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Journal of Applied Mathematics
VolumeΒ 2011, Article IDΒ 352341, 22 pages
http://dx.doi.org/10.1155/2011/352341
Research Article

Positive Solution of Singular Fractional Differential Equation in Banach Space

1Department of Mathematics, Central South University, Changsha, Hunan 410075, China
2Faculty of Science, Hunan Institute of Engineering, Xiangtan, Hunan 411104, China

Received 17 June 2011; Accepted 9 September 2011

Academic Editor: J.Β Biazar

Copyright Β© 2011 Jianxin Cao and Haibo Chen. 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

We investigated a singular multipoint boundary value problem for fractional differential equation in Banach space. The nonlinear term 𝑓(𝑑,π‘₯,𝑦) is positive and singular at π‘₯=πœƒ and (or) 𝑦=πœƒ. Employing regularization, sequential techniques, and diagonalization methods, we obtained some new existence results of positive solution.

1. Introduction

Recently, fractional differential equations have been investigated extensively. The motivation for those works rises from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, chemistry, aerodynamics, and electrodynamics of the complex medium. For examples and details, see [1–5] and the references therein.

Prompted by the application of multipoint boundary value problem (BVP for short) to applied mathematics and physics, these problems have provoked a great deal of attention by many authors. Here, for fractional differential equations, we refer the reader to [6–12]. Rehman and Khan [7] studied the problem𝐷𝛼𝑦(𝑑)+𝑓𝑑,𝑦(𝑑),𝐷𝛽𝑦(𝑑)=0,π‘‘βˆˆ(0,1),𝑦(0)=0,𝐷𝛽𝑦(1)βˆ’π‘šβˆ’2𝑖=1π‘π‘–π·π›½π‘¦ξ€·πœ‰π‘–ξ€Έ=𝑦0,(1.1) where 1<𝛼≀2, 0<π›½β‰€π›Όβˆ’1, 𝑏𝑖β‰₯0, 0<πœ‰π‘–<1, (𝑖=1,2,…,π‘šβˆ’2) with βˆ‘π›Ύ=π‘šβˆ’2𝑖=1π‘π‘–πœ‰π‘–π›Όβˆ’π›½βˆ’1<1, and 𝐷𝛽 represents the standard Riemann-Liouville fractional derivative. The existence and uniqueness of solutions were obtained, by means of Schauder fixed-point theorem and Banach contraction principle. Importantly, they gave the Green function of the multipoint BVP (1.1). But they have not proved the positivity of Green function, so the existence of positive solution is unobtainable. However, only positive solutions are useful for many applications, as some physicists pointed out.

The authors of [13–17] investigated singular problem for fractional differential equations with bounded domain. In particular, Agarwal et al. [13] considered the following Dirichlet problem: 𝐷𝛼𝑦(𝑑)+𝑓(𝑑,𝑦(𝑑),π·πœ‡π‘¦(𝑑))=0,𝑦(0)=𝑦(1)=0,(1.2) where 1<𝛼≀2, 0<πœ‡β‰€π›Όβˆ’1. 𝑓(𝑑,π‘₯,𝑦) satisfies the CarathΓ©odory conditions and is singular at π‘₯=0. In order to overcome the singularity, they used regularization and sequential techniques for the existence of a positive solution.

When the domain where the problem is considered is unbounded, there are few papers about BVP for fractional differential equations in literatures. This situation has changed recently. One can find some works, for example, see [18–22].

In [21], the following BVP:𝐷𝛼𝑦(𝑑)+𝑓(𝑑,𝑦(𝑑))=0,π‘‘βˆˆ(0,+∞),π›Όβˆˆ(1,2),𝑦(0)=0,lim𝑑→+βˆžπ·π›Όβˆ’1𝑦(𝑑)=𝛽𝑦(πœ‰)(1.3) was studied. Using the equicontinuity on any compact intervals and the equiconvergence at infinity of a bounded set, the authors proved that the corresponding operator was completely continuous, then the existence of solutions was obtained by the Leray-Schauder nonlinear alternative theorem.

Let (𝐸,β€–β‹…β€–) be a real Banach space. 𝑃 is a cone in 𝐸 which defines a partial ordering in 𝐸 by π‘₯≀𝑦 if and only if π‘¦βˆ’π‘₯βˆˆπ‘ƒ. 𝑃 is said to be normal if there exists a positive constant 𝑁 such that πœƒβ‰€π‘₯≀𝑦 implies β€–π‘₯‖≀𝑁‖𝑦‖, where πœƒ denotes the zero element of 𝐸, and the smallest 𝑁 is called the normal constant of 𝑃 (it is clear that 𝑁β‰₯1). If π‘₯≀𝑦 and π‘₯≠𝑦, we write π‘₯<𝑦. Let 𝑃+=𝑃⧡{πœƒ}. So, π‘₯βˆˆπ‘ƒ+ if and only if π‘₯>πœƒ. For details on cone theory, see [23].

In this paper, we are concerned with the existence of positive solution of a BVP for fractional differential equation with bounded domain𝐷𝛼𝑦(𝑑)+𝑓𝑑,𝑦(𝑑),𝐷𝛽[],𝑦(𝑑)=πœƒ,a.e.π‘‘βˆˆ0,𝑇(1.4)𝑦(0)=πœƒ,𝐷𝛽𝑦(𝑇)βˆ’π‘šβˆ’2𝑖=1π‘Žπ‘–π‘¦ξ€·πœ‰π‘–ξ€Έβˆ’π‘šβˆ’2𝑖=1π‘π‘–π·π›½π‘¦ξ€·πœ‰π‘–ξ€Έ=𝑦0,(1.5) or with unbounded domain𝐷𝛼𝑦(𝑑)+𝑓𝑑,𝑦(𝑑),𝐷𝛽[𝑦(𝑑)=πœƒ,a.e.π‘‘βˆˆ0,∞),(1.6)𝑦(0)=πœƒ,lim𝑑→+βˆžπ·π›½π‘¦(𝑑)βˆ’π‘šβˆ’2𝑖=1π‘Žπ‘–π‘¦ξ€·πœ‰π‘–ξ€Έβˆ’π‘šβˆ’2𝑖=1π‘π‘–π·π›½π‘¦ξ€·πœ‰π‘–ξ€Έ=𝑦0.(1.7) Here, 1<𝛼≀2, 0<π›½β‰€π›Όβˆ’1, πœ‰π‘–>0, π‘Žπ‘–,𝑏𝑖β‰₯0(𝑖=1,2,…,π‘šβˆ’2), 𝑦0β‰₯πœƒ are real numbers, and 𝐷𝛼 is the standard Riemann-Liouville fractional derivative. And π‘“βˆΆ[0,+∞)×𝑃+×𝑃+→𝑃+ is singular at π‘₯=πœƒ and 𝑦=πœƒ and satisfies other conditions which will be specified later. In addition, 𝑓(𝑑,π‘₯,𝑦) is the CarathΓ©odory function.

We say that 𝑓 satisfies the CarathΓ©odory conditions on [0,+∞)×𝐡, 𝐡=𝑃+×𝑃+(π‘“βˆˆCar([0,+∞)×𝐡)) if(i)𝑓(β‹…,π‘₯,𝑦)∢[0,+∞)→𝐸 is measurable for all (π‘₯,𝑦)∈𝐡,(ii)𝑓(𝑑,β‹…,β‹…)βˆΆπ΅β†’πΈ is continuous for a.e.π‘‘βˆˆ[0,+∞),(iii)for each compact set πΎβŠ‚π΅, there is a function πœ™πΎβˆˆπΏ1[0,+∞) such that ‖𝑓(𝑑,π‘₯,𝑦)β€–β‰€πœ™πΎ[(𝑑),fora.e.π‘‘βˆˆ0,+∞),βˆ€(π‘₯,𝑦)∈𝐾.(1.8)

No contribution exists, as far as we know, concerning the existence of positive solution of the problems (1.4)-(1.5) and (1.6)-(1.7). In the present paper, we consider, firstly, the case of bounded domain, that is, BVP (1.4)-(1.5), and give some existence results by means of regularization process combined with fixed-point theorem due to Krasnosel'skii. Then we investigate the BVP (1.6)-(1.7). As we know, [0,∞) is noncompact. In order to overcome these difficulties, based on the results of BVP (1.4)-(1.5), we use diagonalization process to establish the existence of positive solutions for BVP (1.6)-(1.7). Let us mention that this method was widely used for integer-order differential equations, see, for instance, [5, 22]. Using diagonalization process, Agarwal et al. [20] have considered a class of boundary value problems involving Riemann-Liouville fractional derivative on the half line. And Arara et al. [19] continued this study by considering a BVP with the Caputo fractional derivative.

The remainder of this paper is organized as follows. In Section 2, we introduced some notations, definitions, and preliminary facts about the fractional calculus, which are used in the next two sections. In Section 3, the case with bounded domain is considered. In Section 4, we discuss the existence of a positive solution for the BVP (1.6)-(1.7). We end this paper with giving an example to demonstrate the application of our results in Section 5.

2. Preliminaries

Now, we introduce the Riemann-Liouville fractional- (arbitrary)-order integral and derivative as follows.

Definition 2.1. The fractional- (arbitrary)-order integral of the function 𝑣(𝑑)∈𝐿1([0,𝑏],ℝ) of πœ‡βˆˆβ„+ is defined by πΌπœ‡1𝑣(𝑑)=ξ€œΞ“(πœ‡)𝑑0(π‘‘βˆ’π‘ )πœ‡βˆ’1𝑣(𝑠)𝑑𝑠,𝑑>0.(2.1)

Definition 2.2. The Riemann-Liouville fractional derivative of order πœ‡>0 for a function 𝑣(𝑑) given in the interval [0,∞) is defined by π·πœ‡1𝑣(𝑑)=𝑑Γ(π‘›βˆ’πœ‡)ξ‚π‘‘π‘‘π‘›ξ€œπ‘‘0(π‘‘βˆ’π‘ )π‘›βˆ’πœ‡βˆ’1𝑣(𝑠)𝑑𝑠(2.2) provided that the right hand side is point wise defined. Here, 𝑛=[πœ‡]+1 and [πœ‡] means the integral part of the number πœ‡, and Ξ“ is the Euler gamma function.

The following properties of the fractional calculus theory are well known, see, for example, [2, 4]:(i)𝐷𝛽𝐼𝛽𝑣(𝑑)=𝑣(𝑑) for a.e. π‘‘βˆˆ[0,𝑇], where 𝑣(𝑑)∈𝐿1[0,𝑇], 𝛽>0,(ii)𝐷𝛽𝑣(𝑑)=0 if and only if βˆ‘π‘£(𝑑)=𝑛𝑗=1π‘π‘—π‘‘π›½βˆ’π‘—, where 𝑐𝑗(𝑗=1,2,…,𝑛) are arbitrary constants, 𝑛=[𝛽]+1, 𝛽>0,(iii)πΌπ›½βˆΆπΆ([0,𝑇])→𝐢([0,𝑇]), πΌπ›½βˆΆπΏ1([0,𝑇])→𝐿1([0,𝑇]), 𝛽>0,(iv)𝐷𝛽𝐼𝛼=πΌπ›Όβˆ’π›½ and 𝐷𝛽𝑑𝛼=(Ξ“(𝛽+1)/Ξ“(π›½βˆ’π›Ό+1))π‘‘π›½βˆ’π›Ό for π‘‘βˆˆ[0,𝑇], π›Όβˆ’π›½>0.

More details on fractional derivatives and their properties can be found in [2, 4].

For the sake of convenience, we introduce the following assumptions:(H0)Ξ”=Ξ“(𝛼)π‘‡π›Όβˆ’π›½βˆ’1Ξ“βˆ’(π›Όβˆ’π›½)π‘šβˆ’2𝑖=1π‘Žπ‘–πœ‰π‘–π›Όβˆ’1βˆ’π‘šβˆ’2𝑖=1𝑏𝑖Γ(𝛼)πœ‰π‘–π›Όβˆ’π›½βˆ’1Ξ“(π›Όβˆ’π›½)>0,(2.3)(H1)π‘“βˆˆCar([0,+∞)×𝐡), 𝐡=(0,+∞)Γ—(0,+∞), limβ€–π‘₯β€–β†’0[‖𝑓(𝑑,π‘₯,𝑦)β€–=+∞,fora.e.π‘‘βˆˆ0,+∞)andallπ‘¦βˆˆπ‘ƒ+,lim‖𝑦‖→0[‖𝑓(𝑑,π‘₯,𝑦)β€–=+∞,fora.e.π‘‘βˆˆ0,+∞)andallπ‘₯βˆˆπ‘ƒ+,(2.4) and there exists a positive constant πœ› such that for all 𝑇0β‰₯𝑇, ‖𝑑𝑓(𝑑,π‘₯,𝑦)β€–β‰₯πœ›1βˆ’π‘‡0ξ‚Ά2+π›½βˆ’π›Όξ€Ίfora.e.π‘‘βˆˆ0,𝑇0ξ€»andall(π‘₯,𝑦)∈𝐡,(2.5)(H2)𝑓 fulfills the estimate, 𝛾‖𝑓(𝑑,π‘₯,𝑦)‖≀𝛾(𝑑)0(𝑑)+π‘ž1(β€–π‘₯β€–)+𝑝1()ξ€Έ[‖𝑦‖)+π‘ž(β€–π‘₯β€–)+𝑝(‖𝑦‖fora.e.π‘‘βˆˆ0,+∞),andall(π‘₯,𝑦)∈𝐡,(2.6) where 𝛾,𝛾0∈𝐿1[0,+∞), π‘ž1,𝑝1,π‘ž,π‘βˆˆπΆ((0,+∞),ℝ+), π‘ž1,𝑝1 are nonincreasing, and, for any 𝑇0β‰₯𝑇, ξ€œπ‘‡00𝛾(𝑑)π‘ž1βŽ›βŽœβŽœβŽπΎ1ξ‚€π‘‘π›Όβˆ’1𝑇0ξ€Έβˆ’π‘‘2𝑇0βŽžβŽŸβŽŸβŽ π‘‘π‘‘<+∞,𝐾1=πœ›,ξ€œ2Ξ“(𝛼)𝑇00𝛾(𝑑)𝑝1βŽ›βŽœβŽœβŽπΎ2ξ‚€π‘‘π›Όβˆ’π›½βˆ’1𝑇0ξ€Έβˆ’π‘‘2𝑇0βŽžβŽŸβŽŸβŽ π‘‘π‘‘<+∞,𝐾2=πœ›,2Ξ“(π›Όβˆ’π›½)(2.7) while π‘ž,𝑝 are nondecreasing and limβ€–π‘₯β€–β†’+βˆžπ‘ž(β€–π‘₯β€–)+𝑝(β€–π‘₯β€–)β€–π‘₯β€–=0,(2.8)(H3)for a.e. π‘‘βˆˆ[0,+∞), and for all π·βŠ‚π‘ƒ, 𝑓(𝑑,𝐷,𝐷) is relatively compact.

Remark 2.3. It follows from (2.4) that under condition (H2), limβ€–π‘₯β€–β†’0π‘ž1(β€–π‘₯β€–)=+∞ and lim‖𝑦‖→0𝑝1(‖𝑦‖)=+∞.

In the sequel, 𝐿1([0,𝑇],ℝ) denote the Banach space of functions π‘¦βˆΆ[0,𝑇]→ℝ which are Lebesgue integrable with the norm ‖𝑦‖𝐿1=ξ€œπ‘‡0||||𝑦(𝑑)𝑑𝑑.(2.9)

We give now some auxiliary lemmas in scalar space, which will take an important role throughout the paper.

Lemma 2.4. Suppose that β„Ž(𝑑)∈𝐿1([0,𝑇]) and that (𝐻0) holds, then the unique solution of linear BVP 𝐷𝛼𝑦(𝑑)+β„Ž(𝑑)=0, a.e. π‘‘βˆˆ[0,𝑇] with the boundary condition (1.5) is given by ξ€œπ‘¦(𝑑)=𝑇0𝑦𝐺(𝑑,𝑠)β„Ž(𝑠)𝑑𝑠+0Ξ”π‘‘π›Όβˆ’1,(2.10) where 𝐺1(𝑑,𝑠)=Ξ”βŽ§βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺ⎩(π‘‡βˆ’π‘ )π›Όβˆ’π›½βˆ’1π‘‘π›Όβˆ’1Ξ“βˆ’π‘‘(π›Όβˆ’π›½)π›Όβˆ’1Ξ“(𝛼)π‘šβˆ’2𝑗=π‘–π‘Žπ‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’1βˆ’π‘‘π›Όβˆ’1Ξ“(π›Όβˆ’π›½)π‘šβˆ’2𝑗=π‘–π‘π‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’π›½βˆ’1βˆ’Ξ”(π‘‘βˆ’π‘ )π›Όβˆ’1Ξ“,(𝛼)𝑑β‰₯𝑠,πœ‰π‘–βˆ’1<π‘ β‰€πœ‰π‘–,𝑖=1,2,…,π‘šβˆ’1,(π‘‡βˆ’π‘ )π›Όβˆ’π›½βˆ’1π‘‘π›Όβˆ’1Ξ“βˆ’π‘‘(π›Όβˆ’π›½)π›Όβˆ’1Ξ“(𝛼)π‘šβˆ’2𝑗=π‘–π‘Žπ‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’1βˆ’π‘‘π›Όβˆ’1Ξ“(π›Όβˆ’π›½)π‘šβˆ’2𝑗=π‘–π‘π‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’π›½βˆ’1,𝑑≀𝑠,πœ‰π‘–βˆ’1<π‘ β‰€πœ‰π‘–,𝑖=1,2,…,π‘šβˆ’1.(2.11)

Proof. The proof is similar to that of [7, Lemma 2.2], so we omit it.

Lemma 2.5. Suppose that (𝐻0) holds, then 𝐺(𝑑,𝑠) defined as (2.11) has the following properties:(i)𝐺(𝑑,𝑠) is uniformly continuous about 𝑑 in [0,𝑇],(ii)𝐺(𝑑,𝑠)β‰₯0 for all (𝑑,𝑠)∈[0,𝑇]Γ—[0,𝑇] and 𝐺(𝑑,𝑠)≀𝐸, where 𝑇𝐸=2π›Όβˆ’π›½βˆ’2,ΔΓ(π›Όβˆ’π›½)(2.12)(iii)βˆ«π‘‡0𝐺(𝑑,𝑠)𝑅(𝑠)𝑑𝑠β‰₯π‘‘π›Όβˆ’1(π‘‡βˆ’π‘‘)2/2𝑇Γ(𝛼) if 𝑅(𝑠)β‰₯(1βˆ’π‘‘/𝑇)2+π›½βˆ’π›Ό.

Proof. From (2.3), it is easy to verify (i) and (ii). We now show that (iii) is true. Firstly, if 𝑑β‰₯𝑠, then (2.11) gives 1𝐺(𝑑,𝑠)=Ξ”βŽ§βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩(π‘‡βˆ’π‘ )π›Όβˆ’π›½βˆ’1π‘‘π›Όβˆ’1Ξ“βˆ’π‘‘(π›Όβˆ’π›½)π›Όβˆ’1Ξ“(𝛼)π‘šβˆ’2𝑗=π‘–π‘Žπ‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’1βˆ’π‘‘π›Όβˆ’1Ξ“(π›Όβˆ’π›½)π‘šβˆ’2𝑗=π‘–π‘π‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’π›½βˆ’1βˆ’Ξ”(π‘‘βˆ’π‘ )π›Όβˆ’1,ξ€·πœ‰Ξ“(𝛼)π‘–βˆ’1<π‘ β‰€πœ‰π‘–ξ€Έ.,𝑖=1,2,…,π‘šβˆ’1(2.13) Then, 1𝐺(𝑑,𝑠)β‰₯Ξ”ξƒ―(π‘‡βˆ’π‘ )π›Όβˆ’π›½βˆ’1π‘‘π›Όβˆ’1Ξ“βˆ’π‘‘(π›Όβˆ’π›½)π›Όβˆ’1Ξ“(𝛼)π‘šβˆ’2𝑗=1π‘Žπ‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’1βˆ’π‘‘π›Όβˆ’1Ξ“(π›Όβˆ’π›½)π‘šβˆ’2𝑗=1π‘π‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’π›½βˆ’1βˆ’Ξ”(π‘‘βˆ’π‘ )π›Όβˆ’1ξƒ°β‰₯𝑑Γ(𝛼)π›Όβˆ’1Ξ”ξƒ―(π‘‡βˆ’π‘ )π›Όβˆ’π›½βˆ’1βˆ’1Ξ“(π›Όβˆ’π›½)Ξ“(𝛼)π‘šβˆ’2𝑗=1π‘Žπ‘—πœ‰π‘—π›Όβˆ’1𝑠1βˆ’πœ‰π‘—ξ‚Άπ›Όβˆ’1βˆ’1Ξ“(π›Όβˆ’π›½)π‘šβˆ’2𝑗=1π‘π‘—πœ‰π‘—π›Όβˆ’π›½βˆ’1𝑠1βˆ’πœ‰π‘—ξ‚Άπ›Όβˆ’π›½βˆ’1βˆ’Ξ”(1βˆ’π‘ /𝑑)π›Όβˆ’1ξƒ°β‰₯𝑑Γ(𝛼)π›Όβˆ’1(1βˆ’π‘ /𝑇)π›Όβˆ’π›½βˆ’1Ξ”ξƒ―π‘‡π›Όβˆ’π›½βˆ’1βˆ’1Ξ“(π›Όβˆ’π›½)Ξ“(π›Όβˆ’π›½)π‘šβˆ’2𝑗=1π‘π‘—πœ‰π‘—π›Όβˆ’π›½βˆ’1βˆ’(1βˆ’π‘ /𝑇)𝛽Γ(𝛼)π‘šβˆ’2𝑗=1π‘Žπ‘—πœ‰π‘—π›Όβˆ’1.+Ξ”ξƒͺξƒ°(2.14) From (2.14) and (2.3), we deduce from the Lagrange mean value theorem that 𝑑𝐺(𝑑,𝑠)β‰₯π›Όβˆ’1(1βˆ’π‘ /𝑇)π›Όβˆ’π›½βˆ’1ξ‚€βˆ‘π‘šβˆ’2𝑗=1π‘Žπ‘—πœ‰π‘—π›Όβˆ’1+Δ×𝑠ΔΓ(𝛼)1βˆ’1βˆ’π‘‡ξ‚π›½ξ‚Άβ‰₯π‘‘π›Όβˆ’1(1βˆ’π‘ /𝑇)π›Όβˆ’π›½βˆ’1ξ‚€βˆ‘π‘šβˆ’2𝑗=1π‘Žπ‘—πœ‰π‘—π›Όβˆ’1+ΔΔΓ(𝛼)π›½πœ‰π›½βˆ’1𝑠𝑇.(2.15) In view of (1βˆ’π‘ /𝑇)β‰€πœ‰β‰€1 and 𝛽<1, one can obtain for 𝑑β‰₯𝑠 that 𝑑𝐺(𝑑,𝑠)β‰₯π›Όβˆ’1(1βˆ’π‘ /𝑇)π›Όβˆ’π›½βˆ’1ξ‚€βˆ‘π‘šβˆ’2𝑗=1π‘Žπ‘—πœ‰π‘—π›Όβˆ’1+ΔΔΓ(𝛼)𝑇𝛽𝑠.(2.16) Analogously, if 𝑑≀𝑠, one has 𝑑𝐺(𝑑,𝑠)β‰₯π›Όβˆ’1(1βˆ’π‘ /𝑇)π›Όβˆ’π›½βˆ’1Ξ“.(𝛼)(2.17) It follows from (2.16) and (2.17) that ξ€œπ‘‡0ξ€œπΊ(𝑑,𝑠)𝑅(𝑠)𝑑𝑠=𝑑0ξ€œπΊ(𝑑,𝑠)𝑅(𝑠)𝑑𝑠+𝑇𝑑β‰₯ξ€œπΊ(𝑑,𝑠)𝑅(𝑠)𝑑𝑠𝑑0π‘‘π›Όβˆ’1ξ‚€βˆ‘π‘šβˆ’2𝑗=1π‘Žπ‘—πœ‰π‘—π›Όβˆ’1+Δ𝑠ΔΓ(𝛼)𝑇𝛽𝑠1βˆ’π‘‡ξ‚ξ€œπ‘‘π‘ +π‘‡π‘‘π‘‘π›Όβˆ’1𝑠Γ(𝛼)1βˆ’π‘‡ξ‚β‰₯π‘‘π‘‘π‘ π›Όβˆ’1(π‘‡βˆ’π‘‘)2.2𝑇Γ(𝛼)(2.18) The proof is complete.

3. Existence Results for BVP (1.4)-(1.5)

In this section, we discuss the uniqueness, existence, and continuous dependence of positive solution for problem (1.4)-(1.5). To this end, we introduce some auxiliary technical lemmas.

Let 𝔼={π‘₯∈𝐢([0,𝑇],𝐸)βˆΆπ·π›½π‘₯∈𝐢([0,𝑇],𝐸)} equipped with the norm β€–π‘₯β€–βˆ—=max{β€–π‘₯β€–,‖𝐷𝛽π‘₯β€–}, then 𝔼 is a real Banach space (see [24]).

Since the nonlinear term 𝑓(𝑑,π‘₯,𝑦) is singular at π‘₯=πœƒ and 𝑦=πœƒ, we use the following regularization process. For each π‘šβˆˆπ‘+, define π‘“π‘š by the formulaπ‘“π‘šβŽ§βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽ©π‘“π‘(𝑑,π‘₯,𝑦)=(𝑑,π‘₯,𝑦)ifπ‘₯β‰₯π‘šπ‘,𝑦β‰₯π‘š,𝑓𝑐𝑑,π‘šξ‚π‘,𝑦if0≀π‘₯<π‘šπ‘,𝑦β‰₯π‘š,𝑓𝑐𝑑,π‘₯,π‘šξ‚1ifπ‘₯β‰₯π‘š1,0≀𝑦<π‘š,𝑓𝑐𝑑,π‘š,π‘π‘šξ‚π‘if0≀π‘₯<π‘šπ‘,0≀𝑦<π‘š,(3.1) where 𝑐>πœƒ is a given element of 𝔼 and ‖𝑐‖=1.

Remark 3.1. The function π‘“π‘š defined by (3.1) satisfies π‘“π‘šβˆˆCar([0,𝑇]Γ—π΅βˆ—), π΅βˆ—=𝑃×𝑃. And conditions (H1) and (H2) imply β€–β€–π‘“π‘šβ€–β€–ξ‚€π‘‘(𝑑,π‘₯,𝑦)β‰₯πœ›1βˆ’π‘‡ξ‚1+π›½βˆ’π›Ό[],fora.e.π‘‘βˆˆ0,𝑇andall(π‘₯,𝑦)βˆˆπ΅βˆ—,‖‖𝑓(3.2)π‘šβ€–β€–ξ‚€π›Ύ(𝑑,π‘₯,𝑦)≀𝛾(𝑑)0(𝑑)+π‘ž1ξ‚€β€–β€–β€–1π‘šβ€–β€–β€–ξ‚+𝑝1ξ‚€β€–β€–β€–1π‘šβ€–β€–β€–ξ‚ξ‚,[]+π‘ž(1)+𝑝(1)+π‘ž(β€–π‘₯β€–)+𝑝(‖𝑦‖)fora.e.π‘‘βˆˆ0,𝑇andall(π‘₯,𝑦)βˆˆπ΅βˆ—,‖‖𝑓(3.3)π‘š(‖‖𝛾𝑑,π‘₯,𝑦)≀𝛾(𝑑)0(𝑑)+π‘ž1()β€–π‘₯β€–+𝑝1()ξ€Έ,[]‖𝑦‖+π‘ž(1)+𝑝(1)+π‘ž(β€–π‘₯β€–)+𝑝(‖𝑦‖)fora.e.π‘‘βˆˆ0,𝑇andall(π‘₯,𝑦)∈𝐡.(3.4)

Remark 3.2. The function π‘“π‘š defined by (3.1) satisfies limπ‘šβ†’+βˆžπ‘“π‘š=𝑓.

Define operator π‘„π‘šβˆΆπ‘ƒ+→𝑃+ by the following: ξ€·π‘„π‘šπ‘¦ξ€Έξ€œ(𝑑)=𝑇0𝐺(𝑑,𝑠)π‘“π‘šξ€·π‘ ,𝑦(𝑠),𝐷𝛽𝑦𝑦(𝑠)𝑑𝑠+0Δ𝑑(π›Όβˆ’1).(3.5)

Lemma 3.3. Suppose that (H0) holds, then ξ€·π·π›½π‘„π‘šπ‘¦ξ€Έξ€œ(𝑑)=𝑇0𝐷𝛽𝐺(𝑑,𝑠)π‘“π‘šξ€·π‘ ,𝑦(𝑠),𝐷𝛽𝑦𝑦(𝑠)𝑑𝑠+0Ξ“(𝛼)𝑑ΔΓ(π›Όβˆ’π›½)(π›Όβˆ’π›½βˆ’1),(3.6) where 𝐷𝛽𝐺(𝑑,𝑠)=Ξ“(𝛼)⎧βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽ©Ξ”Ξ“(π›Όβˆ’π›½)(π‘‡βˆ’π‘ )π›Όβˆ’π›½βˆ’1π‘‘π›Όβˆ’π›½βˆ’1Ξ“βˆ’π‘‘(π›Όβˆ’π›½)π›Όβˆ’π›½βˆ’1Ξ“(𝛼)π‘šβˆ’2𝑗=π‘–π‘Žπ‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’1βˆ’π‘‘π›Όβˆ’π›½βˆ’1Ξ“(π›Όβˆ’π›½)π‘šβˆ’2𝑗=π‘–π‘π‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’π›½βˆ’1βˆ’Ξ”(π‘‘βˆ’π‘ )π›Όβˆ’π›½βˆ’1,Ξ“(𝛼)𝑑β‰₯𝑠,πœ‰π‘–βˆ’1<π‘ β‰€πœ‰π‘–,𝑖=1,2,…,π‘šβˆ’1,(π‘‡βˆ’π‘ )π›Όβˆ’π›½βˆ’1π‘‘π›Όβˆ’π›½βˆ’1βˆ’π‘‘Ξ“(π›Όβˆ’π›½)π›Όβˆ’π›½βˆ’1Ξ“(𝛼)π‘šβˆ’2𝑗=π‘–π‘Žπ‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’1βˆ’π‘‘π›Όβˆ’π›½βˆ’1Ξ“(π›Όβˆ’π›½)π‘šβˆ’2𝑗=π‘–π‘π‘—ξ€·πœ‰π‘—ξ€Έβˆ’π‘ π›Όβˆ’π›½βˆ’1,𝑑≀𝑠,πœ‰π‘–βˆ’1<π‘ β‰€πœ‰π‘–,𝑖=1,2,…,π‘šβˆ’1.(3.7)

Proof. The proof is similar to that of [7, Lemma 2.2], and we omit it.

Lemma 3.4. Suppose that (H0) holds, then 𝐷𝛽𝐺(𝑑,𝑠) defined as (3.7) has the following properties:(i)𝐷𝛽𝐺(𝑑,𝑠) is uniformly continuous about 𝑑 in [0,𝑇],(ii)𝐷𝛽𝐺(𝑑,𝑠)β‰₯0 for all (𝑑,𝑠)∈[0,𝑇]Γ—[0,𝑇] and 𝐷𝛽𝐺(𝑑,𝑠)≀𝐸𝐷, where 𝐸𝐷=Ξ“(𝛼)𝑇2π›Όβˆ’2π›½βˆ’2Ξ”[]Ξ“(π›Όβˆ’π›½)2,(3.8)(iii)βˆ«π‘‡0𝐷𝛽𝐺(𝑑,𝑠)𝑅(𝑠)𝑑𝑠β‰₯π‘‘π›Όβˆ’π›½βˆ’1(π‘‡βˆ’π‘‘)2/2𝑇Γ(π›Όβˆ’π›½) if 𝑅(𝑠)β‰₯(1βˆ’π‘‘/𝑇)2+π›½βˆ’π›Ό.

Proof. The proof of this Lemma is similar to that of Lemma 2.5. Hence it is omitted.

Lemma 3.5. Suppose that (H0) and following condition (H4) hold:(H4) there exist positive constants 𝐿,𝐿𝐷 such that ‖‖𝑓𝑠,π‘₯(𝑠),𝐷𝛽π‘₯(𝑠)βˆ’π‘“π‘ ,𝑦(𝑠),𝐷𝛽‖‖𝑦(𝑠)≀𝐿‖π‘₯βˆ’π‘¦β€–+𝐿𝐷‖‖𝐷𝛽π‘₯βˆ’π·π›½π‘¦β€–β€–(3.9) and 𝜏=max{𝐸(𝐿+𝐿𝐷),𝐸𝐷(𝐿+𝐿𝐷)}<1, then π‘„π‘š has a unique fixed point.

Proof. Obviously, π‘“π‘š defined by the formula (3.1) satisfy also the condition (H4). By (3.5) and (3.6), it is easy to show that β€–π‘„π‘šπ‘₯βˆ’π‘„π‘šπ‘¦β€–βˆ—<πœβ€–π‘₯βˆ’π‘¦β€–βˆ—, then Banach contraction principle implies that the operator π‘„π‘š has a unique fixed point, which completes this proof.

The following fixed-point result of cone compression type is due to Krasnosel'skii, which is fundamental to establish another auxiliary existence result (Lemma 3.8).

Lemma 3.6 (see, e.g., [23, 25]). Let π‘Œ be a Banach space, and let π‘ƒβŠ‚π‘Œ be a cone in π‘Œ. Let Ξ©1,Ξ©2 be bounded open balls of π‘Œ centered at the origin with Ξ©1βŠ‚Ξ©2. Suppose that π΄βˆΆπ‘ƒβˆ©(Ξ©2⧡Ω1)→𝑃 is a completely continuous operator such that ‖𝐴π‘₯β€–β‰₯β€–π‘₯β€–forπ‘₯βˆˆπ‘ƒβˆ©πœ•Ξ©1,‖𝐴π‘₯‖≀‖π‘₯β€–forπ‘₯βˆˆπ‘ƒβˆ©πœ•Ξ©2(3.10) hold, then 𝐴 has a fixed point in π‘ƒβˆ©(Ξ©2⧡Ω1).

Lemma 3.7. Let (H0)-(H3) hold, then π‘„π‘šβˆΆπ‘ƒβ†’π‘ƒ and π‘„π‘š is a completely continuous operator.

Proof. Firstly, let π‘¦βˆˆπ‘ƒ, because π‘“π‘šβˆˆCar([0,𝑇]Γ—π΅βˆ—) is positive. It follows from Lemma 2.5 (i) and (ii) that π‘„π‘šπ‘¦βˆˆπΆ([0,𝑇],𝔼) and (π‘„π‘šπ‘¦)(𝑑)β‰₯πœƒ for π‘‘βˆˆ[0,𝑇]. Similarly, from Lemma 3.4 (i) and (ii) we can get that π·π›½π‘„π‘šπ‘¦βˆˆπΆ([0,𝑇],𝔼) and (π·π›½π‘„π‘šπ‘¦)(𝑑)β‰₯πœƒ for π‘‘βˆˆ[0,𝑇]. To summarize, π‘„π‘šβˆΆπ‘ƒβ†’π‘ƒ.
Secondly, we prove that π‘„π‘š is a continuous operator. Let {π‘₯π‘˜}βŠ‚π‘ƒ be a convergent sequence and limπ‘˜β†’+βˆžβ€–π‘₯π‘˜βˆ’π‘₯β€–βˆ—=0, then π‘₯βŠ‚π‘ƒ and β€–π‘₯π‘˜β€–βˆ—β‰€π‘†, where 𝑆 is a positive constant. In view of π‘“π‘šβˆˆCar([0,𝑇]Γ—π΅βˆ—), we have limπ‘˜β†’+βˆžπ‘“π‘š(𝑑,π‘₯π‘˜(𝑑),𝐷𝛽π‘₯π‘˜(𝑑))=π‘“π‘š(𝑑,π‘₯(𝑑),𝐷𝛽π‘₯(𝑑)). Since by (3.2), (3.3), ‖‖𝑓0<π‘šξ€·π‘‘,π‘₯π‘˜(𝑑),𝐷𝛽π‘₯π‘˜ξ€Έβ€–β€–ξ‚€π›Ύ(𝑑)≀𝛾(𝑑)0(𝑑)+π‘ž1ξ‚€1π‘šξ‚+𝑝1ξ‚€1π‘šξ‚ξ‚,+π‘ž(1)+𝑝(1)+π‘ž(𝑆)+𝑝(𝑆)(3.11) the Lebesgue dominated convergence theorem gives limπ‘˜β†’+βˆžξ€œπ‘‡0β€–β€–π‘“π‘šξ€·π‘‘,π‘₯π‘˜(𝑑),𝐷𝛽π‘₯π‘˜ξ€Έ(𝑑)βˆ’π‘“π‘šξ€·π‘‘,π‘₯(𝑑),𝐷𝛽‖‖π‘₯(𝑑)𝑑𝑑=0.(3.12) Now, from (3.12), Lemma 2.5(ii), Lemma 3.4(ii) and from the inequalities (cf. (3.5), (3.6)) β€–β€–ξ€·π‘„π‘šπ‘₯π‘˜ξ€Έξ€·π‘„(𝑑)βˆ’π‘šπ‘₯ξ€Έβ€–β€–ξ€œ(𝑑)≀𝐸𝑇0β€–β€–π‘“π‘šξ€·π‘‘,π‘₯π‘˜(𝑑),𝐷𝛽π‘₯π‘˜ξ€Έ(𝑑)βˆ’π‘“π‘šξ€·π‘‘,π‘₯(𝑑),𝐷𝛽‖‖‖‖𝐷π‘₯(𝑑)𝑑𝑑,π›½π‘„π‘šπ‘₯π‘˜ξ€Έξ€·π·(𝑑)βˆ’π›½π‘„π‘šπ‘₯ξ€Έβ€–β€–(𝑑)β‰€πΈπ·ξ€œπ‘‡0β€–β€–π‘“π‘šξ€·π‘‘,π‘₯π‘˜(𝑑),𝐷𝛽π‘₯π‘˜ξ€Έ(𝑑)βˆ’π‘“π‘šξ€·π‘‘,π‘₯(𝑑),𝐷𝛽π‘₯ξ€Έβ€–β€–(𝑑)𝑑𝑑,(3.13) we have that limπ‘˜β†’+βˆžβ€–π‘„π‘šπ‘₯π‘˜βˆ’π‘„π‘šπ‘₯β€–βˆ—=0, which proves that π‘„π‘š is a continuous operator.
Thirdly, let Ξ©βŠ‚π‘ƒ be bounded in 𝔼 and let β€–π‘₯β€–βˆ—β‰€πΏ for all π‘₯∈Ω, where 𝐿 is a positive constant. We are in position to prove that π‘„π‘š(Ξ©) is bounded. Keeping in mind π‘“π‘šβˆˆCar([0,𝑇]Γ—π΅βˆ—), there exists πœ™βˆˆπΏ1([0,𝑇]) such that ‖‖𝑓0<π‘šξ€·π‘‘,π‘₯π‘˜(𝑑),𝐷𝛽π‘₯π‘˜ξ€Έβ€–β€–[](𝑑)β‰€πœ™(𝑑)fora.e.π‘‘βˆˆ0,𝑇andallπ‘₯∈Ω,(3.14) then (cf. (3.5)) β€–β€–ξ€·π‘„π‘šπ‘₯ξ€Έβ€–β€–ξ€œ(𝑑)≀𝐸𝑇0β€–β€–π‘“π‘šξ€·π‘ ,π‘₯(𝑠),𝐷𝛽‖‖𝑦π‘₯(𝑠)𝑑𝑠+0Δ𝑑(π›Όβˆ’1)β‰€πΈβ€–πœ™β€–πΏ1+‖‖𝑦0‖‖Δ𝑇(π›Όβˆ’1),(3.15) and (cf. (3.6)) β€–β€–ξ€·π·π›½π‘„π‘šπ‘₯ξ€Έβ€–β€–(𝑑)β‰€πΈπ·ξ€œπ‘‡0β€–β€–π‘“π‘šξ€·π‘ ,π‘₯(𝑠),𝐷𝛽‖‖‖‖‖𝑦π‘₯(𝑠)𝑑𝑠+0Ξ“(𝛼)𝑑ΔΓ(π›Όβˆ’π›½)(π›Όβˆ’π›½βˆ’1)β€–β€–β€–β‰€πΈπ·β€–πœ™β€–πΏ1+‖‖𝑦0β€–β€–Ξ“(𝛼)𝑇ΔΓ(π›Όβˆ’π›½)(π›Όβˆ’π›½βˆ’1),(3.16) for π‘‘βˆˆ[0,𝑇] and all π‘₯∈Ω. Therefore, π‘„π‘š(Ξ©) is bounded in 𝔼.
Fourthly, by (H3) and (3.5), it is easy to show that π‘„π‘š(Ξ©)(𝑑) is relatively compact.
Finally, let 0≀𝑑1<𝑑2≀𝑇. From Lemma 2.5(i) and the functions π‘‘π›Όβˆ’1,π‘‘π›Όβˆ’π›½βˆ’1 being uniformly continuous on [0,𝑇], for any arbitrary πœ–>0, there exists a positive number 𝛿(πœ–), such that when |𝑑1βˆ’π‘‘2|<𝛿(πœ–), one has |𝐺𝑇(𝑑1,𝑠)βˆ’πΊπ‘‡(𝑑2,𝑠)|<πœ– and |𝑑1π›Όβˆ’1βˆ’π‘‘2π›Όβˆ’1|<πœ–, then (cf. (3.14)) the inequality β€–β€–ξ€·π‘„π‘šπ‘₯𝑑1ξ€Έβˆ’ξ€·π‘„π‘šπ‘₯𝑑2ξ€Έβ€–β€–<πœ–β€–πœ™β€–πΏ1+‖‖𝑦0β€–β€–Ξ”πœ–(3.17) holds. Hence the set of functions π‘„π‘š(Ξ©) is equicontinuous on [0,𝑇].
Therefore, by the ArzelΓ‘-Ascoli theorem, π‘„π‘š(Ξ©) is relatively compact in 𝔼. We have proved that π‘„π‘š is a completely continuous operator.

Lemma 3.8. Suppose that (H0)-(H3) hold, then the operator π‘„π‘š has at least a fixed point.

Proof. By Lemma 3.7, π‘„π‘šβˆΆπ‘ƒβ†’π‘ƒ is completely continuous. In order to apply Lemma 3.6, we construct two bounded open balls Ξ©1,Ξ©2 and prove that the conditions (3.10) are satisfied with respect to π‘„π‘š.
Firstly, let Ξ©1={π‘¦βˆˆπ”ΌβˆΆβ€–π‘¦β€–βˆ—<π‘Ÿ}, where π‘Ÿ=supπ‘‘βˆˆ[0,𝑇]𝐾1(π‘‘π›Όβˆ’1(π‘‡βˆ’π‘‘)2)/𝑇 and 𝐾1 is defined as in (H2). It follows from Lemma 2.5, (H1), Remark 3.1 and from the definition of π‘„π‘š that β€–(π‘„π‘šπ‘¦)(𝑑)β€–β‰₯𝐾1(π‘‘π›Όβˆ’1(π‘‡βˆ’π‘‘)2/𝑇). Then β€–π‘„π‘šπ‘¦β€–β‰₯π‘Ÿ. Immediately: β€–β€–π‘„π‘šπ‘¦β€–β€–βˆ—β‰₯β€–π‘¦β€–βˆ—,forπ‘¦βˆˆπ‘ƒβˆ©πœ•Ξ©1.(3.18) Secondly, (3.3) and Lemma 2.5(ii) imply that, for π‘₯βˆˆπ‘ƒ, β€–β€–ξ€·π‘„π‘šπ‘¦ξ€Έβ€–β€–ξ€œ(𝑑)≀𝐸𝑇0β€–β€–π‘“π‘šξ€·π‘ ,𝑦(𝑠),𝐷𝛽‖‖‖‖𝑦𝑦(𝑠)𝑑𝑠+0‖‖Δ𝑇(π›Όβˆ’1)ξ€œβ‰€πΈπ‘‡0𝛾𝛾(𝑑)0(𝑑)+π‘ž1ξ‚€1π‘šξ‚+𝑝1ξ‚€1π‘šξ‚ξ€·π·+π‘ž(1)+𝑝(1)+π‘ž(𝑦(𝑠))+𝑝𝛽𝑦‖‖𝑦(𝑠)𝑑𝑠+0‖‖Δ𝑇(π›Όβˆ’1)ξ€œβ‰€πΈπ‘‡0𝛾𝛾(𝑑)0(𝑑)+π‘ž1ξ‚€1π‘šξ‚+𝑝1ξ‚€1π‘šξ‚ξ€·β€–β€–π·+π‘ž(1)+𝑝(1)+π‘ž(‖𝑦‖)+𝑝𝛽𝑦‖‖‖‖𝑦𝑑𝑠+0‖‖Δ𝑇(π›Όβˆ’1)‖‖≀𝐸𝛾𝛾0‖‖𝐿1+ξ‚ƒπ‘ž1ξ‚€1π‘šξ‚+𝑝1ξ‚€1π‘šξ‚ξ€·β€–β€–π·+π‘ž(1)+𝑝(1)+π‘ž(‖𝑦‖)+𝑝𝛽𝑦‖‖‖𝛾‖𝐿1+‖‖𝑦0‖‖Δ𝑇(π›Όβˆ’1),(3.19) because π‘ž,𝑝 are nondecreasing as stated in (H2). Analogously, by (3.3) and Lemma 3.4(ii), one can get that for π‘₯βˆˆπ‘ƒβ€–β€–ξ€·π·π›½π‘„π‘šπ‘¦ξ€Έβ€–β€–(𝑑)≀𝐸𝐷‖‖𝛾𝛾0‖‖𝐿1+ξ‚ƒπ‘ž1ξ‚€1π‘šξ‚+𝑝1ξ‚€1π‘šξ‚ξ€·β€–β€–π·+π‘ž(1)+𝑝(1)+π‘ž(‖𝑦‖)+𝑝𝛽𝑦‖‖×‖𝛾‖𝐿1+‖‖𝑦0β€–β€–Ξ“(𝛼)𝑇ΔΓ(π›Όβˆ’π›½)(π›Όβˆ’π›½βˆ’1).(3.20) Let π‘Š1=max{𝐸,𝐸𝐷} and π‘Š2=max{(‖𝑦0β€–/Ξ”)𝑇(π›Όβˆ’1),(‖𝑦0β€–Ξ“(𝛼)/(ΔΓ(π›Όβˆ’π›½)))𝑇(π›Όβˆ’π›½βˆ’1)}. Hence for π‘₯βˆˆπ‘ƒ, we have the following inequality β€–β€–π‘„π‘šπ‘¦β€–β€–βˆ—β‰€π‘Š1‖‖𝛾𝛾0‖‖𝐿1+ξ‚ƒπ‘ž1ξ‚€1π‘šξ‚+𝑝1ξ‚€1π‘šξ‚ξ€·+π‘ž(1)+𝑝(1)+π‘žβ€–π‘¦β€–βˆ—ξ€Έξ€·+π‘β€–π‘¦β€–βˆ—ξ€Έξ‚„Γ—β€–π›Ύβ€–πΏ1+π‘Š2.(3.21) Since limβ€–π‘₯β€–β†’βˆž(π‘ž(β€–π‘₯β€–)+𝑝(β€–π‘₯β€–))/β€–π‘₯β€–=0 by (H2), there exists a sufficiently large number 𝑅>π‘Ÿ such that π‘Š1‖‖𝛾𝛾0‖‖𝐿1+ξ‚ƒπ‘ž1ξ‚€1π‘šξ‚+𝑝1ξ‚€1π‘šξ‚ξ‚„+π‘ž(1)+𝑝(1)+π‘ž(𝑅)+𝑝(𝑅)‖𝛾‖𝐿1+π‘Š2≀𝑅.(3.22) Let Ξ©2={π‘¦βˆˆπ”ΌβˆΆβ€–π‘¦β€–βˆ—<𝑅}, then (cf. (3.21) and (3.22)) β€–β€–π‘„π‘šπ‘¦β€–β€–βˆ—β‰€β€–π‘¦β€–βˆ—,forπ‘¦βˆˆπ‘ƒβˆ©πœ•Ξ©2.(3.23) Applying Lemma 3.6, we conclude from (3.18) and (3.23) that π‘„π‘š has a fixed point in π‘ƒβˆ©(Ξ©2⧡Ω1).

Lemma 3.9. Suppose that (H0)-(H3) hold, then the sequences {π‘¦π‘š}βˆžπ‘š=1 and {π·π›½π‘¦π‘š}βˆžπ‘š=1 are relatively compact in 𝐢([0,𝑇]), where π‘¦π‘š be a fixed point of operator π‘„π‘š defined by (3.5).

Proof. Let π‘¦π‘š be a fixed point of operator π‘„π‘š, that is, π‘¦π‘šξ€œ(𝑑)=𝑇0𝐺(𝑑,𝑠)π‘“π‘šξ€·π‘ ,π‘¦π‘š(𝑠),π·π›½π‘¦π‘šξ€Έ+𝑦(𝑠)𝑑𝑠0Δ𝑑(π›Όβˆ’1)[],π‘‘βˆˆ0,𝑇,π‘šβˆˆπ‘.(3.24) And consider (cf. (3.6)) π·π›½π‘¦π‘šξ€œ(𝑑)=𝑇0𝐷𝛽𝐺(𝑑,𝑠)π‘“π‘šξ€·π‘ ,π‘¦π‘š(𝑠),π·π›½π‘¦π‘šξ€Έ+𝑦(𝑠)𝑑𝑠0Ξ“(𝛼)𝑑ΔΓ(π›Όβˆ’π›½)(π›Όβˆ’π›½βˆ’1)[],π‘‘βˆˆ0,𝑇,π‘šβˆˆπ‘.(3.25) By Lemma 2.5(iii), Lemma 3.4(iii), and Remark 3.1, we have also β€–β€–π‘¦π‘šβ€–β€–(𝑑)β‰₯𝐾1π‘‘π›Όβˆ’1(π‘‡βˆ’π‘‘)2𝑇+𝑦0Ξ“(𝛼)𝑑ΔΓ(π›Όβˆ’π›½)(π›Όβˆ’π›½βˆ’1)β‰₯𝐾1π‘‘π›Όβˆ’1(π‘‡βˆ’π‘‘)2𝑇[]‖‖𝐷,π‘‘βˆˆ0,𝑇,π‘šβˆˆπ‘,(3.26)π›½π‘¦π‘šβ€–β€–(𝑑)β‰₯𝐾2π‘‘π›Όβˆ’π›½βˆ’1(π‘‡βˆ’π‘‘)2𝑇[],π‘‘βˆˆ0,𝑇,π‘šβˆˆπ‘.(3.27) Hence (cf. (3.4)), β€–β€–π‘“π‘šξ€·π‘‘,π‘¦π‘š(𝑑),π·π›½π‘¦π‘šξ€Έβ€–β€–ξ‚»π›Ύ(𝑑)≀𝛾(𝑑)0(𝑑)+π‘ž1𝐾1π‘‘π›Όβˆ’1(π‘‡βˆ’π‘‘)2𝑇+𝑝1𝐾2π‘‘π›Όβˆ’π›½βˆ’1(π‘‡βˆ’π‘‘)2𝑇𝑦+π‘ž(1)+𝑝(1)+π‘žπ‘šξ€Έξ€·π·(𝑑)+π‘π›½π‘¦π‘šξ€Έξ‚Ό,(𝑑)(3.28) for a.e. π‘‘βˆˆ[0,𝑇], and all π‘šβˆˆπ‘. Therefore, by (3.26), (3.27), Lemma 2.5(ii), Lemma 3.4(ii), and Remark 3.1, β€–β€–π‘¦π‘šβ€–β€–ξ€½β€–β€–(𝑑)≀𝐸𝛾𝛾0‖‖𝐿1+π‘ˆ1+π‘ˆ2+ξ€Ίξ€·β€–β€–π‘¦π‘ž(1)+𝑝(1)+π‘žπ‘šβ€–β€–ξ€Έξ€·β€–β€–π·+π‘π›½π‘¦π‘šβ€–β€–ξ€Έξ€»β€–π›Ύβ€–πΏ1ξ€Ύ+𝑦0Δ𝑇(π›Όβˆ’1),‖‖𝐷(3.29)π›½π‘¦π‘šβ€–β€–(𝑑)≀𝐸𝐷‖‖𝛾𝛾0‖‖𝐿1+π‘ˆ1+π‘ˆ2+ξ€Ίξ€·β€–β€–π‘¦π‘ž(1)+𝑝(1)+π‘žπ‘šβ€–β€–ξ€Έξ€·β€–β€–π·+π‘π›½π‘¦π‘šβ€–β€–ξ€Έξ€»β€–π›Ύβ€–πΏ1ξ€Ύ+𝑦0Ξ“(𝛼)𝑇ΔΓ(π›Όβˆ’π›½)(π›Όβˆ’π›½βˆ’1),(3.30) for π‘‘βˆˆ[0,𝑇], π‘šβˆˆπ‘, where π‘ˆ1=ξ€œπ‘‡0𝛾(𝑑)π‘ž1𝐾1π‘‘π›Όβˆ’1(π‘‡βˆ’π‘‘)2π‘‡ξ‚Άπ‘ˆπ‘‘π‘‘<+∞,(3.31)2=ξ€œπ‘‡0𝛾(𝑑)𝑝1𝐾2π‘‘π›Όβˆ’π›½βˆ’1(π‘‡βˆ’π‘‘)2𝑇𝑑𝑑<+∞.(3.32) In particular, β€–β€–π‘¦π‘šβ€–β€–βˆ—β‰€π‘Š1‖‖𝛾𝛾0‖‖𝐿1+π‘ˆ1+π‘ˆ2+ξ€Ίπ‘žξ€·β€–β€–π‘¦(1)+𝑝(1)+π‘žπ‘šβ€–β€–βˆ—ξ€Έξ€·β€–β€–π‘¦+π‘π‘šβ€–β€–βˆ—ξ€Έξ€»β€–π›Ύβ€–πΏ1ξ€Ύ+π‘Š2,βˆ€π‘šβˆˆπ‘,(3.33) where π‘Š1,π‘Š2 are defined in the proof of Lemma 3.8. Since limπ‘₯β†’+∞(π‘ž(π‘₯)+𝑝(π‘₯))/π‘₯=0 by (H2), there exists a constant π‘Š>0 such that for each π‘₯>π‘Š, π‘Š1‖‖𝛾𝛾0‖‖𝐿1+π‘ˆ1+π‘ˆ2+[π‘ž](1)+𝑝(1)+π‘ž(π‘₯)+𝑝(π‘₯)‖𝛾‖𝐿1ξ€Ύ+π‘Š2<π‘₯.(3.34) Immediately, (cf. (3.33)) β€–β€–π‘¦π‘šβ€–β€–βˆ—β‰€π‘Š,βˆ€π‘šβˆˆπ‘.(3.35) Hence, the sequences {π‘¦π‘š}βˆžπ‘š=1 and {π·π›½π‘¦π‘š}βˆžπ‘š=1 are uniformly bounded.
We will take similar discussions as in Lemma 3.7 to show that {π‘¦π‘š}βˆžπ‘š=1 and {π·π›½π‘¦π‘š}βˆžπ‘š=1 are equicontinuous on [0,𝑇]. Let 0≀𝑑1<𝑑2≀𝑇, then we have β€–β€–ξ€·π‘¦π‘šπ‘‘ξ€Έξ€·1ξ€Έβˆ’ξ€·π‘¦π‘šπ‘‘ξ€Έξ€·2ξ€Έβ€–β€–β‰€ξ€œπ‘‡0||𝐺𝑇𝑑1ξ€Έ,π‘ βˆ’πΊπ‘‡ξ€·π‘‘2ξ€Έ||‖‖𝑓,π‘ π‘šξ€·π‘ ,π‘₯(𝑠),𝐷𝛽‖‖‖‖𝑦π‘₯(𝑠)𝑑𝑠+0β€–β€–Ξ”||𝑑1π›Όβˆ’1βˆ’π‘‘2π›Όβˆ’1||,β€–β€–ξ€·π·π›½π‘¦π‘šπ‘‘ξ€Έξ€·1ξ€Έβˆ’ξ€·π·π›½π‘¦π‘šπ‘‘ξ€Έξ€·2ξ€Έβ€–β€–β‰€ξ€œπ‘‡0||𝐷𝛽𝐺𝑇𝑑1ξ€Έ,π‘ βˆ’π·π›½πΊπ‘‡ξ€·π‘‘2ξ€Έ||‖‖𝑓,π‘ π‘šξ€·π‘ ,π‘₯(𝑠),𝐷𝛽π‘₯ξ€Έβ€–β€–+‖‖𝑦(𝑠)𝑑𝑠0β€–β€–Ξ“(𝛼)||𝑑ΔΓ(π›Όβˆ’π›½)1π›Όβˆ’π›½βˆ’1βˆ’π‘‘2π›Όβˆ’π›½βˆ’1||.(3.36) Using (3.28), (3.35), one can get ‖‖𝑓0<π‘šξ€·π‘‘,π‘¦π‘š(𝑑),π·π›½π‘¦π‘šξ€Έβ€–β€–ξ‚»π›Ύ(𝑑)≀𝛾(𝑑)0(𝑑)+π‘ž1𝐾1π‘‘π›Όβˆ’1(π‘‡βˆ’π‘‘)2𝑇+𝑝1𝐾2π‘‘π›Όβˆ’π›½βˆ’1(π‘‡βˆ’π‘‘)2𝑇.+π‘ž(1)+𝑝(1)+π‘ž(π‘Š)+𝑝(π‘Š)(3.37) From Lemma 2.5 (i), Lemma 3.4 (i), and the functions π‘‘π›Όβˆ’1,π‘‘π›Όβˆ’π›½βˆ’1 being uniformly continuous on [0,𝑇], choosing an arbitrary πœ–>0, there exists a positive number 𝛿(πœ–). When |𝑑1βˆ’π‘‘2|<𝛿(πœ–), we can get |𝐺𝑇(𝑑1,𝑠)βˆ’πΊπ‘‡(𝑑2,𝑠)|<πœ–, |𝐷𝛽𝐺𝑇(𝑑1,𝑠)βˆ’π·π›½πΊπ‘‡(𝑑2,𝑠)|<πœ–, |𝑑1π›Όβˆ’1βˆ’π‘‘2π›Όβˆ’1|<πœ–, and |𝑑1π›Όβˆ’π›½βˆ’1βˆ’π‘‘2π›Όβˆ’π›½βˆ’1|<πœ–. Therefore (cf. (3.36) and (3.37)) the inequalities β€–β€–ξ€·π‘¦π‘šπ‘‘ξ€Έξ€·1ξ€Έβˆ’ξ€·π‘¦π‘šπ‘‘ξ€Έξ€·2ξ€Έβ€–β€–ξ€½β€–β€–<πœ–π›Ύπ›Ύ0‖‖𝐿1+π‘ˆ1+π‘ˆ2+[]π‘ž(1)+𝑝(1)+π‘ž(π‘Š)+𝑝(π‘Š)‖𝛾‖𝐿1ξ€Ύ+‖‖𝑦0β€–β€–Ξ”β€–β€–ξ€·π·πœ–,π›½π‘¦π‘šπ‘‘ξ€Έξ€·1ξ€Έβˆ’ξ€·π·π›½π‘¦π‘šπ‘‘ξ€Έξ€·2ξ€Έβ€–β€–ξ€½β€–β€–<πœ–π›Ύπ›Ύ0‖‖𝐿1+π‘ˆ1+π‘ˆ2+[π‘ž](1)+𝑝(1)+π‘ž(π‘Š)+𝑝(π‘Š)‖𝛾‖𝐿1ξ€Ύ+‖‖𝑦0β€–β€–Ξ“(𝛼)ΔΓ(π›Όβˆ’π›½)πœ–,(3.38) hold, where π‘ˆ1, π‘ˆ2 are defined as (3.31) and (3.32), respectively. As a result, {π‘¦π‘š}βˆžπ‘š=1 and {π·π›½π‘¦π‘š}βˆžπ‘š=1 are equicontinuous on [0,𝑇].
Finally, we prove that {π‘¦π‘š(𝑑)}βˆžπ‘š=1 and {π·π›½π‘¦π‘š(𝑑)}βˆžπ‘š=1 are relatively compact. Because 𝐸 is a Banach space, we need only to show that {π‘¦π‘š(𝑑)}βˆžπ‘š=1 and {π·π›½π‘¦π‘š(𝑑)}βˆžπ‘š=1 are completely bounded. For all πœ€>0, by the Remark 3.2, there exists a sufficiently large positive integer 𝑁, such that if π‘š>𝑁, β€–β€–π‘“π‘šξ€·π‘‘,𝑦(𝑑),𝐷𝛽𝑦(𝑑)βˆ’π‘“π‘‘,𝑦(𝑑),𝐷𝛽𝑦‖‖<πœ€(𝑑)𝐸[],,a.e.π‘‘βˆˆ0,𝑇(3.39) where 𝐸=max{𝐸,𝐸𝐷}.
Hence, by (3.5) and (3.6), we have β€–(π‘„π‘šπ‘¦)(𝑑)βˆ’π‘„0β€–<πœ€ and ‖𝐷𝛽(π‘„π‘šπ‘¦)(𝑑)βˆ’π·π›½π‘„0β€–<πœ€, for π‘š>𝑁, where 𝑄0=ξ€œπ‘‡0𝐺(𝑑,𝑠)𝑓𝑠,𝑦(𝑠),𝐷𝛽𝑦𝑦(𝑠)𝑑𝑠+0Δ𝑑(π›Όβˆ’1),𝐷𝛽𝑄0=ξ€œπ‘‡0𝐷𝛽𝐺(𝑑,𝑠)𝑓𝑠,𝑦(𝑠),𝐷𝛽𝑦𝑦(𝑠)𝑑𝑠+0Ξ“(𝛼)𝑑ΔΓ(π›Όβˆ’π›½)(π›Όβˆ’π›½βˆ’1).(3.40) This implies that {π‘¦π‘š(𝑑)}βˆžπ‘š=1 and {π·π›½π‘¦π‘š(𝑑)}βˆžπ‘š=1 have an πœ€-net constituted by finite elements ({𝑦1(𝑑),𝑦2(𝑑),𝑦𝑁(𝑑),𝑄0} and {𝐷𝛽𝑦1(𝑑),𝐷𝛽𝑦2(𝑑),𝐷𝛽𝑦𝑁(𝑑),𝐷𝛽𝑄0}, resp.) of 𝐸, that is, completely bounded.
Therefore, {π‘¦π‘š}βˆžπ‘š=1 and {π·π›½π‘¦π‘š}βˆžπ‘š=1 are relatively compact in 𝐢([0,𝑇]) by the ArzelΓ‘-Ascoli theorem.
Using above results, we now give the existence of positive solution of singular problem (1.4)-(1.5).

Theorem 3.10. Suppose that (H0)-(H3) hold, then problem (1.4)-(1.5) has a positive solution 𝑦 and ‖𝑦(𝑑)β€–β‰₯𝐾1π‘‘π›Όβˆ’1(π‘‡βˆ’π‘‘)2𝑇,‖‖𝐷𝛽𝑦‖‖(𝑑)β‰₯𝐾2π‘‘π›Όβˆ’π›½βˆ’1(π‘‡βˆ’π‘‘)2𝑇[].,π‘‘βˆˆ0,𝑇(3.41) Moreover, 𝑦 is continuous and β€–π‘¦β€–βˆ—β‰€π‘Š, where π‘Š is a constant as in (3.35).

Proof. From Lemmas 3.8 and 3.9, the operator π‘„π‘š has a fixed point π‘¦π‘š satisfying (3.26), (3.27), (3.35). And {π‘¦π‘š}βˆžπ‘š=1 and {π·π›½π‘¦π‘š}βˆžπ‘š=1 are relatively compact in 𝐢([0,𝑇]). Hence, {π‘¦π‘š}βˆžπ‘š=1 is relatively compact in 𝔼. And therefore, there exist π‘¦βˆˆπ”Ό and a subsequence π‘¦π‘šπ‘˜ of {π‘¦π‘š}βˆžπ‘š=1 such that limπ‘˜β†’βˆžπ‘¦π‘šπ‘˜=𝑦 in 𝔼. Consequently, 𝑦 is positive and continuous. Moreover 𝑦 satisfies (3.47), β€–π‘¦β€–βˆ—β‰€π‘Š. And limπ‘˜β†’βˆžπ‘“π‘šπ‘˜ξ€·π‘‘,π‘¦π‘šπ‘˜(𝑑),π·π›½π‘¦π‘šπ‘˜ξ€Έξ€·(𝑑)=𝑓𝑑,𝑦(𝑑),𝐷𝛽[].𝑦(𝑑),fora.e.π‘‘βˆˆ0,𝑇(3.42) Keeping in mind (3.35) holding, where π‘Š is a positive constant, it follows from inequalities (3.4) and (3.26) and from Lemma 2.5(ii) that β€–β€–0≀𝐺(𝑑,𝑠)π‘“π‘šξ€·π‘ ,π‘¦π‘š(𝑠),π·π›½π‘¦π‘šξ€Έβ€–β€–ξ‚»π›Ύ(𝑠)≀𝐸𝛾(𝑠)0(𝑠)+π‘ž1𝐾1π‘ π›Όβˆ’1(π‘‡βˆ’π‘ )2𝑇+𝑝1𝐾2π‘ π›Όβˆ’π›½βˆ’1(π‘‡βˆ’π‘ )2𝑇,+π‘ž(1)+𝑝(1)+π‘ž(π‘Š)+𝑝(π‘Š)(3.43) for a.e. π‘ βˆˆ[0,𝑇] and all π‘‘βˆˆ[0,𝑇], π‘šβˆˆπ‘. Hence, by the Lebesgue dominated convergence theorem, we have limπ‘˜β†’+βˆžξ€œπ‘‡0𝐺(𝑑,𝑠)π‘“π‘šξ€·π‘ ,π‘¦π‘š(𝑠),π·π›½π‘¦π‘šξ€Έξ€œ(𝑠)𝑑𝑠=𝑇0𝐺(𝑑,𝑠)𝑓𝑠,𝑦(𝑠),𝐷𝛽𝑦(𝑠)𝑑𝑠,(3.44) for π‘‘βˆˆ[0,𝑇]. Now, passing to the limit as π‘˜β†’+∞ in π‘¦π‘šπ‘˜ξ€œ(𝑑)=𝑇0𝐺(𝑑,𝑠)π‘“π‘šπ‘˜ξ€·π‘ ,π‘¦π‘šπ‘˜(𝑠),π·π›½π‘¦π‘šπ‘˜ξ€Έπ‘¦(𝑠)𝑑𝑠+0Δ𝑑(π›Όβˆ’1),(3.45) we have ξ€œπ‘¦(𝑑)=𝑇0𝐺(𝑑,𝑠)𝑓𝑠,𝑦(𝑠),𝐷𝛽𝑦𝑦(𝑠)𝑑𝑠+0Δ𝑑(π›Όβˆ’1)[].,forπ‘‘βˆˆ0,𝑇(3.46) Consequently, 𝑦 is a positive solution of BVP (1.4)-(1.5) by Lemma 2.4.

By Lemmas 3.5 and 3.9, and Theorem 3.10, we give the following unique result without proof.

Theorem 3.11. Suppose that (H0)-(H4) hold, then problem (1.4)-(1.5) has a unique positive solution 𝑦 and ‖𝑦(𝑑)β€–β‰₯𝐾1π‘‘π›Όβˆ’1(π‘‡βˆ’π‘‘)2𝑇,‖‖𝐷𝛽𝑦‖‖(𝑑)β‰₯𝐾2π‘‘π›Όβˆ’π›½βˆ’1(π‘‡βˆ’π‘‘)2𝑇[].,π‘‘βˆˆ0,𝑇(3.47) Moreover, 𝑦 is continuous and β€–π‘¦β€–βˆ—β‰€π‘Š, where π‘Š is a constant as in (3.35).

4. Existence Results for BVP (1.6)-(1.7)

We now give the existence of positive solution of BVP (1.6)-(1.7) by using diagonalization process.

Theorem 4.1. Suppose that (H0)-(H3) hold, then BVP (1.6)-(1.7) has a positive solution 𝑦, and 𝐷𝛽𝑦 is also positive.

Proof. Firstly, choose ξ€½π‘‡π‘›ξ€Ύβˆžπ‘›=1βˆˆπ‘βˆ—tobeasubsequenceofnumbers,suchthat𝑇≀𝑇1<𝑇2<β‹―<𝑇𝑛<β‹―β†‘βˆž,(4.1) then consider the BVP, 𝐷𝛼𝑦(𝑑)+𝑓𝑑,𝑦(𝑑),𝐷𝛽𝑦(𝑑)=πœƒ,a.e.π‘‘βˆˆ0,𝑇𝑛,(4.2) subject to 𝑦(0)=πœƒ,π·π›½π‘¦ξ€·π‘‡π‘›ξ€Έβˆ’π‘šβˆ’2𝑖=1π‘Žπ‘–π‘¦ξ€·πœ‰π‘–ξ€Έβˆ’π‘šβˆ’2𝑖=1π‘π‘–π·π›½π‘¦ξ€·πœ‰π‘–ξ€Έ=𝑦0.(4.3)
Theorem 3.10 guarantees that BVP (4.2)-(4.3) has a positive continuous solution 𝑦𝑛. And for any π‘›βˆˆπ‘, β€–β€–π‘¦π‘›β€–β€–βˆ—β‰€π‘Šπ‘›ξ€Ί,forπ‘‘βˆˆ0,𝑇𝑛,(4.4) where π‘Šπ‘› is a constant defined similarly to π‘Š.
Secondly, we apply the following diagonalization process. For π‘›βˆˆπ‘, let 𝑒𝑛𝑦(𝑑)=𝑛(𝑑),π‘‘βˆˆ0,𝑇𝑛,𝑦𝑛𝑇𝑛𝑇,π‘‘βˆˆπ‘›ξ€Έ.,+∞(4.5) Here, {𝑇𝑛}βˆžπ‘›=1 is defined in (4.1). Notice that 𝑒𝑛(𝑑)∈𝐢[0,+∞) with ‖‖𝑒0≀𝑛‖‖(𝑑)β‰€π‘Š1‖‖𝐷,0≀𝛽𝑒𝑛‖‖(𝑑)β‰€π‘Š1ξ€Ί,forπ‘‘βˆˆ0,𝑇1ξ€».(4.6) Also for π‘›βˆˆπ‘ and π‘‘βˆˆ[0,𝑇1], we get π‘’π‘›ξ€œ(𝑑)=𝑇10𝐺𝑇1ξ€·(𝑑,𝑠)𝑓𝑠,𝑒𝑛(𝑑),𝐷𝛽𝑒𝑛𝑦(𝑑)𝑑𝑠+0Ξ”π‘‘π›Όβˆ’1,(4.7) where 𝐺𝑇𝑛(𝑑,𝑠) are similarly defined as in (2.11), but all of 𝑇 should be replaced by 𝑇𝑛. Then for 𝑑1,𝑑2∈[0,𝑇1], we have ‖‖𝑒𝑛𝑑1ξ€Έβˆ’π‘’π‘›ξ€·π‘‘2ξ€Έβ€–β€–β‰€ξ€œπ‘‡10||𝐺𝑇1𝑑1ξ€Έ,π‘ βˆ’πΊπ‘‡1𝑑2ξ€Έ||‖‖𝑓,𝑠𝑠,π‘’π‘›π‘˜(𝑑),π·π›½π‘’π‘›π‘˜ξ€Έβ€–β€–+𝑦(𝑑)𝑑𝑠0Ξ”||𝑑1π›Όβˆ’1βˆ’π‘‘2π›Όβˆ’1||,‖‖𝐷𝛽𝑒𝑛𝑑1ξ€Έβˆ’π·π›½π‘’π‘›ξ€·π‘‘2ξ€Έβ€–β€–β‰€ξ€œπ‘‡10||𝐷𝛽𝐺𝑇1𝑑1ξ€Έ,π‘ βˆ’π·π›½πΊπ‘‡1𝑑2ξ€Έ||×‖‖𝑓,𝑠𝑠,π‘’π‘›π‘˜(𝑑),π·π›½π‘’π‘›π‘˜ξ€Έβ€–β€–+‖‖𝑦(𝑑)𝑑𝑠0β€–β€–Ξ“(𝛼)||𝑑ΔΓ(π›Όβˆ’π›½)1π›Όβˆ’π›½βˆ’1βˆ’π‘‘2π›Όβˆ’π›½βˆ’1||.(4.8) Thus, when |𝑑1βˆ’π‘‘2|<𝛿(πœ–,1), similarly to (3.38), ‖‖𝑒𝑛𝑑1ξ€Έβˆ’π‘’π‘›ξ€·π‘‘2ξ€Έβ€–β€–ξ€½β€–β€–<πœ–π›Ύπ›Ύ0‖‖𝐿1+π‘ˆ11+π‘ˆ12+ξ€Ίξ€·π‘Šπ‘ž(1)+𝑝(1)+π‘ž1ξ€Έξ€·π‘Š+𝑝1‖𝛾‖𝐿1ξ€Ύ+‖‖𝑦0β€–β€–Ξ”β€–β€–π·πœ–,𝛽𝑒𝑛𝑑1ξ€Έβˆ’π·π›½π‘’π‘›ξ€·π‘‘2ξ€Έβ€–β€–ξ€½β€–β€–<πœ–π›Ύπ›Ύ0‖‖𝐿1+π‘ˆ11+π‘ˆ12+ξ€Ίπ‘žξ€·π‘Š(1)+𝑝(1)+π‘ž1ξ€Έξ€·π‘Š+𝑝1‖𝛾‖𝐿1ξ€Ύ+‖‖𝑦0β€–β€–Ξ“(𝛼)πœ–Ξ”Ξ“(π›Όβˆ’π›½)(4.9) hold for an arbitrary πœ–>0, where 𝛿(πœ–,1) is a suitable positive number and π‘ˆ11,π‘ˆ12 are defined similarly to π‘ˆ1,π‘ˆ2 as in (3.31) and (3.32), respectively. By using (H3), we know that, for a.e. π‘‘βˆˆ[0,+∞), 𝑓(𝑑,π·π‘Š1,π·π‘Š1) is relatively compact, where π·π‘Š1={π‘’βˆˆπΆ([0,𝑇1]βˆΆβ€–π‘’β€–βˆ—β‰€π‘Š1}βˆ©π‘ƒ. Therefore, {𝑒𝑛(𝑑)}βˆžπ‘›=1 and {𝐷𝛽𝑒𝑛(𝑑)}βˆžπ‘›=1 are relatively compact. The ArzelΓ‘-Ascoli theorem guarantees that there is a subsequence π‘βˆ—1 of 𝑁 and a function 𝑧1∈𝐢([0,𝑇1],𝐸) with π‘’π‘›π‘˜β†’π‘§1 in 𝐢([0,𝑇1],𝐸) as π‘˜β†’+∞ through π‘βˆ—1. Obviously, 𝑧1 is positive. Let 𝑁1=π‘βˆ—1⧡{1}, noticing that ‖‖𝑒0≀𝑛‖‖(𝑑)β‰€π‘Š2‖‖𝐷,0≀𝛽𝑒𝑛‖‖(𝑑)β‰€π‘Š2ξ€Ί,forπ‘‘βˆˆ0,𝑇2ξ€».(4.10) Similarly to above argumentation, we have that there is a subsequence π‘βˆ—2 of 𝑁1 and a function 𝑧2∈𝐢([0,𝑇2],𝐸) with π‘’π‘›π‘˜β†’π‘§2 in 𝐢([0,𝑇2],𝐸) as π‘˜β†’+∞ through π‘βˆ—2. Obviously, 𝑧2 is positive. Note that 𝑧1=𝑧2 on [0,𝑇1] since π‘βˆ—2βŠ‚π‘1. Let 𝑁2=π‘βˆ—2⧡{2}. Proceed inductively to obtain for π‘š={2,3,…} a subsequence π‘βˆ—π‘š of π‘π‘šβˆ’1 and a function π‘§π‘šβˆˆπΆ([0,π‘‡π‘š],𝐸) with π‘’π‘›π‘˜β†’π‘§π‘š in 𝐢([0,π‘‡π‘š],𝐸) as π‘˜β†’+∞ through π‘βˆ—π‘š. Also, π‘§π‘š is positive. Let π‘π‘š=π‘βˆ—π‘šβ§΅{π‘š}.
Define a function 𝑦 as follows. Fix π‘‘βˆˆ(0,+∞), and Let π‘šβˆˆπ‘ with π‘ β‰€π‘‡π‘š, then define 𝑦(𝑑)=π‘§π‘š(𝑑). Hence π‘¦βˆˆπΆ([0,+∞),ℝ).
Again fix π‘‘βˆˆ[0,+∞) and Let π‘šβˆˆπ‘ with π‘ β‰€π‘‡π‘š. Then for π‘›βˆˆπ‘π‘š we get π‘’π‘›π‘˜ξ€œ(𝑑)=π‘‡π‘š0πΊπ‘‡π‘šξ€·(𝑑,𝑠)𝑓𝑠,π‘’π‘›π‘˜(𝑠),π·π›½π‘’π‘›π‘˜ξ€Έπ‘¦(𝑠)𝑑𝑠+0Δ𝑑(π›Όβˆ’1).(4.11) Let π‘›π‘˜β†’+∞ through π‘π‘š to obtain π‘§π‘šξ€œ(𝑑)=π‘‡π‘š0πΊπ‘‡π‘šξ€·(𝑑,𝑠)𝑓𝑠,π‘§π‘š(𝑠),π·π›½π‘§π‘šξ€Έπ‘¦(𝑠)𝑑𝑠+0Δ𝑑(π›Όβˆ’1),(4.12) that is, ξ€œπ‘¦(𝑑)=π‘‡π‘š0πΊπ‘‡π‘šξ€·(𝑑,𝑠)𝑓𝑠,𝑦(𝑠),𝐷𝛽𝑦𝑦(𝑠)𝑑𝑠+0Δ𝑑(π›Όβˆ’1).(4.13) We can use this method for each π‘ βˆˆ[0,π‘‡π‘š] and for each π‘šβˆˆπ‘. Hence, 𝐷𝛼𝑦(𝑑)+𝑓𝑑,𝑦(𝑑),𝐷𝛽𝑦(𝑑)=πœƒ,a.e.π‘‘βˆˆ0,π‘‡π‘šξ€»,(4.14) for each π‘šβˆˆπ‘. Consequently, the constructed function 𝑦 is a solution of (1.6)-(1.7). This completes the proof of the theorem.

Remark 4.2. In [21], the authors considered the BVP (1.3). Under some suitable conditions, they obtained the existence result of unbounded solution. In nature, BVP (1.3) is a special form of BVP (1.6)-(1.7). In that scalar situation, π›Όβˆ’π›½=1, 𝑏𝑖=0(𝑖=1,2,…,π‘šβˆ’2), 𝑏1>0, 𝑏𝑖=0(𝑖=2,3,…,π‘šβˆ’2), 𝑦