Positive Solution of Singular Fractional Differential Equation in Banach Space

Abstract

We investigated a singular multipoint boundary value problem for fractional differential equation in Banach space. The nonlinear term 𝑓(𝑡,𝑥,𝑦) is positive and singular at 𝑥=𝜃 and (or) 𝑦=𝜃. 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 problem𝐷𝑦^𝛼(𝑦0()𝑡)=+0𝑓,𝑡,𝐷𝑦(_𝛽𝑡),𝐷^𝛽𝑦(𝑡)=0,𝑡∈(0,1),𝑦(1)−^𝑚−2_𝑖=1𝑏_𝑖𝐷^𝛽𝑦𝜉_𝑖=𝑦₀,(1.1) where 1<𝛼≤2, 0<𝛽≤𝛼−1, 𝑏_𝑖≥0, 0<𝜉_𝑖<1, (𝑖=1,2,…,𝑚−2) with 𝛾=∑^{𝑚−2𝑖=1}𝑏_𝑖𝜉^{𝑖𝛼−𝛽−1}<1, and 𝐷^𝛽 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: 𝐷^𝛼𝑦(𝑡)+𝑦(𝑓0)(𝑡=,𝑦𝑦(𝑡()1,)𝐷=^𝜇0𝑦,(𝑡))=0,(1.2) where 1<𝛼≤2, 0<𝜇≤𝛼−1. 𝑓(𝑡,𝑥,𝑦) satisfies the Carathéodory conditions and is singular at 𝑥=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:𝐷^𝛼𝑦(𝑡)𝑦+(0𝑓)(=𝑡,𝑦0,(𝑡))=l0im,_𝑡→+∞𝑡∈(0,+∞),𝛼∈(1,2),𝐷^𝛼−1𝑦(𝑡)=𝛽𝑦(𝜉)(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,‖⋅‖) be a real Banach space. 𝑃 is a cone in 𝐸 which defines a partial ordering in 𝐸 by 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝑃. 𝑃 is said to be normal if there exists a positive constant 𝑁 such that 𝜃≤𝑥≤𝑦 implies ‖𝑥‖≤𝑁‖𝑦‖, where 𝜃 denotes the zero element of 𝐸, and the smallest 𝑁 is called the normal constant of 𝑃 (it is clear that 𝑁≥1). If 𝑥≤𝑦 and 𝑥≠𝑦, we write 𝑥<𝑦. Let 𝑃₊=𝑃⧵{𝜃}. So, 𝑥∈𝑃₊ if and only if 𝑥>𝜃. 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 domain𝑦(0)𝐷=^𝛼𝑦𝜃(,𝑡)+𝑓𝐷_𝛽𝑡,𝑦(𝑡),𝐷^𝛽𝑦(𝑡)=𝜃,a.e.𝑡∈[0,𝑇],(1.4)𝑦(𝑇)−^𝑚−2_𝑖=1𝑎_𝑖𝑦𝜉_𝑖−^𝑚−2_𝑖=1𝑏_𝑖𝐷^𝛽𝑦𝜉_𝑖=𝑦₀,(1.5) or with unbounded domain𝑦(0)=𝐷𝜃^𝛼𝑦,(𝑡)+li𝑓m_𝑡→+∞𝑡,𝑦(𝑡),𝐷^𝛽𝑦(𝑡)=𝜃,a.e.𝑡∈[0,∞),(1.6)𝐷^𝛽𝑦(𝑡)−^𝑚−2_𝑖=1𝑎_𝑖𝑦𝜉_𝑖−^𝑚−2_𝑖=1𝑏_𝑖𝐷^𝛽𝑦𝜉_𝑖=𝑦₀.(1.7) Here, 1<𝛼≤2, 0<𝛽≤𝛼−1, 𝜉_𝑖>0, 𝑎_𝑖,𝑏_𝑖≥0(𝑖=1,2,…,𝑚−2), 𝑦₀≥𝜃 are real numbers, and 𝐷^𝛼 is the standard Riemann-Liouville fractional derivative. And 𝑓∶[0,+∞)×𝑃₊×𝑃₊→𝑃₊ is singular at 𝑥=𝜃 and 𝑦=𝜃 and satisfies other conditions which will be specified later. In addition, 𝑓(𝑡,𝑥,𝑦) is the Carathéodory function.

We say that 𝑓 satisfies the Carathéodory conditions on [0,+∞)×𝐵, 𝐵=𝑃₊×𝑃₊(𝑓∈Car([0,+∞)×𝐵)) if(i)𝑓(⋅,𝑥,𝑦)∶[0,+∞)→𝐸 is measurable for all (𝑥,𝑦)∈𝐵,(ii)𝑓(𝑡,⋅,⋅)∶𝐵→𝐸 is continuous for a.e.𝑡∈[0,+∞),(iii)for each compact set 𝐾⊂𝐵, there is a function 𝜙_𝐾∈𝐿¹[0,+∞) such that ‖𝑓(𝑡,𝑥,𝑦)‖≤𝜙_𝐾(𝑡),fora.e.𝑡∈[0,+∞),∀(𝑥,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(𝑡)∈𝐿¹([0,𝑏],ℝ) of 𝜇∈ℝ⁺ is defined by 𝐼^𝜇𝑣(𝑡)=1Γ(𝜇)^𝑡0(𝑡−𝑠)^𝜇−1𝑣(𝑠)𝑑𝑠,𝑡>0.(2.1)

Definition 2.2. The Riemann-Liouville fractional derivative of order 𝜇>0 for a function 𝑣(𝑡) given in the interval [0,∞) is defined by 𝐷^𝜇𝑣(𝑡)=1Γ(𝑛−𝜇)𝑑𝑑𝑡^𝑛^𝑡0(𝑡−𝑠)^{𝑛−𝜇−1}𝑣(𝑠)𝑑𝑠(2.2) provided that the right hand side is point wise defined. Here, 𝑛=[𝜇]+1 and [𝜇] means the integral part of the number 𝜇, and Γ is the Euler gamma function.

The following properties of the fractional calculus theory are well known, see, for example, [2, 4]:(i)𝐷^𝛽𝐼^𝛽𝑣(𝑡)=𝑣(𝑡) for a.e. 𝑡∈[0,𝑇], where 𝑣(𝑡)∈𝐿¹[0,𝑇], 𝛽>0,(ii)𝐷^𝛽𝑣(𝑡)=0 if and only if 𝑣(𝑡)=∑^𝑛𝑗=1𝑐_𝑗𝑡^𝛽−𝑗, where 𝑐_𝑗(𝑗=1,2,…,𝑛) are arbitrary constants, 𝑛=[𝛽]+1, 𝛽>0,(iii)𝐼^𝛽∶𝐶([0,𝑇])→𝐶([0,𝑇]), 𝐼^𝛽∶𝐿¹([0,𝑇])→𝐿¹([0,𝑇]), 𝛽>0,(iv)𝐷^𝛽𝐼^𝛼=𝐼^𝛼−𝛽 and 𝐷^𝛽𝑡^𝛼=(Γ(𝛽+1)/Γ(𝛽−𝛼+1))𝑡^𝛽−𝛼 for 𝑡∈[0,𝑇], 𝛼−𝛽>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)Δ=Γ(𝛼)𝑇^{𝛼−𝛽−1}Γ(𝛼−𝛽)−^𝑚−2_𝑖=1𝑎_𝑖𝜉^𝑖𝛼−1−^𝑚−2_𝑖=1𝑏_𝑖Γ(𝛼)𝜉^{𝑖𝛼−𝛽−1}Γ(𝛼−𝛽)>0,(2.3)(H₁)𝑓∈Car([0,+∞)×𝐵), 𝐵=(0,+∞)×(0,+∞), lliimm_{‖𝑥‖→0‖𝑦‖→0}‖𝑓(𝑡,𝑥,𝑦)‖=+∞,fora.e.𝑡∈[0,+∞)andall𝑦∈𝑃₊,‖𝑓(𝑡,𝑥,𝑦)‖=+∞,fora.e.𝑡∈[0,+∞)andall𝑥∈𝑃₊,(2.4) and there exists a positive constant 𝜛 such that for all 𝑇₀≥𝑇, ‖𝑓(𝑡,𝑥,𝑦)‖≥𝜛1−𝑡𝑇₀^{2+𝛽−𝛼}fora.e.𝑡∈0,𝑇₀andall(𝑥,𝑦)∈𝐵,(2.5)(H₂)𝑓 fulfills the estimate, ‖𝑓(𝑡,𝑥,𝑦)‖≤𝛾(𝑡)𝛾₀(f𝑡o)r+a𝑞.e₁.(𝑡‖𝑥∈‖[)0+,+𝑝∞₁()‖,𝑦‖an)d+a𝑞ll(‖(𝑥𝑥‖,𝑦))+∈𝑝𝐵(‖,𝑦‖)(2.6) where 𝛾,𝛾₀∈𝐿¹[0,+∞), 𝑞₁,𝑝₁,𝑞,𝑝∈𝐶((0,+∞),ℝ⁺), 𝑞₁,𝑝₁ are nonincreasing, and, for any 𝑇₀≥𝑇, ^𝑇0₀𝛾(𝑡)𝑞₁⎛⎜⎜⎝𝐾₁𝑡^𝛼−1𝑇₀−𝑡²𝑇₀⎞⎟⎟⎠𝑑𝑡<+∞,𝐾₁=𝜛2Γ(𝛼)_𝑇00,𝛾(𝑡)𝑝₁⎛⎜⎜⎝𝐾₂𝑡^{𝛼−𝛽−1}𝑇₀−𝑡²𝑇₀⎞⎟⎟⎠𝑑𝑡<+∞,𝐾₂=𝜛2Γ(𝛼−𝛽),(2.7) while 𝑞,𝑝 are nondecreasing and lim_{‖𝑥‖→+∞}𝑞(‖𝑥‖)+𝑝(‖𝑥‖)‖𝑥‖=0,(2.8)(H₃)for a.e. 𝑡∈[0,+∞), and for all 𝐷⊂𝑃, 𝑓(𝑡,𝐷,𝐷) is relatively compact.

Remark 2.3. It follows from (2.4) that under condition (H₂), lim_{‖𝑥‖→0}𝑞₁(‖𝑥‖)=+∞ and lim_{‖𝑦‖→0}𝑝₁(‖𝑦‖)=+∞.

In the sequel, 𝐿¹([0,𝑇],ℝ) denote the Banach space of functions 𝑦∶[0,𝑇]→ℝ which are Lebesgue integrable with the norm ‖𝑦‖_𝐿¹=^𝑇0||𝑦(𝑡)||𝑑𝑡.(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 ℎ(𝑡)∈𝐿¹([0,𝑇]) and that (𝐻₀) holds, then the unique solution of linear BVP 𝐷^𝛼𝑦(𝑡)+ℎ(𝑡)=0, a.e. 𝑡∈[0,𝑇] with the boundary condition (1.5) is given by 𝑦(𝑡)=^𝑇0𝐺(𝑡,𝑠)ℎ(𝑠)𝑑𝑠+𝑦⁰Δ𝑡^𝛼−1,(2.10) where 𝐺(𝑡,𝑠)=1Δ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩(𝑇−𝑠)^{𝛼−𝛽−1}𝑡^𝛼−1Γ(𝛼−𝛽)−Γ𝑡(𝛼^𝛼−1)^𝑚−2_𝑗=𝑖𝑎_𝑗𝜉_𝑗−𝑠^𝛼−1−𝑡^𝛼−1Γ(𝛼−𝛽)^𝑚−2_𝑗=𝑖𝑏_𝑗𝜉_𝑗−𝑠^{𝛼−𝛽−1}−Δ(𝑡−𝑠)^𝛼−1𝑡≥𝑠,𝜉_𝑖−1Γ(𝛼),(𝑇−𝑠)_{𝛼−𝛽−1}<𝑠≤𝜉_𝑖,𝑖=1,2,…,𝑚−1,𝑡^𝛼−1Γ(𝛼−𝛽)−Γ𝑡(𝛼^𝛼−1)^𝑚−2_𝑗=𝑖𝑎_𝑗𝜉_𝑗−𝑠^𝛼−1−𝑡^𝛼−1Γ(𝛼−𝛽)^𝑚−2_𝑗=𝑖𝑡≤𝑠,𝜉_𝑖−1𝑏_𝑗𝜉_𝑗−𝑠^{𝛼−𝛽−1},<𝑠≤𝜉_𝑖,𝑖=1,2,…,𝑚−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(𝑡,𝑠) defined as (2.11) has the following properties:(i)𝐺(𝑡,𝑠) is uniformly continuous about 𝑡 in [0,𝑇],(ii)𝐺(𝑡,𝑠)≥0 for all (𝑡,𝑠)∈[0,𝑇]×[0,𝑇] and 𝐺(𝑡,𝑠)≤𝐸, where 𝐸=𝑇^{2𝛼−𝛽−2}ΔΓ(𝛼−𝛽),(2.12)(iii)∫^𝑇0𝐺(𝑡,𝑠)𝑅(𝑠)𝑑𝑠≥𝑡^𝛼−1(𝑇−𝑡)²/2𝑇Γ(𝛼) if 𝑅(𝑠)≥(1−𝑡/𝑇)^{2+𝛽−𝛼}.

Proof. From (2.3), it is easy to verify (i) and (ii). We now show that (iii) is true. Firstly, if 𝑡≥𝑠, then (2.11) gives 𝐺(𝑡,𝑠)=1Δ⎧⎪⎪⎪⎨⎪⎪⎪⎩(𝑇−𝑠)^{𝛼−𝛽−1}𝑡^𝛼−1Γ(𝛼−𝛽)−Γ𝑡(𝛼^𝛼−1)^𝑚−2_𝑗=𝑖𝑎_𝑗𝜉_𝑗−𝑠^𝛼−1−𝑡^𝛼−1Γ(𝛼−𝛽)^𝑚−2_𝑗=𝑖𝑏_𝑗𝜉_𝑗−𝑠^{𝛼−𝛽−1}−Δ(𝑡−𝑠)^𝛼−1𝜉Γ(𝛼)_𝑖−1,<𝑠≤𝜉_𝑖,𝑖=1,2,…,𝑚−1.(2.13) Then, 𝐺(𝑡,𝑠)≥1Δ(𝑇−𝑠)^{𝛼−𝛽−1}𝑡^𝛼−1Γ(𝛼−𝛽)−Γ𝑡(𝛼^𝛼−1)^𝑚−2_𝑗=1𝑎_𝑗𝜉_𝑗−𝑠^𝛼−1−𝑡^𝛼−1Γ(𝛼−𝛽)^𝑚−2_𝑗=1𝑏_𝑗𝜉_𝑗−𝑠^{𝛼−𝛽−1}−Δ(𝑡−𝑠)^𝛼−1≥𝑡Γ(𝛼)_𝛼−1Δ(𝑇Γ−(𝛼𝑠)−𝛽)^{𝛼−𝛽−1}−Γ1(𝛼)^𝑚−2_𝑗=1𝑎_𝑗𝜉^𝑗𝛼−11−𝜉𝑠_𝑗^𝛼−1−1Γ(𝛼−𝛽)^𝑚−2_𝑗=1𝑏_𝑗𝜉^{𝑗𝛼−𝛽−1}1−𝜉𝑠_𝑗^{𝛼−𝛽−1}−Δ(1−𝑠/𝑡)^𝛼−1≥𝑡Γ(𝛼)_𝛼−1(1−𝑠/𝑇)^{𝛼−𝛽−1}ΔΓ𝑇(𝛼−𝛽^{𝛼−𝛽−1})−Γ(𝛼1−𝛽)^𝑚−2_𝑗=1𝑏_𝑗𝜉^{𝑗𝛼−𝛽−1}−(1−𝑠/𝑇)^𝛽Γ(𝛼)^𝑚−2_𝑗=1𝑎_𝑗𝜉^𝑗𝛼−1+Δ.(2.14) From (2.14) and (2.3), we deduce from the Lagrange mean value theorem that 𝐺(𝑡,𝑠)≥𝑡^𝛼−1(1−𝑠/𝑇)^{𝛼−𝛽−1}∑^{𝑚−2𝑗=1}𝑎_𝑗𝜉^𝑗𝛼−1+Δ≥𝑡_𝛼−1ΔΓ(𝛼)×1−1−𝑇𝑠^𝛽(1−𝑠/𝑇)^{𝛼−𝛽−1}∑^{𝑚−2𝑗=1}𝑎_𝑗𝜉^𝑗𝛼−1+ΔΔΓ(𝛼)𝛽𝜉^𝛽−1𝑠𝑇.(2.15) In view of (1−𝑠/𝑇)≤𝜉≤1 and 𝛽<1, one can obtain for 𝑡≥𝑠 that 𝐺(𝑡,𝑠)≥𝑡^𝛼−1(1−𝑠/𝑇)^{𝛼−𝛽−1}∑^{𝑚−2𝑗=1}𝑎_𝑗𝜉^𝑗𝛼−1+ΔΔΓ(𝛼)𝑇𝛽𝑠.(2.16) Analogously, if 𝑡≤𝑠, one has 𝐺(𝑡,𝑠)≥𝑡^𝛼−1(1−𝑠/𝑇)^{𝛼−𝛽−1}Γ(𝛼).(2.17) It follows from (2.16) and (2.17) that ^𝑇0𝐺(𝑡,𝑠)𝑅(𝑠)𝑑𝑠=≥^𝑡0𝐺(𝑡,𝑠)𝑅(𝑠)𝑑𝑠+^𝑇𝑡𝐺(𝑡,𝑠)𝑅(𝑠)𝑑𝑠_𝑡0𝑡^𝛼−1∑^{𝑚−2𝑗=1}𝑎_𝑗𝜉^𝑗𝛼−1+ΔΔΓ(𝛼)𝑇𝛽𝑠1−𝑇𝑠𝑑𝑠+^𝑇𝑡𝑡^𝛼−1≥𝑡Γ(𝛼)1−𝑇𝑠𝑑𝑠_𝛼−1(𝑇−𝑡)²2𝑇Γ(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={𝑥∈𝐶([0,𝑇],𝐸)∶𝐷^𝛽𝑥∈𝐶([0,𝑇],𝐸)} equipped with the norm ‖𝑥‖_∗=max{‖𝑥‖,‖𝐷^𝛽𝑥‖}, then 𝔼 is a real Banach space (see [24]).

Since the nonlinear term 𝑓(𝑡,𝑥,𝑦) is singular at 𝑥=𝜃 and 𝑦=𝜃, we use the following regularization process. For each 𝑚∈𝑁⁺, define 𝑓_𝑚 by the formula𝑓_𝑚(𝑡,𝑥,𝑦)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑓(𝑡,𝑥,𝑦)if𝑥≥𝑐𝑓𝑡,𝑐𝑚,𝑦≥𝑚𝑐,𝑓𝑡,𝑥𝑚,,𝑐𝑦if0≤𝑥<𝑚𝑐,𝑦≥𝑚𝑐,𝑓𝑡,𝑐𝑚if𝑥≥𝑚1,0≤𝑦<𝑚1,𝑚,𝑚𝑐if0≤𝑥<𝑚𝑐,0≤𝑦<𝑚𝑐,(3.1) where 𝑐>𝜃 is a given element of 𝔼 and ‖𝑐‖=1.

Remark 3.1. The function 𝑓_𝑚 defined by (3.1) satisfies 𝑓_𝑚∈Car([0,𝑇]×𝐵_∗), 𝐵_∗=𝑃×𝑃. And conditions (H₁) and (H₂) imply ‖‖𝑓_𝑚(𝑡,𝑥,𝑦)‖‖≥𝜛1−𝑡𝑇^{1+𝛽−𝛼}‖‖𝑓,fora.e.𝑡∈[0,𝑇]andall(𝑥,𝑦)∈𝐵_∗,(3.2)_𝑚(𝑡,𝑥,𝑦)‖‖≤𝛾(𝑡)𝛾₀(𝑡)+𝑞₁‖‖‖1𝑚‖‖‖+𝑝¹‖‖‖𝑚1f‖‖‖ora+.e𝑞.𝑡(1∈)+[0𝑝,𝑇(1])+an𝑞d(‖al𝑥l‖()𝑥+,𝑦𝑝)(∈‖𝑦𝐵‖_∗),‖‖𝑓,(3.3)_𝑚(𝑡,𝑥,𝑦)‖‖≤𝛾(𝑡)𝛾₀(𝑡)+𝑞₁(‖𝑥‖)+𝑝₁(‖𝑦‖fo)r+a.𝑞e.(1𝑡)∈+[𝑝0,(1𝑇)]+a𝑞nd(‖a𝑥l‖l)(𝑥+,𝑝𝑦)(‖∈𝑦‖𝐵).,(3.4)

Remark 3.2. The function 𝑓_𝑚 defined by (3.1) satisfies lim_𝑚→+∞𝑓_𝑚=𝑓.

Define operator 𝑄_𝑚∶𝑃₊→𝑃₊ by the following: 𝑄_𝑚𝑦(𝑡)=^𝑇0𝐺(𝑡,𝑠)𝑓_𝑚𝑠,𝑦(𝑠),𝐷^𝛽𝑦(𝑠)𝑑𝑠+𝑦Δ⁰𝑡^(𝛼−1).(3.5)

Lemma 3.3. Suppose that (H₀) holds, then 𝐷^𝛽𝑄_𝑚𝑦(𝑡)=^𝑇0𝐷^𝛽𝐺(𝑡,𝑠)𝑓_𝑚𝑠,𝑦(𝑠),𝐷^𝛽𝑦(𝑠)𝑑𝑠+𝑦⁰Γ(𝛼)ΔΓ(𝛼−𝛽)𝑡^{(𝛼−𝛽−1)},(3.6) where 𝐷^𝛽𝐺(𝑡,𝑠)=Γ(𝛼)ΔΓ(𝛼−𝛽)⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩(𝑇−𝑠)^{𝛼−𝛽−1}𝑡^{𝛼−𝛽−1}Γ(𝛼−𝛽)−𝑡Γ(𝛼)^{𝛼−𝛽−1}^𝑚−2_𝑗=𝑖𝑎_𝑗𝜉_𝑗−𝑠^𝛼−1−𝑡^{𝛼−𝛽−1}Γ(𝛼−𝛽)^𝑚−2_𝑗=𝑖𝑏_𝑗𝜉_𝑗−𝑠^{𝛼−𝛽−1}−Δ(𝑡−𝑠)^{𝛼−𝛽−1}𝑡≥𝑠,𝜉_𝑖−1Γ(𝛼),(𝑇−𝑠)_{𝛼−𝛽−1}<𝑠≤𝜉_𝑖,𝑖=1,2,…,𝑚−1,𝑡^{𝛼−𝛽−1}Γ(𝛼−𝛽)−𝑡Γ(𝛼)^{𝛼−𝛽−1}^𝑚−2_𝑗=𝑖𝑎_𝑗𝜉_𝑗−𝑠^𝛼−1−𝑡^{𝛼−𝛽−1}Γ(𝛼−𝛽)^𝑚−2_𝑗=𝑖𝑡≤𝑠,𝜉𝑏_𝑗_𝑖−1𝜉_𝑗−𝑠^{𝛼−𝛽−1},<𝑠≤𝜉_𝑖,𝑖=1,2,…,𝑚−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 𝐷^𝛽𝐺(𝑡,𝑠) defined as (3.7) has the following properties:(i)𝐷^𝛽𝐺(𝑡,𝑠) is uniformly continuous about 𝑡 in [0,𝑇],(ii)𝐷^𝛽𝐺(𝑡,𝑠)≥0 for all (𝑡,𝑠)∈[0,𝑇]×[0,𝑇] and 𝐷^𝛽𝐺(𝑡,𝑠)≤𝐸_𝐷, where 𝐸_𝐷=Γ(𝛼)𝑇^{2𝛼−2𝛽−2}Δ[Γ(𝛼−𝛽)]²,(3.8)(iii)∫^𝑇0𝐷^𝛽𝐺(𝑡,𝑠)𝑅(𝑠)𝑑𝑠≥𝑡^{𝛼−𝛽−1}(𝑇−𝑡)²/2𝑇Γ(𝛼−𝛽) if 𝑅(𝑠)≥(1−𝑡/𝑇)^{2+𝛽−𝛼}.

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,𝐿_𝐷 such that ‖‖𝑓𝑠,𝑥(𝑠),𝐷^𝛽𝑥(𝑠)−𝑓𝑠,𝑦(𝑠),𝐷^𝛽𝑦(𝑠)‖‖≤𝐿‖𝑥−𝑦‖+𝐿_𝐷‖‖𝐷^𝛽𝑥−𝐷^𝛽𝑦‖‖(3.9) and 𝜏=max{𝐸(𝐿+𝐿_𝐷),𝐸_𝐷(𝐿+𝐿_𝐷)}<1, then 𝑄_𝑚 has a unique fixed point.

Proof. Obviously, 𝑓_𝑚 defined by the formula (3.1) satisfy also the condition (H₄). By (3.5) and (3.6), it is easy to show that ‖𝑄_𝑚𝑥−𝑄_𝑚𝑦‖_∗<𝜏‖𝑥−𝑦‖_∗, then Banach contraction principle implies that the operator 𝑄_𝑚 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 𝑌 be a Banach space, and let 𝑃⊂𝑌 be a cone in 𝑌. Let Ω₁,Ω₂ be bounded open balls of 𝑌 centered at the origin with Ω₁⊂Ω₂. Suppose that 𝐴∶𝑃∩(Ω₂⧵Ω₁)→𝑃 is a completely continuous operator such that ‖𝐴𝑥‖≥‖𝑥‖for𝑥∈𝑃∩𝜕Ω₁,‖𝐴𝑥‖≤‖𝑥‖for𝑥∈𝑃∩𝜕Ω₂(3.10) hold, then 𝐴 has a fixed point in 𝑃∩(Ω₂⧵Ω₁).

Lemma 3.7. Let (H₀)-(H₃) hold, then 𝑄_𝑚∶𝑃→𝑃 and 𝑄_𝑚 is a completely continuous operator.

Proof. Firstly, let 𝑦∈𝑃, because 𝑓_𝑚∈Car([0,𝑇]×𝐵_∗) is positive. It follows from Lemma 2.5 (i) and (ii) that 𝑄_𝑚𝑦∈𝐶([0,𝑇],𝔼) and (𝑄_𝑚𝑦)(𝑡)≥𝜃 for 𝑡∈[0,𝑇]. Similarly, from Lemma 3.4 (i) and (ii) we can get that 𝐷^𝛽𝑄_𝑚𝑦∈𝐶([0,𝑇],𝔼) and (𝐷^𝛽𝑄_𝑚𝑦)(𝑡)≥𝜃 for 𝑡∈[0,𝑇]. To summarize, 𝑄_𝑚∶𝑃→𝑃.
Secondly, we prove that 𝑄_𝑚 is a continuous operator. Let {𝑥_𝑘}⊂𝑃 be a convergent sequence and lim_𝑘→+∞‖𝑥_𝑘−𝑥‖_∗=0, then 𝑥⊂𝑃 and ‖𝑥_𝑘‖_∗≤𝑆, where 𝑆 is a positive constant. In view of 𝑓_𝑚∈Car([0,𝑇]×𝐵_∗), we have lim_𝑘→+∞𝑓_𝑚(𝑡,𝑥_𝑘(𝑡),𝐷^𝛽𝑥_𝑘(𝑡))=𝑓_𝑚(𝑡,𝑥(𝑡),𝐷^𝛽𝑥(𝑡)). Since by (3.2), (3.3), 0<≤𝛾‖‖𝑓(𝑡_𝑚)𝑡𝛾,₀𝑥_𝑘(𝑡),𝐷^𝛽𝑥_𝑘(𝑡)‖‖(𝑡)+𝑞₁1𝑚+𝑝¹𝑚1+𝑞(1)+𝑝(1)+𝑞(𝑆)+𝑝(𝑆),(3.11) the Lebesgue dominated convergence theorem gives lim_𝑘→+∞^𝑇0‖‖𝑓_𝑚𝑡,𝑥_𝑘(𝑡),𝐷^𝛽𝑥_𝑘(𝑡)−𝑓_𝑚𝑡,𝑥(𝑡),𝐷^𝛽𝑥(𝑡)‖‖𝑑𝑡=0.(3.12) Now, from (3.12), Lemma 2.5(ii), Lemma 3.4(ii) and from the inequalities (cf. (3.5), (3.6)) ‖‖≤𝑄𝐸_𝑚𝑥_𝑘𝑇0(𝑡)−𝑄_𝑚𝑥(𝑡)‖‖‖‖𝐷‖‖𝑓_𝑚𝑡,𝑥_𝑘(𝑡),𝐷^𝛽𝑥_𝑘(𝑡)−𝑓_𝑚𝑡,𝑥(𝑡),𝐷^𝛽𝑥(𝑡)‖‖𝑑𝑡,_𝛽≤𝐸𝑄_𝑚𝐷𝑥_𝑘(𝑡)−𝐷^𝛽𝑄_𝑚𝑥(𝑡)‖‖^𝑇0‖‖𝑓_𝑚𝑡,𝑥_𝑘(𝑡),𝐷^𝛽𝑥_𝑘(𝑡)−𝑓_𝑚𝑡,𝑥(𝑡),𝐷^𝛽𝑥(𝑡)‖‖𝑑𝑡,(3.13) we have that lim_𝑘→+∞‖𝑄_𝑚𝑥_𝑘−𝑄_𝑚𝑥‖_∗=0, which proves that 𝑄_𝑚 is a continuous operator.
Thirdly, let Ω⊂𝑃 be bounded in 𝔼 and let ‖𝑥‖_∗≤𝐿 for all 𝑥∈Ω, where 𝐿 is a positive constant. We are in position to prove that 𝑄_𝑚(Ω) is bounded. Keeping in mind 𝑓_𝑚∈Car([0,𝑇]×𝐵_∗), there exists 𝜙∈𝐿¹([0,𝑇]) such that 0<‖‖𝑓_𝑚𝑡,𝑥_𝑘(𝑡),𝐷^𝛽𝑥_𝑘(𝑡)‖‖≤𝜙(𝑡)fora.e.𝑡∈[0,𝑇]andall𝑥∈Ω,(3.14) then (cf. (3.5)) ‖‖𝑄_𝑚𝑥(𝑡)‖‖≤𝐸^𝑇0‖‖𝑓_𝑚𝑠,𝑥(𝑠),𝐷^𝛽𝑥(𝑠)‖‖𝑑𝑠+𝑦⁰Δ𝑡^(𝛼−1)≤𝐸‖𝜙‖_𝐿¹+‖‖𝑦⁰‖‖Δ𝑇^(𝛼−1),(3.15) and (cf. (3.6)) ‖‖𝐷^𝛽𝑄_𝑚𝑥(𝑡)‖‖≤𝐸_𝐷^𝑇0‖‖𝑓_𝑚𝑠,𝑥(𝑠),𝐷^𝛽𝑥(𝑠)‖‖𝑑𝑠+‖‖‖𝑦⁰Γ(𝛼)ΔΓ(𝛼−𝛽)𝑡^{(𝛼−𝛽−1)}≤𝐸_𝐷‖‖‖‖𝜙‖_𝐿¹+‖‖𝑦⁰‖‖Γ(𝛼)ΔΓ(𝛼−𝛽)𝑇^{(𝛼−𝛽−1)},(3.16) for 𝑡∈[0,𝑇] and all 𝑥∈Ω. Therefore, 𝑄_𝑚(Ω) is bounded in 𝔼.
Fourthly, by (H₃) and (3.5), it is easy to show that 𝑄_𝑚(Ω)(𝑡) is relatively compact.
Finally, let 0≤𝑡₁<𝑡₂≤𝑇. From Lemma 2.5(i) and the functions 𝑡^𝛼−1,𝑡^{𝛼−𝛽−1} being uniformly continuous on [0,𝑇], for any arbitrary 𝜖>0, there exists a positive number 𝛿(𝜖), such that when |𝑡₁−𝑡₂|<𝛿(𝜖), one has |𝐺_𝑇(𝑡₁,𝑠)−𝐺_𝑇(𝑡₂,𝑠)|<𝜖 and |𝑡^1𝛼−1−𝑡^2𝛼−1|<𝜖, then (cf. (3.14)) the inequality ‖‖𝑄_𝑚𝑥𝑡₁−𝑄_𝑚𝑥𝑡₂‖‖<𝜖‖𝜙‖_𝐿¹+‖‖𝑦⁰‖‖Δ𝜖(3.17) holds. Hence the set of functions 𝑄_𝑚(Ω) is equicontinuous on [0,𝑇].
Therefore, by the Arzelá-Ascoli theorem, 𝑄_𝑚(Ω) is relatively compact in 𝔼. We have proved that 𝑄_𝑚 is a completely continuous operator.

Lemma 3.8. Suppose that (H₀)-(H₃) hold, then the operator 𝑄_𝑚 has at least a fixed point.

Proof. By Lemma 3.7, 𝑄_𝑚∶𝑃→𝑃 is completely continuous. In order to apply Lemma 3.6, we construct two bounded open balls Ω₁,Ω₂ and prove that the conditions (3.10) are satisfied with respect to 𝑄_𝑚.
Firstly, let Ω₁={𝑦∈𝔼∶‖𝑦‖_∗<𝑟}, where 𝑟=sup_{𝑡∈[0,𝑇]}𝐾₁(𝑡^𝛼−1(𝑇−𝑡)²)/𝑇 and 𝐾₁ is defined as in (H₂). It follows from Lemma 2.5, (H₁), Remark 3.1 and from the definition of 𝑄_𝑚 that ‖(𝑄_𝑚𝑦)(𝑡)‖≥𝐾₁(𝑡^𝛼−1(𝑇−𝑡)²/𝑇). Then ‖𝑄_𝑚𝑦‖≥𝑟. Immediately: ‖‖𝑄_𝑚𝑦‖‖_∗≥‖𝑦‖_∗,for𝑦∈𝑃∩𝜕Ω₁.(3.18) Secondly, (3.3) and Lemma 2.5(ii) imply that, for 𝑥∈𝑃, ‖‖𝑄_𝑚𝑦(𝑡)‖‖≤𝐸^𝑇0‖‖𝑓_𝑚𝑠,𝑦(𝑠),𝐷^𝛽𝑦(𝑠)‖‖𝑑𝑠+‖‖𝑦⁰‖‖Δ𝑇^(𝛼−1)≤𝐸^𝑇0𝛾(𝑡)𝛾₀(𝑡)+𝑞₁1+𝑞(𝑦(𝑠))+𝑝𝑚𝐷+_𝛽𝑝¹𝑚1+𝑞(1)+𝑝(1)𝑦(𝑠)𝑑𝑠+‖‖𝑦⁰‖‖Δ𝑇^(𝛼−1)≤𝐸^𝑇0𝛾(𝑡)𝛾₀(𝑡)+𝑞₁1+𝑞(‖𝑦‖)+𝑝𝑚‖‖𝐷+_𝛽𝑝¹𝑚1+𝑞(1)+𝑝(1)𝑦‖‖𝑑𝑠+‖‖𝑦⁰‖‖Δ𝑇^(𝛼−1)≤𝐸‖‖𝛾𝛾₀‖‖_𝐿1+𝑞₁1+𝑞(‖𝑦‖𝑚)++𝑝𝑝¹‖‖𝐷𝑚1_𝛽+𝑞(1)+𝑝(1)𝑦‖‖‖𝛾‖_𝐿¹+‖‖𝑦⁰‖‖Δ𝑇^(𝛼−1),(3.19) because 𝑞,𝑝 are nondecreasing as stated in (H₂). Analogously, by (3.3) and Lemma 3.4(ii), one can get that for 𝑥∈𝑃‖‖𝐷^𝛽𝑄_𝑚𝑦(𝑡)‖‖≤𝐸_𝐷‖‖𝛾𝛾×₀‖‖‖𝛾_𝐿1‖+_𝐿1𝑞₁𝑚1+𝑝¹𝑚1+𝑞(1)+𝑝(1)+𝑞(‖𝑦‖)+𝑝‖‖𝐷^𝛽𝑦‖‖+‖‖𝑦⁰‖‖Γ(𝛼)ΔΓ(𝛼−𝛽)𝑇^{(𝛼−𝛽−1)}.(3.20) Let 𝑊₁=max{𝐸,𝐸_𝐷} and 𝑊₂=max{(‖𝑦₀‖/Δ)𝑇^(𝛼−1),(‖𝑦₀‖Γ(𝛼)/(ΔΓ(𝛼−𝛽)))𝑇^{(𝛼−𝛽−1)}}. Hence for 𝑥∈𝑃, we have the following inequality ‖‖𝑄_𝑚𝑦‖‖_∗≤𝑊₁‖‖𝛾𝛾⁰‖‖_𝐿1+𝑞₁1×‖𝛾‖_𝐿1𝑚+𝑝¹𝑚1+𝑞(1)+𝑝(1)+𝑞‖𝑦‖^∗+𝑝‖𝑦‖_∗+𝑊².(3.21) Since lim_{‖𝑥‖→∞}(𝑞(‖𝑥‖)+𝑝(‖𝑥‖))/‖𝑥‖=0 by (H₂), there exists a sufficiently large number 𝑅>𝑟 such that 𝑊₁‖‖𝛾𝛾⁰‖‖_𝐿1+𝑞₁1𝑚+𝑝¹𝑚1+𝑞(1)+𝑝(1)+𝑞(𝑅)+𝑝(𝑅)‖𝛾‖^𝐿1+𝑊²≤𝑅.(3.22) Let Ω₂={𝑦∈𝔼∶‖𝑦‖_∗<𝑅}, then (cf. (3.21) and (3.22)) ‖‖𝑄_𝑚𝑦‖‖_∗≤‖𝑦‖_∗,for𝑦∈𝑃∩𝜕Ω₂.(3.23) Applying Lemma 3.6, we conclude from (3.18) and (3.23) that 𝑄_𝑚 has a fixed point in 𝑃∩(Ω₂⧵Ω₁).

Lemma 3.9. Suppose that (H₀)-(H₃) hold, then the sequences {𝑦_𝑚}^∞𝑚=1 and {𝐷^𝛽𝑦_𝑚}^∞𝑚=1 are relatively compact in 𝐶([0,𝑇]), where 𝑦_𝑚 be a fixed point of operator 𝑄_𝑚 defined by (3.5).

Proof. Let 𝑦_𝑚 be a fixed point of operator 𝑄_𝑚, that is, 𝑦_𝑚(𝑡)=+^𝑇0𝑦𝐺(𝑡,𝑠)𝑓_𝑚𝑠,𝑦_𝑚(𝑠),𝐷^𝛽𝑦_𝑚(𝑠)𝑑𝑠₀Δ𝑡^(𝛼−1),𝑡∈[0,𝑇],𝑚∈𝑁.(3.24) And consider (cf. (3.6)) 𝐷^𝛽𝑦_𝑚(𝑡)=+^𝑇0𝐷𝑦^𝛽𝐺(𝑡,𝑠)𝑓_𝑚𝑠,𝑦_𝑚(𝑠),𝐷^𝛽𝑦_𝑚(𝑠)𝑑𝑠₀Γ(𝛼)ΔΓ(𝛼−𝛽)𝑡^{(𝛼−𝛽−1)},𝑡∈[0,𝑇],𝑚∈𝑁.(3.25) By Lemma 2.5(iii), Lemma 3.4(iii), and Remark 3.1, we have also ‖‖𝑦_𝑚(𝑡)‖‖≥𝐾₁𝑡^𝛼−1(𝑇−𝑡)²𝑇+𝑦⁰Γ(𝛼)ΔΓ(𝛼−𝛽)𝑡^{(𝛼−𝛽−1)}≥𝐾₁𝑡^𝛼−1(𝑇−𝑡)²‖‖𝐷𝑇,𝑡∈[0,𝑇],𝑚∈𝑁,(3.26)_𝛽𝑦_𝑚(𝑡)‖‖≥𝐾₂𝑡^{𝛼−𝛽−1}(𝑇−𝑡)²𝑇,𝑡∈[0,𝑇],𝑚∈𝑁.(3.27) Hence (cf. (3.4)), ‖‖𝑓_𝑚𝑡,𝑦_𝑚(𝑡),𝐷^𝛽𝑦_𝑚(𝑡)‖‖≤𝛾(𝑡)𝛾₀(𝑡)+𝑞₁𝐾¹𝑡^𝛼−1(𝑇−𝑡)²𝑇+𝑝¹𝐾²𝑡^{𝛼−𝛽−1}(𝑇−𝑡)²+𝑞(1)+𝑝(1)+𝑞𝑦_𝑚𝑇(𝑡)+𝑝𝐷^𝛽𝑦_𝑚(𝑡),(3.28) for a.e. 𝑡∈[0,𝑇], and all 𝑚∈𝑁. Therefore, by (3.26), (3.27), Lemma 2.5(ii), Lemma 3.4(ii), and Remark 3.1, ‖‖𝑦_𝑚(𝑡)‖‖≤+𝐸𝑦₀‖‖𝛾𝛾₀‖‖_𝐿1+𝑈₁+𝑈₂+𝑞(1)+𝑝(1)+𝑞‖‖𝑦_𝑚‖‖+𝑝‖‖𝐷^𝛽𝑦_𝑚‖‖‖𝛾‖_𝐿1Δ𝑇^(𝛼−1)‖‖𝐷,(3.29)_𝛽𝑦_𝑚(𝑡)‖‖≤+𝐸_𝐷𝑦₀‖‖𝛾𝛾₀‖‖_𝐿1+𝑈₁+𝑈₂+𝑞(1)+𝑝(1)+𝑞‖‖𝑦_𝑚‖‖+𝑝‖‖𝐷^𝛽𝑦_𝑚‖‖‖𝛾‖_𝐿1Γ(𝛼)ΔΓ(𝛼−𝛽)𝑇^{(𝛼−𝛽−1)},(3.30) for 𝑡∈[0,𝑇], 𝑚∈𝑁, where 𝑈₁=^𝑇0𝛾(𝑡)𝑞₁𝐾¹𝑡^𝛼−1(𝑇−𝑡)²𝑈𝑇𝑑𝑡<+∞,(3.31)₂=^𝑇0𝛾(𝑡)𝑝₁𝐾²𝑡^{𝛼−𝛽−1}(𝑇−𝑡)²𝑇𝑑𝑡<+∞.(3.32) In particular, ‖‖𝑦_𝑚‖‖_∗≤𝑊₁‖‖𝛾𝛾₀‖‖_𝐿1+𝑈₁+𝑈₂+𝑞(1)+𝑝(1)+𝑞‖‖𝑦_𝑚‖‖_∗+𝑝‖‖𝑦_𝑚‖‖_∗‖∀𝛾𝑚‖∈_𝐿¹𝑁+,𝑊₂,(3.33) where 𝑊₁,𝑊₂ are defined in the proof of Lemma 3.8. Since lim_𝑥→+∞(𝑞(𝑥)+𝑝(𝑥))/𝑥=0 by (H₂), there exists a constant 𝑊>0 such that for each 𝑥>𝑊, 𝑊₁‖‖𝛾𝛾₀‖‖_𝐿1+𝑈₁+𝑈₂+[𝑞(1)+𝑝(1)+𝑞(𝑥)+𝑝(𝑥)]‖𝛾‖_𝐿¹+𝑊₂<𝑥.(3.34) Immediately, (cf. (3.33)) ‖‖𝑦_𝑚‖‖_∗≤𝑊,∀𝑚∈𝑁.(3.35) Hence, the sequences {𝑦_𝑚}^∞𝑚=1 and {𝐷^𝛽𝑦_𝑚}^∞𝑚=1 are uniformly bounded.
We will take similar discussions as in Lemma 3.7 to show that {𝑦_𝑚}^∞𝑚=1 and {𝐷^𝛽𝑦_𝑚}^∞𝑚=1 are equicontinuous on [0,𝑇]. Let 0≤𝑡₁<𝑡₂≤𝑇, then we have ‖‖≤𝑦_𝑚_𝑇0𝑡₁−𝑦_𝑚𝑡₂‖‖||𝐺_𝑇𝑡₁,𝑠−𝐺_𝑇𝑡₂,𝑠||‖‖𝑓_𝑚𝑠,𝑥(𝑠),𝐷^𝛽𝑥(𝑠)‖‖𝑑𝑠+‖‖𝑦⁰‖‖Δ||𝑡^1𝛼−1‖‖𝐷_𝛽−𝑡^2𝛼−1||,≤𝑦_𝑚𝑇0𝑡₁−𝐷^𝛽𝑦_𝑚𝑡₂‖‖+‖‖||𝑦𝐷^𝛽𝐺_𝑇𝑡₁,𝑠−𝐷^𝛽𝐺_𝑇𝑡₂,𝑠||‖‖𝑓_𝑚𝑠,𝑥(𝑠),𝐷^𝛽𝑥(𝑠)‖‖𝑑𝑠₀‖‖Γ(𝛼)ΔΓ(𝛼−𝛽)||𝑡^{1𝛼−𝛽−1}−𝑡^{2𝛼−𝛽−1}||.(3.36) Using (3.28), (3.35), one can get 0<≤𝛾‖‖𝑓(𝑡_𝑚)𝑡,𝛾𝑦_𝑚0(𝑡),𝐷^𝛽𝑦_𝑚(𝑡)‖‖(𝑡)+𝑞₁𝐾¹𝑡^𝛼−1(𝑇−𝑡)²𝑇+𝑝¹𝐾²𝑡^{𝛼−𝛽−1}(𝑇−𝑡)²+𝑞(1)+𝑝(1)+𝑞(𝑊)+𝑝(𝑊).𝑇(3.37) From Lemma 2.5 (i), Lemma 3.4 (i), and the functions 𝑡^𝛼−1,𝑡^{𝛼−𝛽−1} being uniformly continuous on [0,𝑇], choosing an arbitrary 𝜖>0, there exists a positive number 𝛿(𝜖). When |𝑡₁−𝑡₂|<𝛿(𝜖), we can get |𝐺_𝑇(𝑡₁,𝑠)−𝐺_𝑇(𝑡₂,𝑠)|<𝜖, |𝐷^𝛽𝐺_𝑇(𝑡₁,𝑠)−𝐷^𝛽𝐺_𝑇(𝑡₂,𝑠)|<𝜖, |𝑡^1𝛼−1−𝑡^2𝛼−1|<𝜖, and |𝑡^{1𝛼−𝛽−1}−𝑡^{2𝛼−𝛽−1}|<𝜖. Therefore (cf. (3.36) and (3.37)) the inequalities ‖‖𝑦_𝑚<𝜖𝑡₁‖‖𝛾−𝛾₀𝑦_𝑚𝑡₂‖‖‖‖_𝐿1+𝑈₁+𝑈₂+[𝑞(1)+𝑝(1)+𝑞(𝑊)+𝑝(𝑊)]‖𝛾‖_𝐿¹+‖‖𝑦⁰‖‖‖‖𝐷Δ𝜖,_𝛽<𝑦𝜖_𝑚‖‖𝛾𝑡₁𝛾₀−𝐷^𝛽𝑦_𝑚𝑡₂‖‖‖‖_𝐿1+𝑈₁+𝑈₂+[𝑞(1)+𝑝(1)+𝑞(𝑊)+𝑝(𝑊)]‖𝛾‖_𝐿¹+‖‖𝑦⁰‖‖Γ(𝛼)ΔΓ(𝛼−𝛽)𝜖,(3.38) hold, where 𝑈₁, 𝑈₂ are defined as (3.31) and (3.32), respectively. As a result, {𝑦_𝑚}^∞𝑚=1 and {𝐷^𝛽𝑦_𝑚}^∞𝑚=1 are equicontinuous on [0,𝑇].
Finally, we prove that {𝑦_𝑚(𝑡)}^∞𝑚=1 and {𝐷^𝛽𝑦_𝑚(𝑡)}^∞𝑚=1 are relatively compact. Because 𝐸 is a Banach space, we need only to show that {𝑦_𝑚(𝑡)}^∞𝑚=1 and {𝐷^𝛽𝑦_𝑚(𝑡)}^∞𝑚=1 are completely bounded. For all 𝜀>0, by the Remark 3.2, there exists a sufficiently large positive integer 𝑁, such that if 𝑚>𝑁, ‖‖𝑓_𝑚𝑡,𝑦(𝑡),𝐷^𝛽𝑦(𝑡)−𝑓𝑡,𝑦(𝑡),𝐷^𝛽𝑦(𝑡)‖‖<𝜀𝐸,a.e.𝑡∈[0,𝑇],(3.39) where 𝐸=max{𝐸,𝐸_𝐷}.
Hence, by (3.5) and (3.6), we have ‖(𝑄_𝑚𝑦)(𝑡)−𝑄₀‖<𝜀 and ‖𝐷^𝛽(𝑄_𝑚𝑦)(𝑡)−𝐷^𝛽𝑄₀‖<𝜀, for 𝑚>𝑁, where 𝑄₀=^𝑇0𝐺(𝑡,𝑠)𝑓𝑠,𝑦(𝑠),𝐷^𝛽𝑦(𝑠)𝑑𝑠+𝑦⁰Δ𝑡^(𝛼−1)𝐷_𝛽,𝑄₀=^𝑇0𝐷^𝛽𝐺(𝑡,𝑠)𝑓𝑠,𝑦(𝑠),𝐷^𝛽𝑦(𝑠)𝑑𝑠+𝑦⁰Γ(𝛼)ΔΓ(𝛼−𝛽)𝑡^{(𝛼−𝛽−1)}.(3.40) This implies that {𝑦_𝑚(𝑡)}^∞𝑚=1 and {𝐷^𝛽𝑦_𝑚(𝑡)}^∞𝑚=1 have an 𝜀-net constituted by finite elements ({𝑦₁(𝑡),𝑦₂(𝑡),𝑦_𝑁(𝑡),𝑄₀} and {𝐷^𝛽𝑦₁(𝑡),𝐷^𝛽𝑦₂(𝑡),𝐷^𝛽𝑦_𝑁(𝑡),𝐷^𝛽𝑄₀}, resp.) of 𝐸, that is, completely bounded.
Therefore, {𝑦_𝑚}^∞𝑚=1 and {𝐷^𝛽𝑦_𝑚}^∞𝑚=1 are relatively compact in 𝐶([0,𝑇]) by the Arzelá-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(𝑡)‖≥𝐾₁𝑡^𝛼−1(𝑇−𝑡)²‖‖𝐷_𝛽𝑇,𝑦(𝑡)‖‖≥𝐾₂𝑡^{𝛼−𝛽−1}(𝑇−𝑡)²𝑇,𝑡∈[0,𝑇].(3.41) Moreover, 𝑦 is continuous and ‖𝑦‖_∗≤𝑊, where 𝑊 is a constant as in (3.35).

Proof. From Lemmas 3.8 and 3.9, the operator 𝑄_𝑚 has a fixed point 𝑦_𝑚 satisfying (3.26), (3.27), (3.35). And {𝑦_𝑚}^∞𝑚=1 and {𝐷^𝛽𝑦_𝑚}^∞𝑚=1 are relatively compact in 𝐶([0,𝑇]). Hence, {𝑦_𝑚}^∞𝑚=1 is relatively compact in 𝔼. And therefore, there exist 𝑦∈𝔼 and a subsequence 𝑦_𝑚𝑘 of {𝑦_𝑚}^∞𝑚=1 such that lim_𝑘→∞𝑦_𝑚𝑘=𝑦 in 𝔼. Consequently, 𝑦 is positive and continuous. Moreover 𝑦 satisfies (3.47), ‖𝑦‖_∗≤𝑊. And lim_𝑘→∞𝑓_𝑚𝑘𝑡,𝑦_𝑚𝑘(𝑡),𝐷^𝛽𝑦_𝑚𝑘(𝑡)=𝑓𝑡,𝑦(𝑡),𝐷^𝛽𝑦(𝑡),fora.e.𝑡∈[0,𝑇].(3.42) Keeping in mind (3.35) holding, where 𝑊 is a positive constant, it follows from inequalities (3.4) and (3.26) and from Lemma 2.5(ii) that 0≤≤𝐸‖‖𝐺𝛾((𝑠𝑡),𝑠)𝛾𝑓_𝑚0𝑠,𝑦_𝑚(𝑠),𝐷^𝛽𝑦_𝑚(𝑠)‖‖(𝑠)+𝑞₁𝐾¹𝑠^𝛼−1(𝑇−𝑠)²𝑇+𝑝¹𝐾²𝑠^{𝛼−𝛽−1}(𝑇−𝑠)²+𝑞(1)+𝑝(1)+𝑞(𝑊)+𝑝(𝑊),𝑇(3.43) for a.e. 𝑠∈[0,𝑇] and all 𝑡∈[0,𝑇], 𝑚∈𝑁. Hence, by the Lebesgue dominated convergence theorem, we have lim_𝑘→+∞^𝑇0𝐺(𝑡,𝑠)𝑓_𝑚𝑠,𝑦_𝑚(𝑠),𝐷^𝛽𝑦_𝑚(𝑠)𝑑𝑠=^𝑇0𝐺(𝑡,𝑠)𝑓𝑠,𝑦(𝑠),𝐷^𝛽𝑦(𝑠)𝑑𝑠,(3.44) for 𝑡∈[0,𝑇]. Now, passing to the limit as 𝑘→+∞ in 𝑦_𝑚𝑘(𝑡)=^𝑇0𝐺(𝑡,𝑠)𝑓_𝑚𝑘𝑠,𝑦_𝑚𝑘(𝑠),𝐷^𝛽𝑦_𝑚𝑘(𝑠)𝑑𝑠+𝑦⁰Δ𝑡^(𝛼−1),(3.45) we have 𝑦(𝑡)=^𝑇0𝐺(𝑡,𝑠)𝑓𝑠,𝑦(𝑠),𝐷^𝛽𝑦(𝑠)𝑑𝑠+𝑦⁰Δ𝑡^(𝛼−1),for𝑡∈[0,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(𝑡)‖≥𝐾₁𝑡^𝛼−1(𝑇−𝑡)²‖‖𝐷_𝛽𝑇,𝑦(𝑡)‖‖≥𝐾₂𝑡^{𝛼−𝛽−1}(𝑇−𝑡)²𝑇,𝑡∈[0,𝑇].(3.47) Moreover, 𝑦 is continuous and ‖𝑦‖_∗≤𝑊, where 𝑊 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 𝐷^𝛽𝑦 is also positive.

Proof. Firstly, choose 𝑇_𝑛suc^∞𝑛=1h∈th𝑁at^∗𝑇to≤b𝑇e₁asubsequenceofnumbers,<𝑇₂<⋯<𝑇_𝑛<⋯↑∞,(4.1) then consider the BVP, 𝐷^𝛼𝑦(𝑡)+𝑓𝑡,𝑦(𝑡),𝐷^𝛽𝑦(𝑡)=𝜃,a.e.𝑡∈0,𝑇_𝑛,(4.2) subject to 𝑦(0)=𝜃,𝐷^𝛽𝑦𝑇_𝑛−^𝑚−2_𝑖=1𝑎_𝑖𝑦𝜉_𝑖−^𝑚−2_𝑖=1𝑏_𝑖𝐷^𝛽𝑦𝜉_𝑖=𝑦₀.(4.3)
Theorem 3.10 guarantees that BVP (4.2)-(4.3) has a positive continuous solution 𝑦_𝑛. And for any 𝑛∈𝑁, ‖‖𝑦_𝑛‖‖_∗≤𝑊^𝑛,for𝑡∈0,𝑇_𝑛,(4.4) where 𝑊^𝑛 is a constant defined similarly to 𝑊.
Secondly, we apply the following diagonalization process. For 𝑛∈𝑁, let 𝑢_𝑛(𝑡)=𝑦^𝑛𝑦_𝑛(𝑡),𝑡∈0,𝑇_𝑛,𝑇_𝑛,𝑡∈𝑇_𝑛,+∞.(4.5) Here, {𝑇_𝑛}^∞𝑛=1 is defined in (4.1). Notice that 𝑢_𝑛(𝑡)∈𝐶[0,+∞) with 0≤‖‖𝑢_𝑛(𝑡)‖‖≤𝑊¹,0≤‖‖𝐷^𝛽𝑢_𝑛(𝑡)‖‖≤𝑊¹,for𝑡∈0,𝑇₁.(4.6) Also for 𝑛∈𝑁 and 𝑡∈[0,𝑇₁], we get 𝑢_𝑛(𝑡)=^𝑇1₀𝐺_𝑇1(𝑡,𝑠)𝑓𝑠,𝑢_𝑛(𝑡),𝐷^𝛽𝑢_𝑛(𝑡)𝑑𝑠+𝑦⁰Δ𝑡^𝛼−1,(4.7) where 𝐺_𝑇𝑛(𝑡,𝑠) are similarly defined as in (2.11), but all of 𝑇 should be replaced by 𝑇_𝑛. Then for 𝑡₁,𝑡₂∈[0,𝑇₁], we have ‖‖𝑢_𝑛𝑡₁−𝑢_𝑛𝑡₂‖‖≤+^𝑇1₀𝑦||𝐺_𝑇1𝑡₁,𝑠−𝐺_𝑇1𝑡₂,𝑠||‖‖𝑓𝑠,𝑢_𝑛𝑘(𝑡),𝐷^𝛽𝑢_𝑛𝑘(𝑡)‖‖𝑑𝑠₀Δ||𝑡^1𝛼−1‖‖𝐷_𝛽−𝑡^2𝛼−1||,𝑢_𝑛𝑡₁−𝐷^𝛽𝑢_𝑛𝑡₂‖‖≤+^𝑇1₀‖‖𝑦||𝐷^𝛽𝐺_𝑇1𝑡₁,𝑠−𝐷^𝛽𝐺_𝑇1𝑡₂,𝑠||×‖‖𝑓𝑠,𝑢_𝑛𝑘(𝑡),𝐷^𝛽𝑢_𝑛𝑘(𝑡)‖‖𝑑𝑠₀‖‖Γ(𝛼)ΔΓ(𝛼−𝛽)||𝑡^{1𝛼−𝛽−1}−𝑡^{2𝛼−𝛽−1}||.(4.8) Thus, when |𝑡₁−𝑡₂|<𝛿(𝜖,1), similarly to (3.38), ‖‖𝑢_𝑛<𝑡₁𝜖−‖‖𝑢𝛾_𝑛𝛾₀𝑡₂‖‖‖‖_𝐿1+𝑈¹¹+𝑈¹²+𝑞(1)+𝑝(1)+𝑞𝑊¹+𝑝𝑊¹‖𝛾‖_𝐿¹+‖‖𝑦⁰‖‖‖‖𝐷Δ𝜖,_𝛽<𝑢_𝑛𝜖𝑡₁‖‖𝛾−𝛾₀𝐷^𝛽𝑢_𝑛𝑡₂‖‖‖‖_𝐿1+𝑈¹¹+𝑈¹²+𝑞(1)+𝑝(1)+𝑞𝑊¹+𝑝𝑊¹‖𝛾‖_𝐿¹+‖‖𝑦⁰‖‖Γ(𝛼)ΔΓ(𝛼−𝛽)𝜖(4.9) hold for an arbitrary 𝜖>0, where 𝛿(𝜖,1) is a suitable positive number and 𝑈¹¹,𝑈¹² are defined similarly to 𝑈₁,𝑈₂ as in (3.31) and (3.32), respectively. By using (H₃), we know that, for a.e. 𝑡∈[0,+∞), 𝑓(𝑡,𝐷_𝑊¹,𝐷_𝑊¹) is relatively compact, where 𝐷_𝑊¹={𝑢∈𝐶([0,𝑇₁]∶‖𝑢‖_∗≤𝑊¹}∩𝑃. Therefore, {𝑢_𝑛(𝑡)}^∞𝑛=1 and {𝐷^𝛽𝑢_𝑛(𝑡)}^∞𝑛=1 are relatively compact. The Arzelá-Ascoli theorem guarantees that there is a subsequence 𝑁^∗1 of 𝑁 and a function 𝑧₁∈𝐶([0,𝑇₁],𝐸) with 𝑢_𝑛𝑘→𝑧₁ in 𝐶([0,𝑇₁],𝐸) as 𝑘→+∞ through 𝑁^∗1. Obviously, 𝑧₁ is positive. Let 𝑁₁=𝑁^∗1⧵{1}, noticing that 0≤‖‖𝑢_𝑛(𝑡)‖‖≤𝑊²,0≤‖‖𝐷^𝛽𝑢_𝑛(𝑡)‖‖≤𝑊²,for𝑡∈0,𝑇₂.(4.10) Similarly to above argumentation, we have that there is a subsequence 𝑁^∗2 of 𝑁₁ and a function 𝑧₂∈𝐶([0,𝑇₂],𝐸) with 𝑢_𝑛𝑘→𝑧₂ in 𝐶([0,𝑇₂],𝐸) as 𝑘→+∞ through 𝑁^∗2. Obviously, 𝑧₂ is positive. Note that 𝑧₁=𝑧₂ on [0,𝑇₁] since 𝑁^∗2⊂𝑁₁. Let 𝑁₂=𝑁^∗2⧵{2}. Proceed inductively to obtain for 𝑚={2,3,…} a subsequence 𝑁^∗𝑚 of 𝑁_𝑚−1 and a function 𝑧_𝑚∈𝐶([0,𝑇_𝑚],𝐸) with 𝑢_𝑛𝑘→𝑧_𝑚 in 𝐶([0,𝑇_𝑚],𝐸) as 𝑘→+∞ through 𝑁^∗𝑚. Also, 𝑧_𝑚 is positive. Let 𝑁_𝑚=𝑁^∗𝑚⧵{𝑚}.
Define a function 𝑦 as follows. Fix 𝑡∈(0,+∞), and Let 𝑚∈𝑁 with 𝑠≤𝑇_𝑚, then define 𝑦(𝑡)=𝑧_𝑚(𝑡). Hence 𝑦∈𝐶([0,+∞),ℝ).
Again fix 𝑡∈[0,+∞) and Let 𝑚∈𝑁 with 𝑠≤𝑇_𝑚. Then for 𝑛∈𝑁_𝑚 we get 𝑢_𝑛𝑘(𝑡)=^𝑇𝑚₀𝐺_𝑇𝑚(𝑡,𝑠)𝑓𝑠,𝑢_𝑛𝑘(𝑠),𝐷^𝛽𝑢_𝑛𝑘(𝑠)𝑑𝑠+𝑦⁰Δ𝑡^(𝛼−1).(4.11) Let 𝑛_𝑘→+∞ through 𝑁_𝑚 to obtain 𝑧_𝑚(𝑡)=^𝑇𝑚₀𝐺_𝑇𝑚(𝑡,𝑠)𝑓𝑠,𝑧_𝑚(𝑠),𝐷^𝛽𝑧_𝑚(𝑠)𝑑𝑠+𝑦⁰Δ𝑡^(𝛼−1),(4.12) that is, 𝑦(𝑡)=^𝑇𝑚₀𝐺_𝑇𝑚(𝑡,𝑠)𝑓𝑠,𝑦(𝑠),𝐷^𝛽𝑦(𝑠)𝑑𝑠+𝑦⁰Δ𝑡^(𝛼−1).(4.13) We can use this method for each 𝑠∈[0,𝑇_𝑚] and for each 𝑚∈𝑁. Hence, 𝐷^𝛼𝑦(𝑡)+𝑓𝑡,𝑦(𝑡),𝐷^𝛽𝑦(𝑡)=𝜃,a.e.𝑡∈0,𝑇_𝑚,(4.14) for each 𝑚∈𝑁. 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, 𝛼−𝛽=1, 𝑏_𝑖=0(𝑖=1,2,…,𝑚−2), 𝑏₁>0, 𝑏_𝑖=0(𝑖=2,3,…,𝑚−2), 𝑦₀=0, and 𝑓=𝑓(𝑡,𝑦(𝑡)) are not singular, then our Theorem 4.1 includes the result from [21]. But our approach is different from those of [21].

Proceeding the similar arguments above, we list the following unique result for BVP (1.6)-(1.7), and the proof will be omitted.

Theorem 4.3. Suppose that (H₀)-(H₄) hold, then BVP (1.6)-(1.7) has a unique positive solution 𝑦, and 𝐷^𝛽𝑦 is also positive.

5. Application

We end this paper with giving an example to demonstrate the application of our existence result.

Example 5.1. Consider the following BVP in scalar space: 𝐷^𝛼𝑦(𝑡)+2−sin13𝑡−1𝑒^−𝑡+1𝑦^𝜂1+1𝐷^𝛽𝑦^𝜅1+𝑦^𝜂+𝐷^𝛽𝑦^𝜅𝑦(0)=0,lim_𝑡→+∞=0,𝐷^𝛽𝑦(𝑡)−14𝑦(4)−2𝑦(5)−𝐷^𝛽𝑦(6)=1,(5.1) where 1<𝛼≤2, 0<𝛽<𝛼−1. We will apply Theorem 4.1 with 𝜛=1, 𝛾(𝑡)=2−sin(1/(3𝑡−1)), 𝛾₀(𝑡)=𝑒^−𝑡, 𝑞₁(𝑥)=1/𝑥^𝜂1, 𝑝₁(𝑥)=1/(𝑥)^𝜅1, 𝑞(𝑥)=𝑥^𝜂, 𝑝(𝑡)=(𝑥)^𝜅. Clearly (H₁) holds because 𝑓(𝑡,𝑥,𝑦)≥1 for 𝑡∈[0,+∞)⧵{1/3}, (𝑥,𝑦)∈(0,+∞)×(0,+∞). And also (H₂) holds for 𝜂,𝜅∈(0,1), 𝜂₁∈(0,1/2), and 𝜅₁∈(0,1/2). Hence, Theorem 4.1 guarantees the existence of positive solution of (5.1).