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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 605614, 9 pages
http://dx.doi.org/10.1155/2011/605614
Research Article

Existence Results for Singular Boundary Value Problem of Nonlinear Fractional Differential Equation

Department of Mathematics, Shandong University of Science and Technology, Qingdao 266510, China

Received 27 December 2010; Revised 2 March 2011; Accepted 3 March 2011

Academic Editor: ElenaΒ Braverman

Copyright Β© 2011 Yujun Cui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By applying a fixed point theorem for mappings that are decreasing with respect to a cone, this paper investigates the existence of positive solutions for the nonlinear fractional boundary value problem: 𝐷𝛼0+𝑒(𝑑)+𝑓(𝑑,𝑒(𝑑))=0, 0<𝑑<1, 𝑒(0)=𝑒′(0)=𝑒′(1)=0, where 2<𝛼<3, 𝐷𝛼0+ is the Riemann-Liouville fractional derivative.

1. Introduction

Many papers and books on fractional calculus differential equation have appeared recently. Most of them are devoted to the solvability of the linear initial fractional equation in terms of a special function [1–4]. Recently, there has been significant development in the existence of solutions and positive solutions to boundary value problems for fractional differential equations by the use of techniques of nonlinear analysis (fixed point theorems, Leray-Schauder theory, etc.), see [5, 6] and the references therein.

In this paper, we consider the following boundary value problems of the nonlinear fractional differential equation 𝐷𝛼0+𝑒(𝑑)+𝑓(𝑑,𝑒(𝑑))=0,0<𝑑<1,2<𝛼<3,𝑒(0)=π‘’ξ…ž(0)=π‘’ξ…ž(1)=0,(1.1) where 𝐷𝛼0+ is the standard Riemann-Liouville fractional derivative and 𝑓(𝑑,π‘₯) is singular at π‘₯=0. Our assumptions throughout are(H1)𝑓(𝑑,π‘₯)∢(0,1)Γ—(0,∞)β†’[0,∞) is continuous,(H2)𝑓(𝑑,π‘₯) is decreasing in π‘₯, for each fixed 𝑑,(H3)limπ‘₯β†’0+𝑓(𝑑,π‘₯)=∞ and limπ‘₯β†’βˆžπ‘“(𝑑,π‘₯)=0, uniformly on compact subsets of (0,1), and(H4)∫0<10𝑓(𝑑,π‘žπœƒ(𝑑))𝑑𝑑<∞ for all πœƒ>0 and π‘žπœƒ as defined in (3.1).

The seminal paper by Gatica et al. [7] in 1989 has had a profound impact on the study of singular boundary value problems for ordinary differential equations (ODEs). They studied singularities of the type in (H1)–(H4) for second order Sturm-Louiville problems, and their key result hinged on an application of a particular fixed point theorem for operators which are decreasing with respect to a cone. Various authors have used these techniques to study singular problems of various types. For example, Henderson and Yin [8] as well as Eloe and Henderson [9, 10] have studied right focal, focal, conjugate, and multipoint singular boundary value problems for ODEs. However, as far as we know, no paper is concerned with boundary value problem for fractional differential equation by using this theorem. As a result, the goal of this paper is to fill the gap in this area.

Motivated by the above-mentioned papers and [11], the purpose of this paper is to establish the existence of solutions for the boundary value problem (1.1) by the use of a fixed point theorem used in [7, 11]. The paper has been organized as follows. In Section 2, we give basic definitions and provide some properties of the corresponding Green's function which are needed later. We also state the fixed point theorem from [7] for mappings that are decreasing with respect to a cone. In Section 3, we formulate two lemmas which establish a priori upper and lower bounds on solutions of (1.1). We then state and prove our main existence theorem.

For fractional differential equation and applications, we refer the reader to [1–3]. Concerning boundary value problems (1.1) with ordinary derivative (not fractional one), we refer the reader to [12, 13].

2. Some Preliminaries and a Fixed Point Theorem

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the literature.

Definition 2.1 (see [3]). The Riemann-Liouville fractional integral of order 𝛼>0 of a function π‘“βˆΆ(0,∞)→𝑅 is given by 𝐼𝛼0+1𝑓(𝑑)=ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓(𝑠)𝑑𝑠,(2.1) provided that the right-hand side is pointwise defined on (0,∞).

Definition 2.2 (see [3]). The Riemann-Liouville fractional derivative of order 𝛼>0 of a continuous function π‘“βˆΆ(0,∞)→𝑅 is given by 𝐷𝛼0+1𝑓(𝑑)=𝑑Γ(π‘›βˆ’π›Ό)ξ‚π‘‘π‘‘π‘›ξ€œπ‘‘0𝑓(𝑠)(π‘‘βˆ’π‘ )π›Όβˆ’π‘›+1𝑑𝑠,(2.2) where π‘›βˆ’1≀𝛼<𝑛, provided that the right-hand side is pointwise defined on (0,∞).

Definition 2.3. By a solution of the boundary value problem (1.1) we understand a function π‘’βˆˆπΆ[0,1] such that 𝐷𝛼0+𝑒 is continuous on (0, 1) and 𝑒 satisfies (1.1).

Lemma 2.4 (see [3]). Assume that π‘’βˆˆπΆ(0,1)∩𝐿(0,1) with a fractional derivative of order 𝛼>0 that belongs to 𝐢(0,1)∩𝐿(0,1). Then 𝐼𝛼0+𝐷𝛼0+𝑒(𝑑)=𝑒(𝑑)+𝑐1π‘‘π›Όβˆ’1+𝑐2π‘‘π›Όβˆ’2+β‹―+π‘π‘π‘‘π›Όβˆ’π‘(2.3) for some π‘π‘–βˆˆπ‘…,𝑖=1,…,𝑁, 𝑁=[𝛼].

Lemma 2.5. Given π‘“βˆˆπΆ[0,1], and 2<𝛼<3, the unique solution of 𝐷𝛼0+𝑒(𝑑)+𝑓(𝑑)=0,0<𝑑<1,𝑒(0)=π‘’ξ…ž(0)=π‘’ξ…ž(1)=0(2.4) is ξ€œπ‘’(𝑑)=10𝐺(𝑑,𝑠)𝑓(𝑠)𝑑𝑠,(2.5) where ⎧βŽͺ⎨βŽͺβŽ©π‘‘πΊ(𝑑,𝑠)=π›Όβˆ’1(1βˆ’π‘ )π›Όβˆ’2βˆ’(π‘‘βˆ’π‘ )π›Όβˆ’1Γ𝑑(𝛼),0≀𝑠≀𝑑≀1,π›Όβˆ’1(1βˆ’π‘ )π›Όβˆ’2Ξ“(𝛼),0≀𝑑≀𝑠≀1.(2.6)

Proof. We may apply Lemma 2.4 to reduce (2.4) to an equivalent integral equation 𝑒(𝑑)=βˆ’πΌπ›Ό0+𝑓(𝑑)+𝑐1π‘‘π›Όβˆ’1+𝑐2π‘‘π›Όβˆ’2+𝑐3π‘‘π›Όβˆ’3(2.7) for some π‘π‘–βˆˆπ‘…,𝑖=1,2,3. From 𝑒(0)=π‘’ξ…ž(0)=π‘’ξ…ž(1)=0, one has 𝑐1=ξ€œ10(1βˆ’π‘ )π›Όβˆ’2Ξ“(𝛼)𝑓(𝑠)𝑑𝑠,𝑐2=𝑐3=0.(2.8) Therefore, the unique solution of problem (2.4) is ξ€œπ‘’(𝑑)=10π‘‘π›Όβˆ’1(1βˆ’π‘ )π›Όβˆ’21Ξ“(𝛼)𝑓(𝑠)π‘‘π‘ βˆ’ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1=ξ€œπ‘“(𝑠)𝑑𝑠𝑑0ξ‚Έπ‘‘π›Όβˆ’1(1βˆ’π‘ )π›Όβˆ’2βˆ’(π‘‘βˆ’π‘ )π›Όβˆ’1ξ‚Ήπ‘“ξ€œΞ“(𝛼)(𝑠)𝑑𝑠+1π‘‘π‘‘π›Όβˆ’1(1βˆ’π‘ )π›Όβˆ’2𝑓=ξ€œΞ“(𝛼)(𝑠)𝑑𝑠10𝐺(𝑑,𝑠)𝑓(𝑠)𝑑𝑠.(2.9)

Lemma 2.6. The function 𝐺(𝑑,𝑠) defined by (2.6) satisfies the following conditions: (i)𝐺(𝑑,𝑠)>0, 0<𝑑,𝑠<1,(ii)π‘ž(𝑑)𝐺(1,𝑠)≀𝐺(𝑑,𝑠)≀𝐺(1,𝑠)=𝑠(1βˆ’π‘ )π›Όβˆ’2/(Ξ“(𝛼)) for 0≀𝑑,𝑠≀1, where π‘ž(𝑑)=π‘‘π›Όβˆ’1.

Proof. Observing the expression of 𝐺(𝑑,𝑠), it is clear that 𝐺(𝑑,𝑠)>0 for 0<𝑑,𝑠<1. For given π‘ βˆˆ(0,1), 𝐺(𝑑,𝑠) is increasing with respect to 𝑑. Consequently, 𝐺(𝑑,𝑠)≀𝐺(1,𝑠) for 0≀𝑑,𝑠≀1.
If 𝑠≀𝑑, we have 𝐺(𝑑,𝑠)=𝑑(π‘‘βˆ’π‘‘π‘ )π›Όβˆ’2βˆ’(π‘‘βˆ’π‘ )(π‘‘βˆ’π‘ )π›Όβˆ’2Ξ“β‰₯(𝛼)𝑑(π‘‘βˆ’π‘‘π‘ )π›Όβˆ’2βˆ’(π‘‘βˆ’π‘ )(π‘‘βˆ’π‘‘π‘ )π›Όβˆ’2=Ξ“(𝛼)π‘ π‘‘π›Όβˆ’2(1βˆ’π‘ )π›Όβˆ’2β‰₯Ξ“(𝛼)π‘ π‘‘π›Όβˆ’1(1βˆ’π‘ )π›Όβˆ’2Ξ“(𝛼)=π‘ž(𝑑)𝐺(1,𝑠).(2.10) If 𝑑≀𝑠, we have 𝑑𝐺(𝑑,𝑠)=π›Όβˆ’1(1βˆ’π‘ )π›Όβˆ’2Ξ“β‰₯(𝛼)π‘ π‘‘π›Όβˆ’1(1βˆ’π‘ )π›Όβˆ’2Ξ“(𝛼)=π‘ž(𝑑)𝐺(1,𝑠).(2.11)

Let 𝐸 be a Banach space, π‘ƒβŠ‚πΈ be a cone in 𝐸. Every cone 𝑃 in 𝐸 defines a partial ordering in 𝐸 given by π‘₯≀𝑦 if and only if π‘¦βˆ’π‘₯βˆˆπ‘ƒ. If π‘₯≀𝑦 and π‘₯≠𝑦, we write π‘₯<𝑦. A cone 𝑃 is said to be normal if there exists a constant 𝑁>0 such that πœƒβ‰€π‘₯≀𝑦 implies β€–π‘₯‖≀𝑁‖𝑦‖. If 𝑃 is normal, then every order interval [π‘₯,𝑦]={π‘§βˆˆπΈβˆ£π‘₯≀𝑧≀𝑦} is bounded. For the concepts and properties about the cone theory we refer to [14, 15].

Next we state the fixed point theorem due to Gatica et al. [7] which is instrumental in proving our existence results.

Theorem 2.7 (Gatica-Oliker-Waltman fixed point theorem). Let 𝐸 be a Banach space, π‘ƒβŠ‚πΈ be a normal cone, and π·βŠ‚π‘ƒ be such that if π‘₯,π‘¦βˆˆπ· with π‘₯≀𝑦, then [π‘₯,𝑦]βŠ‚π·. Let π‘‡βˆΆπ·β†’π‘ƒ be a continuous, decreasing mapping which is compact on any closed order interval contained in 𝐷, and suppose there exists an π‘₯0βŠ‚π· such that 𝑇2π‘₯0 is defined (where 𝑇2π‘₯0=𝑇(𝑇π‘₯0)) and 𝑇π‘₯0, 𝑇2π‘₯0 are order comparable to π‘₯0. Then 𝑇 has a fixed point in 𝐷 provided that either: (i)𝑇π‘₯0≀π‘₯0 and 𝑇2π‘₯0≀π‘₯0;(ii)π‘₯0≀𝑇π‘₯0 and π‘₯0≀𝑇2π‘₯0; or(iii)The complete sequence of iterates {𝑇𝑛π‘₯0}βˆžπ‘›=0 is defined and there exists 𝑦0∈𝐷 such that 𝑇𝑦0∈𝐷 with 𝑦0≀𝑇𝑛π‘₯0 for all π‘›βˆˆN

3. Main Results

In this section, we apply Theorem 2.7 to a sequence of operators that are decreasing with respect to a cone. These obtained fixed points provide a sequence of iterates which converges to a solution of (1.1).

Let the Banach space 𝐸=𝐢[0,1] with the maximum norm ‖𝑒‖=maxπ‘‘βˆˆ[0,1]|𝑒(𝑑)|, and let 𝑃={π‘’βˆˆπΈβˆ£π‘’(𝑑)β‰₯0,π‘‘βˆˆ[0,1]}. 𝑃 is a norm cone in 𝐸. For πœƒ>0, let π‘žπœƒ(𝑑)=πœƒβ‹…π‘ž(𝑑),(3.1) where π‘ž(𝑑) is given in Lemma 2.6. Define π·βŠ‚π‘ƒ by 𝐷=π‘’βˆˆπ‘ƒβˆ£βˆƒπœƒ(𝑒)>0suchthat𝑒(𝑑)β‰₯π‘žπœƒ[]ξ€Ύ(𝑑),π‘‘βˆˆ0,1,(3.2) and the integral operator π‘‡βˆΆπ·β†’π‘ƒ by ξ€œ(𝑇𝑒)(𝑑)=10𝐺(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠,(3.3) where 𝐺(𝑑,𝑠) is given in (2.6). It suffices to define 𝐷 as above, since the singularity in 𝑓 precludes us from defining 𝑇 on all of 𝑃. Furthermore, it can easily be verified that 𝑇 is well defined. In fact, note that for π‘’βˆˆπ· there exists πœƒ(𝑒)>0 such that 𝑒(𝑑)β‰₯π‘žπœƒ(𝑑) for all π‘‘βˆˆ[0,1]. Since 𝑓(𝑑,π‘₯) decreases with respect to π‘₯, we see 𝑓(𝑑,𝑒(𝑑))≀𝑓(𝑑,π‘žπœƒ(𝑑)) for π‘‘βˆˆ[0,1]. Thus, ξ€œ0≀10ξ€œπΊ(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠≀10𝑓𝑠,π‘žπœƒξ€Έ(𝑠)𝑑𝑠<∞.(3.4) Similarly, 𝑇 is decreasing with respect to 𝐷.

Lemma 3.1. π‘’βˆˆπ· is a solution of (1.1) if and only if 𝑇𝑒=𝑒.

Proof. One direction of the lemma is obviously true. To see the other direction, let π‘’βˆˆπ·. Then ∫(𝑇𝑒)(𝑑)=10𝐺(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠, and 𝑇𝑒 satisfies (1.1). Moreover, by Lemma 2.6, we have ξ€œ(𝑇𝑒)(𝑑)=10ξ€œπΊ(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠β‰₯π‘ž(𝑑)10𝐺[].(1,𝑠)𝑓(𝑠,𝑒(𝑠))𝑑𝑠=π‘ž(𝑑)‖𝑇𝑒‖,βˆ€π‘‘βˆˆ0,1(3.5) Thus, there exists some πœƒ(𝑇𝑒) such that (𝑇𝑒)(𝑑)β‰₯π‘žπœƒ(𝑑), which implies that π‘‡π‘’βˆˆπ·. That is, π‘‡βˆΆπ·β†’π·.
We now present two lemmas that are required in order to apply Theorem 2.7. The first establishes a priori upper bound on solutions, while the second establishes a priori lower bound on solutions.

Lemma 3.2. If 𝑓 satisfies (H1)–(H4), then there exists an 𝑆>0 such that ‖𝑒‖≀𝑆 for any solution π‘’βˆˆπ· of (1.1).

Proof. For the sake of contradiction, suppose that the conclusion is false. Then there exists a sequence {𝑒𝑛}βˆžπ‘›=1 of solutions to (1.1) such that ‖𝑒𝑛‖≀‖𝑒𝑛+1β€– with limπ‘›β†’βˆžβ€–π‘’π‘›β€–=∞. Note that for any solution π‘’π‘›βˆˆπ· of (1.1), by (3.5), we have 𝑒𝑛(𝑑)=𝑇𝑒𝑛‖‖𝑒(𝑑)β‰₯π‘ž(𝑑)𝑛‖‖[],π‘‘βˆˆ0,1,𝑛β‰₯1.(3.6) Then, assumptions (H2) and (H4) yield, for 0≀𝑑≀1 and all 𝑛β‰₯1, 𝑒𝑛(𝑑)=π‘‡π‘’π‘›ξ€Έξ€œ(𝑑)=10𝐺(𝑑,𝑠)𝑓𝑠,𝑒𝑛≀1(𝑠)π‘‘π‘ ξ€œΞ“(𝛼)10𝑠(1βˆ’π‘ )π›Όβˆ’2𝑓𝑠,π‘žβ€–π‘’1β€–ξ€Έ(𝑠)𝑑𝑠=𝑁,(3.7) for some 0<𝑁<+∞. In particular, ‖𝑒𝑛‖≀𝑁, for all 𝑛β‰₯1, which contradicts limπ‘›β†’βˆžβ€–π‘’π‘›β€–=∞.

Lemma 3.3. If 𝑓 satisfies (H1)–(H4), then there exists an 𝑅>0 such that ‖𝑒‖β‰₯𝑅 for any solution π‘’βˆˆπ· of (1.1).

Proof. For the sake of contradiction, suppose 𝑒𝑛(𝑑)β†’0 uniformly on [0,1] as π‘›β†’βˆž. Let 𝑀=inf{𝐺(𝑑,𝑠)∢(𝑑,𝑠)∈[1/4,3/4]Γ—[1/4,3/4]}>0. From (H3), we see that limπ‘₯β†’0+𝑓(𝑑,π‘₯)=∞ uniformly on compact subsets of (0,1). Hence, there exists some 𝛿>0 such that for π‘‘βˆˆ[1/4,3/4] and 0<π‘₯<𝛿, we have 𝑓(𝑑,π‘₯)β‰₯2/𝑀. On the other hand, there exists an 𝑛0βˆˆπ‘ such that 𝑛β‰₯𝑛0 implies 0<𝑒𝑛(𝑑)<𝛿/2, for π‘‘βˆˆ(0,1). So, for π‘‘βˆˆ[1/4,3/4] and 𝑛β‰₯𝑛0, 𝑒𝑛(𝑑)=π‘‡π‘’π‘›ξ€Έξ€œ(𝑑)=10𝐺(𝑑,𝑠)𝑓𝑠,π‘’π‘›ξ€Έξ€œ(𝑠)𝑑𝑠β‰₯3/41/4𝐺(𝑑,𝑠)𝑓𝑠,π‘’π‘›ξ€Έξ€œ(𝑠)𝑑𝑠β‰₯𝑀3/41/4𝑓𝛿𝑠,2ξ‚ξ€œπ‘‘π‘ β‰₯𝑀3/41/42𝑀𝑑s=1.(3.8) But this contradicts the assumption that ‖𝑒𝑛‖→0 uniformly on [0,1] as π‘›β†’βˆž. Hence, there exists an 𝑅>0 such that 𝑅≀‖𝑒‖.

We now present the main result of the paper.

Theorem 3.4. If 𝑓 satisfies (H1)–(H4), then (1.1) has at least one positive solution.

Proof. For each 𝑛β‰₯1, defined π‘£π‘›βˆΆ[0,1]β†’[0,+∞) by π‘£π‘›ξ€œ(𝑑)=10𝐺(𝑑,𝑠)𝑓(𝑠,𝑛)𝑑𝑠.(3.9) By conditions (H1)–(H4), for 𝑛β‰₯1, 0<𝑣𝑛+1(𝑑)≀𝑣𝑛],(𝑑),on(0,1(3.10)limπ‘›β†’βˆžπ‘£π‘›[].(𝑑)=0uniformlyon0,1(3.11) Now define a sequence of functions π‘“π‘›βˆΆ(0,1)Γ—[0,+∞), 𝑛β‰₯1, by 𝑓𝑛(𝑑,π‘₯)=𝑓𝑑,maxπ‘₯,𝑣𝑛(𝑑)ξ€Ύξ€Έ.(3.12) Then, for each 𝑛β‰₯1, 𝑓𝑛 is continuous and satisfies (H2). Furthermore, for 𝑛β‰₯1, 𝑓𝑛𝑓(𝑑,π‘₯)≀𝑓(𝑑,π‘₯)on(0,1)Γ—(0,+∞),𝑛(𝑑,π‘₯)≀𝑓𝑑,𝑣𝑛(𝑑)on(0,1)Γ—(0,+∞).(3.13) Note that 𝑓𝑛 has effectively β€œremoved the singularity” in 𝑓(𝑑,π‘₯) at π‘₯=0, then we define a sequence of operators π‘‡π‘›βˆΆπ‘ƒβ†’π‘ƒ, 𝑛β‰₯1, by ξ€·π‘‡π‘›π‘’ξ€Έξ€œ(𝑑)=10𝐺(𝑑,𝑠)𝑓𝑛(𝑠,𝑒(𝑠))𝑑𝑠,π‘’βˆˆπ‘ƒ.(3.14) From standard arguments involving the Arzela-Ascoli Theorem, we know that each 𝑇𝑛 is in fact a compact mapping on 𝑃. Furthermore, 𝑇𝑛(0)β‰₯0 and 𝑇2𝑛(0)β‰₯0. By Theorem 2.7, for each 𝑛β‰₯1, there exists π‘’π‘›βˆˆπ‘ƒ such that 𝑇𝑛𝑒𝑛(π‘₯)=𝑒𝑛(𝑑) for π‘‘βˆˆ[0,1]. Hence, for each 𝑛β‰₯1, 𝑒𝑛 satisfies the boundary conditions of the problem. In addition, for each 𝑒𝑛, ξ€·π‘‡π‘›π‘’π‘›ξ€Έξ€œ(𝑑)=10𝐺(𝑑,𝑠)𝑓𝑛𝑠,π‘’π‘›ξ€Έξ€œ(𝑠)𝑑𝑠=10𝐺(𝑑,𝑠)𝑓𝑛𝑒𝑠,max𝑛(𝑠),π‘£π‘›β‰€ξ€œ(𝑠)𝑑𝑠10𝐺(𝑑,𝑠)𝑓𝑛𝑠,𝑣𝑛(𝑠)𝑑𝑠≀𝑇𝑣𝑛(𝑑),(3.15) which implies 𝑒𝑛𝑇(𝑑)=𝑛𝑒𝑛(𝑑)≀𝑇𝑣𝑛[](𝑑),π‘‘βˆˆ0,1,π‘›βˆˆN.(3.16) Arguing as in Lemma 3.2 and using (3.11), it is fairly straightforward to show that there exists an 𝑆>0 such that ‖𝑒𝑛‖≀𝑆 for all π‘›βˆˆπ‘. Similarly, we can follow the argument of Lemma 3.3 and (3.5) to show that there exists an 𝑅>0 such that 𝑒𝑛[](𝑑)β‰₯π‘ž(𝑑)𝑅,on0,1,for𝑛β‰₯1.(3.17) Since π‘‡βˆΆπ·β†’π· is a compact mapping, there is a subsequence of {𝑇𝑒𝑛} which converges to some π‘’βˆ—βˆˆπ·. We relabel the subsequence as the original sequence so that π‘‡π‘’π‘›β†’π‘’βˆ— as π‘›β†’βˆž.
To conclude the proof of this theorem, we need to show that limπ‘›β†’βˆžβ€–β€–π‘‡π‘’π‘›βˆ’π‘’π‘›β€–β€–=0.(3.18) To that end, fixed πœƒ=𝑅, and let πœ€>0 be give. By the integrability condition (H4), there exists 0<𝛿<1 such that ξ€œπ›Ώ0𝑠(1βˆ’π‘ )π›Όβˆ’2𝑓𝑠,π‘žπœƒξ€Έ(𝑠)𝑑𝑠<Ξ“(𝛼)2πœ€.(3.19) Further, by (3.11), there exists an 𝑛0 such that, for 𝑛β‰₯𝑛0, 𝑣𝑛(𝑑)β‰€π‘žπœƒ[],(𝑑)on𝛿,1(3.20) so that 𝑣𝑛(𝑑)β‰€π‘žπœƒ(𝑑)≀𝑒𝑛[].(𝑑)on𝛿,1(3.21) Thus, for π‘ βˆˆ[𝛿,1] and 𝑛β‰₯𝑛0, 𝑓𝑛𝑠,𝑒𝑛𝑒(𝑠)=𝑓𝑠,max𝑛(𝑠),𝑣𝑛(𝑠)ξ€Ύξ€Έ=𝑓𝑠,𝑒𝑛(𝑠),(3.22) and for π‘‘βˆˆ[0,1], 𝑇𝑒𝑛(𝑑)βˆ’π‘’π‘›(𝑑)=𝑇𝑒𝑛(𝑑)βˆ’π‘‡π‘›π‘’π‘›=ξ€œ(𝑑)10𝑓𝐺(𝑑,𝑠)𝑠,𝑒𝑛(𝑠)βˆ’π‘“π‘›ξ€·π‘ ,𝑒𝑛(𝑠)𝑑𝑠.(3.23) Thus, for π‘‘βˆˆ[0,1], ||𝑇𝑒𝑛(𝑑)βˆ’π‘’π‘›||≀1(𝑑)ξ‚Έξ€œΞ“(𝛼)𝛿0𝑠(1βˆ’π‘ )π›Όβˆ’2𝑓𝑠,π‘’π‘›ξ€Έξ€œ(𝑠)𝑑𝑠+𝛿0𝑠(1βˆ’π‘ )π›Όβˆ’2𝑓𝑒𝑠,max𝑛(𝑠),𝑣𝑛≀1(𝑠)ξ€Ύξ€Έπ‘‘π‘ ξ‚Έξ€œΞ“(𝛼)𝛿0𝑠(1βˆ’π‘ )π›Όβˆ’2𝑓𝑠,π‘’π‘›ξ€Έξ€œ(𝑠)𝑑𝑠+𝛿0𝑠(1βˆ’π‘ )π›Όβˆ’2𝑓𝑠,𝑒𝑛≀2(𝑠)π‘‘π‘ Ξ“ξ€œ(𝛼)𝛿0𝑠(1βˆ’π‘ )π›Όβˆ’2𝑓𝑠,π‘žπœƒξ€Έ(𝑠)𝑑𝑠<πœ€.(3.24) Since π‘‘βˆˆ[0,1] was arbitrary, we conclude that β€–π‘‡π‘’π‘›βˆ’π‘’π‘›β€–β‰€πœ€ for all 𝑛β‰₯𝑛0. Hence, π‘’βˆ—βˆˆ[π‘žπ‘…,𝑆] and for π‘‘βˆˆ[0,1]π‘‡π‘’βˆ—ξ‚΅(𝑑)=𝑇limπ‘›β†’βˆžπ‘‡π‘’π‘›ξ‚Άξ‚΅(𝑑)=𝑇limπ‘›β†’βˆžπ‘’π‘›ξ‚Ά(𝑑)=limπ‘›β†’βˆžπ‘‡π‘’π‘›=π‘’βˆ—(𝑑).(3.25)

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. The Project Supported by the National Science Foundation of China (10971179) and Research Award Fund for Outstanding Young Scientists of Shandong Province (BS2010SF023).

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