Positive Solution to a Fractional Boundary Value Problem

Abstract

A fractional boundary value problem is considered. By means of Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green function, and Guo-Krasnosel'skii fixed point theorem on cone, some results on the existence, uniqueness, and positivity of solutions are obtained.

1. Introduction

Fractional differential equations are a natural generalization of ordinary differential equations. In the last few decades many authors pointed out that differential equations of fractional order are suitable for the metallization of various physical phenomena and that they have numerous applications in viscoelasticity, electrochemistry, control and electromagnetic, and so forth, see [1–4].

This work is devoted to the study of the following fractional boundary value problem (P1):^𝑐𝐷^𝑞0₊𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),^𝑐𝐷^𝜎0₊𝑢(𝑡),0<𝑡<1,(1.1)𝑢(0)=𝑢^(0)=0,𝑢^(1)=𝛼𝑢^(1),(1.2) where 𝑓∶[0,1]×ℝ×ℝ→ℝ is a given function, 2<𝑞<3, 1<𝜎<2 and ^𝑐𝐷^𝑞0₊ denotes the Caputo's fractional derivative. Our results allow the function 𝑓 to depend on the fractional derivative ^𝑐𝐷^𝜎0₊𝑢(𝑡) which leads to extra difficulties. No contributions exist, as far as we know, concerning the existence of positive solutions of the fractional differential equation (1.1) jointly with the nonlocal condition (1.2).

Our mean objective is to investigate the existence, uniqueness, and existence of positive solutions for the fractional boundary value problem (P1), by using Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cone.

The research in this area has grown significantly and many papers appeared on this subject, using techniques of nonlinear analysis, see [5–14].

In [6], El-Shahed considered the following nonlinear fractional boundary value problem𝐷^𝑞0₊𝑢(𝑡)𝑢+(0𝜆)𝑎=(𝑡)𝑢𝑓_(𝑢(𝑡))=0,0<𝑡<1,(0)=𝑢^(1)=0,(1.3) where 2<𝑞≤3 and 𝐷^𝑞0₊ is the Riemann-Liouville fractional derivative. Using the Krasnoselskii's fixed-point theorem on cone, he proved the existence and nonexistence of positive solutions for the above fractional boundary value problem.

Liang and Zhang in [9] studied the existence and uniqueness of positive solutions by the properties of the Green function, the lower and upper solution method and fixed point theorem for the fractional boundary value problem 𝑢(0𝐷)=^𝑞0₊𝑢𝑢(_𝑡)+𝑓(𝑡,𝑢(𝑡))=0,0<𝑡<1,(0)=0,𝑢^(1)=^𝑚−2_𝑖=1𝛽_𝑖𝑢^𝜁_𝑖,(1.4) where 2<𝑞≤3 and 𝐷^𝑞0₊ is the Riemann-Liouville fractional derivative.

In [5] Bai and Lü investigated the existence and multiplicity of positive solutions for nonlinear fractional differential equation boundary value problem:𝐷^𝑞0₊𝑢(𝑡)+𝑢𝑓(0(𝑡),𝑢=(𝑡𝑢))(1=)=0,0,0<𝑡<1,(1.5) where 1<𝑞≤1 and 𝐷^𝑞0₊ is the Riemann-Liouville fractional derivative. Applying fixed-point theorems on cone, they prove some existence and multiplicity results of positive solutions.

This paper is organized as follows, in the Section 2 we cite some definitions and lemmas needed in our proofs. Section 3 treats the existence and uniqueness of solution by using Banach contraction principle, Leray Schauder nonlinear alternative. Section 4 is devoted to prove the existence of positive solutions with the help of Guo-Krasnoselskii Theorem, then we give some examples illustrating the previous results.

2. Preliminaries and Lemmas

In this section, we present some lemmas and definitions from fractional calculus theory which will be needed later.

Definition 2.1. If 𝑔∈𝐶([𝑎,𝑏]) and 𝛼>0, then the Riemann-Liouville fractional integral is defined by 𝐼^𝛼𝑎₊𝑔(𝑡)=1Γ(𝛼)^𝑡𝑎𝑔(𝑠)(𝑡−𝑠)^1−𝛼𝑑𝑠.(2.1)

Definition 2.2. Let 𝛼≥0, 𝑛=[𝛼]+1. If 𝑓∈𝐴𝐶^𝑛[𝑎,𝑏] then the Caputo fractional derivative of order 𝛼 of 𝑓 defined by ^𝑐𝐷^𝛼𝑎₊𝑔(𝑡)=1Γ(𝑛−𝛼)^𝑡𝑎𝑔^(𝑛)(𝑠)(𝑡−𝑠)^{𝛼−𝑛+1}𝑑𝑠,(2.2) exists almost everywhere on [𝑎,𝑏] ([𝛼] is the entire part of 𝛼).

Lemma 2.3 (see [15]). Let 𝛼,𝛽>0 and 𝑛=[𝛼]+1, then the following relations hold: ^𝑐𝐷^𝛼0₊𝑡^𝛽−1=(Γ(𝛽)/Γ(𝛽−𝛼))𝑡^𝛽−1, 𝛽>𝑛 and ^𝑐𝐷^𝛼0₊𝑡^𝑘=0, 𝑘=0,1,2,…,𝑛−1.

Lemma 2.4 (see [15]). For 𝛼>0, 𝑔(𝑡)∈𝐶(0,1), the homogenous fractional differential equation ^𝑐𝐷^𝛼𝑎₊𝑔(𝑡)=0(2.3) has a solution 𝑔(𝑡)=𝑐₁+𝑐₂𝑡+𝑐₃𝑡²+⋯+𝑐_𝑛𝑡^𝑛−1,(2.4) where, 𝑐_𝑖∈𝑅, 𝑖=0,…,𝑛, and 𝑛=[𝛼]+1.

Denote by  𝐿¹([0,1],ℝ) the Banach space of Lebesgue integrable functions from [0,1] into ℝ with the norm ||𝑦||_𝐿¹=∫¹⁰|𝑦(𝑡)|𝑑𝑡.

The following Lemmas gives some properties of Riemann-Liouville fractional integrals and Caputo fractional derivative.

Lemma 2.5 (see [16]). Let 𝑝,𝑞≥0, 𝑓∈𝐿₁[𝑎,𝑏]. Then 𝐼^𝑝0₊𝐼^𝑞0₊𝑓(𝑡)=𝐼^0𝑝+𝑞₊𝑓(𝑡)=𝐼^𝑞0₊𝐼^𝑝0₊𝑓(𝑡) and ^𝑐𝐷^𝑞0₊𝐼^𝑞0₊𝑓(𝑡)=𝑓(𝑡), for all 𝑡∈𝑎,𝑏].

Lemma 2.6 (see [15]). Let 𝛽>𝛼>0. Then the formula ^𝑐𝐷^𝛼0₊𝐼^𝛽0₊𝑓(𝑡)=𝐼^0𝛽−𝛼₊𝑓(𝑡), holds almost everywhere on 𝑡∈𝑎,𝑏], for 𝑓∈𝐿₁[𝑎,𝑏] and it is valid at any point 𝑥∈[𝑎,𝑏] if 𝑓∈𝐶[𝑎,𝑏].

Now we start by solving an auxiliary problem.

Lemma 2.7. Let 2<𝑞<3, 1<𝜎<2 and 𝑦∈𝐶[0,1]. The unique solution of the fractional boundary value problem 𝑢(0)=𝐷𝑢^𝑐𝑞0₊𝑢_(𝑡)=𝑦(𝑡),0<𝑡<1,(0)=0,𝑢^(1)=𝛼𝑢^(1)(2.5) is given by 𝑢(𝑡)=1Γ(𝑞−2)¹⁰𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠,(2.6) where 𝐺(𝑡,𝑠)=⎧⎪⎪⎨⎪⎪⎩(𝑡−𝑠)^𝑞−1𝑞(𝑞−1)+(1−𝛼𝑠𝑡)^3−𝑞−𝑡(1−𝑠)^𝑞−2𝛼𝑡𝑞−1,𝑠≤𝑡(1−𝑠)^3−𝑞−𝑡(1−𝑠)^𝑞−2𝑞−1,𝑡≤𝑠.(2.7)

Proof. Applying Lemmas 2.4 and 2.5 to (2.5) we get 𝑢(𝑡)=𝐼^𝑞0₊𝑦(𝑡)+𝑐₁+𝑐₂𝑡+𝑐₃𝑡².(2.8) Differentiating both sides of (2.8) and using Lemma 2.6 it yields 𝑢𝑢^(𝑡)=𝐼^0𝑞−1₊𝑦(𝑡)+𝑐₂+𝑐₃𝑡,_(𝑡)=𝐼^0𝑞−2₊𝑦(𝑡)+𝑐₃.(2.9) The first condition in (2.5) implies 𝑐₁=𝑐₃=0, the second one gives 𝑐₂=𝛼𝐼^0𝑞−2₊𝑦(1)−𝐼^0𝑞−1₊𝑦(1). Substituting 𝑐₂ by its value in (2.8), we obtain 𝑢(𝑡)=𝐼^𝑞0₊𝑦(𝑡)+𝑡𝛼𝐼^0𝑞−2₊𝑦(1)−𝐼^0𝑞−1₊𝑦(1)(2.10) that can be written as 𝑢(𝑡)=1Γ(𝑞−2)^𝑡0(𝑞𝑡(−𝑞𝑠−)1)^𝑞−1+(1−𝛼𝑠𝑡)^3−𝑞−𝑡(1−𝑠)^𝑞−2+1𝑞−1𝑦(𝑠)𝑑𝑠Γ(𝑞−2)^1𝑡(1−𝛼𝑠𝑡)^3−𝑞−𝑡(1−𝑠)^𝑞−2𝑞−1𝑦(𝑠)𝑑𝑠(2.11) that is equivalent to 𝑢(𝑡)=1Γ(𝑞−2)¹⁰𝐺(𝑡,𝑠)𝑦(𝑠)𝑑𝑠,(2.12) where 𝐺 is defined by (2.7). The proof is complete.

3. Existence and Uniqueness Results

In this section we prove the existence and uniqueness of solutions in the Banach space 𝐸 of all functions 𝑢∈𝐶²[0,1] into ℝ, with the norm ||𝑢||=max_𝑡∈[0,1]|𝑢(𝑡)|+max_𝑡∈[0,1]|^𝑐𝐷^𝜎0₊𝑢(𝑡)|. We know that ^𝑐𝐷^𝜎0₊𝑢∈𝐶[0,1], 1<𝜎<2, see [15]. Denote by 𝐸⁺={𝑢∈𝐸,𝑢(𝑡)≥0,𝑡∈[0,1]}. Throughout this section, we suppose that 𝑓∈𝐶([0,1]×ℝ×ℝ,ℝ). Define the integral operator 𝑇∶𝐸→𝐸 by𝑇𝑢(𝑡)=1Γ(𝑞−2)¹⁰𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠)𝑑𝑠.(3.1)

Lemma 3.1. The function 𝑢∈𝐸 is solution of the fractional boundary value problem (P1) if and only if 𝑇𝑢(𝑡)=𝑢(𝑡), for all 𝑡∈[0,1].

Proof. Let 𝑢 be solution of (P1) and 𝑣(𝑡)=∫¹⁰𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠))𝑑𝑠. In view of (2.10) we have 𝑣(𝑡)=+𝐼𝛼^𝑞0₊𝑡𝑓𝐼_0𝑞−2+𝑡,𝑢(𝑡),^𝑐𝐷^𝜎0₊𝑢(𝑡)𝑓1,𝑢(1),^𝑐𝐷^𝜎0₊𝑢(1)−𝑡𝐼^0𝑞−1₊𝑓1,𝑢(1),^𝑐𝐷^𝜎0₊𝑢(1).(3.2) With the help of Lemma 2.6 we obtain ^𝑐𝐷^𝑞0₊𝑣(𝑡)=+^𝑐𝐷𝑡𝛼𝐼^𝑞0₊𝐼^𝑞0_+0𝑞−2+𝑓𝑡,𝑢(𝑡),^𝑐𝐷^𝜎0₊𝑢(𝑡)=𝑓𝑡,𝑢(𝑡𝑓),_𝑐1,𝑢(1),^𝑐𝐷^𝜎0₊𝑢(1)−𝑡𝐼^0𝑞−1₊𝑓1,𝑢(1),^𝑐𝐷^𝜎0₊𝑢(1)𝐷^𝜎0₊𝑢(𝑡).(3.3) It is clear that 𝑣 satisfies conditions (1.2), then it is a solution for the problem (P1). The proof is complete.

Theorem 3.2. Assume that there exist nonnegative functions 𝑔,ℎ∈𝐿¹([0,1],ℝ₊) such that for all 𝑥,𝑦∈ℝ and 𝑡∈[0,1], one has ||𝑓𝑡,𝑥,𝑥−𝑓𝑡,𝑦,𝑦||≤𝑔(𝑡)||𝑥−𝑦||+ℎ(𝑡)||𝑥−𝑦||,(3.4)𝐶_𝑔+𝐶_ℎ<1,𝐴_𝑔+𝐴_ℎ<(2−𝜎)Γ(2−𝜎),(3.5) where 𝐶𝐶_𝑔=‖‖𝐼^0𝑞−1₊𝑔‖‖_𝐿1+|𝛼|𝐼^0𝑞−2₊𝑔(1)+𝐼^0𝑞−1₊𝑔(1),𝐴_𝑔=2𝐼^0𝑞−1₊𝑔(1)+|𝛼|𝐼^0𝑞−2₊𝑔(1),_ℎ=‖‖𝐼^0𝑞−1₊ℎ‖‖_𝐿1+|𝛼|𝐼^0𝑞−2₊ℎ(1)+𝐼^0𝑞−1₊ℎ(1),𝐴_ℎ=2𝐼^0𝑞−1₊ℎ(1)+|𝛼|𝐼^0𝑞−2₊ℎ(1).(3.6) Then the fractional boundary value problem (P1) has a unique solution 𝑢 in 𝐸.

To prove Theorem 3.2, we use the following property of Riemann-Liouville fractional integrals.

Lemma 3.3. Let 𝑞>0, 𝑓∈𝐿₁([𝑎,𝑏],ℝ₊). Then, for all 𝑡∈𝑎,𝑏] we have 𝐼^0𝑞+1₊𝑓(𝑡)≤‖‖𝐼^𝑞0₊𝑓‖‖_𝐿1.(3.7)

Proof. Let 𝑓∈𝐿₁([𝑎,𝑏],ℝ₊), then ‖‖𝐼^𝑞0₊𝑓‖‖_𝐿1=¹⁰𝐼^𝑞0₊𝑓(𝑟)𝑑𝑟≥1Γ(𝑞)^𝑡𝑎^𝑟𝑎𝑓(𝑠)(𝑟−𝑠)^1−𝑞=1𝑑𝑠𝑑𝑟Γ(𝑞)^𝑡𝑎^𝑡𝑠𝑓(𝑠)(𝑟−𝑠)^1−𝑞𝑑𝑟𝑑𝑠=𝐼^0𝑞+1₊𝑓(𝑡).(3.8)

Now we prove Theorem 3.2.

Proof. We transform the fractional boundary value problem to a fixed point problem. By Lemma 3.1, the fractional boundary value problem (P1) has a solution if and only if the operator 𝑇 has a fixed point in 𝐸. Now we will prove that 𝑇 is a contraction. Let 𝑢,𝑣∈𝐸, in view of (2.10) we get 𝑇𝑢(𝑡)−𝑇𝑣(𝑡)=1Γ(𝑞−2)¹⁰=𝐼_𝑞0+𝐺(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠)−𝑓𝑠,𝑣(𝑠),^𝑐𝐷^𝜎0₊𝑣(𝑠)𝑑𝑠+𝑡𝛼𝑓𝐼_0𝑞−2+𝑡,𝑢(𝑡),^𝑐𝐷^𝜎0₊𝑢(𝑡)−𝑓𝑠,𝑣(𝑡),^𝑐𝐷^𝜎0₊𝑣(𝑡)−𝑡𝐼_0𝑞−1+𝑓1,𝑢(1),^𝑐𝐷^𝜎0₊𝑢(1)−𝑓1,𝑣(1),^𝑐𝐷^𝜎0₊𝑣(1)𝑓1,𝑢(1),^𝑐𝐷^𝜎0₊𝑢(1)−𝑓1,𝑣(1),^𝑐𝐷^𝜎0₊𝑣(1),(3.9) with the help of (3.4) we obtain ||𝑇𝑢(𝑡)−𝑇𝑣(𝑡)||≤+mmaxax||𝑢||(_𝑐𝑡)−𝑣(𝑡)||𝐼^𝑞0₊𝑔(𝑡)+|𝛼|𝐼^0𝑞−2₊𝑔(1)+𝐼^0𝑞−1₊𝑔(1)𝐷^𝜎0₊𝑢(𝑡)−^𝑐𝐷^𝜎0₊𝑣(𝑡)||𝐼^𝑞0₊ℎ(𝑡)+|𝛼|𝐼^0𝑞−2₊ℎ(1)+𝐼^0𝑞−1₊ℎ(1).(3.10) Lemma 3.3 implies ||𝑇𝑢(𝑡)−𝑇𝑣(𝑡)||≤‖𝑢−𝑣+‖‖‖𝐼‖‖𝐼^0𝑞−1₊𝑔‖‖_𝐿1+|𝛼|𝐼^0𝑞−2₊𝑔(1)+𝐼^0𝑞−1₊𝑔(1)_0𝑞−1+ℎ‖‖_𝐿1+|𝛼|𝐼^0𝑞−2₊ℎ(1)+𝐼^0𝑞−1₊ℎ(1)=‖𝑢−𝑣‖𝐶_𝑔+𝐶_ℎ.(3.11) In view of (3.5) it yields ||𝑇𝑢−𝑇𝑣||<‖𝑢−𝑣‖.(3.12) On the other hand we have ^𝑐𝐷^𝜎0₊𝑇𝑢−^𝑐𝐷^𝜎0₊𝑇𝑣=1Γ(2−𝜎)^𝑡0(𝑇𝑢)^(𝑠)−(𝑇𝑣)^(𝑠)(𝑡−𝑠)^𝜎−1𝑑𝑠,(3.13) where (𝑇𝑢)^(𝑡)=1Γ(𝑞−2)¹⁰𝐺𝐺₁(𝑡,𝑠)𝑓𝑠,𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠)𝑑𝑠,₁(𝑡,𝑠)=𝜕𝐺(𝑡,𝑠)𝜕𝑡=⎧⎪⎪⎨⎪⎪⎩(𝑡−𝑠)^𝑞−2𝑞+(1−𝛼𝑠)^3−𝑞−(1−𝑠)^𝑞−2𝛼𝑞−1,0≤𝑠≤𝑡≤1(1−𝑠)^3−𝑞−(1−𝑠)^𝑞−2𝑞−1,0≤𝑡≤𝑠≤1.(3.14)
Therefore ^𝑐𝐷^𝜎0₊𝑇𝑢−^𝑐𝐷^𝜎0₊𝑇𝑣=1Γ(𝑞−2)Γ(2−𝜎)^𝑡0¹⁰×𝑓𝑟,𝑢(𝑟),_𝑐(𝑡−𝑠)^−𝜎+1𝐺₁(𝑠,𝑟)𝐷^𝜎0₊𝑢(𝑟)−𝑓𝑟,𝑣(𝑟),^𝑐𝐷^𝜎0₊𝑣(𝑟)𝑑𝑟𝑑𝑠.(3.15) Applying hypothesis (3.4) we get ||^𝑐𝐷^𝜎0₊𝑇𝑢−^𝑐𝐷^𝜎0₊𝑇𝑣||≤max|𝑢−𝑣|Γ(𝑞−2)Γ(2−𝜎)^𝑡0¹⁰+max||_𝑐(𝑡−𝑠)^−𝜎+1||𝐺₁(𝑠,𝑟)||𝑔(𝑟)𝑑𝑟𝑑𝑠𝐷^𝜎0₊𝑢−^𝑐𝐷^𝜎0₊𝑣||Γ(𝑞−2)Γ(2−𝜎)^𝑡0¹⁰(𝑡−𝑠)^−𝜎+1||𝐺₁(𝑠,𝑟)||ℎ(𝑟)𝑑𝑟𝑑𝑠.(3.16) Let us estimate the term ∫¹⁰(𝜕𝐺(𝑠,𝑟)/𝜕𝑠)𝑔(𝑟)𝑑𝑟. We have ¹⁰||𝐺₁(𝑠,𝑟)||𝑔(𝑟)𝑑𝑟≤Γ(𝑞−2)2(𝑞−2)𝐼^0𝑞−1+𝑔(1)𝑞−1+|𝛼|𝐼^0𝑞−2+=Γ(𝑞−2)2𝐼_0𝑞−1+𝑔(1)𝑔(1)+|𝛼|𝐼^0𝑞−2₊𝑔(1)=Γ(𝑞−2)𝐴_𝑔.(3.17) Consequently (3.16) becomes ||^𝑐𝐷^𝜎0₊𝑇𝑢−^𝑐𝐷^𝜎0₊𝑇𝑣||≤‖𝑢−𝑣‖1(2−𝜎)Γ(2−𝜎)𝐴^𝑔+𝐴_ℎ.(3.18)
With the help of hypothesis (3.5) it yields ||^𝑐𝐷^𝜎0₊𝑇𝑢−^𝑐𝐷^𝜎0₊𝑇𝑣||≤‖𝑢−𝑣‖.(3.19)
Taking into account (3.12)–(3.19) we obtain ‖𝑇𝑢−𝑇𝑣‖<‖𝑢−𝑣‖,(3.20) from here, the contraction principle ensures the uniqueness of solution for the fractional boundary value problem (P1). This finishes the proof.

Now we give an existence result for the fractional boundary value problem (P1).

Theorem 3.4. Assume that 𝑓(𝑡,0,0)≠0 and there exist nonnegative functions 𝑘,ℎ,𝑔∈𝐿¹([0,1],ℝ₊), 𝜙,𝜓∈𝐶(ℝ₊,ℝ^∗+) nondecreasing on ℝ₊ and 𝑟>0, such that ||𝑓𝑡,𝑥,𝑥||≤𝑘(𝑡)𝜓(|𝑥|)+ℎ(𝑡)𝜙||𝑥||+𝑔(𝑡),a.e.(𝑡,𝑥)∈[0,1]×ℝ.(3.21)(𝜓(𝑟)+𝜙(𝑟)+1)𝐶¹Γ(𝑞−2)+(2−𝜎)𝐶Γ²(2−𝜎)<𝑟,(3.22) where 𝐶₁=max{𝐶_𝑘,𝐶_ℎ,𝐶_𝑔}, 𝐶₂=max{𝐴_𝑘,𝐴_ℎ,𝐴_𝑔}, 𝐶_ℎ and 𝐶_𝑔 are defined as in Theorem 3.2 and 𝐶_𝑘=‖‖𝐴𝐼^0𝑞−1₊𝑘‖‖_𝐿1+|𝛼|𝐼^0𝑞−2₊𝑘(1)+𝐼^0𝑞−1₊𝑘(1),_𝑘=2𝐼^0𝑞−1₊𝑘(1)+|𝛼|𝐼^0𝑞−2₊𝑘(1).(3.23) Then the fractional boundary value problem (P1) has at least one nontrivial solution 𝑢^∗∈𝐸.

To prove this Theorem, we apply Leray-Schauder nonlinear alternative.

Lemma 3.5 (see [17]). Let 𝐹 be a Banach space and Ω a bounded open subset of 𝐹, 0∈Ω. 𝑇∶Ω→𝐹 be a completely continuous operator. Then, either there exists 𝑥∈𝜕Ω, 𝜆>1 such that 𝑇(𝑥)=𝜆𝑥, or there exists a fixed point 𝑥^∗∈Ω.

Proof. First let us prove that 𝑇 is completely continuous. It is clear that 𝑇 is continuous since 𝑓 and 𝐺 are continuous. Let 𝐵_𝑟={𝑢∈𝐸,‖𝑢‖≤𝑟} be a bounded subset in 𝐸. We shall prove that 𝑇(𝐵_𝑟) is relatively compact.
(i) For 𝑢∈𝐵_𝑟 and using (3.21) we get ||𝑇𝑢(𝑡)||≤1Γ(𝑞−2)¹⁰||𝐺(𝑡,𝑠)||𝑘(𝑠)𝜓||𝑢(𝑠)||+ℎ(𝑠)𝜙||^𝑐𝐷^𝜎0₊𝑢(𝑠)||+𝑔(𝑠)𝑑𝑠.(3.24) Since 𝜓 and 𝜙 are nondecreasing then (3.24) implies ||𝑇𝑢(𝑡)||≤1Γ(𝑞−2)¹⁰≤1||𝐺(𝑡,𝑠)||[𝑘(𝑠)𝜓(‖𝑢‖)+ℎ(𝑠)𝜙(‖𝑢‖)+𝑔(𝑠)]𝑑𝑠Γ(𝑞−2)¹⁰||𝐺(𝑡,𝑠)||[𝑘(𝑠)𝜓(𝑟)+ℎ(𝑠)𝜙(𝑟)+𝑔(𝑠)]𝑑𝑠,(3.25) using similar techniques as to get (3.12) it yields ||𝑇𝑢(𝑡)||≤1Γ(𝑞−2)𝜓(𝑟)‖‖𝐼^0𝑞−1++𝜙(𝑟)‖‖𝐼_0𝑞−1+𝑘‖‖_𝐿1+|𝛼|𝐼^0𝑞−2₊𝑘(1)+𝐼^0𝑞−1₊𝑘(1)+‖‖𝐼ℎ‖‖_𝐿1+|𝛼|𝐼^0𝑞−2₊ℎ(1)+𝐼^0𝑞−1₊ℎ(1)_0𝑞−1+=1𝑔‖‖_𝐿1+|𝛼|𝐼^0𝑞−2₊𝑔(1)+𝐼^0𝑞−1₊𝑔(1)Γ(𝑞−2)𝐶^𝑘𝜓(𝑟)+𝐶_ℎ𝜙(𝑟)+𝐶_𝑔.(3.26) Hence ||𝑇𝑢(𝑡)||≤𝐶¹Γ(𝑞−2)[𝜓(𝑟)+𝜙(𝑟)+1].(3.27)
Moreover, we have ||(𝑇𝑢(𝑡))^||≤1Γ(𝑞−2)𝜓(𝑟)¹⁰|||𝜕𝐺𝜕𝑡(𝑡,𝑠)|||𝑘(𝑠)𝑑𝑠+𝜙(𝑟)¹⁰|||𝜕𝐺𝜕𝑡(𝑡,𝑠)|||ℎ(𝑠)+¹⁰|||𝜕𝐺𝜕𝑡(𝑡,𝑠)|||𝑔(𝑠)𝑑𝑠.(3.28) Using (3.17) we obtain ||^𝑐𝐷^𝜎0₊𝑇𝑢||≤𝐶²(2−𝜎)Γ(2−𝜎)(𝜓(𝑟)+𝜙(𝑟)+1).(3.29) From (3.27) and (3.29) we get ‖𝑇𝑢‖=(𝜓(𝑟)+𝜙(𝑟)+1)𝐶¹Γ(𝑞−2)+(2−𝜎)𝐶Γ²(2−𝜎),(3.30) then 𝑇(𝐵_𝑟) is uniformly bounded.
(ii) 𝑇(𝐵_𝑟) is equicontinuous. Indeed for all 𝑡₁, 𝑡₂∈[0,1], 𝑡₁<𝑡₂, 𝑢∈𝐵_𝑟, let 𝐶=max(|𝑓(𝑡,𝑢(𝑡),^𝑐𝐷^𝜎0₊𝑢(𝑡))|, 0≤𝑡≤1,‖𝑢‖<𝑟), therefore ||𝑇𝑢𝑡₁−𝑇𝑢𝑡₂||≤𝐶Γ(𝑞−2)^𝑡1₀+||𝐺𝑡₁,𝑠−𝐺𝑡₂,𝑠||𝑑𝑠_𝑡2𝑡1||𝐺𝑡₁,𝑠−𝐺𝑡₂,𝑠||𝑑𝑠+^1𝑡₂||𝐺𝑡₁,𝑠−𝐺𝑡₂,𝑠||𝑑𝑠,(3.31) that implies ||𝑇𝑢𝑡₁−𝑇𝑢𝑡₂||≤𝐶Γ(𝑞−2)^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1+𝑡𝑞(𝑞−1)₂−𝑡₁|𝛼|(1−𝑠)^3−𝑞−(1−𝑠)^𝑞−2𝑞−1𝑑𝑠+^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1+𝑡𝑞(𝑞−1)₂−𝑡₁|𝛼|(1−𝑠)^3−𝑞−(1−𝑠)^𝑞−2+𝑞−1𝑑𝑠_1𝑡2𝑡₁−𝑡₂|𝛼|(1−𝑠)^3−𝑞−(1−𝑠)^𝑞−2𝑞−1𝑑𝑠.(3.32)
Let us consider the function Φ(𝑥)=𝑥^𝑞−1−(𝑞−1)𝑥, we see that Φ is decreasing on [0,1], consequently (𝑡₂−𝑠)^𝑞−1−(𝑡₁−𝑠)^𝑞−1≤(𝑞−1)(𝑡₂−𝑡₁), from which we deduce ||𝑇𝑢𝑡₁−𝑇𝑢𝑡₂||≤𝐶Γ(𝑞−2)𝑡²−𝑡₁^𝑡1₀1𝑞(𝑞−1)+||||(1−𝛼𝑠)^3−𝑞−(1−𝑠)^𝑞−2+𝑞−1||||𝑑𝑠_𝑡2𝑡1𝑡₂−𝑠^𝑞−1𝑞(𝑞−1)+𝑡²−𝑡₁|||||𝛼|(1−𝑠)^3−𝑞−(1−𝑠)^𝑞−2+𝑡𝑞−1||||𝑑𝑠₁−𝑡₂^1𝑡₂|||||𝛼|(1−𝑠)^3−𝑞−(1−𝑠)^𝑞−2𝑞−1||||𝑑𝑠.(3.33) Some computations give ||𝑇𝑢𝑡₁−𝑇𝑢𝑡₂||≤𝐶𝑡²−𝑡₁+Γ(𝑞𝐶−2)𝑞(𝑞1−1)+(𝑞3−|𝛼2|)+𝑞−31Γ(𝑞−2)^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝑞(𝑞−1)𝑑𝑠.(3.34) On the other hand we have ||^𝑐𝐷^𝜎0₊𝑇𝑢𝑡₁−^𝑐𝐷^𝜎0₊𝑇𝑢𝑡₂||≤+1Γ(2−𝜎)^𝑡1₀𝑡¹+1−𝑠^−𝜎+1−𝑡₂−𝑠^−𝜎+1||(𝑇𝑢(𝑠))^||𝑑𝑠Γ(2−𝜎)^𝑡2_𝑡1𝑡₂−𝑠^−𝜎+1||(𝑇𝑢(𝑠))^||𝑑𝑠.(3.35) Using (3.17) and (3.28) it yields ||(𝑇𝑢(𝑡))^||≤[𝜓(𝑟)+𝜙(𝑟)+1]𝐶₂,(3.36) then ||^𝑐𝐷^𝜎0₊𝑇𝑢𝑡₁−^𝑐𝐷^𝜎0₊𝑇𝑢𝑡₂||≤[𝜓(𝑟)+𝜙(𝑟)+1]𝐶²(2−𝜎)Γ(2−𝜎)2𝑡²−𝑡₁^2−𝜎+𝑡^22−𝜎−𝑡_12−𝜎,(3.37) when 𝑡₁→𝑡₂, in (3.34) and (3.37) then |𝑇𝑢(𝑡₁)−𝑇𝑢(𝑡₂)| and |^𝑐𝐷^𝜎0₊𝑇𝑢(𝑡₁)−^𝑐𝐷^𝜎0₊𝑇𝑢(𝑡₂)| tend to 0. Consequently 𝑇(𝐵_𝑟) is equicontinuous. From Arzelá-Ascoli Theorem we deduce that 𝑇 is completely continuous operator.
Now we apply Leray Schauder nonlinear alternative to prove that 𝑇 has at least one nontrivial solution in 𝐸.
Letting Ω={𝑢∈𝐸∶‖𝑢‖<𝑟}, for any 𝑢∈𝜕Ω, such that 𝑢=𝜆𝑇𝑢, 0<𝜆<1, we get, with the help of (3.27), ||𝑢(𝑡)||=𝜆||𝑇𝑢(𝑡)||≤||𝑇𝑢(𝑡)||≤𝐶¹Γ(𝑞−2)[𝜓(𝑟)+𝜙(𝑟)+1].(3.38) Taking into account (3.29) we obtain ||^𝑐𝐷^𝜎0₊𝑢(𝑡)||≤𝐶²(2−𝜎)Γ(2−𝜎)(𝜓(𝑟)+𝜙(𝑟)+1).(3.39) From (3.38), (3.39), and (3.22) we deduce that ‖𝑢‖≤(𝜓(𝑟)+𝜙(𝑟)+1)𝐶¹Γ(𝑞−2)+(2−𝜎)𝐶Γ²(2−𝜎)<𝑟,(3.40) this contradicts the fact that 𝑢∈𝜕Ω. Lemma 3.5 allows us to conclude that the operator 𝑇 has a fixed point 𝑢^∗∈Ω and then the fractional boundary value problem (P1) has a nontrivial solution 𝑢^∗∈𝐸. The proof is complete.

4. Existence of Positive Solutions

In this section we investigate the positivity of solution for the fractional boundary value problem (P1), for this we make the following hypotheses.(H1)𝑓(𝑡,𝑢,𝑣)=𝑎(𝑡)𝑓₁(𝑢,𝑣) where 𝑎∈𝐶((0,1),(0,∞)) and 𝑓₁∈𝐶(ℝ₊×ℝ,ℝ₊).(H2)0<∫¹⁰𝐺(𝑠,𝑠)𝑎(𝑠)𝑑𝑠<∞.

Now we give the properties of the Green function.

Lemma 4.1. Let 𝐺(𝑡,𝑠) be the function defined by (2.7). If 𝛼≥1 then 𝐺(𝑡,𝑠) has the following properties: (i)𝐺(𝑡,𝑠)∈𝐶([0,1]×[0,1]), 𝐺(𝑡,𝑠)>0, for all 𝑡, 𝑠∈]0,1[.(ii)If 𝑡,𝑠∈(𝜏,1), 𝜏>0, then 0<𝜏𝐺(𝑠,𝑠)≤𝐺(𝑡,𝑠)≤2𝜏𝐺(𝑠,𝑠).(4.1)

Proof. (i) It is obvious that 𝐺(𝑡,𝑠)∈𝐶([0,1]×[0,1]), moreover, we have 𝛼𝑡(1−𝑠)^3−𝑞−𝑡(1−𝑠)^𝑞−2𝑞−1=𝑞−1(1𝑡−𝑠)^3−𝑞≥𝑡[(𝑞−1)𝛼−(1−𝑠)]𝑞−1(1−𝑠)^3−𝑞[𝛼−1+𝑠],(4.2) which is positive if 𝛼≥1. Hence 𝐺(𝑡,𝑠) is nonnegative for all 𝑡,𝑠∈]0,1[.
(ii) Let 𝑡,𝑠∈(𝜏,1), it is easy to see that 𝐺(𝑠,𝑠)≠0, then we have 𝐺(𝑡,𝑠)𝐺(𝑠,𝑠)=(𝑡−𝑠)^𝑞−1(1−𝑠)^3−𝑞≤𝑞1𝑠+[((𝑞1−−1𝑠))𝛼₂−(1−𝑠)]+𝑠𝑡𝐺(𝑡,𝑠)𝑠≤𝜏2,0<𝜏≤𝑠≤𝑡<1,𝐺(𝑠,𝑠)=𝑠𝑡≤𝜏2,0<𝜏≤𝑡≤𝑠≤1.(4.3) Now we look for lower bounds of 𝐺(𝑡,𝑠)𝐺(𝑡,𝑠)𝐺(𝑠,𝑠)≥𝑠𝑡≥𝜏𝑠≥𝜏,0<𝜏≤𝑠≤𝑡<1,0<𝜏≤𝑡≤𝑠≤1.(4.4) Finally, since 𝐺(𝑠,𝑠) is nonnegative we obtain 0<𝜏𝐺(𝑠,𝑠)≤𝐺(𝑡,𝑠)≤(2/𝜏)𝐺(𝑠,𝑠).

We recall the definition of positive of solution.

Definition 4.2. A function 𝑢(𝑡) is called positive solution of the fractional boundary value problem (P1) if 𝑢(𝑡)≥0, for all 𝑡∈[0,1].

Lemma 4.3. If 𝑢∈𝐸⁺ and 𝛼≥1, then the solution of the fractional boundary value problem (P1) is positive and satisfies min_{𝑡∈(𝜏,1)}𝑢(𝑡)+^𝑐𝐷^𝜎0₊𝑢(𝑡)≥𝜏²2‖𝑢‖.(4.5)

Proof. First let us remark that under the assumptions on 𝑢 and 𝑓, the function ^𝑐𝐷^𝜎0₊𝑢 is nonnegative. From Lemma 3.1 we have 𝑢(𝑡)=1Γ(𝑞−2)¹⁰𝐺(𝑡,𝑠)𝑎(𝑠)𝑓₁𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠)𝑑𝑠.(4.6) Applying the right-hand side of inequality (4.1) we get 𝑢(𝑡)≤2𝜏Γ(𝑞−2)¹⁰𝐺(𝑠,𝑠)𝑎(𝑠)𝑓₁𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠)𝑑𝑠.(4.7) Moreover, (4.1) gives ^𝑐𝐷^𝜎0₊𝑢(𝑡)=1Γ(𝑞−2)Γ(2−𝜎)^𝑡0¹⁰×𝑎(𝑟)𝑓₁(𝑡−𝑠)^−𝜎+1𝐺₁(𝑠,𝑟)≤𝑢(𝑟1),^𝑐𝐷^𝜎0₊𝑢(𝑟)𝑑𝑠𝑑𝑟𝜏(2−𝜎)Γ(𝑞−2)Γ(2−𝜎)¹⁰𝐺₁(𝑟,𝑟)𝑎(𝑟)𝑓₁𝑢(𝑟),^𝑐𝐷^𝜎0₊𝑢(𝑟)𝑑𝑟.(4.8) Combining (4.7) and (4.8) yields ‖𝑢‖≤2𝜏Γ(𝑞−2)¹⁰𝐺(𝑠,𝑠)+𝐺¹(𝑠,𝑠)(2−𝜎)Γ(2−𝜎)𝑎(𝑠)𝑓¹𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠)𝑑𝑠,(4.9) hence ¹⁰𝐺(𝑠,𝑠)+𝐺¹(𝑠,𝑠)(2−𝜎)Γ(2−𝜎)𝑎(𝑠)𝑓¹𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠)𝑑𝑠≥𝜏Γ(𝑞−2)2‖𝑢‖.(4.10) In view of the-left hand side of (4.1), we obtain for all 𝑡∈(𝜏,1)𝑢(𝑡)≥𝜏Γ(𝑞−2)¹⁰𝐺(𝑠,𝑠)𝑎(𝑠)𝑓₁𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠)𝑑𝑠,(4.11) on the other hand we have ^𝑐𝐷^𝜎0₊𝑢(𝑡)≥𝜏^2−𝜎(2−𝜎)Γ(𝑞−2)Γ(2−𝜎)¹⁰𝐺₁(𝑟,𝑟)𝑎(𝑟)𝑓₁𝑢(𝑟),^𝑐𝐷^𝜎0₊𝑢(𝑟)𝑑𝑟.(4.12)
From (4.11) and (4.12) we get min_{𝑡∈(𝜏,1)}𝑢(𝑡)+^𝑐𝐷^𝜎0₊𝑢(𝑡)≥𝜏Γ(𝑞−2)¹⁰𝐺(𝑠,𝑠)+𝐺¹(𝑠,𝑠)×𝑎(𝑠)𝑓₁(2−𝜎)Γ(2−𝜎)𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠)𝑑𝑠,(4.13)
with the help of (4.10) we deduce min_{𝑡∈(𝜏,1)}𝑢(𝑡)+^𝑐𝐷^𝜎0₊𝑢(𝑡)≥𝜏²2‖𝑢‖.(4.14)
The proof is complete.

Define the quantities 𝐴₀ and 𝐴_∞ by 𝐴₀=lim_{(|𝑢|+|𝑣|)→0}𝑓₁(𝑢,𝑣)|𝑢|+|𝑣|,𝐴^∞=lim_{(|𝑢|+|𝑣|)→∞}𝑓₁(𝑢,𝑣)|𝑢|+|𝑣|.(4.15) The case 𝐴₀=0 and 𝐴_∞=∞ is called superlinear case and the case 𝐴₀=∞ and 𝐴_∞=0 is called sublinear case.

The main result of this section is as follows.

Theorem 4.4. Under the assumption of Lemma 4.3, the fractional boundary value problem (P1) has at least one positive solution in the both cases superlinear as well as sublinear.

To prove Theorem 4.4 we apply the well-known Guo-Krasnosel'skii fixed point theorem on cone.

Theorem 4.5 (see [18]). Let 𝐸 be a Banach space, and let 𝐾⊂𝐸, be a cone. Assume Ω₁ and Ω₂ are open subsets of 𝐸 with 0∈Ω₁, Ω₁⊂Ω₂ and let 𝒜∶𝐾∩Ω₂⧵Ω₁⟶𝐾(4.16) be a completely continuous operator such that (i)||𝒜𝑢||≤||𝑢||, 𝑢∈𝐾∩𝜕Ω₁, and ||𝒜𝑢||≥||𝑢||, 𝑢∈𝐾∩𝜕Ω₂, or(ii)||𝒜𝑢||≥||𝑢||, 𝑢∈𝐾∩𝜕Ω₁, and ||𝒜𝑢||≤||𝑢||, 𝑢∈𝐾∩𝜕Ω₂.Then 𝒜 has a fixed point in 𝐾∩(Ω₂⧵Ω₁).

Proof. To prove Theorem 4.4 we define the cone 𝐾 by 𝐾=𝑢∈𝐸⁺,min_{𝑡∈(𝜏,1)}𝑢(𝑡)+^𝑐𝐷^𝜎0₊𝑢(𝑡)≥𝜏²2‖𝑢‖.(4.17) It is easy to check that 𝐾 is a nonempty closed and convex subset of 𝐸, hence it is a cone. Using Lemma 4.3 we see that 𝑇𝐾⊂𝐾. From the prove of Theorem 3.4, we know that 𝑇 is completely continuous in 𝐸.
Let us prove the superlinear case. First, since 𝐴₀=0, for any 𝜀>0, there exists 𝑅₁>0, such that 𝑓₁(𝑢,𝑣)≤𝜀(|𝑢|+|𝑣|)(4.18) for 0<|𝑢|+|𝑣|≤𝑅₁. Letting Ω₁={𝑢∈𝐸,‖𝑢‖<𝑅₁}, for any 𝑢∈𝐾∩𝜕Ω₁, it yields 𝑇𝑢(𝑡)=1Γ(𝑞−2)¹⁰≤2𝜀‖𝑢‖𝐺(𝑡,𝑠)𝑎(𝑠)𝑓₁𝑢(𝑠),^𝑐𝐷^𝜎0₊𝑢(𝑠)𝑑𝑠𝜏Γ(𝑞−2)¹⁰𝐺(𝑠,𝑠)𝑎(𝑠)𝑑𝑠.(4.19) Moreover, we have ^𝑐𝐷^𝜎0₊𝑇𝑢(𝑡)=1Γ(𝑞−2)Γ(2−𝜎)^𝑡0¹⁰×𝑎(𝑟)𝑓₁(𝑡−𝑠)^−𝜎+1𝐺₁(𝑠,𝑟)≤𝑢(𝑟1),^𝑐𝐷^𝜎0₊𝑢(𝑟)𝑑𝑠𝑑𝑟𝜏(2−𝜎)Γ(𝑞−2)Γ(2−𝜎)¹⁰≤𝜀‖𝑢‖𝐺₁(𝑟,𝑟)𝑎(𝑟)||𝑢(𝑟)||+||^𝑐𝐷^𝜎0₊𝑢(𝑟)||𝑑𝑟𝜏(2−𝜎)Γ(𝑞−2)Γ(2−𝜎)¹⁰𝐺₁(𝑠,𝑠)𝑎(𝑠)𝑑𝑠.(4.20) From (4.19) and (4.20) we conclude ‖𝑇𝑢‖≤2𝜀‖𝑢‖𝜏Γ(𝑞−2)¹⁰𝐺(𝑠,𝑠)+𝐺¹(𝑠,𝑠)(2−𝜎)Γ(2−𝜎)𝑎(𝑠)𝑑𝑠.(4.21)
In view of hypothesis (H2), one can choose 𝜀 such that 𝜀≤𝜏Γ(𝑞−2)2∫¹⁰𝐺(𝑠,𝑠)+𝐺₁(𝑠,𝑠)/(2−𝜎)Γ(2−𝜎)𝑎(𝑠)𝑑𝑠.(4.22) The inequalities (4.21) and (4.22) imply that ||𝑇𝑢||≤||𝑢||, for all 𝑢∈𝐾∩𝜕Ω₁. Second, in view of 𝐴_∞=∞, then for any 𝑀>0, there exists 𝑅₂>0, such that 𝑓₁(𝑢,𝑣)≥𝑀(|𝑢|+|𝑣|) for |𝑢|+|𝑣|≥𝑅₂. Let 𝑅=max{2𝑅₁,(2𝑅₂/𝜏²)} and denote by Ω₂ the open set {𝑢∈𝐸/||𝑢||<𝑅}. If 𝑢∈𝐾∩𝜕Ω₂ then min_{𝑡∈(𝜏,1)}𝑢(𝑡)+^𝑐𝐷^𝜎0₊𝑢(𝑡)≥𝜏²2‖𝑢‖=𝜏2²𝑅≥𝑅².(4.23) Using the left-hand side of (4.1) and Lemma 4.3, we obtain 𝑇𝑢(𝑡)≥𝜏𝑀Γ(𝑞−2)¹⁰≥𝜏𝐺(𝑠,𝑠)𝑎(𝑠)||𝑢(𝑠)||+||^𝑐𝐷^𝜎0₊𝑢(𝑠)||𝑑𝑠₃𝑀|||𝑢|||2Γ(𝑞−2)¹⁰𝐺(𝑠,𝑠)𝑎(𝑠)𝑑𝑠.(4.24) Moreover, we get with the help of (4.12) ^𝑐𝐷^𝜎0₊𝑇𝑢(𝑡)≥𝜏^4−𝜎𝑀|||𝑢|||2(2−𝜎)Γ(𝑞−2)Γ(2−𝜎)¹⁰𝐺₁(𝑠,𝑠)𝑎(𝑠)𝑑𝑠.(4.25) In view of (4.26) and (4.24) we can write 𝑇𝑢(𝑡)+^𝑐𝐷^𝜎0₊𝑇𝑢(𝑡)≥𝜏³𝑀|||𝑢|||2Γ(𝑞−2)¹⁰𝐺(𝑠,𝑠)+𝜏^1−𝜎𝐺₁(𝑠,𝑠)≥𝜏(2−𝜎)Γ(2−𝜎)𝑎(𝑠)𝑑𝑠₃𝑀|||𝑢|||2Γ(𝑞−2)¹⁰𝐺(𝑠,𝑠)+𝐺¹(𝑠,𝑠)(2−𝜎)Γ(2−𝜎)𝑎(𝑠)𝑑𝑠.(4.26) Let us choose 𝑀 such that 𝑀≥2Γ(𝑞−2)𝜏³∫¹⁰𝐺(𝑠,𝑠)+𝐺₁(𝑠,𝑠)/(2−𝜎)Γ(2−𝜎)𝑎(𝑠)𝑑𝑠,(4.27) then we get 𝑇𝑢(𝑡)+^𝑐𝐷^𝜎0₊𝑇𝑢(𝑡)≥||𝑢||. Hence, ||||𝑇𝑢||||≥|||𝑢|||,∀𝑢∈𝐾∩𝜕Ω₂.(4.28) The first part of Theorem 4.5 implies that 𝑇 has a fixed point in 𝐾∩(Ω₂⧵0𝑥00082Ω₁) such that 𝑅₂≤||𝑢||≤𝑅. To prove the sublinear case we apply similar techniques. The proof is complete.

In order to illustrate our results, we give the following examples.

Example 4.6. The fractional boundary value problem ^𝑐𝐷^05/2₊𝑢=𝑡−110³𝑢(0)=𝑢_𝑢+𝑡²𝐷^05/4₊𝑢+ln𝑡,0<𝑡<1,(0)=0,𝑢^(1)=−12𝑢^(1)(4.29) has a unique solution in 𝐸.

Proof. In this case we have 𝑓(𝑡,𝑥,𝑦)=((𝑡−1)/10)³𝑥+𝑡²𝑦+ln𝑡, 2<𝑞=5/2<3, 𝜎=(5/4)<2, 𝛼=−1/2 and ||𝑓𝑡,𝑥,𝑥−𝑓𝑡,𝑦,𝑦||≤1−𝑡10³||𝑥−𝑥||+𝑡²||𝑦−𝑦||,(4.30) then 𝑔(𝑡)=((1−𝑡)/10)³ and ℎ(𝑡)=𝑡². Some calculus give 𝐼_0𝑞−2+‖‖𝐼^0𝑞−1₊𝑔‖‖_𝐿1=0.17846×10⁻³,𝐼^0𝑞−1₊𝑔(1)=0.48001×10⁻³,𝑔(1)=01‖‖6𝐼12_0𝑞−1+0×10⁻³,𝐶_𝑔=7.3907×10⁻⁴,𝐴_𝑔=1.0406×10⁻³,𝐼ℎ‖‖_𝐿1=0.038210,𝐼^0𝑞−1₊ℎ(1)=0.17194,_0𝑞−2+ℎ(1)=0.601𝐶8,𝐶_ℎ=0.49747,𝐴_ℎ=0.42448,_𝑔𝐴+𝐶_ℎ=0.49821<1,_𝑔+𝐴_ℎ=0.42552<(2−𝜎)Γ(2−𝜎)=0.91906.(4.31) Thus Theorem 3.2 implies that fractional boundary value problem (4.29) has a unique in 𝐸.

Example 4.7. The fractional boundary value problem ^𝑐𝐷^07/3₊𝑢=(1−𝑡)²⎛⎜⎜⎜⎜⎝𝑢₄1001+𝑢²+ln1+^𝑐𝐷^06/5₊𝑢²+1𝑢(0)=𝑢_9⎞⎟⎟⎟⎟⎠=0,0<𝑡<1,(0)=0,𝑢^(1)=32𝑢^(1)(4.32) has at least one nontrivial solution in 𝐸.

Proof. We apply Theorem 3.4 to prove that the fractional boundary value problem (4.32) has at least one nontrivial solution. We have 𝑞=7/3, 𝜎=6/5, 𝛼=3/2, and ||𝑓𝑡,𝑥,𝑥||=(1−𝑡)²𝑥⁴1001+𝑥²+(1−𝑡)²ln1+𝑥²9+(1−𝑡)²≤|𝑥|²100(1−𝑡)²+(1−𝑡)²ln1+𝑥²9+(1−𝑡)²≤𝑘(𝑡)𝜓(|𝑥|)+ℎ(𝑡)𝜙||𝑥||+𝑔(𝑡),(4.33) where 𝑘(𝑡)=ℎ(𝑡)=𝑔(𝑡)=(1−𝑡)², 𝜓(𝑥)=(𝑥/10)², 𝜙(𝑥)=ln(1+𝑥²)/9, 𝑓(𝑡,0,0)≠0. Let us find 𝑟 such that (3.22) holds, for this we have ‖‖𝐼^0𝑞−1₊𝑔‖‖𝐶_𝐿1=0.19382,𝐼^0𝑞−2₊𝑔(1)=0.15998,𝐼^0𝑞−1₊𝑔(1)=0.33595,_𝑔=𝐶₁=0.76974,𝐴_𝑔=𝐶₂=0.91187.(4.34) We see that (3.22) is equivalent to 1.2664(((𝑟²)/100)+(ln(1+𝑟²)/9)+1)−𝑟 which is negative for 𝑟=6.