Positive Solutions of Nonlinear Eigenvalue Problems for a Nonlocal Fractional Differential Equation

Abstract

By using the fixed point theorem, positive solutions of nonlinear eigenvalue problems for a nonlocal fractional differential equation 𝐷^𝛼₀₊𝑢(𝑡)+𝜆𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0,0<𝑡<1,𝑢(0)=0,𝑢(1)=Σ^∞_𝑖=1𝛼_𝑖𝑢(𝜉_𝑖) are considered, where 1<𝛼≤2 is a real number, 𝜆 is a positive parameter, 𝐷^𝛼₀₊ is the standard Riemann-Liouville differentiation, and 𝜉_𝑖∈(0,1),𝛼_𝑖∈[0,∞) with Σ^∞_𝑖=1𝛼_𝑖𝜉_𝑖^𝛼−1<1, 𝑎(𝑡)∈𝐶([0,1],[0,∞)),𝑓(𝑡,𝑢)∈𝐶([0,∞),[0,∞)).

1. Introduction

Fractional differential equations have been of great interest recently. This is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. For details, see [1–6] and references therein.

Recently, many results were obtained dealing with the existence and multiplicity of solutions of nonlinear fractional differential equations by the use of techniques of nonlinear analysis, see [7–21] and the reference therein. Bai and Lü [7] studied the existence of positive solutions of nonlinear fractional differential equation 𝐷^𝛼₀₊𝑢(𝑡)((𝑡𝑢(0)(1)+𝑓𝑡,𝑢))=0,0<𝑡<1,=𝑢=0, (1.1) where 1<𝛼≤2 is a real number, 𝐷^𝛼₀₊ is the standard Riemann-Liouville differentiation, and 𝑓∶[0,1]×[0,∞)→[0,∞) is continuous. They derived the corresponding Green function and obtained some properties as follows.

Proposition 1.1. Green’s function 𝐺(𝑡,𝑠) satisfies the following conditions:(R1)𝐺(𝑡,𝑠)∈𝐶([0,1]×[0,1]), and 𝐺(𝑡,𝑠)>0 for 𝑡,𝑠∈(0,1);(R2) there exists a positive function 𝛾∈𝐶(0,1) such thatmin_{1/4≤𝑡≤3/4}𝐺(𝑡,𝑠)≥𝛾(𝑠)max_0≤𝑡≤1𝐺(𝑡,𝑠)≥𝛾(𝑠)𝐺(𝑠,𝑠),𝑠∈(0,1), (1.2) where 𝐺()=[𝑡(𝑡,𝑠1−𝑠)]^𝛼−1−(𝑡−𝑠)^𝛼−1Γ(𝛼)[𝑡(,0≤𝑠≤𝑡≤1,1−𝑠)]^𝛼−1Γ(𝛼),0≤𝑡≤𝑠≤1. (1.3)

It is well known that the cone plays a very important role in applying Green’s function in research area. In [7], the authors cannot acquire a positive constant taken instead of the role of positive function 𝛾(𝑠) with 1<𝛼<2 in (1.2). In [9], Jiang and Yuan obtained some new properties of the Green function and established a new cone. The results can be stated as follows.

Proposition 1.2. Green’s function 𝐺(𝑡,𝑠) defined by (1.3) has the following properties: 𝐺(𝑡,𝑠)=𝐺(1−𝑠,1−𝑡) and 𝛼−1Γ(𝛼)𝑡^𝛼−1(1−𝑡)(1−𝑠)^𝛼−1()≤1𝑠≤𝐺𝑡,𝑠Γ(𝛼)𝑡^𝛼−1(1−𝑡)(1−𝑠)^𝛼−2,∀𝑡,𝑠∈(0,1). (1.4)

Proposition 1.3. The function 𝐺^∗(𝑡,𝑠)∶=𝑡^2−𝛼𝐺(𝑡,𝑠) has the following properties: 𝑞(𝑡)Φ(𝑠)≤𝐺^∗(𝑡,𝑠)≤Φ(𝑠),∀𝑡,𝑠∈[0,1], (1.5) where 𝑞(𝑡)=(𝛼−1)𝑡(1−𝑡),Φ(𝑠)=(1/Γ(𝛼))𝑠(1−𝑠)^𝛼−1.

In this paper, we study the existence of positive solutions of nonlinear eigenvalue problems for a nonlocal fractional differential equation𝐷^𝛼₀₊𝑢(𝑡)(𝑡)𝑓((𝑡𝑢(0)(1)=+𝜆𝑎𝑡,𝑢))=0,0<𝑡<1,=0,𝑢^∞_𝑖=1𝛼_𝑖𝑢𝜉_𝑖, (1.6) where 1<𝛼≤2 is a real number, 𝜆 is a positive parameter, 𝐷^𝛼₀₊ is the standard Riemann-Liouville differentiation, and 𝜉_𝑖∈(0,1),𝛼_𝑖∈[0,∞) with Σ^∞_𝑖=1𝛼_𝑖𝜉_𝑖^𝛼−1<1, 𝑎(𝑡)∈𝐶([0,1],[0,∞)),𝑓(𝑡,𝑢)∈𝐶([0,∞),[0,∞)).

We assume the following conditions hold throughout the paper:(H1)𝜉_𝑖∈(0,1),𝛼_𝑖∈[0,∞) are both constants with Σ^∞_𝑖=1𝛼_𝑖𝜉_𝑖^𝛼−1<1;(H2)𝑎(𝑡)∈𝐶([0,1],[0,∞)), 𝑎(𝑡)≢0;(H3)𝑓(𝑡,𝑢)∈𝐶([0,1]×[0,∞),[0,∞)), and there exist 𝑔∈𝐶([0,+∞),[0,+∞)),𝑞₁,𝑞₂∈𝐶((0,1),(0,+∞)) such that𝑞₁(𝑡)𝑔𝑦≤𝑓𝑡,𝑡^𝛼−2𝑦≤𝑞₂(𝑡)𝑔𝑦,𝑡∈(0,1),𝑦∈[0,+∞), (1.7) where ∫¹₀𝑞_𝑖(𝑠)𝑑𝑠<+∞,𝑖=1,2.

2. The Preliminary Lemmas

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

Definition 2.1. The fractional integral of order 𝛼>0 of a function 𝑦∶(0,∞)→𝑅 is given by 𝐼^𝛼₀₊𝑦(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠 (2.1) provided the right side is pointwise defined on (0,∞).

Definition 2.2. The fractional derivative of order 𝛼>0 of a function 𝑦∶(0,∞)→𝑅 is given by 𝐷^𝛼₀₊𝑦(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡₀𝑦(𝑠)(𝑡−𝑠)^{𝛼−𝑛+1}𝑑𝑠, (2.2) where 𝑛=[𝛼]+1, provided the right side is pointwise defined on (0,∞).

Lemma 2.3. Let 𝛼>0. If one assumes 𝑢∈𝐶(0,1)∩𝐿(0,1), then the fractional differential equation 𝐷^𝛼₀₊𝑢(𝑡)=0 (2.3) has 𝑢(𝑡)=𝐶₁𝑡^𝛼−1+𝐶₂𝑡^𝛼−2+⋯+𝐶_𝑁𝑡^𝛼−𝑁,𝐶_𝑖∈𝑅,𝑖=1,2,…,𝑁, where 𝑁 is the smallest integer greater than or equal to 𝛼, as unique solutions.

Lemma 2.4. Assume that 𝑢∈𝐶(0,1)∩𝐿(0,1) with a fractional derivative of order 𝛼>0 that belongs to 𝑢∈𝐶(0,1)∩𝐿(0,1). Then 𝐼^𝛼₀₊𝐷^𝛼₀₊𝑢(𝑡)=𝑢(𝑡)+𝐶₁𝑡^𝛼−1+𝐶₂𝑡^𝛼−2+⋯+𝐶_𝑁𝑡^𝛼−𝑁 (2.4) for some 𝐶_𝑖∈𝑅,𝑖=1,2,…,𝑁.

Lemma 2.5 (see [7]). Given 𝑦∈𝐶[0,1] and 1<𝛼≤2, the unique solution of 𝐷^𝛼₀₊𝑢(𝑡)(𝑡)𝑢(0)(1)+𝑦=0,0<𝑡<1,=𝑢=0 (2.5) is 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠, (2.6) where 𝐺()=[𝑡(𝑡,𝑠1−𝑠)]^𝛼−1−(𝑡−𝑠)^𝛼−1Γ(𝛼)[𝑡(,0≤𝑠≤𝑡≤1,1−𝑠)]^𝛼−1Γ(𝛼),0≤𝑡≤𝑠≤1. (2.7)

Lemma 2.6. Suppose (H1) holds. Given 𝑦∈𝐶[0,1] and 1<𝛼≤2, the unique solution of 𝐷^𝛼₀₊𝑢(𝑡)(𝑡)𝑢(0)(1)=+𝑦=0,0<𝑡<1,=0,𝑢^∞_𝑖=1𝛼_𝑖𝑢𝜉_𝑖 (2.8) is 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠+𝐴𝑦𝑡^𝛼−1, (2.9) where 𝐺()=[𝑡(𝑡,𝑠1−𝑠)]^𝛼−1−(𝑡−𝑠)^𝛼−1Γ(𝛼)[𝑡(,0≤𝑠≤𝑡≤1,1−𝑠)]^𝛼−1Γ(𝛼)𝐴𝑦=Σ,0≤𝑡≤𝑠≤1,_∞𝑖=1𝛼_𝑖∫¹₀𝐺𝜉_𝑖,𝑠𝑦(𝑠)𝑑𝑠1−Σ^∞_𝑖=1𝛼_𝑖𝜉_𝑖^𝛼−1. (2.10)

Proof. By applying Lemmas 2.4 and 2.5, we have 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠+𝐶₁𝑡^𝛼−1+𝐶₂𝑡^𝛼−2. (2.11) Because ^∞_𝑖=1𝛼_𝑖¹₀𝐺𝜉_𝑖Σ,𝑠𝑑𝑠=^∞𝑖=1𝛼_𝑖𝜉_𝑖^𝛼−11−𝜉_𝑖𝛼Γ(𝛼),𝛼_𝑖𝜉_𝑖^𝛼−11−𝜉_𝑖<𝛼_𝑖𝜉_𝑖^𝛼−1, (2.12) by (H1), Σ^∞_𝑖=1𝛼_𝑖𝜉_𝑖^𝛼−1(1−𝜉_𝑖) is convergent; therefore, Σ^∞_𝑖=1𝛼_𝑖∫¹₀𝐺(𝜉_𝑖,𝑠)𝑑𝑠 is convergent. Note that 𝑦(𝑡) is continuous function on [0,1], so Σ^∞_𝑖=1𝛼_𝑖∫¹₀𝐺(𝜉_𝑖,𝑠)𝑦(𝑠)𝑑𝑠 is convergent.
From 𝑢(0)=0,𝑢(1)=Σ^∞_𝑖=1𝛼_𝑖𝑢(𝜉_𝑖), we have 𝐶₂=0,𝐶₁=Σ^∞_𝑖=1𝛼_𝑖∫¹₀𝐺(𝜉_𝑖,𝑠)𝑦(𝑠)𝑑𝑠/(1−Σ^∞_𝑖=1𝛼_𝑖𝜉_𝑖^𝛼−1). Therefore, 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠+𝐴𝑦𝑡^𝛼−1,𝐴𝑦=Σ^∞_𝑖=1𝛼_𝑖∫¹₀𝐺𝜉_𝑖,𝑠𝑦(𝑠)𝑑𝑠1−Σ^∞_𝑖=1𝛼_𝑖𝜉_𝑖^𝛼−1. (2.13)

Lemma 2.7 (see [7]). Let 𝐸 be a Banach space, 𝑃⊆𝐸 a cone, and Ω₁,Ω₂ two bounded open sets of 𝐸 with 0∈Ω₁⊂Ω₁⊂Ω₂. Suppose that 𝑇∶𝑃∩(Ω₂⧵Ω₁)→𝑃 is a completely continuous operator such that either(i)‖𝑇𝑥‖≤‖𝑥‖,𝑥∈𝑃∩𝜕Ω₁, and ‖𝑇𝑥‖≥‖𝑥‖,𝑥∈𝑃∩𝜕Ω₂, or(ii)‖𝑇𝑥‖≥‖𝑥‖,𝑥∈𝑃∩𝜕Ω₁, and ‖𝑇𝑥‖≤‖𝑥‖,𝑥∈𝑃∩𝜕Ω₂,holds. Then 𝑇 has a fixed point in 𝑃∩(Ω₂⧵Ω₁).

3. The Main Results

Let𝐺₀()()+𝑡𝑡,𝑠=𝐺𝑡,𝑠^𝛼−1Σ^∞_𝑖=1𝛼_𝑖𝐺𝜉_𝑖,𝑠1−Σ^∞_𝑖=1𝛼_𝑖𝜉_𝑖^𝛼−1. (3.1) Then 𝑢(𝑡) is the solution of BVP (1.6) if and only if 𝑇𝑢(𝑡)=𝑢(𝑡), where 𝑇 is the operator defined by𝑇𝑢(𝑡)∶=𝜆¹₀𝐺₀(𝑡,𝑠)𝑎(𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠. (3.2) By similar arguments to Proposition 1.3, we obtain the following result.

Lemma 3.1. Suppose (H1) holds. The function 𝐺(𝑡,𝑠)∶=𝑡^2−𝛼𝐺₀(𝑡,𝑠) has the following properties: 𝑞(𝑡)Ψ(𝑠)≤𝐺(𝑡,𝑠)≤Ψ(𝑠),∀𝑡,𝑠∈[0,1], (3.3) where 𝑞(𝑡)=(𝛼−1)𝑡(1−𝑡),Ψ(𝑠)=Φ(𝑠)+Σ^∞_𝑖=1𝛼_𝑖𝐺(𝜉_𝑖,𝑠)/(1−Σ^∞_𝑖=1𝛼_𝑖𝜉_𝑖^𝛼−1).

Let 𝐸=𝐶[0,1] be endowed with the ordering 𝑢≤𝑣 if 𝑢(𝑡)≤𝑣(𝑡) for all 𝑡∈[0,1], and the maximum norm ‖𝑢‖=max_0≤𝑡≤1|𝑢(𝑡)|. Define the cone 𝑃⊂𝐸 by 𝑃={𝑢∈𝐸∣𝑢(𝑡)≥0}, and𝐾=𝑢∈𝑃∣𝑢(𝑡)≥𝑞(𝑡)‖𝑢‖, (3.4) where 𝑞(𝑡) is defined by (3.3).

It is easy to see that 𝑃 and 𝐾 are cones in 𝐸. For any 0<𝑟<𝑅<+∞, let 𝐾_𝑟={𝑢∈𝐾‖𝑢‖<𝑟},𝜕𝐾_𝑟={𝑢∈𝐾‖𝑢‖=𝑟},𝐾_𝑟={𝑢∈𝐾‖𝑢‖≤𝑟}, and 𝐾_𝑅⧵𝐾_𝑟={𝑢∈𝐾∣𝑟≤‖𝑢‖≤𝑅}.

For convenience, we introduce the following notations:𝑔₀=lim_𝑢→0+𝑔(𝑢)𝑢,𝑔_∞=lim_𝑢→+∞𝑔(𝑢)𝑢. (3.5) By similar arguments to Lemma 4.1 of [9], we obtain the following result.

Lemma 3.2. Assume that (H1)–(H3) hold. Let 𝑇∶𝐾→𝐸 be the operator defined by 𝑇𝑢(𝑡)∶=𝜆¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)𝑑𝑠. (3.6) Then 𝑇∶𝐾→𝐾 is completely continuous.

Theorem 3.3. Assuming (H1)–(H3) hold, 𝑔₀,𝑔_∞ exist. Then, for each 𝜆 satisfying 1((𝛼−1)/16)²∫^3/4_1/4Ψ(𝑠)𝑎(𝑠)𝑞₁(𝑠)𝑑𝑠𝑔₀1<𝜆<∫¹₀Ψ(𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑑𝑠𝑔_∞, (3.7) there exists at least one positive solution of BVP (1.6) in 𝑃.

Theorem 3.4. Assuming (H1)–(H3) hold, 𝑔₀,𝑔_∞ exist. Then, for each 𝜆 satisfying 1((𝛼−1)/16)²∫^3/4_1/4Ψ(𝑠)𝑎(𝑠)𝑞₁(𝑠)𝑑𝑠𝑔_∞1<𝜆<∫¹₀Ψ(𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑑𝑠𝑔₀, (3.8) there exists at least one positive solution of BVP (1.6) in 𝑃.

Proof of Theorem 3.3. Let 𝜆 be given as in (3.7), and choose 𝜀>0 such that 1((𝛼−1)/16)²∫^3/4_1/4Ψ(𝑠)𝑎(𝑠)𝑞₁(𝑠)𝑑𝑠𝑔₀1−𝜀≤𝜆≤∫¹₀Ψ(𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑑𝑠𝑔_∞+𝜀. (3.9) Beginning with 𝑔₀, there exists an 𝐻₁>0 such that 𝑔(𝑢)≥(𝑔₀−𝜀)𝑢, for 0<𝑢≤𝐻₁. So 𝑢∈𝐾 and ‖𝑢‖=𝐻₁. For 𝑡∈[1/4,3/4], we have 𝑇𝑢(𝑡)=𝜆¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)𝑑𝑠≥𝜆¹₀𝑞(𝑡)Ψ(𝑠)𝑎(𝑠)𝑞₁(𝑠)𝑔(𝑢(𝑠))𝑑𝑠≥𝜆¹₀𝑞(𝑡)Ψ(𝑠)𝑎(𝑠)𝑞₁(𝑠)𝑔₀−𝜀𝑢(𝑠)𝑑𝑠≥𝜆𝛼−116^23/4_1/4Ψ(𝑠)𝑎(𝑠)𝑞₁(𝑠)𝑔₀‖𝑢‖≥‖𝑢‖−𝜀𝑑𝑠=𝐻₁. (3.10) Thus, ‖𝑇𝑢‖≥‖𝑢‖. So, if we let Ω₁=𝑢∈𝐾∣‖𝑢‖<𝐻₁, (3.11) then 𝑇𝑢≥‖𝑢‖,𝑢∈𝜕Ω₁. (3.12)
It remains to consider 𝑔_∞. There exists an 𝐻₂ such that 𝑔(𝑢)≤(𝑔_∞+𝜀)𝑢, for all 𝑢≥𝐻₂. There are the two cases, (a), where 𝑔 is bounded, and (b), where 𝑔 is unbounded.
Case a. Suppose 𝑁>0 is such that 𝑔(𝑢)≤𝑁, for all 0<𝑢<∞.
Let 𝐻₂=max{2𝐻₁,𝑁𝜆∫¹₀Ψ(𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑑𝑠}. Then, for 𝑢∈𝐾 with ‖𝑢‖=𝐻₂, we have 𝑇𝑢(𝑡)≤𝜆¹₀Ψ(𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑔(𝑢(𝑠))𝑑𝑠≤𝜆𝑁¹₀Ψ(𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑑𝑠≤𝐻₂=‖𝑢‖. (3.13) So, if we let Ω₂=𝑢∈𝐾∣‖𝑢‖<𝐻₂, (3.14) then 𝑇𝑢≤‖𝑢‖,𝑢∈𝜕Ω₂. (3.15)
Case b. Let 𝐻₂>max{2𝐻₁,𝐻₂} be such that 𝑔(𝑢)≤𝑔(𝐻₂),0<𝑢<𝐻₂. Choosing 𝑢∈𝐾 with ‖𝑢‖=𝐻₂, 𝑇𝑢(𝑡)≤𝜆¹₀Ψ(𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑔𝐻₂𝑑𝑠≤𝜆¹₀Ψ(𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑔_∞+𝜀𝑑𝑠𝐻₂≤𝐻₂=‖𝑢‖, (3.16) and so ‖𝑇𝑢‖≤‖𝑢‖. For this case, if we let Ω₂=𝑢∈𝐾∣‖𝑢‖<𝐻₂, (3.17) then 𝑇𝑢≤‖𝑢‖,𝑢∈𝜕Ω₂. (3.18) Therefore, by (ii) of Lemma 2.7, 𝑇 has a fixed point 𝑢 such that 𝐻₁≤‖𝑢‖≤𝐻₂ and satisfies 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)𝑑𝑠. (3.19) It is obvious that 𝑦(𝑡)=𝑡^𝛼−2𝑢(𝑡) is solution of (1.6) for 𝑡∈(0,1], and 𝑦(𝑡)=¹₀𝐺₀(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝑦(𝑠)𝑑𝑠,𝑡∈(0,1]. (3.20) Next, we will prove 𝑦(0)=0. From 𝑢∈𝐶[0,1] and (H1)–(H3), we have lim_𝑡→0+𝑦(𝑡)=lim_𝑡→0+¹₀𝐺₀(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝑦(𝑠)𝑑𝑠=lim_𝑡→0+¹₀𝐺₀(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)𝑑𝑠≤lim_𝑡→0+¹₀𝐺₀(𝑡,𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑔(𝑢(𝑠))𝑑𝑠≤lim_𝑡→0+¹₀𝐺₀(𝑡,𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑑𝑠max_{‖𝑢‖≤𝐻2}𝑔(𝑢)≤0. (3.21) Thus, 𝑦(0)=0. then 𝑦(𝑡)=𝑡^𝛼−2𝑢(𝑡) is solution of (1.6) for 𝑡∈[0,1].

Proof of Theorem 3.4. Let 𝜆 be given as in (3.8), and choose 𝜀>0 such that 1((𝛼−1)/16)²∫^3/4_1/4Ψ(𝑠)𝑎(𝑠)𝑞₁(𝑠)𝑑𝑠𝑔_∞1−𝜀≤𝜆≤∫¹₀Ψ(𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑑𝑠𝑔₀+𝜀. (3.22) Beginning with 𝑔₀, there exists an 𝐻₁>0 such that 𝑔(𝑢)≤(𝑔₀+𝜀)𝑢, for 0<𝑢≤𝐻₁. So, for 𝑢∈𝐾 and ‖𝑢‖=𝐻₁, we have 𝑇𝑢(𝑡)=𝜆¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)𝑑𝑠≤𝜆¹₀Ψ(𝑠)𝑎(𝑠)𝑞₂(𝑠)𝑑𝑠𝑔₀‖𝑢‖≤‖𝑢‖.+𝜀 (3.23) Thus, ‖𝑇𝑢‖≤‖𝑢‖. So, if we let Ω₁=𝑢∈𝐾∣‖𝑢‖<𝐻₁, (3.24) then 𝑇𝑢≤‖𝑢‖,𝑢∈𝜕Ω₁. (3.25)
Next, considering 𝑔_∞, there exists an 𝐻₂ such that 𝑔(𝑢)≥(𝑔_∞−𝜀)𝑢, for all 𝑢≥𝐻₂. Let 𝐻₂=max{2𝐻₁,(16/(𝛼−1))𝐻₂}. Then, 𝑢∈𝐾‖𝑢‖=𝐻₂. For 𝑡∈[1/4,3/4], we have 𝑇𝑢(𝑡)=𝜆¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)𝑑𝑠≥𝜆¹₀𝑞(𝑡)Ψ(𝑠)𝑎(𝑠)𝑞₁(𝑠)𝑔(𝑢(𝑠))𝑑𝑠≥𝜆¹₀𝑞(𝑡)Ψ(𝑠)𝑎(𝑠)𝑞₁(𝑠)𝑔_∞−𝜀𝑢(𝑠)𝑑𝑠≥𝜆𝛼−116^23/4_1/4Ψ(𝑠)𝑎(𝑠)𝑞₁(𝑠)𝑔_∞‖𝑢‖≥‖𝑢‖−𝜀𝑑𝑠=𝐻₂, (3.26) and so ‖𝑇𝑢‖≥‖𝑢‖. For this case, if we let Ω₂=𝑢∈𝐾∣‖𝑢‖<𝐻₂, (3.27) then 𝑇𝑢≥‖𝑢‖,𝑢∈𝜕Ω₂. (3.28)
Therefore, by (i) of Lemma 2.7, 𝑇 has a fixed point 𝑢 such that 𝐻₁≤‖𝑢‖≤𝐻₂ and satisfies 𝑢(𝑡)=¹₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑓𝑠,𝑠^𝛼−2𝑢(𝑠)𝑑𝑠. (3.29) By similar method to Theorem 3.3, we can get 𝑦(0)=0, then 𝑦(𝑡)=𝑡^𝛼−2𝑢(𝑡) is solution of (1.6) for 𝑡∈[0,1]. We complete the proof.

Acknowledgments

This work is supported by the NSFC (11061030, 11101335, 11026060), Gansu Provincial Department of Education Fund (No. 1101-02), and Science and Technology Bureau of Lanzhou City (No. 2011-2-72). The authors are very grateful to the anonymous referees for their valuable suggestions.