Existence Results for a New Class of Boundary Value Problems of Nonlinear Fractional Differential Equations

Meysam Alvan 1, Rahmat Darzi 2 and Amin Mahmoodi 3,* 1 Department of mathematics, Central Tehran Branch Islamic Azad university, Tehran 13185/768, Iran; m.alvan.r.math.t@gmail.com 2 Department of Mathematics, Neka Branch Islamic Azad University, Neka 48411-86114, Iran; r.darzi@iauneka.ac.ir 3 Department of mathematics, Central Tehran Branch Islamic Azad univesity, Tehran 13185/768, Iran * Correspondence: a_mahmoodi@iauctb.ac.ir; Tel.: +98-21-6693-3501


Introduction
In the recent years, fractional calculus has been one of the most interesting issues that have attracted many scientists, specially in mathematics and engineering sciences.Many natural phenomena can be presented by boundary value problems of fractional differential equations.Many authors in different fields such as chemical physics, fluid flows, electrical networks, viscoelasticity, try to model these phenomena by boundary value problems of fractional differential equations [1][2][3][4].To achieve extra information in fractional calculus, specially boundary value problems, readers can refer to valuable papers or books .
In this paper, we investigate the existence and uniqueness of solution for the following new class of fractional boundary value problem c D α 0 `u ptq `2r c D α´1 0 `u ptq `r2 c D α´2 0 `u ptq " f pt, u ptqq , r ą 0, 0 ă t ă 1 (1) with the boundary conditions u p0q " u p1q , u 1 p0q " u 1 p1q , u 1 pξq `ru pξq " η, ξ P p0, 1q where c D α´i 0 `pi " 0, 1, 2q are the standard Caputo derivative and f : r0, 1s ˆR Ñ R is a continuously differentiable function satisfying the following assumptions: pA 0 q f P C pr0, 1s ˆR, Rq and there exists a constant L ą 0 so that | f pt, uq ´f pt, vq| ď L |u ´v| , t P p0, 1q , @u, v P R, in which L satisfies the condition L ă r 2 G pα ´1q 2e r .
pA 1 q f P C pr0, 1s ˆR, Rq , p P C pr0, 1sq and A is a constant, so that | f pt, uq| ď p ptq `A |u| , t P p0, 1q , @u P R, Because the boundary conditions u p0q " u p1q and u 1 p0q " u 1 p1q in (1.2) involve periodicity, it is not possible to directly transform the boundary value into integral equation.To overcome this problem, presenting a suitable substitution is needed.It is worth saying that Lemma 2.7 (see Lemma 2.3 in [17] and Lemma 2.6 in [21]) is an important and valuable tool to achieve the new result.The contraction mapping principle and fixed point theorem play the main role in finding new existence results for the problem.
The main result of this paper can be seen in two Theorems; 3.1 and 3.2.In Theorem 3.1, the uniqueness of solution is proved by using Banach contraction principle.In Theorem3.2,we present an existence theorem by means of Schauder fixed point theorem.
We can extend the result even for the following boundary value problem where n ´1 ď αă n, n ą4, with the boundary conditions The plan of this paper is as follows: In Section 2, we give some basic definitions and technical lemmas.Section 3 contains the proofs of our main results.Finally, we provide two examples to show the applicability of the results.

Basic Definitions and Preliminaries
In this section, we present some definitions and technical lemmas which will be used in the remainder of this paper.These and the related results and proofs can be found in the literature [6][7][8]17,21].
Definition 2.1.( [7,8]) The Riemann-Liouville fractional integral of order α ą 0, of a function u : R `Ñ R is defined by whenever the right-hand side is defined on R `.
Definition 2.2.( [7,8]) The Riemann-Liouville fractional derivative of order α ą 0, of a function u : R `Ñ R is given by where n " rαs `1, and rαs denotes the integer part of real number α. Definition 2.3.( [7,8]) The Caputo fractional derivative of order α ą 0, of a function u : R `Ñ R is defined by whenever the right-hand side is defined on R `.
Definition 2.4. ([7,8])The Caputo fractional derivative of order α ą 0, of a function u : R `Ñ R is defined via the Riemann-Liouville fractional derivative by Where n " α for α P N; n " rαs `1 for α R N.
Then, u P C 2 r0, 1s is a solution of BVP p12q ´p13q .Thus, this ends the proof.

Main Result
Let U " C r0, 1s be a Banach space with the norm ||u|| " max tPr0,1s tu ptqu .Consider the space U with the norm ||u|| ˚" max tPr0,1s u ptq e ´rt ( in which r is described as in p1q.It is well known that the norm ||u|| ˚is equivalent to the norm ||u||. For the forthcoming analysis, we need the assumptions pA0q and pA1q .where the function g ptq " f `t, v ptq e ´rt ˘is continuous on r0, 1s , for any v P U (from pA 0 q).It is easy to see that the operator T maps U into U.
In view of Lemma p2.10q , the operator T has a fixed point v P V if and only if u " ve ´rt is a solution of BVP p1.1q ´p1.2q with u P C 2 r0, 1s .So, it is sufficient to show that the operator T has a fixed point on U.For v 1 , v 2 P U and for s P C r0, 1s , we obtain ˇˇf `s, v 2 psq e ´rs ˘´f `s, v 1 psq e ´rs ˘ˇď L ˇˇv 2 psq e ´rs ´v1 psq e ´rs ˇď L||v 2 ´v1 || ˚(39) Hence, from p39q , we have the following inequality Consequently, By the Banach contraction principle, it follows that T has an unique fixed point v P U. Therefore, u " e ´rt v is a unique solution of FBVP p1q ´p2q .Now, we prove the existence of solutions of p1q ´p2q by applying Schauder fixed point theorem.
Thus, T maps B R into B R , i.e.T pB R q Ď B R .Now, we prove that T is completely continuous on B R .We will give the proof in the case that U is equipped with the usual norm, since the norm ||u|| ˚is equivalent to the usual norm.Since T pB R q Ď B R , we have ||Tu|| ď ||Tu|| ˚er ď pv 0 `Rq e r for any u P B R , and so tz; z P T pB R qu is uniformly bounded.On the other hand, for any v P B R , it follows from p40q and pA1q that pTvq 1 ptq ď ¨η `P `A ´η r `Rr G pα ´1q ‚e r , t P r0, 1s and this shows that T pB R q is equicontinuous.Thus, by Arzella-Ascoli theorem, it implies that T pB R q is relatively compact.Finally, we show that T is continuous on B R .Let pv n q be an arbitrary sequence in B R and v P B R so that ||v n ´v|| Ñ 0 as n Ñ 8. Therefore, ||v n ´v|| ˚Ñ 0, as n Ñ 8 and so there exists two constants k 1 , k 2 so that v n ptq e ´rt pn " 1, 2, . ..q and v ptq e ´rt P rk 1 , k 2 s , for each t P r0, 1s .Since f is uniformly continuous on r0, 1s ˆrk 1 , k 2 s , it follows that for any ą 0, there exists δ ą 0 whenever |u 1 ´u2 | ă δ, u 1 , u 2 P rk 1 , k 2 s then, where ϑ " r 2 Γ pα ´1q 2e r .Since v n Ñ v, there exists N ě 1, such that the following relation Thus, all the assumptions of the Schauder fixed point theorem are satisfied.Then, there exists a point v P B R with v " Tv In view of Lemma p14q , we conclude that u " ve ´rt `u P C 2 r0, 1s ˘is a solution of boundary value problem p1q ´p2q .As a result, the proof is complete.`u ptq `r2 u ptq " f pt, u ptqq , 0 ă t ă 1, u p0q " u p1q , u 1 p0q " u 1 p1q , u 1 pξq `ru pξq " η, (44) where r ą 0, f pt, uq " p 1 ptq `p2 ptq u with p 1 , p 2 P C r0, 1s and max |p 2 ptq| tPr0,1s ď r 2 ?π 4

Illustrative Examples
. Thus, the conclusion of Theorem 3.2 applies to the problem.

Proof.
Let u P C 2 r0, 1s be a solution of BVP p12q ´p13q .Since u 2 P C r0, 1s , Def. p2.3q show that c D α´2 0 `u P C r0, 1s and c D α´1 0 `u P C 1 r0, 1s From the relation c D α 0 `u " g ptq ´2r 2 c D α´1 0 `u ´r c D α´2 0 `u and g P C r0, 1s , we have c D α 0 `u P C p0, 1q .Thus, by Lemma 2.7, we have the following relations