Positive Solutions for Three-Point Boundary Value Problem of Fractional Differential Equation with 𝑝-Laplacian Operator

Abstract

We investigate the existence of multiple positive solutions for three-point boundary value problem of fractional differential equation with 𝑝-Laplacian operator −𝒟_𝐭^𝛽(𝜑_𝑝(𝒟_𝐭^𝛼𝑥))(𝑡)=ℎ(𝑡)𝑓(𝑡,𝑥(𝑡)),𝑡∈(0,1),𝑥(0)=0,𝒟_𝐭^𝛾𝑥(1)=𝑎𝒟_𝐭^𝛾𝑥(𝜉),𝒟_𝐭^𝛼𝑥(0)=0, where 𝒟_𝐭^𝛽,𝒟_𝐭^𝛼,𝒟_𝐭^𝛾 are the standard Riemann-Liouville derivatives with 1<𝛼≤2,0<𝛽≤1,0<𝛾≤1,0≤𝛼−𝛾−1, 𝜉∈(0,1) and the constant 𝑎 is a positive number satisfying 𝑎𝜉^{𝛼−𝛾−2}≤1−𝛾; 𝑝-Laplacian operator is defined as 𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠,𝑝>1. By applying monotone iterative technique, some sufficient conditions for the existence of multiple positive solutions are established; moreover iterative schemes for approximating these solutions are also obtained, which start off a known simple linear function. In the end, an example is worked out to illustrate our main results.

1. Introduction

In this paper, we study the existence of multiple positive solutions for the following three-point boundary value problem of fractional differential equation with 𝑝-Laplacian operator −𝒟_𝐭^𝛽𝜑_𝑝𝒟_𝐭^𝛼𝑥(𝑡)(𝑡)𝑓((𝑡(),𝑥(0)=ℎ𝑡,𝑥)),𝑡∈0,1=0,𝒟_𝐭^𝛾𝑥(1)=𝑎𝒟_𝐭^𝛾𝑥(𝜉),𝒟_𝐭^𝛼𝑥(0)=0, (1) where 𝒟_𝐭^𝛽,𝒟_𝐭^𝛼,𝒟_𝐭^𝛾 are the standard Riemann-Liouville derivatives with 1<𝛼≤2,0<𝛽≤1,0<𝛾≤1,0≤𝛼−𝛾−1,𝜉∈(0,1) and the constant 𝑎 is a positive number satisfying 𝑎𝜉^{𝛼−𝛾−2}≤1−𝛾; 𝑝-Laplacian operator is defined as 𝜑_𝑝(𝑠)=|𝑠|^𝑝−2𝑠,𝑝>1.

Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, and engineering. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], and the papers [4–14] and the references therein.

In [15], Li et al. were concerned with the nonlinear differential equation of fractional order 𝒟_𝐭^𝛼𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡))=0,𝑡∈(0,1),1<𝛼≤2 (2) subject to the boundary conditions 𝑥(0)=0,𝒟_𝐭^𝛽𝑥(1)=𝑎𝒟_𝐭^𝛽𝑥(𝜉). (3) By using some fixed point theorems, the existence and multiplicity results of positive solutions were established.

On the other hand, the differential equations with 𝑝-Laplacian have also been widely studied owing to the fact that 𝑝-Laplacian boundary value problems have important application in theory and application of mathematics and physics. For example, in [16], by using the fixed point index, Yang and Yan investigated the existence of positive solution for the third-order Sturm-Liouville boundary value problems with 𝑝-Laplacian operator: 𝜑_𝑝𝑥^′′(𝑡)^′((𝑡(),(0)+𝑓𝑡,𝑥))=0,𝑡∈0,1𝑎𝑥−𝑏𝑥^′(0)(1)=0𝑐𝑥+𝑑𝑥^′(1)=0,𝑥^′′(0)=0. (4) However, there are few articles dealing with the existence of solutions to boundary value problems for fractional differential equation with 𝑝-Laplacian operator. In [17], the authors investigated the nonlinear nonlocal problem 𝒟_𝐭^𝛽𝜑_𝑝𝒟_𝐭^𝛼𝑥(𝑡)((𝑡(),𝑥(0)(1)(𝜉),𝒟+𝑓𝑡,𝑥))=0,𝑡∈0,1=0,𝑥=𝑎𝑥_𝐭𝛼𝑥(0)=0, (5) where 0<𝛼≤2,0<𝛽≤1,0≤𝑎≤1,0<𝜉<1. By using Krasnoselskii’s fixed point theorem and Leggett-Williams theorem, some sufficient conditions for the existence of positive solutions to the above BVP are obtained. In [18], by using upper and lower solutions method, under suitable monotone conditions, Wang et al. investigated the existence of positive solutions to the following nonlocal problem: 𝒟_𝐭^𝛽&𝜑_𝑝𝒟_𝐭^𝛼𝑥(𝑡)((𝑡()𝑥(0)(1)(𝜉),𝒟+𝑓𝑡,𝑥))=0,𝑡∈0,1=0,𝑥=𝑎𝑥_𝐭𝛼𝑥(0)=0,𝒟_𝐭^𝛼𝑥(1)=𝑏𝒟_𝐭^𝛼𝑥𝜂, (6) where 1<𝛼,𝛽≤2,0≤𝑎,𝑏≤1,0<𝜉,𝜂<1. Recently, Chai [19] investigated the two-point boundary value problem of fractional differential equation with 𝑝-Laplacian operator: 𝒟_𝐭^𝛽𝜑_𝑝𝒟_𝐭^𝛼𝑥(𝑡)((𝑡(),𝑥(0)(1)+𝑓𝑡,𝑥))=0,𝑡∈0,1=0,𝑥+𝑎𝒟_𝐭^𝛾𝑥(1)𝒟=0,_𝐭𝛼𝑥(0)=0. (7) By means of the fixed point theorem on cones, some existence and multiplicity results of positive solutions are obtained.

Motivated by the above mentioned works, in this paper, we consider the multiplicity results of positive solutions for the three point boundary value problem of fractional differential equation with 𝑝-Laplacian operator. Difference to [15–19], by using monotone iterative technique, we not only establish the existence of multiple positive solutions but also obtain the iterative sequences of these positive solutions.

2. Preliminaries and Lemmas

In this section, we introduce some preliminary facts which are used throughout this paper.

Definition 1 (see [1–3]). The Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑥∶(0,+∞)→ℝ is given by 𝐼^𝛼𝑥(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑥(𝑠)𝑑𝑠 (8) provided that the right-hand side is pointwise defined on (0,+∞).

Definition 2 (see [1–3]). The Riemann-Liouville fractional derivative of order 𝛼>0 of a function 𝑥∶(0,+∞)→ℝ is given by 𝒟_𝐭^𝛼𝑥(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛×^𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑥(𝑠)𝑑𝑠, (9) where 𝑛=[𝛼]+1, [𝛼] denotes the integer part of number 𝛼, provided that the right-hand side is pointwise defined on (0,+∞).

Proposition 3 (see [1–3]). (1) If 𝑥∈𝐿¹(0,1),𝜈>𝜎>0, then 𝐼^𝜈𝐼^𝜎𝑥(𝑡)=𝐼^𝜈+𝜎𝑥(𝑡),𝒟_𝐭^𝜎𝐼^𝜈𝑥(𝑡)=𝐼^𝜈−𝜎𝑥(𝑡),𝒟_𝐭^𝜎𝐼^𝜎𝑥(𝑡)=𝑥(𝑡). (10)
(2) If 𝜈>0,𝜎>0, then 𝒟_𝐭^𝜈𝑡^𝜎−1=Γ(𝜎)Γ()𝑡𝜎−𝜈^{𝜎−𝜈−1}. (11)

Proposition 4 (see [1–3]). Let 𝛼>0, and 𝑓(𝑥) is integrable, then 𝐼^𝛼𝒟_𝐭^𝛼𝑥(𝑡)=𝑓(𝑥)+𝑐₁𝑥^𝛼−1+𝑐₂𝑥^𝛼−2+⋯+𝑐_𝑛𝑥^𝛼−𝑛, (12) where 𝑐_𝑖∈ℝ(𝑖=1,2,…,𝑛), 𝑛 is the smallest integer greater than or equal to 𝛼.

Definition 5. A function 𝑢∈𝐶(𝐼,𝑅) is called a nonnegative solution of BVP (1), if 𝑢≥0 on [0,1] and satisfies (1). Moreover, if 𝑢(𝑡)>0,𝑡∈(0,1), then 𝑢 is said to be a positive solution of BVP (1).
For forthcoming analysis, we first consider the following fractional differential equation: 𝒟_𝐭^𝛼𝑥(𝑡)(𝑡)(),𝑥(0)+𝑔=0,𝑡∈0,1=0,𝒟_𝐭^𝛾𝑥(1)=𝑎𝒟_𝐭^𝛾𝑥(𝜉). (13)

Lemma 6 (see [15]). If 1<𝛼≤2 and 𝑔∈𝐿¹[0,1], then the boundary value problem (13) has the unique solution 𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝑔(𝑠)𝑑𝑠, (14) where 𝐺()=𝑡,𝑠𝑑𝑡^𝛼−1(1−𝑠)^{𝛼−𝛾−1}−𝑎𝑑𝑡^𝛼−1(𝜉−𝑠)^{𝛼−𝛾−1}−(𝑡−𝑠)^𝛼−1Γ(𝛼),0≤𝑠≤min{𝑡,𝜉}≤1,𝑑𝑡^𝛼−1(1−𝑠)^{𝛼−𝛾−1}−(𝑡−𝑠)^𝛼−1Γ(𝛼),0<𝜉≤𝑠≤𝑡≤1,𝑑𝑡^𝛼−1(1−𝑠)^{𝛼−𝛾−1}−𝑎𝑑𝑡^𝛼−1(𝜉−𝑠)^{𝛼−𝛾−1}Γ(𝛼),0≤𝑡≤𝑠≤𝜉<1,𝑑𝑡^𝛼−1(1−𝑠)^{𝛼−𝛾−1}Γ(𝛼),max{𝑡,𝜉}≤𝑠≤1, (15) where 𝑑=(1−𝑎𝜉^{𝛼−𝛾−1})⁻¹.

Lemma 7 (see [15]). The Green function 𝐺(𝑡,𝑠) in Lemma 6 has the following properties:(i)𝐺(𝑡,𝑠) is continuous on [0,1]×[0,1],(ii)𝐺(𝑡,𝑠)>0 for any 𝑠,𝑡∈(0,1). And if 𝑎𝜉^{𝛼−𝛾−2}≤1−𝛾, the Green function 𝐺(𝑡,𝑠) also satisfies(iii)𝐺(𝑡,𝑠)≤𝐺(𝑠,𝑠) for any 𝑠,𝑡∈[0,1],(iv)there exists a positive function 𝜎(𝑠)∈𝐶(0,1) such that min_{𝜉≤𝑡≤1}𝐺(𝑡,𝑠)≥𝜎(𝑠)𝐺(𝑠,𝑠), (16)where 𝐺(𝑠,𝑠)=𝑑𝑠^𝛼−1(1−𝑠)^{𝛼−𝛾−1}−𝑎𝑑𝑠^𝛼−1(𝜉−𝑠)^{𝛼−𝛾−1}Γ(𝛼),0≤𝑠≤𝜉,𝑑𝑠^𝛼−1(1−𝑠)^{𝛼−𝛾−1}Γ(𝛼),𝜉≤𝑠≤1. (17)

Let ℕ be the set of positive integers, let ℝ be the set of real numbers, and let ℝ₊ be the set of nonnegative real numbers. Let 𝐼=[0,1]. Denote by 𝐶(𝐼,ℝ) the Banach space of all continuous functions from 𝐼 into ℝ with the norm ‖𝑥‖=max{𝑥(𝑡)∶𝑡∈𝐼}. (18)

Define the cone 𝑃 in 𝐶(𝐼,ℝ₊) as 𝑃=𝑢∈𝐶𝐼,ℝ₊∶min_{𝜉≤𝑡≤1}𝑥(𝑡)≥𝜎(𝑡)‖𝑥‖,𝑡∈𝐼. (19) Let 𝑞>1 satisfy the relation (1/𝑞)+(1/𝑝)=1, where 𝑝 is given by (1).

To study BVP (1), we first consider the associated linear BVP: 𝒟_𝐭^𝛽𝜑_𝑝𝒟_𝐭^𝛼𝑥(𝑡)(𝑡)()𝑥(0)+𝑔=0,𝑡∈0,1=0,𝒟_𝐭^𝛾𝑥(1)=𝑎𝒟_𝐭^𝛾𝑥(𝜉),𝒟_𝐭^𝛼𝑥(0)=0, (20) for 𝑔∈𝐿¹[0,1] and 𝑔≥0.

Let 𝑤=𝒟_𝐭^𝛼𝑥,𝑣=𝜑_𝑝(𝑤). By Proposition 4, the solution of initial value problem 𝒟_𝐭^𝛽𝑣(𝑡)+𝑔(𝑡)=0,𝑡∈(0,1),𝑣(0)=0, (21) is given by 𝑣(𝑡)=𝐶₁𝑡^𝛽−1−𝐼^𝛽𝑔(𝑡),𝑡∈[0,1]. From the relations 𝑣(0)=0,0<𝛽≤1, it follows that 𝐶₁=0, and so 𝑣(𝑡)=−𝐼^𝛽𝑔(𝑡),𝑡∈[0,1]. (22) Noting that 𝒟_𝐭^𝛼𝑥=𝑤,𝑤=𝜑_𝑝⁻¹(𝑣), from (22), we know that the solution of (20) satisfies 𝒟_𝐭^𝛼𝑥(𝑡)=𝜑_𝑝⁻¹−𝐼^𝛽𝑔(𝑡)()𝑥(0),𝑡∈0,1=0,𝒟_𝐭^𝛾𝑥(1)=𝑎𝒟_𝐭^𝛾𝑥(𝜉). (23) By Lemma 6, the solution of (23) can be written as 𝑥(𝑡)=−¹₀𝐺(𝑡,𝑠)𝜑_𝑝⁻¹−𝐼^𝛽𝑔(𝑠)𝑑𝑠,𝑡∈𝐼. (24) Since ℎ(𝑠)≥0, 𝑠∈𝐼, we have 𝜑_𝑝⁻¹(−𝐼^𝛽𝑔(𝑠))=−(𝐼^𝛽𝑔(𝑠))^𝑞−1, 𝑠∈𝐼, and so 𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝐼^𝛽𝑔(𝑠)^𝑞−1=Γ𝛽𝑑𝑠_1−𝑞10𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑔(𝜏)𝑑𝜏^𝑞−1𝑑𝑠,𝑡∈𝐼, (25) which implies that the solution of (23) is given by 𝑥(𝑡)=¹₀𝐺(𝑡,𝑠)𝐼^𝛽𝑔(𝑠)^𝑞−1=Γ𝛽𝑑𝑠_1−𝑞10𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1𝑔(𝜏)𝑑𝜏^𝑞−1𝑑𝑠. (26) For the convenience, we make the following assumptions.(H1)𝑓∶[0,1]×[0,+∞)→(0,+∞) is continuous and nondecreasing, and there exists a constant 𝜆>0 such that, for any 𝑡∈[0,1],𝑥∈[0,+∞), 𝑓(𝑡,𝑐𝑥)≥𝑐^𝜆𝑓(𝑡,𝑥),∀0<𝑐≤1. (27)(H2)ℎ(𝑡)∈𝐿¹(0,1) is nonnegative on (0, 1), and 0<¹₀𝐺(𝑠,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑑𝜏^𝑞−1𝑑𝑠<+∞. (28)

Remark 8. By (27), for any 𝑐≥1,(𝑡,𝑥)∈[0,1]×[0,+∞), clearly, 𝑓(𝑡,𝑐𝑥)≤𝑐^𝜆𝑓(𝑡,𝑥). (29) Now, for any 𝑥∈𝑃, define one operator 𝑇 as follows: (𝑇𝑥)(𝑡)=Γ𝛽^1−𝑞1₀𝐺()×𝑡,𝑠_𝑠0(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1𝑑𝑠,𝑡∈𝐼. (30) Then by (20) and (23), the BVP (1) is equivalent to the fixed point problems of the operators 𝑇.

Lemma 9. Assume that (H1) and (H2) hold. Then 𝑇∶𝑃→𝑃 are continuous, compact, and nondecreasing.

Proof. In fact, for any 𝑥∈𝑃, ‖𝑇𝑥‖=max_𝑡∈𝐼Γ𝛽^1−𝑞1₀𝐺()×𝑡,𝑠_𝑠0(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1≤Γ𝛽𝑑𝑠_1−𝑞×¹₀𝐺(𝑠,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1𝑑𝑠. (31) On the other hand, by Lemma 7, min_{𝜉≤𝑡≤1}𝑇𝑥(𝑡)=min_{𝜉≤𝑡≤1}Γ𝛽^1−𝑞1₀𝐺()×𝑡,𝑠_𝑠0(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1(𝑡)Γ𝛽𝑑𝑠≥𝜎^1−𝑞1₀𝐺()×𝑠,𝑠_𝑠0(𝑠−𝜏)^𝛽−1×ℎ(𝜏)𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1(𝑡‖.𝑑𝑠=𝜎)‖𝑇𝑥 (32) So 𝑇(𝑃)⊂𝑃.
Next, supposing 𝐷⊂𝑃 is a bounded set, then for any 𝑥∈𝐷, there exists a constant 𝑀>0 such that ||𝑥||≤𝑀. Thus for any 𝑥∈𝐷, we have ‖𝑇𝑥‖=max_𝑡∈𝐼Γ𝛽^1−𝑞×¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1×ℎ(𝜏)𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1≤Γ𝛽𝑑𝑠_1−𝑞×¹₀𝐺(𝑠,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1≤Γ𝛽𝑑𝑠_1−𝑞10𝐺(𝑠,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑑𝜏^𝑞−1×𝑑𝑠sup_{[]×[]0,10,𝑀}𝑓(𝑡,𝑥)^𝑞−1<+∞, (33) which implies 𝑇(𝐷) is bounded. On the other hand, according to the Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we easily see 𝑇∶𝑃→𝑃 is completely continuous. In the end, noticing the monotonicity of 𝑓 on 𝑥 and the definition of 𝑇, we also have that the operator 𝑇 is nondecreasing.

3. Main Results

Define two constants 𝐵=Γ𝛽^1−𝑞1₀𝐺(𝑠,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑑𝜏^𝑞−1𝑑𝑠,𝑙=max_𝑡∈𝐼𝑓(𝑡,1). (34)

Theorem 10. Suppose conditions (H1) and (H2) hold. If there exists a positive constant 𝑏>1 such that 𝑙^𝑞−1𝐵≤𝑏^{1−𝜆(𝑞−1)}, (35) where 𝑙 and 𝐵 are defined by (34), then the BVP (1) has the maximal and minimal solutions 𝑥^∗ and 𝑦^∗, which are positive, and there exist two positive constants 𝑚₁≤𝑚₂ such that 𝑚₂𝜎(𝑡)≤𝑥^∗(𝑡)𝑚≤𝑏,₁𝜎(𝑡)≤𝑦^∗(𝑡)≤𝑏,𝑡∈𝐼. (36) Moreover for initial values 𝑥₀=𝑏,𝑦₀=0, define the iterative sequences 𝑥_𝑛,𝑦_𝑛 by 𝑥_𝑛(𝑡)=𝑇𝑥_𝑛−1(𝑡)=𝑇^𝑛𝑥₀(𝑡),𝑦_𝑛(𝑡)=𝑇𝑦_𝑛−1(𝑡)=𝑇^𝑛𝑦₀(𝑡). (37) Then lim_𝑛→+∞𝑥_𝑛=𝑥^∗,lim_𝑛→+∞𝑦_𝑛=𝑦^∗ (38) for 𝑡∈𝐼 uniformly, respectively.

Proof. Let 𝑃[0,𝑏]={𝑥∈𝑃∶0≤||𝑥||≤𝑏}; we firstly prove 𝑇(𝑃[0,𝑏])⊂𝑃[0,𝑏]. In fact, for any 𝑥∈𝑃[0,𝑏], we have 0≤𝑥(𝑡)≤max_𝑡∈𝐼𝑥(𝑡)=‖𝑥‖≤𝑏. (39) By the assumption (H1), we have 0≤𝑓(𝑡,𝑥(𝑡))≤𝑓(𝑡,𝑏)≤𝑏^𝜆𝑓(𝑡,1)≤𝑏^𝜆max_𝑡∈𝐼𝑓(𝑡,1)=𝑙𝑏^𝜆. (40) It follows from Lemma 9 that 𝑇∶𝑃→𝑃 is completely continuous operator; thus by (35) and (40), we have ‖𝑇𝑥‖=max_𝑡∈𝐼Γ𝛽^1−𝑞×¹₀𝐺(𝑡,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1×ℎ(𝜏)𝑓(𝜏,𝑥(𝜏))𝑑𝑡^𝑞−1≤Γ𝛽𝑑𝑠_1−𝑞×¹₀𝐺(𝑠,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑓(𝜏,𝑥(𝜏))𝑑𝜏^𝑞−1≤Γ𝛽𝑑𝑠_1−𝑞×¹₀𝐺(𝑠,𝑠)^𝑠₀(𝑠−𝜏)^𝛽−1ℎ(𝜏)𝑑𝜏^𝑞−1×𝑑𝑠𝑙𝑏^𝜆𝑞−1=𝐵𝑙𝑏^𝜆𝑞−1≤𝑏, (41) which implies that 𝑇(𝑃[0,𝑏])⊂𝑃[0,𝑏].
Let 𝑦₀(𝑡)=0,𝑡∈[0,1]; then 𝑦₀(𝑡)∈𝑃[0,𝑏]. Leting 𝑦₁(𝑡)=(𝑇𝑦₀)(𝑡), we have 𝑦₁∈𝑃[0,𝑏]. Denote 𝑦_𝑛+1=𝑇𝑦_𝑛=𝑇^𝑛+1𝑦₀,𝑛=1,2,…. (42) It follows from 𝑇(𝑃[0,𝑏])⊂𝑃[0,𝑏] that 𝑦_𝑛∈𝑃[0,𝑏]. Since 𝑇 is compact, we obtain that {𝑦_𝑛} is a sequentially compact set.
Since 𝑦₁=𝑇𝑦₀=𝑇0∈𝑃[0,𝑏], we have 𝑦₁(𝑡)=𝑇𝑦₀(𝑡)=(𝑇0)(𝑡)≥0=𝑦₀(𝑡),𝑡∈[0,1]. (43) By the induction, we get 𝑦_𝑛+1≥𝑦_𝑛,𝑛=0,1,2,…. (44) Consequently, there exists 𝑦^∗∈𝑃[0,𝑏] such that 𝑦_𝑛→𝑦^∗. Letting 𝑛→+∞, from the continuity of 𝑇 and 𝑇𝑦_𝑛=𝑦_𝑛−1, we obtain 𝑇𝑦∗=𝑦^∗, which implies that 𝑦^∗ is a nonnegative solution of boundary value problem (1). Since 𝑓∶[0,1]×[0,+∞)→(0,+∞), we know the zero function is not the solution of boundary value problem (1), thus max_0≤𝑡≤1|𝑦^∗(𝑡)|>0; by 𝑦^∗∈𝑃, we have 𝑦^∗(𝑡)≥𝑦^∗𝜎(𝑡)>0,𝑡∈(0,1), (45) that is, 𝑦^∗ is a positive solution of boundary value problem (1).
On the other hand, let 𝑥₀(𝑡)=𝑏,𝑡∈[0,1]; then 𝑥₀(𝑡)∈𝑃[0,𝑏]. Leting 𝑥₁=𝑇𝑥₀, from the previous expressions, we have 𝑥₁∈𝑃[0,𝑏]. Thus let us denote 𝑥_𝑛+1=𝑇𝑥_𝑛=𝑇^𝑛+1𝑥₀,𝑛=1,2,…. (46) It follows from 𝑇(𝑃[0,𝑏])⊂𝑃[0,𝑏] that 𝑥_𝑛∈𝑃[0,𝑏]𝑛=0,1,2,…. (47) Since 𝑇 is compact by Lemma 9, we can assert that {𝑥_𝑛} is a sequentially compact set.
Now, since 𝑥₁∈𝑃[0,𝑏], we have 0≤𝑥₁(𝑡)≤𝑥₁≤𝑏=𝑥₀(𝑡). (48) It follows from Lemma 9 that 𝑇∶𝑃→𝑃 is nondecreasing, so 𝑥₂=𝑇𝑥₁≤𝑇𝑥₀=𝑥₁. (49) By the induction, we have 𝑥_𝑛+1≤𝑥_𝑛,𝑛=0,1,2,…. (50) Consequently, there exists 𝑥^∗∈𝑃[0,𝑏] such that 𝑥_𝑛→𝑥^∗. Letting 𝑛→+∞, from the continuity of 𝑇 and 𝑇𝑥_𝑛=𝑥_𝑛−1, we obtain 𝑇𝑥^∗=𝑥^∗, which implies that 𝑥^∗ is a nonnegative solution of boundary value problem (1).
Next, noting 𝑥₀(𝑡)=𝑏≥𝑦₀(𝑡)=0,𝑡∈[0,1], thus it follows from monotonicity of 𝑇 that 𝑇𝑥₀≥𝑇𝑦₀; by the induction, we have 𝑥_𝑛≥𝑦_𝑛,𝑛=0,1,2,…, which implies that 𝑥^∗≥𝑦^∗. Thus by (45) we have 𝑥^∗(𝑡)≥𝑦^∗(𝑡)≥𝑥^∗𝜎(𝑡)>0,𝑡∈(0,1). (51) This means that 𝑥^∗ is also a positive solution of boundary value problem (1).
In the end, let 𝑢^∗ be any fixed point of 𝑇 in 𝑃[0,𝑏], then 𝑦₀=0≤𝑢^∗≤𝑏=𝑥₀, (52) and then 𝑦₁=𝑇𝑦₀≤𝑇𝑢^∗=𝑢^∗≤𝑇𝑏=𝑥₁. (53) By induction, we have 𝑦_𝑛≤𝑢^∗≤𝑥_𝑛,𝑛=1,2,3,…. (54) Taking the limit, we have 𝑦^∗≤𝑢^∗≤𝑥^∗. (55) This implies that 𝑥^∗ and 𝑦^∗ are maximal and minimal solutions of the BVP (1).Let 𝑚₁=||𝑦^∗||,𝑚₂=||𝑥^∗||, then we have 𝑚₂𝜎(𝑡)≤𝑥^∗(𝑡)𝑚≤𝑏,₁𝜎(𝑡)≤𝑦^∗(𝑡)≤𝑏,𝑡∈𝐼. (56) The proof is completed.

Remark 11. If ℎ(𝑡)≡1, then (H2) holds naturally, and in this case we take 𝐵^∗=Γ𝛽^1−𝑞1₀𝑑Γ(𝛼)^𝑠₀(𝑠−𝜏)^𝛽−1𝑑𝜏^𝑞−1=𝑑𝛽𝑑𝑠𝛽Γ^1−𝑞Γ(𝛼)𝛽.𝑞−1+1 (57) Thus we have the following Corollary 12.

Corollary 12. Suppose condition (H1) holds and ℎ(𝑡)≡1. If there exists a positive constant 𝑏>1 such that 𝑙^𝑞−1𝐵^∗≤𝑏^{1−𝜆(𝑞−1)}, (58) where 𝑙 is defined by (34), then the BVP (1) has the maximal and minimal solutions 𝑥^∗ and 𝑦^∗, which are positive, and there exist two positive constants 𝑚₁≤𝑚₂ such that 𝑚₂𝜎(𝑡)≤𝑥^∗(𝑡)𝑚≤𝑏,₁𝜎(𝑡)≤𝑦^∗(𝑡)≤𝑏,𝑡∈𝐼. (59) Moreover for initial values 𝑥₀=𝑏,𝑦₀=0, define the iterative sequences 𝑥_𝑛,𝑦_𝑛 by 𝑥_𝑛(𝑡)=𝑇𝑥_𝑛−1(𝑡)=𝑇^𝑛𝑥₀(𝑡),𝑦_𝑛(𝑡)=𝑇𝑦_𝑛−1(𝑡)=𝑇^𝑛𝑦₀(𝑡). (60) Then lim_𝑛→+∞𝑥_𝑛=𝑥^∗,lim_𝑛→+∞𝑦_𝑛=𝑦^∗ (61) for 𝑡∈𝐼 uniformly, respectively.

Corollary 13. Suppose conditions (H1)and(H2) hold. If 𝜆<𝑝−1. (62) Then there exists a constant 𝑏>1 such that the BVP (1) has the maximal and minimal solutions 𝑥^∗ and 𝑦^∗, which are positive, and there exist two positive constants 𝑚₁≤𝑚₂ such that 𝑚₂𝜎(𝑡)≤𝑥^∗(𝑡)𝑚≤𝑏,₁𝜎(𝑡)≤𝑦^∗(𝑡)≤𝑏,𝑡∈𝐼. (63) Moreover for initial values 𝑥₀=𝑏,𝑦₀=0, define the iterative sequences 𝑥_𝑛,𝑦_𝑛 by 𝑥_𝑛(𝑡)=𝑇𝑥_𝑛−1(𝑡)=𝑇^𝑛𝑥₀(𝑡),𝑦_𝑛(𝑡)=𝑇𝑦_𝑛−1(𝑡)=𝑇^𝑛𝑦₀(𝑡). (64) Then lim_𝑛→+∞𝑥_𝑛=𝑥^∗,lim_𝑛→+∞𝑦_𝑛=𝑦^∗ (65) for 𝑡∈𝐼 uniformly, respectively.

Proof . It follows from 𝜆<𝑝−1 that lim_𝑢→+∞𝑢^𝜆𝑢^𝑝−1=0, (66) which implies that there exists 𝑏>1 large enough such that 𝑏^𝜆𝑏^𝑝−1<1𝐵^∗𝑝−1𝑙. (67) Notice that (1/𝑝)+(1/𝑞)=1; (67) is equivalent to 𝑙^𝑞−1𝐵^∗≤𝑏^{1−𝜆(𝑞−1)}. (68) By Theorem 10, the conclusion of Corollary 13 holds.

Remark 14. In Corollary 13, we obtain that the BVP (1) has the maximal and minimal solutions 𝑥^∗ and 𝑦^∗ only by comparing 𝑝−1 to 𝜆. But note that 𝑝 and 𝜆 are irrelative, so (62) is easy to be satisfied; this implies that Corollary 13 is very interesting.

Example 15. Consider the following boundary value problem: −𝒟_𝐭^1/2𝜑₆𝒟_𝐭^3/2𝑥(𝑡)=sin𝑡𝑥⁴(𝑡)+𝑡²𝑥^3/4(𝑡)+2𝑡³(),𝑥(0)+5𝑡+2,𝑡∈0,1=0,𝒟_𝐭^1/6𝑥(1)=16√³𝒟16^𝐭1/6𝑥14,𝒟_𝐭^3/2𝑥(0)=0. (69)
Let 𝛼=3/2,𝛽=1/2,𝛾=1/6,𝑝=6, and 𝑓(𝑡,𝑥)=sin𝑡𝑥⁴+𝑡²𝑥^3/4+2𝑡³+5𝑡+2,ℎ(𝑡)≡1, then 𝑎𝜉^{𝛼−𝛾−2}=16√³1164^−2/3=165<1−𝛾=6. (70) For any 0<𝑐≤1 and 𝑥∈[0,+∞), we have 𝑓(𝑡,𝑐𝑥)=sin𝑡(𝑐𝑥)⁴+𝑡²(𝑐𝑥)^3/4+2𝑡³()+5𝑡+2≥sin𝑡𝑐𝑥⁴+𝑡²𝑐⁴𝑥^3/4+𝑐⁴2𝑡³+5𝑡+2=𝑐⁴𝑓(𝑡,𝑥). (71) Taking 𝜆=4, then 𝜆=4<𝑝−1=5, (72) which implies that (62) holds. By Corollary 13, we know the BVP (69) has at least two positive solutions.