact in X. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M228">View MathML</a> is precompact in X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M229">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M202">View MathML</a>, using the same argument as above for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M200">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M232">View MathML</a>

(3.13)

By (3.13) and Lemma 2.17, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M223">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M234">View MathML</a>. Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M235">View MathML</a>. Continuing such a process interval by interval to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M204">View MathML</a>, we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M223">View MathML</a> on every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M239">View MathML</a>. This means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M240">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M133">View MathML</a>) is precompact in X for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M240">View MathML</a> has a convergent subsequence in X. In view of (3.10), we can easily prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M240">View MathML</a> itself is convergent in X. That is, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M245">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M246">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M247">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60">View MathML</a>. By (3.4) and (3.9), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M249">View MathML</a>

(3.14)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M247">View MathML</a>, then by Lebesgue-dominated convergence theorem, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M251">View MathML</a>

(3.15)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M253">View MathML</a>. Similarly, we can prove that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M254">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M255">View MathML</a>. By (3.6), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M188">View MathML</a>, and u is a fixed point of Q, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M257">View MathML</a>. By induction, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M258">View MathML</a>. By (3.10) and taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M247">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M260">View MathML</a>. That means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M261">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M262">View MathML</a> are the minimal and maximal fixed points of Q on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M263">View MathML</a>, respectively. By (3.5), they are the minimal and maximal mild solutions of the Cauchy problem (1.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M263">View MathML</a>, respectively. □

Corollary 3.2LetXbe an ordered Banach space, whose positive conePis regular. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>) is positive, the Cauchy problem (1.1) has a lower solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M147">View MathML</a>and an upper solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M148">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M149">View MathML</a>, (H1), (H2), and (H3) hold. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169">View MathML</a>, which can be obtained by a monotone iterative procedure starting from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169">View MathML</a>, respectively.

Proof Since P is regular, any ordered-monotonic and ordered-bounded sequence in X is convergent. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M60">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M275">View MathML</a> be an increasing or decreasing sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M276">View MathML</a>. By (H1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M277">View MathML</a> is an ordered-monotonic and ordered-bounded sequence in X. Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M278">View MathML</a>. By the properties of the measure of noncompactness, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M279">View MathML</a>

(3.16)

So, (H4) holds. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M280">View MathML</a> be an increasing or decreasing sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M263">View MathML</a>. By (H2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M165">View MathML</a> is an ordered-monotonic and ordered-bounded sequence in X. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M165">View MathML</a> is precompact in X. Thus, (H5) holds. By Theorem 3.1, the proof is then complete. □

Corollary 3.3LetXbe an ordered and weakly sequentially complete Banach space, whose positive conePis normal with normal constantN. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>) is positive, the Cauchy problem (1.1) has a lower solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M147">View MathML</a>and an upper solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M148">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M149">View MathML</a>, (H1), (H2), and (H3) hold. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169">View MathML</a>, which can be obtained by a monotone iterative procedure starting from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169">View MathML</a>, respectively.

Proof In an ordered and weakly sequentially complete Banach space, the normal cone P is regular. Then the proof is complete. □

Corollary 3.4LetXbe an ordered and reflective Banach space, whose positive conePis normal with normal constantN. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>) is positive, the Cauchy problem (1.1) has a lower solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M147">View MathML</a>and an upper solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M148">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M149">View MathML</a>, (H1), (H2), and (H3) hold. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169">View MathML</a>, which can be obtained by a monotone iterative procedure starting from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M169">View MathML</a>, respectively.

Proof In an ordered and reflective Banach space, the normal cone P is regular. Then the proof is complete. □

4 Examples

Example 4.1 In order to illustrate our results, we consider the following impulsive fractional partial differential equation with nonlocal initial condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M302">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M304">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M305">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M35">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M307">View MathML</a> is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M308">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M98">View MathML</a>) is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M310">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M311">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M312">View MathML</a>. Then X is a Banach space, and P is a regular cone in X. Define the operator A as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M313">View MathML</a>

then −A generate an analytic semigroup of uniformly bounded linear operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>) in X (see [11]). By the maximum principle, we can easily find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M6">View MathML</a>) is a positive semigroup. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M320">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M321">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M322">View MathML</a>, then the system (4.1) can be reformulated as the problem (1.1) in X. It is easy to find that (H2) holds. Moreover, we assume that the following conditions hold:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M323">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M324">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M325">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M326">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M327">View MathML</a>.

(b) There exists w such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M328">View MathML</a>

(4.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M329">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M327">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M324">View MathML</a>), w is continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M332">View MathML</a>, left continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M18">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M334">View MathML</a> exists, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M336">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M337">View MathML</a> are continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M332">View MathML</a>.

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M339">View MathML</a> is continuous on any bounded and ordered interval.

(d) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M341">View MathML</a> on a bounded and ordered interval, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M191">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/71/mathml/M343">View MathML</a>

(4.3)

Theorem 4.2If (a)-(d) are satisfied, then the system (4.1) has the minimal and maximal mild solutions between 0 andw.

Proof By (a) and (b), we know 0 and w are the lower and upper solutions of the problem (1.1), respectively. (c) implies that (H1) are satisfied. (d) implies that (H3) are satisfied. Then by Corollary 3.2, the system (4.1) has the minimal and maximal mild solutions between 0 and w. □

Competing interests

The author declares that she has no competing interests.

Acknowledgements

This research was supported by Talent Introduction Scientific Research Foundation of Northwest University for Nationalities.

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