# Positive Solutions for a Fractional Boundary Value Problem with a Perturbation Term

Authors: Yumei Zou
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We obtain some new upper and lower estimates for the Green’s function associated with a fractional boundary value problem with a perturbation term. Criteria for the existence of positive solutions of the problem are then obtained based on it.


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## Body

## 1. Introduction

In this paper, we are investigating the existence of positive solution for fractional differential equation with a perturbation term(1)-Dαxt+atxt=ft,xt,t∈0,1with the boundary condition (BC)(2)x0=x′0=x′1=0,where 2<α<3, a∈C[0,1], and f∈C([0,1]×[0,+∞),R). Here, Dαx is the standard Riemann-Liouville derivative of order α>0 of a continuous function x:(0,∞)→R is given by (3)Dαxt=1Γn-αddtn∫0txst-sα-n+1ds,n-1≤α<nprovided that the right-hand side is pointwise defined on (0,∞).Spurred by the extensively applicability of fractional derivatives in a variety of mathematical models in science and engineering [1–3], the subject of fractional differential equations with boundary value problems, which emerged as a new branch of differential equations, have attracted a great deal of attention for decades. As a small sampling of recent development, we refer the reader to [4–14]. When one seeks the existence of solution of boundary value problems for fractional differential equations, the usual method is converted to a Fredholm integral equation and find the fixed points by using various techniques of nonlinear analysis such as Banach contraction map principle [13, 15], linear operator theory [16, 17], Leggett-Williams fixed point theorem [12, 18], Schauder fixed point theorem and Leray-Schauder nonlinear alternative theory [19], and Krasnosel’skii fixed point theorem [20]. It should be noted that the Green’s functions play a vital role in the construction of an appropriate Fredholm integral equation. However, as a result of the unusual feature of the fractional calculus, the investigation on the Green’s functions for fractional boundary value problems is still in the initial stage. Recently, based on the spectral theory, the authors in [21] give an associated Green’s function for BVP (1) (2) as series of functions. This idea was also used in [22–24].In the next section, we will study some new sharper upper and lower estimates for the Green’s function of BVP (1) (2) than the ones given in [21]. In Section 3, we employ the new estimate to obtain the existence of a positive solution of BVP (1) (2). The idea of this paper may trace to [21–27].


## 2. Some New Upper and Lower Estimates for the Green’s Function

Firstly, we present the Green’s function for BVP (1) (2) which is given in [21]. Let G0:[0,1]×[0,1]→R be defined by (4)G0t,s=tα-11-sα-2-t-sα-1Γα,0≤s≤t≤1,tα-11-sα-2Γα,0≤t≤s≤1.It is well known that the function G0(t,s) is Green’s function for BVP (1) (2) with a(t)≡0. In the following lemma we present some properties of Green’s function G0(t,s), see [28] for details. Listed properties will be used later for estimating the upper bound and the lower bound on Green’s function G(t,s) of BVP (1) (2).Lemma 1 (see [28]).
The functionG0(t,s) defined by (4) satisfies the following conditions:(i)
0≤G0t,s≤tα-11-sα-2/Γα, 0≤t,  s≤1,(ii)
tα-1G0(1,s)≤G0(t,s)≤G0(1,s)=s1-sα-2/Γα,  for 0≤t,  s≤1.LetM=∫01as1-sα-2/Γαsα-1ds,  M1=∫01as1-sα-2/Γ(α)ds, A=maxt∈[0,1]at. For  n=1,2,…, define (5)Gnt,s=∫01aτG0t,τGn-1τ,sdτ=∫01⋯∫01ar1G0t,r1·ar2G0r1,r2⋯arnG0rn-1,rn·G0rn,sdr1⋯drnand(6)Gt,s=∑n=0+∞-1nGnt,s.It follows from Theorem  2.1 in [21] that the function G(t,s) defined by (6) as a series of functions that converge uniformly is the Green’s function for BVP (1) (2) if A<(α-1)Γ(α+1) holds. Furthermore, the function G(t,s) satisfies the following property: (7)1-δG0t,s≤Gt,s≤1+δG0t,s,t,s∈0,1provided A<(α-1)Γ(α+1)α+1-1, where δ=αA/(α-1)Γ(α+1)-A<1.The uniform convergence of (6) follows from the fact that T<1, where the operator T is defined by the following form: (8)Txt=∫01G0t,sasxsds,x∈C0,1.Indeed, the uniform convergence of (6) can be obtained by r(T)<1 (see [21, 29]), where r(T) is the spectral radius of T.Lemma 2.
IfM<1 holds, then r(T)<1.Proof.
By Lemma1, for any x∈C[0,1], we have (9)Txt≤∫01G0t,sasxsds≤tα-1AxΓα∫011-sα-2ds=Axα-1Γαtα-1.Hence, we conclude that (10)T2xt≤∫01G0t,sasTxsds≤Axα-1Γα∫01tα-1as1-sα-2Γαsα-1ds=AMxα-1Γαtα-1.By induction, one has (11)Tnxt≤AMn-1xα-1Γαtα-1,which implies that (12)Tn≤AMn-1α-1Γα.Note that M<1. Then by the Gelfand formula, we get (13)rT=limn→+∞Tnn≤M<1.Lemma 3.
IfM+M1<1 holds, then for any t,s∈[0,1], (14)1-M-M11-MG0t,s≤Gt,s≤1-M+M11-MG0t,s.Proof.
By Lemma1, for n≥1 and t,s∈[0,1], we have(15)Gnt,s≤∫01⋯∫01ar1G0t,r1·ar2G0r1,r2⋯arnG0rn-1,rn·G0rn,sdr1⋯drn≤∫01⋯∫01ar1tα-11-r1α-2Γα·ar2r1α-11-r2α-2Γα⋯arnrn-1α-11-rnα-2Γα·s1-sα-2Γαdr1⋯drn=tα-1s1-sα-2Γα∫01ar1r1α-11-r1α-2Γαdr1·∫01ar2r2α-11-r2α-2Γαdr2⋯∫01arn-1rn-1α-11-rn-1α-2Γαdrn-1·∫01arn1-rnα-2Γαdrn=M1Mn-1tα-1s1-sα-2Γα≤M1Mn-1G0t,s,t,s∈0,1.Then introducing the above inequality into (6) can lead to (16)∑n=1+∞-1nGnt,s≤∑n=1+∞Gnt,s≤∑n=1+∞M1Mn-1G0t,s=M11-MG0t,s,t,s∈0,1.It is easy to verify that if M+M1<1, then M1/1-M<1. Therefore, (14) follows from (6) and (16).Similar to the proof of Lemmas2 and 3, we can obtain the following results.Lemma 4.
IfA<Γ2α-1/Γα-1 holds, then r(T)<1.Lemma 5.
IfA/(α-1)Γ(α)+AΓ(α-1)/Γ(2α-1)<1 holds, then for any t,s∈[0,1], (17)1-γG0t,s≤Gt,s≤1+γG0t,s,where γ=A/(α-1)Γ(α)1-AΓ(α-1)/Γ(2α-1)-1.By the properties of definite integral andα∈(2,3), we assert that (18)∫01τα-11-τα-2Γαdτ<∫01τ1-τα-2Γαdτ,that is (19)Γα-1Γ2α-1<1α-1Γα+1.Thus, we obtain that (20)γ=A/α-1Γα1-AΓα-1/Γ2α-1<αA/α-1Γα+11-A/α-1Γα+1=δ.This means that Lemmas 2–5 is more general and complements many known results.Combining Lemmas1 and 3, we obtain the following result.Theorem 6.
IfM+M1<1 holds. Then for any t,s∈[0,1], (21)1-M-M11-Mtα-1G01,s≤Gt,s≤1-M+M11-MG01,s.Theorem 7.
Ifx(t) satisfies the boundary conditions (2), (22)-Dαxt+atxt≥0.If M+M1<1 holds, then (23)xt≥1-M-M11-M+M1tα-1x,t∈0,1.


## 3. Existence Theorems

In this section, we shall employ Theorem6 to investigate the existence results for BVP (1) (2). Let C[0,1] be the Banach space endowed with the maximum norm x=maxt∈[0,1]|x(t)|.Theorem 8.
Assume that there existc2>c1>0 such that (24)infx∈Ω∫01G01,sfs,xsds≥c11-M1-M-M1and (25)supx∈Ω∫01G01,sfs,xsds≤c21-M1-M+M1,where (26)Ω=x∈C0,1:c1tα-1≤xt≤c2,t∈0,1.Then BVP (1) (2) has at least one positive solution in Ω.Proof.
Define an operatorT by (27)Sxt=∫01Gt,sfs,xsds,x∈C0,1,where G(t,s) is given by (6). Obviously, x(t) is a solution of BVP (1)(2) if and only if x∈C[0,1] is a fixed point of S. Moreover, we can show that S:C[0,1]→C[0,1] is completely continuous.
For any givenx∈Ω, by (24) and (25), we conclude that (28)Sxt≥1-M-M11-Mtα-1infx∈S∫01G01,sfs,xsds≥c1tα-1and (29)Sxt≤1-M+M11-Msupx∈S∫01G01,sfs,xsds≤c2.Therefore, S(Ω)⊂Ω. By Schauder’s fixed point theorem, S has a fixed point x in Ω which implies that BVP (1) (2) has at least one positive solution in Ω.The following corollaries are direct results of Theorem8.Corollary 9.
Assume that there existc2>c1>0 such that for any t∈[0,1], f(t,·) is nondecreasing on [0,c2], (30)∫01G01,sfs,c1sα-1ds≥c11-M1-M-M1and (31)∫01G01,sfs,c2ds≤c21-M1-M+M1.Then BVP (1) (2) has at least one positive solution in Ω.Corollary 10.
Assume that there existc2>c1>0 such that for any t∈[0,1], f(t,·) is nonincreasing on [0,c2], (32)∫01G01,sfs,c1sα-1ds≤c21-M1-M+M1and (33)∫01G01,sfs,c2ds≥c11-M1-M-M1.Then BVP (1) (2) has at least one positive solution in Ω.Example 11.
Consider the BVP(34)-D5/2xt+t1-txt=xt,t∈0,1,x0=x′0=x′1=0.After simple computation, we have M=Γ(α+1)/Γ(2α+1)=π/64, M1=1/Γ(α+2)=16π/105π.Letf(t,x)=x. It is easy to see that (30) and (31) hold when c1 is small enough and c2 is large enough. Then, by Corollary 9, BVP (34) has at least one solution.



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*Source: 9070247-2018-08-07.xml*