Positive Solutions for Multipoint Boundary Value Problems for Singular Fractional Differential Equations

Abstract

A class of nonlinear multipoint boundary value problems for singular fractional differential equations is considered. By means of a coupled fixed point theorem on ordered sets, some results on the existence and uniqueness of positive solutions are obtained.

1. Introduction

Fractional calculus is used to formulate different phenomena in physics, biology, medicine, and so forth. For more details on the applications of fractional calculus, we refer the reader to [1–4]. On the other hand, some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator has been discussed by Lakshmikantham [5–7], Bai and Lü [8], El-Sayed et al. [9, 10], Bai [11, 12], Zhang [13], and so forth.

Very recently, Liang and Zhang [14] considered the following m-point boundary value problem: 𝐷^𝛼₀⁺𝑢(𝑡)((𝑡𝑢(0)+𝑓𝑡,𝑢))=0,0<𝑡<1,2<𝛼≤3,=𝑢^′(0)=0,𝑢^′(1)=^𝑚−2_𝑖=1𝛽_𝑖𝑢^′𝜉_𝑖, (1) where 𝐷^𝛼₀⁺ is the Riemann-Liouville fractional derivative, 0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<1 satisfies 0<∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2<1, and 𝑓∶[0,1]×[0,+∞)→[0,+∞) is continuous and nondecreasing with respect to the second variable. Under some hypotheses, using a fixed point theorem on ordered sets, the authors established the existence and uniqueness of solution to such problem.

Motivated by the abovementioned work, in this paper, we deal with the following multipoint boundary value problem: 𝐷^𝛼₀⁺𝑢(𝑡)((𝑡)(𝑡𝑢(0)+𝑓𝑡,𝑢,𝑢))=0,0<𝑡<1,2<𝛼≤3,=𝑢^′(0)=0,𝑢^′(1)=^𝑚−2_𝑖=1𝛽_𝑖𝑢^′𝜉_𝑖, (2)(3) where 𝑓∶(0,1]×[0,+∞)×[0,+∞)→[0,+∞) is continuous and lim_𝑡→0⁺𝑓(𝑡,⋅,⋅)=+∞ (𝑓 is singular at 𝑡=0).

2. Preliminaries

The following preliminaries will be useful later.

Definition 1. The Riemann-Liouville fractional derivative of order 𝛼>0 of a continuous function 𝜑∶(0,+∞)→ℝ is given by 𝐷^𝛼₀⁺𝜑(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^(𝑛)𝑡₀𝜑(𝑠)(𝑡−𝑠)^{𝛼−𝑛+1}𝑑𝑠, (4) where 𝑛=[𝛼]+1, [𝛼] denotes the integer part of number 𝛼, provided that the right side is pointwise defined on (0,+∞). Here, Γ is the Euler gamma function defined by Γ(𝛼)=₀^+∞𝑡^𝛼−1𝑒^−𝑡𝑑𝑡. (5)

Definition 2. The Riemann-Liouville fractional integral of order 𝛼>0 of a given function 𝜑∶(0,+∞)→ℝ is defined by 𝐼^𝛼₀⁺𝜑(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝜑(𝑠)𝑑𝑠, (6) provided that the right side is defined on (0,+∞).

From the definition of the Riemann-Liouville derivative, we can obtain the following statement.

Lemma 3 (see [15]). Let 𝛼>0. If one assumes that 𝑢∈𝐶(0,1)∩𝐿(0,1), then the fractional differential equation 𝐷^𝛼₀⁺𝑢(𝑡)=0 (7) has 𝑢(𝑡)=𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑁𝑡^𝛼−𝑁, 𝑐_𝑖∈ℝ, 𝑖=1,2,…,𝑁 as unique solutions, where 𝑁 is the smallest integer greater than or equal to 𝛼.

Lemma 4 (see [15]). Assume that 𝑢∈𝐶(0,1)∩𝐿(0,1) with a fractional derivative of order 𝛼>0 that belongs to 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, (8) for some 𝑐_𝑖∈ℝ, 𝑖=1,2,…,𝑁, where 𝑁 is the smallest integer greater than or equal to 𝛼.

Lemma 5 (see [14]). Assume that ∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2≠1. If ℎ∈𝐶([0,1]), then the boundary value problem 𝐷^𝛼₀⁺𝑢(𝑡)(𝑡)𝑢(0)+ℎ=0,0<𝑡<1,2<𝛼≤3,=𝑢^′(0)=0,𝑢^′(1)=^𝑚−2_𝑖=1𝛽_𝑖𝑢^′𝜉_𝑖, (9) has a unique solution 𝑢(𝑡)=¹₀𝐺()ℎ(𝑠)𝑡𝑡,𝑠𝑑𝑠+^𝛼−1(𝛼−1)1−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2×^𝑚−2_𝑖=1𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠ℎ(𝑠)𝑑𝑠, (10) where 𝐺()=𝑡𝑡,𝑠^𝛼−1(1−𝑠)^𝛼−2−(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑡,𝑖𝑓0≤𝑠≤𝑡≤1,_𝛼−1(1−𝑠)^𝛼−2Γ(𝛼)𝐻()=(),𝑖𝑓0≤𝑡≤𝑠≤1,𝑡,𝑠𝜕𝐺𝑡,𝑠.𝜕𝑡 (11)

It is clear that, for all 0≤𝑡≤𝑠≤1, the function 𝐻(𝑡,𝑠)≥0. For 0≤𝑠≤𝑡≤1, we have 𝑡𝑠≤𝑠; then 𝑡^𝛼−2(1−𝑠)^𝛼−2−(𝑡−𝑠)^𝛼−2≥0. Hence for 0≤𝑡≤𝑠≤1 one has 𝐻(𝑡,𝑠)≥0. The following properties of 𝐺 will be used later.

Lemma 6 (see [14]). The following properties hold.(i)𝐺is a nonnegative continuous function on [0,1]×[0,1];(ii)𝐺(⋅,𝑠) is strictly increasing for all 𝑠∈[0,1].

Let (𝑋,⪯) be a partially ordered set endowed with a metric 𝑑. Let 𝐹∶𝑋×𝑋→𝑋 be a given mapping.

Definition 7. One says that (𝑋,⪯) is directed if, for every (𝑥,𝑦)∈𝑋×𝑋, there exists 𝑧∈𝑋 such that 𝑥⪯𝑧, 𝑦⪯𝑧 and there exists 𝑤∈𝑋 such that 𝑥≽𝑤, 𝑦≽𝑤.

Definition 8. We say that (𝑋,⪯,𝑑) is regular if the following conditions hold. (𝑐₁)if {𝑥_𝑛} is a nondecreasing sequence in 𝑋 such that 𝑥_𝑛→𝑥∈𝑋, then 𝑥_𝑛⪯𝑥 for all 𝑛;(𝑐₂) if {𝑦_𝑛} is a decreasing sequence in 𝑋 such that 𝑦_𝑛→𝑦∈𝑋, then 𝑦_𝑛≽𝑦 for all 𝑛.

Example 9. Let 𝑋=𝐶([0,𝑇]), 𝑇>0 be the set of real continuous functions on [0,𝑇]. We endow 𝑋 with the standard metric 𝑑 given by 𝑑(𝑢,𝑣)=max_{0≤𝑡≤𝑇}|𝑢(𝑡)−𝑣(𝑡)|,𝑢,𝑣∈𝑋. (12) We define the partial order ⪯ on 𝑋 by 𝑢,𝑣∈𝑋,𝑢⪯𝑣⟺𝑢(𝑡)≤𝑣(𝑡)∀𝑡∈[0,𝑇]. (13) Let 𝑥,𝑦∈𝑋. For 𝑧=max{𝑥,𝑦}, that is, 𝑧(𝑡)=max{𝑥(𝑡),𝑦(𝑡)}, for all 𝑡∈[0,𝑇], we have 𝑥⪯𝑧 and 𝑦⪯𝑧. For 𝑤=min{𝑥,𝑦}, that is, 𝑤(𝑡)=min{𝑥(𝑡),𝑦(𝑡)}, for all 𝑡∈[0,𝑇], we have 𝑥≽𝑤 and 𝑦≽𝑤. This implies that (𝑋,⪯) is directed. Now, let {𝑥_𝑛} be a nondecreasing sequence in 𝑋 such that 𝑑(𝑥_𝑛,𝑥)→0 as 𝑛→∞, for some 𝑥∈𝑋. Then, for all 𝑡∈[0,𝑇], {𝑥_𝑛(𝑡)} is a nondecreasing sequence of real numbers converging to 𝑥(𝑡). Thus, we have 𝑥_𝑛(𝑡)≤𝑥(𝑡), for all 𝑛, that is, 𝑥_𝑛⪯𝑥 for all 𝑛. Similarly, if {𝑦_𝑛} is a decreasing sequence in 𝑋 such that 𝑑(𝑦_𝑛,𝑦)→0 as 𝑛→∞, for some 𝑦∈𝑋, we get that 𝑦_𝑛≽𝑦 for all 𝑛. Then, we proved that (𝑋,⪯,𝑑) is regular.

Definition 10 (see [16]). An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled fixed point of 𝐹 if 𝐹𝑥,𝑦=𝑥,𝐹𝑦,𝑥=𝑦. (14)

Definition 11 (see [16]). One says that 𝐹 has the mixed monotone property if 𝑥,𝑦,(𝑢,𝑣)∈𝑋×𝑋,𝑥⪯𝑢,𝑦≽𝑣⟹𝐹𝑥,𝑦⪯𝐹(𝑢,𝑣). (15)

Denote by Φ the set of functions 𝜑∶[0,+∞)→[0,+∞) satisfying the following.(Φ₁)𝜑 is continuous;(Φ₂)𝜑 is nondecreasing;(Φ₃)𝜑⁻¹({0})={0}.

The following two lemmas are fundamental for the proof of our main result.

Lemma 12 (see [17]). Let (𝑋,⪯) be a partially ordered set and suppose that there exists a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping having the mixed monotone property on 𝑋 such that 𝜓𝑑𝐹()𝑑()𝑑(),𝑥,𝑦,𝐹𝑢,𝑣≤𝜓max𝑥,𝑢,𝑑𝑦,𝑣−𝜑max𝑥,𝑢,𝑑𝑦,𝑣 (16) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 with 𝑥≽𝑢 and 𝑦⪯𝑣, where 𝜓,𝜑∈Φ. Suppose also that (𝑋,⪯,𝑑) is regular and there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀⪯𝐹𝑥₀,𝑦₀,𝑦₀≽𝐹𝑦₀,𝑥₀. (17) Then, 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Moreover, if {𝑥_𝑛} and {𝑦_𝑛} are the sequences in 𝑋 defined by 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,𝑛=0,1,…, (18) then lim_𝑛→∞𝑑𝑥_𝑛,𝑥^∗=lim_𝑛→∞𝑑𝑦_𝑛,𝑦^∗=0. (19)

Lemma 13 (see [17]). Adding to the hypotheses of Lemma 12 the condition (𝑋,⪯) is directed; one obtains uniqueness of the coupled fixed point. Moreover, one has the equality 𝑥^∗=𝑦^∗.

3. Main Result

Let Banach space 𝐸=𝐶([0,1]) be endowed with the norm ‖𝑢‖_∞=max_0≤𝑡≤1|𝑢(𝑡)|. We define the partial order ⪯ on 𝐸 by 𝑢,𝑣∈𝐸,𝑢⪯𝑣⟺𝑢(𝑡)≤𝑣(𝑡)∀𝑡∈[0,1]. (20) In Example 9, we proved that (𝐸,⪯) with the classic metric given by 𝑑(𝑢,𝑣)=‖𝑢−𝑣‖_∞,𝑢,𝑣∈𝐸 (21) satisfies the following properties: (𝐸,⪯) is directed and (𝐸,⪯,𝑑) is regular.

Define the closed cone 𝑃⊂𝐸 by 𝑃={𝑢∈𝐸∶𝑢≽0}, (22) where 0 denotes the zero function.

Definition 14. One says that (𝑢⁻,𝑢⁺)∈𝐶([0,1])×𝐶([0,1]) is a coupled lower and upper solution to (2)-(3) if, for all 𝑡∈[0,1], one has 𝑢⁻(𝑡)≤¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢⁻(𝑠),𝑢⁺(𝑠)+𝑡𝑑𝑠_𝛼−1(𝛼−1)1−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2×^𝑚−2_𝑖=1𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑓𝑠,𝑢⁻(𝑠),𝑢⁺(𝑠)𝑢𝑑𝑠,₊(𝑡)≥¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢⁺(𝑠),𝑢⁻(𝑠)+𝑡𝑑𝑠_𝛼−1(𝛼−1)1−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2×^𝑚−2_𝑖=1𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑓𝑠,𝑢⁺(𝑠),𝑢⁻(𝑠)𝑑𝑠. (23)

The main result of this paper is the following.

Theorem 15. Suppose that the following conditions hold. (i)0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<1 satisfies 0<∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2<1 with 𝛽_𝑖>0 for 𝑖=1,…,𝑚−2;(ii)𝑓∶(0,1]×[0,+∞)×[0,+∞)→[0,+∞) is continuous, lim_𝑡→0⁺𝑓(𝑡,⋅,⋅)=+∞;(iii)there exists 𝜎∈(0,1) such that 𝑡↦𝑡^𝜎𝑓(𝑡,𝑥,𝑦) is continuous on [0,1] for all 𝑥,𝑦∈[0,+∞);(iv)there exists ≤0<𝜆1−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^{𝛼−𝜎−1}1−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2−𝛼−1𝛼−𝜎Γ(1−𝜎)(𝛼−1)Γ(𝛼−𝜎)⁻¹ (24) such that for all 𝑥,𝑦,𝑧,𝑤∈[0,+∞) with 𝑥≥𝑧, 𝑦≤𝑤 and 𝑡∈[0,1], 0≤𝑡^𝜎𝑓(),,𝑡,𝑥,𝑦−𝑓𝑡,𝑧,𝑤)≤𝜆𝜂max(𝑥−𝑧𝑤−𝑦 (25) where 𝜂∶[0,+∞)→[0,+∞) is nondecreasing, 𝛽∶𝑢↦𝑢−𝜂(𝑢)∈Φ;(v)equations (2)-(3) has a coupled lower and upper solution (𝑢⁻,𝑢⁺)∈𝑃×𝑃.
Then, (I)the boundary value problem (2)-(3) has a unique positive solution 𝑢^∗∈𝐶([0,1]);(II)the sequences {𝑢_𝑛} and {𝑣_𝑛} defined by 𝑢₀=𝑢⁻,𝑢_𝑛+1=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑢_𝑛(𝑠),𝑣_𝑛(𝑠+𝑡)𝑑𝑠_𝛼−1(𝛼−1)1−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2×^𝑚−2_𝑖=1𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑓𝑠,𝑢_𝑛(𝑠),𝑣_𝑛(𝑠)𝑣𝑑𝑠,𝑛=0,1,…,₀=𝑢⁺,𝑣_𝑛+1=¹₀𝐺(𝑡,𝑠)𝑓𝑠,𝑣_𝑛(𝑠),𝑢_𝑛(𝑠)+𝑡𝑑𝑠_𝛼−1(𝛼−1)1−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2×^𝑚−2_𝑖=1𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑓𝑠,𝑣_𝑛(𝑠),𝑢_𝑛(𝑠)𝑑𝑠,𝑛=0,1,…, (26)converge uniformly to 𝑢^∗.

Proof. Suppose that 𝑢 is a solution to the boundary value problem (2)-(3). Then, from Lemma 5, we have 𝑢(𝑡)=¹₀𝐺()𝑓((𝑠)(𝑠+𝑡𝑡,𝑠𝑠,𝑢,𝑢))𝑑𝑠_𝛼−1(𝛼−1)1−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2×^𝑚−2_𝑖=1𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑓(𝑠,𝑢(𝑠),𝑢(𝑠))𝑑𝑠, (27) for all 𝑡∈[0,1].
Consider the operator 𝐹 defined by 𝐹(𝑢,𝑣)(𝑡)=¹₀𝐺()𝑓((𝑠)(𝑠+𝑡𝑡,𝑠𝑠,𝑢,𝑣))𝑑𝑠_𝛼−1(𝛼−1)1−∑^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2×^𝑚−2_𝑖=1𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑓(𝑠,𝑢(𝑠),𝑣(𝑠))𝑑𝑠, (28) for all 𝑡∈[0,1], for all 𝑢,𝑣∈𝑃. From (iii) and Lemma 6, we have that 𝐹(𝑃×𝑃)⊂𝑃.
Let (𝑥,𝑦),(𝑢,𝑣)∈𝑃×𝑃 such that 𝑥⪯𝑢 and 𝑦≽𝑣. From (25), we have 𝑠^𝜎𝑓𝑠,𝑥(𝑠),𝑦(𝑠)≤𝑠^𝜎𝑓(𝑠,𝑢(𝑠),𝑣(𝑠)),∀𝑠∈[0,1]. (29) Since the operator 𝐹 is linear and increasing with respect to the function 𝑓, we deduce that 𝐹𝑥,𝑦(𝑡)≤𝐹(𝑢,𝑣)(𝑡),∀𝑠∈[0,1]. (30) This implies that 𝐹 has the mixed monotone property with respect to the partial order ⪯ given by (20).
In the sequel, we denote 𝜉=^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^𝛼−2. (31) Let (𝑥,𝑦),(𝑢,𝑣)∈𝑃×𝑃 such that 𝑥≽𝑢 and 𝑦⪯𝑣. For all 𝑡∈[0,1], using (25) and Lemma 6, we have 𝐹(𝑡)(𝑡=𝑥,𝑦−𝐹𝑢,𝑣)()₁₀𝐺(𝑓(𝑠)(𝑠)((𝑠)(𝑠+𝑡𝑡,𝑠)𝑠,𝑥,𝑦−𝑓𝑠,𝑢,𝑣))𝑑𝑠_𝛼−1()×𝛼−1)(1−𝜉_{𝑚−2𝑖=1}𝛽_𝑖¹₀𝐻𝜉_𝑖𝑓(𝑠)(𝑠)((𝑠)(𝑠=,𝑠𝑠,𝑥,𝑦−𝑓𝑠,𝑢,𝑣))𝑑𝑠₁₀𝐺(𝑡,𝑠)𝑠^−𝜎𝑠^𝜎𝑓(𝑠)(𝑠)((𝑠)(𝑠+𝑡𝑠,𝑥,𝑦−𝑓𝑠,𝑢,𝑣))𝑑𝑠_𝛼−1()×𝛼−1)(1−𝜉_{𝑚−2𝑖=1}𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑠^−𝜎𝑠^𝜎𝑓(𝑠)(𝑠)((𝑠)(𝑠≤𝑠,𝑥,𝑦−𝑓𝑠,𝑢,𝑣))𝑑𝑠₁₀𝐺(𝑡,𝑠)𝑠^−𝜎𝑥(𝑠)(𝑠)(𝑠)(𝑠+𝑡𝜆𝜂max−𝑢,𝑣−𝑦)𝑑𝑠_𝛼−1()×𝛼−1)(1−𝜉_{𝑚−2𝑖=1}𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑠^−𝜎×𝑥(𝑠)(𝑠)(𝑠)(𝑠𝑑()𝜆𝜂max−𝑢,𝑣−𝑦)𝑑𝑠≤𝜆𝜂max𝑥,𝑢,𝑑𝑦,𝑣×max_[]𝑧∈0,1¹₀𝐺(𝑧,𝑠)𝑠^−𝜎𝑧𝑑𝑠+^𝛼−1()×𝛼−1)(1−𝜉_{𝑚−2𝑖=1}𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑠^−𝜎𝑑𝑠. (32) Thus, for all (𝑥,𝑦),(𝑢,𝑣)∈𝑃×𝑃 such that 𝑥≽𝑢 and 𝑦⪯𝑣, we have 𝑑𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝜆𝜒𝜂max𝑑(𝑥,𝑢),𝑑𝑦,𝑣, (33) where 𝜒=max_𝑧∈[0,1]¹₀𝐺(𝑧,𝑠)𝑠^−𝜎𝑧𝑑𝑠+^𝛼−1()×𝛼−1)(1−𝜉_{𝑚−2𝑖=1}𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑠^−𝜎𝑑𝑠 (34) Now, let 𝑧∈[0,1]. We have ¹₀𝐺(𝑧,𝑠)𝑠^−𝜎=𝑑𝑠_𝑧0𝑧^𝛼−1(1−𝑠)^𝛼−2−(𝑧−𝑠)^𝛼−1Γ(𝛼)𝑠^−𝜎+𝑑𝑠_1𝑧𝑧^𝛼−1(1−𝑠)^𝛼−2Γ(𝛼)𝑠^−𝜎=1𝑑𝑠Γ(𝛼)𝑧^𝛼−11₀(1−𝑠)^𝛼−2𝑠^−𝜎−𝑑𝑠_𝑧0(𝑧−𝑠)^𝛼−1𝑠^−𝜎=𝑧𝑑𝑠_𝛼−1Γ(𝛼)¹₀(1−𝑠)^𝛼−2𝑠^−𝜎𝑑𝑠−𝑧^1−𝜎1₀(1−𝑠)^𝛼−1𝑠^−𝜎=𝑧𝑑𝑠_𝛼−1Γ(𝛼)𝐵(1−𝜎,𝛼−1)−𝑧^1−𝜎𝐵(1−𝜎,𝛼), (35) where 𝐵 denotes the beta function. Recall that beta and gamma functions satisfy the following properties: 𝐵(𝑎,𝑏)=¹₀(1−𝑠)^𝑏−1𝑠^𝑎−1Γ(𝑎)Γ(𝑏)𝑑𝑠=Γ(),Γ()(𝑎)𝑎+𝑏𝑎+1=𝑎Γfor𝑎,𝑏>0. (36) Thus we have ¹₀𝐺(𝑧,𝑠)𝑠^−𝜎𝑧𝑑𝑠=^𝛼−1Γ(𝛼)1−𝛼−1𝑧𝛼−𝜎^1−𝜎𝐵(1−𝜎,𝛼−1). (37) Using the same computation as above, we can show that, for all 𝑖=1,…,𝑚−2, we have ¹₀𝐻𝜉_𝑖,𝑠𝑠^−𝜎𝑑𝑠=𝛼−1Γ(𝛼)𝜉_𝑖^𝛼−21−𝜉_𝑖^1−𝜎𝐵(1−𝜎,𝛼−1). (38) Now, (37) and (38) give us that ¹₀𝐺(𝑧,𝑠)𝑠^−𝜎𝑧𝑑𝑠+^𝛼−1(𝛼−1)(1−𝜉)^𝑚−2_𝑖=1𝛽_𝑖¹₀𝐻𝜉_𝑖,𝑠𝑠^−𝜎=𝑧𝑑𝑠_𝛼−1Γ(𝛼)1−𝜉̃−1−𝜉𝛼−1𝑧𝛼−𝜎^1−𝜎𝐵(1−𝜎,𝛼−1), (39) where 𝜉̃=^𝑚−2_𝑖=1𝛽_𝑖𝜉_𝑖^{𝛼−𝜎−1}≤𝜉. (40) But the maximum of the above function depending in 𝑧 is attained at 𝑧=1. Then 𝜉̃𝜒=1−−1−𝜉𝛼−1𝐵()𝛼−𝜎1−𝜎,𝛼−1Γ(𝛼)=1−𝜉̃−1−𝜉𝛼−1Γ()𝛼−𝜎1−𝜎()Γ().𝛼−1𝛼−𝜎 (41) Now, it follows from (33), (41), and condition (iv) that 𝜓𝑑𝐹()𝑑()𝑑(),𝑥,𝑦,𝐹𝑢,𝑣≤𝜓max𝑥,𝑢,𝑑𝑦,𝑣−𝜑max𝑥,𝑢,𝑑𝑦,𝑣 (42) with 𝜓(𝑡)=𝑡 and 𝜑≡𝛽.
Finally, taking (𝑥₀,𝑦₀)=(𝑢⁻,𝑢⁺)∈𝑃×𝑃, we have from condition (v) that 𝑥₀⪯𝐹(𝑥₀,𝑦₀) and 𝑦₀≽𝐹(𝑦₀,𝑥₀).
Now, from Lemmas 12 and 13, there exists a unique 𝑢^∗∈𝑃 such that 𝑢^∗=𝐹(𝑢^∗,𝑢^∗); that is, 𝑢^∗ is the unique positive solution to (2)-(3). The convergence of the sequences {𝑢_𝑛} and {𝑣_𝑛} to 𝑢^∗ follows immediately from (19). This makes end to the proof.

Example 16. Consider the fractional boundary value problem 𝐷₀^5/2₊𝑢(𝑡)+(𝑡−1/2)²2√𝑡𝑢(𝑡)+1𝑢(𝑡)𝑢(0)+1=0,0<𝑡<1,=𝑢^′(0)=0,2𝑢^′(1)=𝑢^′14. (43)(44) In this case, we have 52<𝛼=2<3,𝛽₁=12,𝜉₁=14,1𝜉=4,̃1𝜉=8,𝑓=()𝑡,𝑥,𝑦𝑡−1/2²2√𝑡1𝑥+,(]𝑦+1∀𝑡∈0,1,𝑥,𝑦≥0. (45) Note that 𝑓 is continuous on (0,1]×[0,+∞)×[0,+∞) and lim_𝑡→0⁺𝑓(𝑡,⋅,⋅)=+∞. Let 𝜎=𝜆=1/2 and 𝜂(𝑡)=(1/2)𝑡. For all 𝑥,𝑦,𝑧,𝑤∈[0,+∞) with 𝑥≥𝑧, 𝑦≤𝑤 and 𝑡∈[0,1], we have 0≤𝑡^1/2𝑓()=()𝑡,𝑥,𝑦−𝑓𝑡,𝑧,𝑤𝑡−1/2²2(𝑥−𝑧)+𝑤−𝑦)≤1𝑦+1(𝑤+1𝑡−2²≤1max𝑥−𝑧,𝑤−𝑦4.max𝑥−𝑧,𝑤−𝑦=𝜆𝜂max𝑥−𝑧,𝑤−𝑦 (46) On the other hand, we have 10<𝜆=2<1−𝛽₁𝜉₁^{𝛼−𝜎−1}1−𝛽₁𝜉₁^𝛼−2−𝛼−1𝛼−𝜎Γ(1−𝜎)(𝛼−1)Γ(𝛼−𝜎)⁻¹=185√𝜋. (47)
Now, since 𝐺, 𝐻, and 𝑓 are nonnegative continuous functions, it is easy to prove that 𝑢⁻≡0 is a lower solution to (43)-(44). Moreover, for all 𝑐>0, we use (37), (38), and (41) to obtain ¹₀𝐺()𝑓()+𝑡𝑡,𝑠𝑠,𝑐,0𝑑𝑠_𝛼−1()𝛽𝛼−1)(1−𝜉¹¹₀𝐻𝜉₁𝑓()≤,𝑠𝑠,𝑐,0𝑑𝑠𝑐+18¹₀𝐺(𝑡,𝑠)𝑠^−𝜎+𝑡𝑑𝑠_𝛼−1()𝛽𝛼−1)(1−𝜉¹¹₀𝐻𝜉_𝑖,𝑠𝑠^−𝜎≤𝑑𝑠𝑐+181−𝜉̃−1−𝜉𝛼−1Γ()𝛼−𝜎1−𝜎()Γ()=5√𝛼−1𝛼−𝜎𝜋().144𝑐+1 (48) Then, an upper solution 𝑢⁺ to (43)-(44) will be any constant 𝑐>0 satisfying 5√𝜋()144𝑐+1≤𝑐. (49) Finally, from Theorem 15, Problems (43) and (44) have a unique positive solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.

Acknowledgments

This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.