International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article
!A Erratum for this article has been published. To view the article details, please click the β€˜Erratum’ tab above.

Research Article | Open Access

Volume 2011 |Article ID 763456 | 19 pages | https://doi.org/10.1155/2011/763456

Positive Solution to a Fractional Boundary Value Problem

Academic Editor: Eduardo Liz
Received09 May 2011
Accepted26 Jun 2011
Published11 Aug 2011

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):π‘π·π‘ž0+𝑒(𝑑)=𝑓𝑑,𝑒(𝑑),π‘π·πœŽ0+𝑒(𝑑),0<𝑑<1,(1.1)𝑒(0)=π‘’ξ…žξ…ž(0)=0,π‘’ξ…ž(1)=π›Όπ‘’ξ…žξ…ž(1),(1.2) where π‘“βˆΆ[0,1]×ℝ×ℝ→ℝ is a given function, 2<π‘ž<3, 1<𝜎<2 and π‘π·π‘ž0+ denotes the Caputo's fractional derivative. Our results allow the function 𝑓 to depend on the fractional derivative π‘π·πœŽ0+𝑒(𝑑) 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 problemπ·π‘ž0+𝑒(𝑑)+πœ†π‘Ž(𝑑)𝑓(𝑒(𝑑))=0,0<𝑑<1,𝑒(0)=π‘’ξ…ž(0)=π‘’ξ…ž(1)=0,(1.3) where 2<π‘žβ‰€3 and π·π‘ž0+ 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 π·π‘ž0+𝑒𝑒(𝑑)+𝑓(𝑑,𝑒(𝑑))=0,0<𝑑<1,(0)=π‘’ξ…ž(0)=0,π‘’ξ…ž(1)=π‘šβˆ’2𝑖=1π›½π‘–π‘’ξ…žξ€·πœπ‘–ξ€Έ,(1.4) where 2<π‘žβ‰€3 and π·π‘ž0+ is the Riemann-Liouville fractional derivative.

In [5] Bai and LΓΌ investigated the existence and multiplicity of positive solutions for nonlinear fractional differential equation boundary value problem:π·π‘ž0+𝑒(𝑑)+𝑓(𝑑,𝑒(𝑑))=0,0<𝑑<1,𝑒(0)=𝑒(1)=0,(1.5) where 1<π‘žβ‰€1 and π·π‘ž0+ 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 π‘”βˆˆπΆ([π‘Ž,𝑏]) and 𝛼>0, then the Riemann-Liouville fractional integral is defined by πΌπ›Όπ‘Ž+1𝑔(𝑑)=ξ€œΞ“(𝛼)π‘‘π‘Žπ‘”(𝑠)(π‘‘βˆ’π‘ )1βˆ’π›Όπ‘‘π‘ .(2.1)

Definition 2.2. Let 𝛼β‰₯0, 𝑛=[𝛼]+1. If π‘“βˆˆπ΄πΆπ‘›[π‘Ž,𝑏] then the Caputo fractional derivative of order 𝛼 of 𝑓 defined by π‘π·π›Όπ‘Ž+1𝑔(𝑑)=ξ€œΞ“(π‘›βˆ’π›Ό)π‘‘π‘Žπ‘”(𝑛)(𝑠)(π‘‘βˆ’π‘ )π›Όβˆ’π‘›+1𝑑𝑠,(2.2) exists almost everywhere on [π‘Ž,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: 𝑐𝐷𝛼0+π‘‘π›½βˆ’1=(Ξ“(𝛽)/Ξ“(π›½βˆ’π›Ό))π‘‘π›½βˆ’1, 𝛽>𝑛 and 𝑐𝐷𝛼0+π‘‘π‘˜=0, π‘˜=0,1,2,…,π‘›βˆ’1.

Lemma 2.4 (see [15]). For 𝛼>0, 𝑔(𝑑)∈𝐢(0,1), the homogenous fractional differential equation π‘π·π›Όπ‘Ž+𝑔(𝑑)=0(2.3) has a solution 𝑔(𝑑)=𝑐1+𝑐2𝑑+𝑐3𝑑2+β‹―+π‘π‘›π‘‘π‘›βˆ’1,(2.4) where, π‘π‘–βˆˆπ‘…, 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(𝑑)|𝑑𝑑.

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

Lemma 2.5 (see [16]). Let 𝑝,π‘žβ‰₯0, π‘“βˆˆπΏ1[π‘Ž,𝑏]. Then 𝐼𝑝0+πΌπ‘ž0+𝑓(𝑑)=𝐼0𝑝+π‘ž+𝑓(𝑑)=πΌπ‘ž0+𝐼𝑝0+𝑓(𝑑) and π‘π·π‘ž0+πΌπ‘ž0+𝑓(𝑑)=𝑓(𝑑), for all π‘‘βˆˆπ‘Ž,𝑏].

Lemma 2.6 (see [15]). Let 𝛽>𝛼>0. Then the formula 𝑐𝐷𝛼0+𝐼𝛽0+𝑓(𝑑)=𝐼0π›½βˆ’π›Ό+𝑓(𝑑), holds almost everywhere on π‘‘βˆˆπ‘Ž,𝑏], for π‘“βˆˆπΏ1[π‘Ž,𝑏] and it is valid at any point π‘₯∈[π‘Ž,𝑏] if π‘“βˆˆπΆ[π‘Ž,𝑏].

Now we start by solving an auxiliary problem.

Lemma 2.7. Let 2<π‘ž<3, 1<𝜎<2 and π‘¦βˆˆπΆ[0,1]. The unique solution of the fractional boundary value problem π‘π·π‘ž0+𝑒(𝑑)=𝑦(𝑑),0<𝑑<1,𝑒(0)=π‘’ξ…žξ…ž(0)=0,π‘’ξ…ž(1)=π›Όπ‘’ξ…žξ…ž(1)(2.5) is given by 1𝑒(𝑑)=ξ€œΞ“(π‘žβˆ’2)10𝐺(𝑑,𝑠)𝑦(𝑠)𝑑𝑠,(2.6) where ⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩𝐺(𝑑,𝑠)=(π‘‘βˆ’π‘ )π‘žβˆ’1π‘ž+(π‘žβˆ’1)𝛼𝑑(1βˆ’π‘ )3βˆ’π‘žβˆ’π‘‘(1βˆ’π‘ )π‘žβˆ’2π‘žβˆ’1,𝑠≀𝑑𝛼𝑑(1βˆ’π‘ )3βˆ’π‘žβˆ’π‘‘(1βˆ’π‘ )π‘žβˆ’2π‘žβˆ’1,𝑑≀𝑠.(2.7)

Proof. Applying Lemmas 2.4 and 2.5 to (2.5) we get 𝑒(𝑑)=πΌπ‘ž0+𝑦(𝑑)+𝑐1+𝑐2𝑑+𝑐3𝑑2.(2.8) Differentiating both sides of (2.8) and using Lemma 2.6 it yields π‘’ξ…ž(𝑑)=𝐼0π‘žβˆ’1+𝑦(𝑑)+𝑐2+𝑐3𝑒𝑑,ξ…žξ…ž(𝑑)=𝐼0π‘žβˆ’2+𝑦(𝑑)+𝑐3.(2.9) The first condition in (2.5) implies 𝑐1=𝑐3=0, the second one gives 𝑐2=𝛼𝐼0π‘žβˆ’2+𝑦(1)βˆ’πΌ0π‘žβˆ’1+𝑦(1). Substituting 𝑐2 by its value in (2.8), we obtain 𝑒(𝑑)=πΌπ‘ž0+𝑦(𝑑)+𝑑𝛼𝐼0π‘žβˆ’2+𝑦(1)βˆ’πΌ0π‘žβˆ’1+𝑦(1)(2.10) that can be written as 1𝑒(𝑑)=Ξ“ξ€œ(π‘žβˆ’2)𝑑0ξ‚Έ(π‘‘βˆ’π‘ )π‘žβˆ’1π‘ž+(π‘žβˆ’1)𝛼𝑑(1βˆ’π‘ )3βˆ’π‘žβˆ’π‘‘(1βˆ’π‘ )π‘žβˆ’2ξ‚Ή+1π‘žβˆ’1𝑦(𝑠)π‘‘π‘ ξ€œΞ“(π‘žβˆ’2)1𝑑𝛼𝑑(1βˆ’π‘ )3βˆ’π‘žβˆ’π‘‘(1βˆ’π‘ )π‘žβˆ’2ξ‚Ήπ‘žβˆ’1𝑦(𝑠)𝑑𝑠(2.11) that is equivalent to 1𝑒(𝑑)=ξ€œΞ“(π‘žβˆ’2)10𝐺(𝑑,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 π‘’βˆˆπΆ2[0,1] into ℝ, with the norm ||𝑒||=maxπ‘‘βˆˆ[0,1]|𝑒(𝑑)|+maxπ‘‘βˆˆ[0,1]|π‘π·πœŽ0+𝑒(𝑑)|. We know that π‘π·πœŽ0+π‘’βˆˆπΆ[0,1], 1<𝜎<2, see [15]. Denote by 𝐸+={π‘’βˆˆπΈ,𝑒(𝑑)β‰₯0,π‘‘βˆˆ[0,1]}. Throughout this section, we suppose that π‘“βˆˆπΆ([0,1]×ℝ×ℝ,ℝ). Define the integral operator π‘‡βˆΆπΈβ†’πΈ by1𝑇𝑒(𝑑)=ξ€œΞ“(π‘žβˆ’2)10𝐺(𝑑,𝑠)𝑓𝑠,𝑒(𝑠),π‘π·πœŽ0+𝑒(𝑠)𝑑𝑠.(3.1)

Lemma 3.1. The function π‘’βˆˆπΈ is solution of the fractional boundary value problem (P1) if and only if 𝑇𝑒(𝑑)=𝑒(𝑑), for all π‘‘βˆˆ[0,1].

Proof. Let 𝑒 be solution of (P1) and βˆ«π‘£(𝑑)=10𝐺(𝑑,𝑠)𝑓(𝑠,𝑒(𝑠),π‘π·πœŽ0+𝑒(𝑠))𝑑𝑠. In view of (2.10) we have 𝑣(𝑑)=πΌπ‘ž0+𝑓𝑑,𝑒(𝑑),π‘π·πœŽ0+𝑒(𝑑)+𝛼𝑑𝐼0π‘žβˆ’2+𝑓1,𝑒(1),π‘π·πœŽ0+𝑒(1)βˆ’π‘‘πΌ0π‘žβˆ’1+𝑓1,𝑒(1),π‘π·πœŽ0+.𝑒(1)(3.2) With the help of Lemma 2.6 we obtain π‘π·π‘ž0+𝑣(𝑑)=π‘π·π‘ž0+πΌπ‘ž0+𝑓𝑑,𝑒(𝑑),π‘π·πœŽ0+𝑒(𝑑)+𝑑𝛼𝐼0π‘žβˆ’2+𝑓1,𝑒(1),π‘π·πœŽ0+𝑒(1)βˆ’π‘‘πΌ0π‘žβˆ’1+𝑓1,𝑒(1),π‘π·πœŽ0+𝑒(1)=𝑓𝑑,𝑒(𝑑),π‘π·πœŽ0+.𝑒(𝑑)(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,β„ŽβˆˆπΏ1([0,1],ℝ+) such that for all π‘₯,π‘¦βˆˆβ„ and π‘‘βˆˆ[0,1], one has ||𝑓𝑑,π‘₯,π‘₯ξ€Έξ€·βˆ’π‘“π‘‘,𝑦,𝑦||||||||≀𝑔(𝑑)π‘₯βˆ’π‘¦+β„Ž(𝑑)π‘₯βˆ’π‘¦||,(3.4)𝐢𝑔+πΆβ„Ž<1,𝐴𝑔+π΄β„Ž<(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ),(3.5) where 𝐢𝑔=‖‖𝐼0π‘žβˆ’1+𝑔‖‖𝐿1+|𝛼|𝐼0π‘žβˆ’2+𝑔(1)+𝐼0π‘žβˆ’1+𝑔(1),𝐴𝑔=2𝐼0π‘žβˆ’1+𝑔(1)+|𝛼|𝐼0π‘žβˆ’2+𝐢𝑔(1),β„Ž=‖‖𝐼0π‘žβˆ’1+β„Žβ€–β€–πΏ1+|𝛼|𝐼0π‘žβˆ’2+β„Ž(1)+𝐼0π‘žβˆ’1+β„Ž(1),π΄β„Ž=2𝐼0π‘žβˆ’1+β„Ž(1)+|𝛼|𝐼0π‘žβˆ’2+β„Ž(1).(3.6) Then the fractional boundary value problem (P1) has a unique solution 𝑒 in 𝐸.

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

Lemma 3.3. Let π‘ž>0, π‘“βˆˆπΏ1([π‘Ž,𝑏],ℝ+). Then, for all π‘‘βˆˆπ‘Ž,𝑏] we have 𝐼0π‘ž+1+‖‖𝐼𝑓(𝑑)β‰€π‘ž0+𝑓‖‖𝐿1.(3.7)

Proof. Let π‘“βˆˆπΏ1([π‘Ž,𝑏],ℝ+), then β€–β€–πΌπ‘ž0+𝑓‖‖𝐿1=ξ€œ10πΌπ‘ž0+1𝑓(π‘Ÿ)π‘‘π‘Ÿβ‰₯ξ€œΞ“(π‘ž)π‘‘π‘Žξ€œπ‘Ÿπ‘Žπ‘“(𝑠)(π‘Ÿβˆ’π‘ )1βˆ’π‘ž=1π‘‘π‘ π‘‘π‘ŸΞ“ξ€œ(π‘ž)π‘‘π‘Žξ‚΅ξ€œπ‘‘π‘ π‘“(𝑠)(π‘Ÿβˆ’π‘ )1βˆ’π‘žξ‚Άπ‘‘π‘Ÿπ‘‘π‘ =𝐼0π‘ž+1+𝑓(𝑑).(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 𝑒,π‘£βˆˆπΈ, in view of (2.10) we get 1𝑇𝑒(𝑑)βˆ’π‘‡π‘£(𝑑)=ξ€œΞ“(π‘žβˆ’2)10𝑓𝐺(𝑑,𝑠)𝑠,𝑒(𝑠),π‘π·πœŽ0+𝑒(𝑠)βˆ’π‘“π‘ ,𝑣(𝑠),π‘π·πœŽ0+𝑣(𝑠)𝑑𝑠=πΌπ‘ž0+𝑓𝑑,𝑒(𝑑),π‘π·πœŽ0+𝑒(𝑑)βˆ’π‘“π‘ ,𝑣(𝑑),π‘π·πœŽ0+𝐼𝑣(𝑑)+𝑑𝛼0π‘žβˆ’2+𝑓1,𝑒(1),π‘π·πœŽ0+𝑒(1)βˆ’π‘“1,𝑣(1),π‘π·πœŽ0+𝑣(1)ξ‚ξ‚βˆ’π‘‘πΌ0π‘žβˆ’1+𝑓1,𝑒(1),π‘π·πœŽ0+𝑒(1)βˆ’π‘“1,𝑣(1),π‘π·πœŽ0+,𝑣(1)(3.9) with the help of (3.4) we obtain ||||||||𝐼𝑇𝑒(𝑑)βˆ’π‘‡π‘£(𝑑)≀max𝑒(𝑑)βˆ’π‘£(𝑑)π‘ž0+𝑔(𝑑)+|𝛼|𝐼0π‘žβˆ’2+𝑔(1)+𝐼0π‘žβˆ’1+||𝑔(1)+maxπ‘π·πœŽ0+𝑒(𝑑)βˆ’π‘π·πœŽ0+||𝐼𝑣(𝑑)π‘ž0+β„Ž(𝑑)+|𝛼|𝐼0π‘žβˆ’2+β„Ž(1)+𝐼0π‘žβˆ’1+.β„Ž(1)(3.10) Lemma 3.3 implies ||||‖‖𝐼𝑇𝑒(𝑑)βˆ’π‘‡π‘£(𝑑)β‰€β€–π‘’βˆ’π‘£β€–0π‘žβˆ’1+𝑔‖‖𝐿1+|𝛼|𝐼0π‘žβˆ’2+𝑔(1)+𝐼0π‘žβˆ’1++‖‖𝐼𝑔(1)0π‘žβˆ’1+β„Žβ€–β€–πΏ1+|𝛼|𝐼0π‘žβˆ’2+β„Ž(1)+𝐼0π‘žβˆ’1+ξ‚„=ξ€·πΆβ„Ž(1)β€–π‘’βˆ’π‘£β€–π‘”+πΆβ„Žξ€Έ.(3.11) In view of (3.5) it yields ||||π‘‡π‘’βˆ’π‘‡π‘£<β€–π‘’βˆ’π‘£β€–.(3.12) On the other hand we have π‘π·πœŽ0+π‘‡π‘’βˆ’π‘π·πœŽ0+1𝑇𝑣=ξ€œΞ“(2βˆ’πœŽ)𝑑0(𝑇𝑒)ξ…ž(𝑠)βˆ’(𝑇𝑣)ξ…ž(𝑠)(π‘‘βˆ’π‘ )πœŽβˆ’1𝑑𝑠,(3.13) where (𝑇𝑒)ξ…ž1(𝑑)=ξ€œΞ“(π‘žβˆ’2)10𝐺1ξ‚€(𝑑,𝑠)𝑓𝑠,𝑒(𝑠),π‘π·πœŽ0+𝐺𝑒(𝑠)𝑑𝑠,1(𝑑,𝑠)=πœ•πΊ(𝑑,𝑠)=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©πœ•π‘‘(π‘‘βˆ’π‘ )π‘žβˆ’2π‘ž+𝛼(1βˆ’π‘ )3βˆ’π‘žβˆ’(1βˆ’π‘ )π‘žβˆ’2π›Όπ‘žβˆ’1,0≀𝑠≀𝑑≀1(1βˆ’π‘ )3βˆ’π‘žβˆ’(1βˆ’π‘ )π‘žβˆ’2π‘žβˆ’1,0≀𝑑≀𝑠≀1.(3.14)
Therefore π‘π·πœŽ0+π‘‡π‘’βˆ’π‘π·πœŽ0+1𝑇𝑣=ξ€œΞ“(π‘žβˆ’2)Ξ“(2βˆ’πœŽ)𝑑0ξ€œ10(π‘‘βˆ’π‘ )βˆ’πœŽ+1𝐺1×𝑓(𝑠,π‘Ÿ)π‘Ÿ,𝑒(π‘Ÿ),π‘π·πœŽ0+𝑒(π‘Ÿ)βˆ’π‘“π‘Ÿ,𝑣(π‘Ÿ),π‘π·πœŽ0+𝑣(π‘Ÿ)ξ‚ξ‚π‘‘π‘Ÿπ‘‘π‘ .(3.15) Applying hypothesis (3.4) we get ||π‘π·πœŽ0+π‘‡π‘’βˆ’π‘π·πœŽ0+||≀𝑇𝑣max|π‘’βˆ’π‘£|ξ€œΞ“(π‘žβˆ’2)Ξ“(2βˆ’πœŽ)𝑑0ξ€œ10(π‘‘βˆ’π‘ )βˆ’πœŽ+1||𝐺1||+||(𝑠,π‘Ÿ)𝑔(π‘Ÿ)π‘‘π‘Ÿπ‘‘π‘ maxπ‘π·πœŽ0+π‘’βˆ’π‘π·πœŽ0+𝑣||ξ€œΞ“(π‘žβˆ’2)Ξ“(2βˆ’πœŽ)𝑑0ξ€œ10(π‘‘βˆ’π‘ )βˆ’πœŽ+1||𝐺1||(𝑠,π‘Ÿ)β„Ž(π‘Ÿ)π‘‘π‘Ÿπ‘‘π‘ .(3.16) Let us estimate the term ∫10(πœ•πΊ(𝑠,π‘Ÿ)/πœ•π‘ )𝑔(π‘Ÿ)π‘‘π‘Ÿ. We have ξ€œ10||𝐺1||(𝑠,π‘Ÿ)𝑔(π‘Ÿ)π‘‘π‘Ÿβ‰€Ξ“(π‘žβˆ’2)2(π‘žβˆ’2)𝐼0π‘žβˆ’1+𝑔(1)π‘žβˆ’1+|𝛼|𝐼0π‘žβˆ’2+ξƒͺ𝑔(1)=Ξ“(π‘žβˆ’2)2𝐼0π‘žβˆ’1+𝑔(1)+|𝛼|𝐼0π‘žβˆ’2+𝑔(1)=Ξ“(π‘žβˆ’2)𝐴𝑔.(3.17) Consequently (3.16) becomes ||π‘π·πœŽ0+π‘‡π‘’βˆ’π‘π·πœŽ0+||≀1π‘‡π‘£β€–π‘’βˆ’π‘£β€–ξ€·π΄(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)𝑔+π΄β„Žξ€Έ.(3.18)
With the help of hypothesis (3.5) it yields ||π‘π·πœŽ0+π‘‡π‘’βˆ’π‘π·πœŽ0+||π‘‡π‘£β‰€β€–π‘’βˆ’π‘£β€–.(3.19)
Taking into account (3.12)–(3.19) we obtain β€–π‘‡π‘’βˆ’π‘‡π‘£β€–<β€–π‘’βˆ’π‘£β€–,(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(𝑑,0,0)β‰ 0 and there exist nonnegative functions π‘˜,β„Ž,π‘”βˆˆπΏ1([0,1],ℝ+), πœ™,πœ“βˆˆπΆ(ℝ+,β„βˆ—+) nondecreasing on ℝ+ and π‘Ÿ>0, such that ||𝑓𝑑,π‘₯,π‘₯ξ€Έ||ξ€·||β‰€π‘˜(𝑑)πœ“(|π‘₯|)+β„Ž(𝑑)πœ™π‘₯||ξ€Έ+𝑔(𝑑),a.e.[](𝑑,π‘₯)∈0,1×ℝ.(3.21)𝐢(πœ“(π‘Ÿ)+πœ™(π‘Ÿ)+1)1+𝐢Γ(π‘žβˆ’2)2ξ‚Ά(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)<π‘Ÿ,(3.22) where 𝐢1=max{πΆπ‘˜,πΆβ„Ž,𝐢𝑔}, 𝐢2=max{π΄π‘˜,π΄β„Ž,𝐴𝑔}, πΆβ„Ž and 𝐢𝑔 are defined as in Theorem 3.2 and πΆπ‘˜=‖‖𝐼0π‘žβˆ’1+π‘˜β€–β€–πΏ1+|𝛼|𝐼0π‘žβˆ’2+π‘˜(1)+𝐼0π‘žβˆ’1+π΄π‘˜(1),π‘˜=2𝐼0π‘žβˆ’1+π‘˜(1)+|𝛼|𝐼0π‘žβˆ’2+π‘˜(1).(3.23) Then the fractional boundary value problem (P1) has at least one nontrivial solution π‘’βˆ—βˆˆπΈ.

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

Lemma 3.5 (see [17]). Let 𝐹 be a Banach space and Ξ© a bounded open subset of 𝐹, 0∈Ω. π‘‡βˆΆΞ©β†’πΉ be a completely continuous operator. Then, either there exists π‘₯βˆˆπœ•Ξ©, πœ†>1 such that 𝑇(π‘₯)=πœ†π‘₯, or there exists a fixed point π‘₯βˆ—βˆˆΞ©.

Proof. First let us prove that 𝑇 is completely continuous. It is clear that 𝑇 is continuous since 𝑓 and 𝐺 are continuous. Let π΅π‘Ÿ={π‘’βˆˆπΈ,β€–π‘’β€–β‰€π‘Ÿ} be a bounded subset in 𝐸. We shall prove that 𝑇(π΅π‘Ÿ) is relatively compact.
(i) For π‘’βˆˆπ΅π‘Ÿ and using (3.21) we get ||||≀1𝑇𝑒(𝑑)ξ€œΞ“(π‘žβˆ’2)10||||||||ξ€Έξ‚€||𝐺(𝑑,𝑠)π‘˜(𝑠)πœ“π‘’(𝑠)+β„Ž(𝑠)πœ™π‘π·πœŽ0+||𝑒(𝑠)+𝑔(𝑠)𝑑𝑠.(3.24) Since πœ“ and πœ™ are nondecreasing then (3.24) implies ||||≀1𝑇𝑒(𝑑)ξ€œΞ“(π‘žβˆ’2)10||||[]≀1𝐺(𝑑,𝑠)π‘˜(𝑠)πœ“(‖𝑒‖)+β„Ž(𝑠)πœ™(‖𝑒‖)+𝑔(𝑠)π‘‘π‘ ξ€œΞ“(π‘žβˆ’2)10||𝐺||[π‘˜](𝑑,𝑠)(𝑠)πœ“(π‘Ÿ)+β„Ž(𝑠)πœ™(π‘Ÿ)+𝑔(𝑠)𝑑𝑠,(3.25) using similar techniques as to get (3.12) it yields ||||≀1𝑇𝑒(𝑑)Γ‖‖𝐼(π‘žβˆ’2)πœ“(π‘Ÿ)0π‘žβˆ’1+π‘˜β€–β€–πΏ1+|𝛼|𝐼0π‘žβˆ’2+π‘˜(1)+𝐼0π‘žβˆ’1+ξ‚ξ‚€β€–β€–πΌπ‘˜(1)+πœ™(π‘Ÿ)0π‘žβˆ’1+β„Žβ€–β€–πΏ1+|𝛼|𝐼0π‘žβˆ’2+β„Ž(1)+𝐼0π‘žβˆ’1++ξ‚€β€–β€–πΌβ„Ž(1)0π‘žβˆ’1+𝑔‖‖𝐿1+|𝛼|𝐼0π‘žβˆ’2+𝑔(1)+𝐼0π‘žβˆ’1+𝑔=1(1)Γ𝐢(π‘žβˆ’2)π‘˜πœ“(π‘Ÿ)+πΆβ„Žπœ™(π‘Ÿ)+𝐢𝑔.(3.26) Hence ||||≀𝐢𝑇𝑒(𝑑)1[].Ξ“(π‘žβˆ’2)πœ“(π‘Ÿ)+πœ™(π‘Ÿ)+1(3.27)
Moreover, we have ||(𝑇𝑒(𝑑))ξ…ž||≀1ξ‚Έξ€œΞ“(π‘žβˆ’2)πœ“(π‘Ÿ)10|||πœ•πΊ|||ξ€œπœ•π‘‘(𝑑,𝑠)π‘˜(𝑠)𝑑𝑠+πœ™(π‘Ÿ)10|||πœ•πΊ|||ξ€œπœ•π‘‘(𝑑,𝑠)β„Ž(𝑠)+10|||πœ•πΊ|||ξ‚Ή.πœ•π‘‘(𝑑,𝑠)𝑔(𝑠)𝑑𝑠(3.28) Using (3.17) we obtain ||π‘π·πœŽ0+||≀𝐢𝑇𝑒2(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)(πœ“(π‘Ÿ)+πœ™(π‘Ÿ)+1).(3.29) From (3.27) and (3.29) we get 𝐢‖𝑇𝑒‖=(πœ“(π‘Ÿ)+πœ™(π‘Ÿ)+1)1+𝐢Γ(π‘žβˆ’2)2ξ‚Ά,(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)(3.30) then 𝑇(π΅π‘Ÿ) is uniformly bounded.
(ii) 𝑇(π΅π‘Ÿ) is equicontinuous. Indeed for all 𝑑1, 𝑑2∈[0,1], 𝑑1<𝑑2, π‘’βˆˆπ΅π‘Ÿ, let 𝐢=max(|𝑓(𝑑,𝑒(𝑑),π‘π·πœŽ0+𝑒(𝑑))|, 0≀𝑑≀1,‖𝑒‖<π‘Ÿ), therefore ||𝑑𝑇𝑒1ξ€Έξ€·π‘‘βˆ’π‘‡π‘’2ξ€Έ||β‰€πΆξ‚΅ξ€œΞ“(π‘žβˆ’2)𝑑10||𝐺𝑑1𝑑,π‘ βˆ’πΊ2ξ€Έ||+ξ€œ,𝑠𝑑𝑠𝑑2𝑑1||𝐺𝑑1𝑑,π‘ βˆ’πΊ2ξ€Έ||ξ€œ,𝑠𝑑𝑠+1𝑑2||𝐺𝑑1𝑑,π‘ βˆ’πΊ2ξ€Έ||ξ‚Ά,,𝑠𝑑𝑠(3.31) that implies ||𝑑𝑇𝑒1ξ€Έξ€·π‘‘βˆ’π‘‡π‘’2ξ€Έ||β‰€πΆξ€œΞ“(π‘žβˆ’2)𝑑10𝑑2ξ€Έβˆ’π‘ π‘žβˆ’1βˆ’ξ€·π‘‘1ξ€Έβˆ’π‘ π‘žβˆ’1+ξ€·π‘‘π‘ž(π‘žβˆ’1)2βˆ’π‘‘1ξ€Έξ‚΅|𝛼|(1βˆ’π‘ )3βˆ’π‘žβˆ’(1βˆ’π‘ )π‘žβˆ’2ξ‚Άξ€œπ‘žβˆ’1𝑑𝑠+𝑑2𝑑1𝑑2ξ€Έβˆ’π‘ π‘žβˆ’1+ξ€·π‘‘π‘ž(π‘žβˆ’1)2βˆ’π‘‘1ξ€Έξ‚΅|𝛼|(1βˆ’π‘ )3βˆ’π‘žβˆ’(1βˆ’π‘ )π‘žβˆ’2ξ‚Ά+ξ€œπ‘žβˆ’1𝑑𝑠1𝑑2𝑑1βˆ’π‘‘2ξ€Έξ‚΅|𝛼|(1βˆ’π‘ )3βˆ’π‘žβˆ’(1βˆ’π‘ )π‘žβˆ’2ξ‚Άπ‘žβˆ’1𝑑𝑠.(3.32)
Let us consider the function Ξ¦(π‘₯)=π‘₯π‘žβˆ’1βˆ’(π‘žβˆ’1)π‘₯, we see that Ξ¦ is decreasing on [0,1], consequently (𝑑2βˆ’π‘ )π‘žβˆ’1βˆ’(𝑑1βˆ’π‘ )π‘žβˆ’1≀(π‘žβˆ’1)(𝑑2βˆ’π‘‘1), from which we deduce ||𝑑𝑇𝑒1ξ€Έξ€·π‘‘βˆ’π‘‡π‘’2ξ€Έ||≀𝐢𝑑Γ(π‘žβˆ’2)2βˆ’π‘‘1ξ€Έξ€œπ‘‘101+||||π›Όπ‘ž(π‘žβˆ’1)(1βˆ’π‘ )3βˆ’π‘žβˆ’(1βˆ’π‘ )π‘žβˆ’2||||+ξ€œπ‘žβˆ’1𝑑𝑠𝑑2𝑑1𝑑2ξ€Έβˆ’π‘ π‘žβˆ’1+ξ€·π‘‘π‘ž(π‘žβˆ’1)2βˆ’π‘‘1ξ€Έ|||||𝛼|(1βˆ’π‘ )3βˆ’π‘žβˆ’(1βˆ’π‘ )π‘žβˆ’2||||+ξ€·π‘‘π‘žβˆ’1𝑑𝑠1βˆ’π‘‘2ξ€Έξ€œ1𝑑2|||||𝛼|(1βˆ’π‘ )3βˆ’π‘žβˆ’(1βˆ’π‘ )π‘žβˆ’2||||ξƒ­.π‘žβˆ’1𝑑𝑠(3.33) Some computations give ||𝑑𝑇𝑒1ξ€Έξ€·π‘‘βˆ’π‘‡π‘’2ξ€Έ||≀𝐢𝑑2βˆ’π‘‘1ξ€Έξ‚΅1Ξ“(π‘žβˆ’2)+π‘ž(π‘žβˆ’1)3|𝛼|+3(π‘žβˆ’2)ξ‚Ά+πΆπ‘žβˆ’1ξ€œΞ“(π‘žβˆ’2)𝑑2𝑑1𝑑2ξ€Έβˆ’π‘ π‘žβˆ’1π‘ž(π‘žβˆ’1)𝑑𝑠.(3.34) On the other hand we have ||π‘π·πœŽ0+𝑑𝑇𝑒1ξ€Έβˆ’π‘π·πœŽ0+𝑑𝑇𝑒2ξ€Έ||1≀+ξ€œΞ“(2βˆ’πœŽ)𝑑10𝑑1ξ€Έβˆ’π‘ βˆ’πœŽ+1βˆ’ξ€·π‘‘2ξ€Έβˆ’π‘ βˆ’πœŽ+1||(𝑇𝑒(𝑠))ξ…ž||+1π‘‘π‘ ξ€œΞ“(2βˆ’πœŽ)𝑑2𝑑1𝑑2ξ€Έβˆ’π‘ βˆ’πœŽ+1||(𝑇𝑒(𝑠))ξ…ž||𝑑𝑠.(3.35) Using (3.17) and (3.28) it yields ||(𝑇𝑒(𝑑))ξ…ž||≀[]πΆπœ“(π‘Ÿ)+πœ™(π‘Ÿ)+12,(3.36) then ||π‘π·πœŽ0+𝑑𝑇𝑒1ξ€Έβˆ’π‘π·πœŽ0+𝑑𝑇𝑒2ξ€Έ||≀[]πΆπœ“(π‘Ÿ)+πœ™(π‘Ÿ)+12(2𝑑2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)2βˆ’π‘‘1ξ€Έ2βˆ’πœŽ+𝑑22βˆ’πœŽβˆ’π‘‘12βˆ’πœŽξ‚„,(3.37) when 𝑑1→𝑑2, in (3.34) and (3.37) then |𝑇𝑒(𝑑1)βˆ’π‘‡π‘’(𝑑2)| and |π‘π·πœŽ0+𝑇𝑒(𝑑1)βˆ’π‘π·πœŽ0+𝑇𝑒(𝑑2)| tend to 0. Consequently 𝑇(π΅π‘Ÿ) is equicontinuous. From ArzelΓ‘-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 Ξ©={π‘’βˆˆπΈβˆΆβ€–π‘’β€–<π‘Ÿ}, for any π‘’βˆˆπœ•Ξ©, such that 𝑒=πœ†π‘‡π‘’, 0<πœ†<1, we get, with the help of (3.27), ||||||||≀||||≀𝐢𝑒(𝑑)=πœ†π‘‡π‘’(𝑑)𝑇𝑒(𝑑)1[].Ξ“(π‘žβˆ’2)πœ“(π‘Ÿ)+πœ™(π‘Ÿ)+1(3.38) Taking into account (3.29) we obtain ||π‘π·πœŽ0+||≀𝐢𝑒(𝑑)2(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)(πœ“(π‘Ÿ)+πœ™(π‘Ÿ)+1).(3.39) From (3.38), (3.39), and (3.22) we deduce that 𝐢‖𝑒‖≀(πœ“(π‘Ÿ)+πœ™(π‘Ÿ)+1)1+𝐢Γ(π‘žβˆ’2)2ξ‚Ά(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)<π‘Ÿ,(3.40) this contradicts the fact that π‘’βˆˆπœ•Ξ©. Lemma 3.5 allows us to conclude that the operator 𝑇 has a fixed point π‘’βˆ—βˆˆΞ© and then the fractional boundary value problem (P1) has a nontrivial solution π‘’βˆ—βˆˆπΈ. 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(𝑑,𝑒,𝑣)=π‘Ž(𝑑)𝑓1(𝑒,𝑣) where π‘ŽβˆˆπΆ((0,1),(0,∞)) and 𝑓1∈𝐢(ℝ+×ℝ,ℝ+).(H2)∫0<10𝐺(𝑠,𝑠)π‘Ž(𝑠)𝑑𝑠<∞.

Now we give the properties of the Green function.

Lemma 4.1. Let 𝐺(𝑑,𝑠) be the function defined by (2.7). If 𝛼β‰₯1 then 𝐺(𝑑,𝑠) has the following properties: (i)𝐺(𝑑,𝑠)∈𝐢([0,1]Γ—[0,1]), 𝐺(𝑑,𝑠)>0, for all 𝑑, π‘ βˆˆ]0,1[.(ii)If 𝑑,π‘ βˆˆ(𝜏,1), 𝜏>0, then 20<𝜏𝐺(𝑠,𝑠)≀𝐺(𝑑,𝑠)β‰€πœπΊ(𝑠,𝑠).(4.1)

Proof. (i) It is obvious that 𝐺(𝑑,𝑠)∈𝐢([0,1]Γ—[0,1]), moreover, we have 𝛼𝑑(1βˆ’π‘ )3βˆ’π‘žβˆ’π‘‘(1βˆ’π‘ )π‘žβˆ’2=π‘‘π‘žβˆ’1π‘žβˆ’1(1βˆ’π‘ )3βˆ’π‘ž[]β‰₯𝑑(π‘žβˆ’1)π›Όβˆ’(1βˆ’π‘ )π‘žβˆ’1(1βˆ’π‘ )3βˆ’π‘ž[],π›Όβˆ’1+𝑠(4.2) which is positive if 𝛼β‰₯1. Hence 𝐺(𝑑,𝑠) is nonnegative for all 𝑑,π‘ βˆˆ]0,1[.
(ii) Let 𝑑,π‘ βˆˆ(𝜏,1), it is easy to see that 𝐺(𝑠,𝑠)β‰ 0, then we have 𝐺(𝑑,𝑠)𝐺=(𝑠,𝑠)(π‘‘βˆ’π‘ )π‘žβˆ’1(1βˆ’π‘ )3βˆ’π‘ž[]+π‘‘π‘žπ‘ (π‘žβˆ’1)π›Όβˆ’(1βˆ’π‘ )𝑠≀1+(1βˆ’π‘ )2𝑠≀2𝜏,0<πœβ‰€π‘ β‰€π‘‘<1,𝐺(𝑑,𝑠)=𝑑𝐺(𝑠,𝑠)𝑠≀2𝜏,0<πœβ‰€π‘‘β‰€π‘ β‰€1.(4.3) Now we look for lower bounds of 𝐺(𝑑,𝑠)𝐺(𝑑,𝑠)β‰₯𝑑𝐺(𝑠,𝑠)𝑠β‰₯πœπ‘ β‰₯𝜏,0<πœβ‰€π‘ β‰€π‘‘<1,0<πœβ‰€π‘‘β‰€π‘ β‰€1.(4.4) Finally, since 𝐺(𝑠,𝑠) is nonnegative we obtain 0<𝜏𝐺(𝑠,𝑠)≀𝐺(𝑑,𝑠)≀(2/𝜏)𝐺(𝑠,𝑠).

We recall the definition of positive of solution.

Definition 4.2. A function 𝑒(𝑑) is called positive solution of the fractional boundary value problem (P1) if 𝑒(𝑑)β‰₯0, for all π‘‘βˆˆ[0,1].

Lemma 4.3. If π‘’βˆˆπΈ+ and 𝛼β‰₯1, then the solution of the fractional boundary value problem (P1) is positive and satisfies minπ‘‘βˆˆ(𝜏,1)𝑒(𝑑)+π‘π·πœŽ0+β‰₯πœπ‘’(𝑑)22‖𝑒‖.(4.5)

Proof. First let us remark that under the assumptions on 𝑒 and 𝑓, the function π‘π·πœŽ0+𝑒 is nonnegative. From Lemma 3.1 we have 1𝑒(𝑑)=ξ€œΞ“(π‘žβˆ’2)10𝐺(𝑑,𝑠)π‘Ž(𝑠)𝑓1𝑒(𝑠),π‘π·πœŽ0+𝑒(𝑠)𝑑𝑠.(4.6) Applying the right-hand side of inequality (4.1) we get 2𝑒(𝑑)β‰€ξ€œπœΞ“(π‘žβˆ’2)10𝐺(𝑠,𝑠)π‘Ž(𝑠)𝑓1𝑒(𝑠),π‘π·πœŽ0+𝑒(𝑠)𝑑𝑠.(4.7) Moreover, (4.1) gives π‘π·πœŽ0+1𝑒(𝑑)=ξ€œΞ“(π‘žβˆ’2)Ξ“(2βˆ’πœŽ)𝑑0ξ€œ10(π‘‘βˆ’π‘ )βˆ’πœŽ+1𝐺1(𝑠,π‘Ÿ)Γ—π‘Ž(π‘Ÿ)𝑓1𝑒(π‘Ÿ),π‘π·πœŽ0+≀1𝑒(π‘Ÿ)π‘‘π‘ π‘‘π‘Ÿξ€œπœ(2βˆ’πœŽ)Ξ“(π‘žβˆ’2)Ξ“(2βˆ’πœŽ)10𝐺1(π‘Ÿ,π‘Ÿ)π‘Ž(π‘Ÿ)𝑓1𝑒(π‘Ÿ),π‘π·πœŽ0+𝑒(π‘Ÿ)π‘‘π‘Ÿ.(4.8) Combining (4.7) and (4.8) yields 2β€–π‘’β€–β‰€ξ€œπœΞ“(π‘žβˆ’2)10𝐺𝐺(𝑠,𝑠)+1(𝑠,𝑠)ξ‚Ή(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)π‘Ž(𝑠)𝑓1𝑒(𝑠),π‘π·πœŽ0+𝑒(𝑠)𝑑𝑠,(4.9) hence ξ€œ10𝐺𝐺(𝑠,𝑠)+1(𝑠,𝑠)ξ‚Ή(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)π‘Ž(𝑠)𝑓1𝑒(𝑠),π‘π·πœŽ0+𝑒(𝑠)𝑑𝑠β‰₯πœΞ“(π‘žβˆ’2)2‖𝑒‖.(4.10) In view of the-left hand side of (4.1), we obtain for all π‘‘βˆˆ(𝜏,1)πœπ‘’(𝑑)β‰₯ξ€œΞ“(π‘žβˆ’2)10𝐺(𝑠,𝑠)π‘Ž(𝑠)𝑓1𝑒(𝑠),π‘π·πœŽ0+𝑒(𝑠)𝑑𝑠,(4.11) on the other hand we have π‘π·πœŽ0+πœπ‘’(𝑑)β‰₯2βˆ’πœŽξ€œ(2βˆ’πœŽ)Ξ“(π‘žβˆ’2)Ξ“(2βˆ’πœŽ)10𝐺1(π‘Ÿ,π‘Ÿ)π‘Ž(π‘Ÿ)𝑓1𝑒(π‘Ÿ),π‘π·πœŽ0+𝑒(π‘Ÿ)π‘‘π‘Ÿ.(4.12)
From (4.11) and (4.12) we get minπ‘‘βˆˆ(𝜏,1)𝑒(𝑑)+π‘π·πœŽ0+β‰₯πœπ‘’(𝑑)ξ€œΞ“(π‘žβˆ’2)10𝐺𝐺(𝑠,𝑠)+1(𝑠,𝑠)ξ‚Ή(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)Γ—π‘Ž(𝑠)𝑓1𝑒(𝑠),π‘π·πœŽ0+𝑒(𝑠)𝑑𝑠,(4.13)
with the help of (4.10) we deduce minπ‘‘βˆˆ(𝜏,1)𝑒(𝑑)+π‘π·πœŽ0+β‰₯πœπ‘’(𝑑)22‖𝑒‖.(4.14)
The proof is complete.

Define the quantities 𝐴0 and 𝐴∞ by 𝐴0=lim(|𝑒|+|𝑣|)β†’0𝑓1(𝑒,𝑣)|𝑒|+|𝑣|,𝐴∞=lim(|𝑒|+|𝑣|)β†’βˆžπ‘“1(𝑒,𝑣)|.𝑒|+|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 πΎβŠ‚πΈ, be a cone. Assume Ξ©1 and Ξ©2 are open subsets of 𝐸 with 0∈Ω1, Ξ©1βŠ‚Ξ©2 and let ξ‚€π’œβˆΆπΎβˆ©Ξ©2⧡Ω1ξ‚βŸΆπΎ(4.16) be a completely continuous operator such that (i)||π’œπ‘’||≀||𝑒||, π‘’βˆˆπΎβˆ©πœ•Ξ©1, and ||π’œπ‘’||β‰₯||𝑒||, π‘’βˆˆπΎβˆ©πœ•Ξ©2, or(ii)||π’œπ‘’||β‰₯||𝑒||, π‘’βˆˆπΎβˆ©πœ•Ξ©1, and ||π’œπ‘’||≀||𝑒||, π‘’βˆˆπΎβˆ©πœ•Ξ©2.Then π’œ has a fixed point in 𝐾∩(Ξ©2⧡Ω1).

Proof. To prove Theorem 4.4 we define the cone 𝐾 by 𝐾=π‘’βˆˆπΈ+,minπ‘‘βˆˆ(𝜏,1)𝑒(𝑑)+π‘π·πœŽ0+β‰₯πœπ‘’(𝑑)22ξ‚Ό.‖𝑒‖(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 π‘‡πΎβŠ‚πΎ. 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 πœ€>0, there exists 𝑅1>0, such that 𝑓1(𝑒,𝑣)β‰€πœ€(|𝑒|+|𝑣|)(4.18) for 0<|𝑒|+|𝑣|≀𝑅1. Letting Ξ©1={π‘’βˆˆπΈ,‖𝑒‖<𝑅1}, for any π‘’βˆˆπΎβˆ©πœ•Ξ©1, it yields 1𝑇𝑒(𝑑)=ξ€œΞ“(π‘žβˆ’2)10𝐺(𝑑,𝑠)π‘Ž(𝑠)𝑓1𝑒(𝑠),π‘π·πœŽ0+≀𝑒(𝑠)𝑑𝑠2πœ€β€–π‘’β€–ξ€œπœΞ“(π‘žβˆ’2)10𝐺(𝑠,𝑠)π‘Ž(𝑠)𝑑𝑠.(4.19) Moreover, we have π‘π·πœŽ0+1𝑇𝑒(𝑑)=ξ€œΞ“(π‘žβˆ’2)Ξ“(2βˆ’πœŽ)𝑑0ξ€œ10(π‘‘βˆ’π‘ )βˆ’πœŽ+1𝐺1(𝑠,π‘Ÿ)Γ—π‘Ž(π‘Ÿ)𝑓1𝑒(π‘Ÿ),π‘π·πœŽ0+≀1𝑒(π‘Ÿ)π‘‘π‘ π‘‘π‘Ÿξ€œπœ(2βˆ’πœŽ)Ξ“(π‘žβˆ’2)Ξ“(2βˆ’πœŽ)10𝐺1ξ‚€||||+||(π‘Ÿ,π‘Ÿ)π‘Ž(π‘Ÿ)𝑒(π‘Ÿ)π‘π·πœŽ0+||≀𝑒(π‘Ÿ)π‘‘π‘Ÿπœ€β€–π‘’β€–ξ€œπœ(2βˆ’πœŽ)Ξ“(π‘žβˆ’2)Ξ“(2βˆ’πœŽ)10𝐺1(𝑠,𝑠)π‘Ž(𝑠)𝑑𝑠.(4.20) From (4.19) and (4.20) we conclude ‖𝑇𝑒‖≀2πœ€β€–π‘’β€–ξ€œπœΞ“(π‘žβˆ’2)10𝐺𝐺(𝑠,𝑠)+1(𝑠,𝑠)ξ‚Ή(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)π‘Ž(𝑠)𝑑𝑠.(4.21)
In view of hypothesis (H2), one can choose πœ€ such that πœ€β‰€πœΞ“(π‘žβˆ’2)2∫10𝐺𝐺(𝑠,𝑠)+1.(𝑠,𝑠)/(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)ξ€Έξ€»π‘Ž(𝑠)𝑑𝑠(4.22) The inequalities (4.21) and (4.22) imply that ||𝑇𝑒||≀||𝑒||, for all π‘’βˆˆπΎβˆ©πœ•Ξ©1. Second, in view of 𝐴∞=∞, then for any 𝑀>0, there exists 𝑅2>0, such that 𝑓1(𝑒,𝑣)β‰₯𝑀(|𝑒|+|𝑣|) for |𝑒|+|𝑣|β‰₯𝑅2. Let 𝑅=max{2𝑅1,(2𝑅2/𝜏2)} and denote by Ξ©2 the open set {π‘’βˆˆπΈ/||𝑒||<𝑅}. If π‘’βˆˆπΎβˆ©πœ•Ξ©2 then minπ‘‘βˆˆ(𝜏,1)𝑒(𝑑)+π‘π·πœŽ0+β‰₯πœπ‘’(𝑑)22β€–πœπ‘’β€–=22𝑅β‰₯𝑅2.(4.23) Using the left-hand side of (4.1) and Lemma 4.3, we obtain 𝑇𝑒(𝑑)β‰₯πœπ‘€ξ€œΞ“(π‘žβˆ’2)10ξ‚€||||+||𝐺(𝑠,𝑠)π‘Ž(𝑠)𝑒(𝑠)π‘π·πœŽ0+||β‰₯πœπ‘’(𝑠)𝑑𝑠3𝑀|||||𝑒|ξ€œ2Ξ“(π‘žβˆ’2)10𝐺(𝑠,𝑠)π‘Ž(𝑠)𝑑𝑠.(4.24) Moreover, we get with the help of (4.12) π‘π·πœŽ0+πœπ‘‡π‘’(𝑑)β‰₯4βˆ’πœŽπ‘€|||||𝑒|ξ€œ2(2βˆ’πœŽ)Ξ“(π‘žβˆ’2)Ξ“(2βˆ’πœŽ)10𝐺1(𝑠,𝑠)π‘Ž(𝑠)𝑑𝑠.(4.25) In view of (4.26) and (4.24) we can write 𝑇𝑒(𝑑)+π‘π·πœŽ0+πœπ‘‡π‘’(𝑑)β‰₯3𝑀|||||𝑒|ξ€œ2Ξ“(π‘žβˆ’2)10ξ‚ΈπœπΊ(𝑠,𝑠)+1βˆ’πœŽπΊ1(𝑠,𝑠)ξ‚Ήβ‰₯𝜏(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)π‘Ž(𝑠)𝑑𝑠3𝑀|||||𝑒|ξ€œ2Ξ“(π‘žβˆ’2)10𝐺𝐺(𝑠,𝑠)+1(𝑠,𝑠)ξ‚Ή(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)π‘Ž(𝑠)𝑑𝑠.(4.26) Let us choose 𝑀 such that 𝑀β‰₯2Ξ“(π‘žβˆ’2)𝜏3∫10𝐺𝐺(𝑠,𝑠)+1,(𝑠,𝑠)/(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)ξ€Έξ€»π‘Ž(𝑠)𝑑𝑠(4.27) then we get 𝑇𝑒(𝑑)+π‘π·πœŽ0+𝑇𝑒(𝑑)β‰₯||𝑒||. Hence, ||||||||β‰₯||||𝑇𝑒|𝑒|,βˆ€π‘’βˆˆπΎβˆ©πœ•Ξ©2.(4.28) The first part of Theorem 4.5 implies that 𝑇 has a fixed point in 𝐾∩(Ξ©2⧡0π‘₯00082Ξ©1) such that 𝑅2≀||𝑒||≀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 𝑐𝐷05/2+𝑒=π‘‘βˆ’1103𝑒+𝑑2𝐷05/4+𝑒𝑒+ln𝑑,0<𝑑<1,(0)=π‘’ξ…žξ…ž(0)=0,π‘’ξ…ž(1)=βˆ’12π‘’ξ…žξ…ž(1)(4.29) has a unique solution in 𝐸.

Proof. In this case we have 𝑓(𝑑,π‘₯,𝑦)=((π‘‘βˆ’1)/10)3π‘₯+𝑑2𝑦+ln𝑑, 2<π‘ž=5/2<3, 𝜎=(5/4)<2, 𝛼=βˆ’1/2 and ||𝑓𝑑,π‘₯,π‘₯ξ€Έξ€·βˆ’π‘“π‘‘,𝑦,𝑦||≀1βˆ’π‘‘ξ‚103||π‘₯βˆ’π‘₯||+𝑑2||π‘¦βˆ’π‘¦||,(4.30) then 𝑔(𝑑)=((1βˆ’π‘‘)/10)3 and β„Ž(𝑑)=𝑑2. Some calculus give ‖‖𝐼0π‘žβˆ’1+𝑔‖‖𝐿1=0.17846Γ—10βˆ’3,𝐼0π‘žβˆ’1+𝑔(1)=0.48001Γ—10βˆ’3,𝐼0π‘žβˆ’2+𝑔(1)=016120Γ—10βˆ’3,𝐢𝑔=7.3907Γ—10βˆ’4,𝐴𝑔=1.0406Γ—10βˆ’3,‖‖𝐼0π‘žβˆ’1+β„Žβ€–β€–πΏ1=0.038210,𝐼0π‘žβˆ’1+β„ŽπΌ(1)=0.17194,0π‘žβˆ’2+β„Ž(1)=0.6018,πΆβ„Ž=0.49747,π΄β„ŽπΆ=0.42448,𝑔+πΆβ„Žπ΄=0.49821<1,𝑔+π΄β„Ž=0.42552<(2βˆ’πœŽ)Ξ“(2βˆ’πœŽ)=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 𝑐𝐷07/3+𝑒=(1βˆ’π‘‘)2βŽ›βŽœβŽœβŽœβŽœβŽπ‘’4ξ€·1001+𝑒2ξ€Έ+ξ‚΅ξ‚€ln1+𝑐𝐷06/5+𝑒2ξ‚Ά+19⎞⎟⎟⎟⎟⎠=0,0<𝑑<1,𝑒(0)=π‘’ξ…žξ…ž(0)=0,π‘’ξ…ž3(1)=2π‘’ξ…žξ…ž(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 π‘ž=7/3, 𝜎=6/5, 𝛼=3/2, and ||𝑓𝑑,π‘₯,π‘₯ξ€Έ||=(1βˆ’π‘‘)2π‘₯4ξ€·1001+π‘₯2ξ€Έ+(1βˆ’π‘‘)2ξ‚€ln1+π‘₯29+(1βˆ’π‘‘)2≀|π‘₯|2100(1βˆ’π‘‘)2+(1βˆ’π‘‘)2ξ‚€ln1+π‘₯29+(1βˆ’π‘‘)2ξ€·||β‰€π‘˜(𝑑)πœ“(|π‘₯|)+β„Ž(𝑑)πœ™π‘₯||ξ€Έ+𝑔(𝑑),(4.33) where π‘˜(𝑑)=β„Ž(𝑑)=𝑔(𝑑)=(1βˆ’π‘‘)2, πœ“(π‘₯)=(π‘₯/10)2, πœ™(π‘₯)=ln(1+π‘₯2)/9, 𝑓(𝑑,0,0)β‰ 0. Let us find π‘Ÿ such that (3.22) holds, for this we have ‖‖𝐼0π‘žβˆ’1+𝑔‖‖𝐿1=0.19382,𝐼0π‘žβˆ’2+𝑔(1)=0.15998,𝐼0π‘žβˆ’1+𝐢𝑔(1)=0.33595,𝑔=𝐢1=0.76974,𝐴𝑔=𝐢2=0.91187.(4.34) We see that (3.22) is equivalent to 1.2664(((π‘Ÿ2)/100)+(ln(1+π‘Ÿ2)/9)+1)βˆ’π‘Ÿ which is negative for π‘Ÿ=6.

References

  1. R. L. Bagley and P. J. Torvik, β€œA theoretical basis for the application of fractional calculus to viscoelasticity,” Journal of Rheology, vol. 27, no. 3, pp. 201–210, 1983. View at: Publisher Site | Google Scholar
  2. R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. View at: Publisher Site
  3. K. B. Oldham, β€œFractional differential equations in electrochemistry,” Advances in Engineering Software, vol. 41, no. 1, pp. 9–12, 2010. View at: Publisher Site | Google Scholar
  4. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  5. Z. Bai and H. LΓΌ, β€œPositive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. M. El-Shahed, β€œPositive solutions for boundary value problem of nonlinear fractional differential equation,” Abstract and Applied Analysis, vol. 2007, Article ID 10368, 8 pages, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. A. Guezane-Lakoud and S. Kelaiaia, β€œSolvability of a three-point nonlinear boundary-value problem,” Electronic Journal of Differential Equations, vol. 2010, no. 139, pp. 1–9, 2010. View at: Google Scholar | Zentralblatt MATH
  8. H. Jafari and V. Daftardar-Gejji, β€œPositive solutions of nonlinear fractional boundary value problems using Adomian decomposition method,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 700–706, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. S. Liang and J. Zhang, β€œExistence and uniqueness of positive solutions to m-point boundary value problem for nonlinear fractional differential equation,” Journal of Applied Mathematics and Computing. In press. View at: Publisher Site | Google Scholar
  10. T. Qiu and Z. Bai, β€œExistence of positive solutions for singular fractional differential equations,” Electronic Journal of Differential Equations, vol. 2008, article 146, 2008. View at: Google Scholar
  11. X. Su and L. Liu, β€œExistence of solution for boundary value problem of nonlinear fractional differential equation,” Applied Mathematics, vol. 22, no. 3, pp. 291–298, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. X. Xu, D. Jiang, and C. Yuan, β€œMultiple positive solutions for the boundary value problem of a nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applicationss, vol. 71, no. 10, pp. 4676–4688, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. Y. Zhao, S. Sun, Z. Han, and M. Zhang, β€œPositive solutions for boundary value problems of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6950–6958, 2011. View at: Publisher Site | Google Scholar
  14. Y. Zhao, S. Sun, Z. Han, and Q. Li, β€œThe existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2086–2097, 2011. View at: Publisher Site | Google Scholar
  15. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006.
  16. G. A. Anastassiou, β€œOn right fractional calculus,” Chaos, Solitons and Fractals, vol. 42, no. 1, pp. 365–376, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  17. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
  18. D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.

Copyright Β© 2011 A. Guezane-Lakoud and R. Khaldi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

619Β Views | 414Β Downloads | 1Β Citation
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder