Weighted fractional differential equations with infinite delay in Banach spaces

Abstract This paper is devoted to the study of fractional differential equations with Riemann-Liouville fractional derivatives and infinite delay in Banach spaces. The weighted delay is developed to deal with the case of non-zero initial value, which leads to the unboundedness of the solutions. Existence and uniqueness results are obtained based on the theory of measure of non-compactness, Schaude’s and Banach’s fixed point theorems. As auxiliary results, a fractional Gronwall type inequality is proved, and the comparison property of fractional integral is discussed.


Introduction
In this paper, we study nonlinear functional fractional differential equation with weighted infinite delay in an abstract Banach space X , of the form where 0 <˛Ä 1, D˛is the Riemann-Liouville fractional derivative, Q y.t / D t 1 ˛y .t /, f W .0; b B ! B is a given function satisfying some assumptions, and B is the phase space that will be specified later. We give the definition of solutions, and investigate the existence and continuous dependence of solutions to such equations in the space C 1 ˛. .a; bI X /.
In the past several decades fractional differential equations have attracted a considerable interest in both mathematics and applications, since they have been proved to be valuable tools in modeling many physical phenomena. There has been a significant development in fractional differential equations in the past decades. See, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein. Fractional differential equations in Banach spaces are also wildly studied [1,[11][12][13][14]. Among these works, some authors study functional fractional differential equations [2,3,6,8,11]. For example, in [2], Benchohra et al. studied fractional order differential equations  However, it is known that the Riemann-Liouville fractional derivative of a function y is unbounded at some neighborhoods of the initial point 0, except that y.0/ D 0. For this reason, when y.0/ ¤ 0, the solutions to the functional fractional differential equations given in the mentioned papers may not be well-defined. An example can be found in [6]. In order to remedy this defect, in [6], the author modified the model and studied the weighted fractional differential equations with infinite delay. Existence and continuous dependence results of solutions are obtained in finite dimensional spaces.
In this paper, we continue the work of [6], to study the weighted fractional differential equations with infinite delay in Banach spaces. The difficulty is that the bounded subsets in Banach spaces are not compact in general. To get a compact subset in the space of continuous functions, we have to suppose some conditions involving the measure of non-compactness on the nonlinear term. The function space C 1 ˛. .0; b W X / we encountered in this paper is a space of unbounded functions, which is different from C.OE0; b W X /. We need some further discussion on the relevant subsets. As auxiliary results, we also prove a Gronwall type inequality for fractional differential equations, and discuss the comparison property of fractional integral.

Preliminaries and lemmas
In this section we collect some definitions and results needed in our further investigations.
Let .X; k k/ be a Banach space. Denote by C.OEa; bI X / the space of all continuous X valued functions defined on OEa; b with the supremum norm, and L 1 loc ..a; b/I X / the space of Bochner integrable functions u W .a; b/ ! X with the norm kuk D R b a ku.t /kdt. We also consider the space C r ..a; bI X / consisting of all continuous functions f W .a; b ! X such that lim t !a .t a/ r f .t / exists, with the norm kf k C r D supfk.t a/ r f .t /kI t 2 .a; bg.
provided the right side is pointwisely defined, where . / denotes the well-known gamma function, i.e., .z/ D R 1 0 e t t z 1 dt.

Definition 2.2 ([5]
). Let˛> 0 be fixed and n D OE˛ C 1. The Riemann-Liouville fractional derivative of order˛of h W .a; b ! X at the point t is defined by .t s/ n ˛ 1 h.s/ds; t 2 OEa; b provided the right side is pointwisely defined, where OE˛ denotes the integer part of the real number˛.
When 0 <˛< 1, then For simplicity, when a D 0, we denote D0 and I0 by D˛and I˛, respectively.
for t > 0 and some constant C 2 R.
In the literature devoted to equations with infinite delay, the selection of the state space B plays an important role in the study of both qualitative and quantitative theory. A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato [15]. For a detailed discussion on the topic, we refer to the book by Hino et al [16]. , and x. / is continuous on OEa; b, then for every t 2 OEa; b, the following conditions hold: The theory of the measure of noncompactness was initiated by Kuratowski in 1930s and has played a very important role in nonlinear analysis in recent decades. It is often applied to the theories of differential and integral equations as well as to the operator theory and geometry of Banach spaces [17][18][19][20][21][22]. One of the most important examples of measure of noncompactness is the Hausdorff's measure of noncompactnessˇY , which is defined by Y .B/ D inffr > 0I B can be covered with a finite number of balls of radius equal to rg for bounded set B in a Banach space Y .
The following properties of Hausdorff's measure of noncompactness are well known.
where d Y .B; C / means the nonsymmetric (or symmetric) Hausdorff distance between B and C in Y .
nD1 is a decreasing sequence of bounded closed nonempty subsets of Y and lim n!C1ˇY .W n / D 0, then T C1 nD1 W n is nonempty and compact in Y .
In this paper we denote byˇthe Hausdorff measure of noncompactness of X and byˇc the Hausdorff measure of noncompactness of C.OEa; bI X /. To discuss the existence we need the following lemmas in this paper.

Auxiliary results
In the qualitative theory of differential and Volterra integral equations, especially in establishing of boundedness and stability, the Gronwall type inequalities play a very important role. See, for example, [23][24][25][26]. In 1919, T. H. Gronwall proved a remarkable inequality which has attracted and continues to attract considerable attention in the literature.
holds for all t 2 OEa; b. Then we have In 1967, S. C. Chu and F. T. Metcalf proved in [23] a generalized Gronwall inequality for convolution type integral equations as follows.
, is the resolvent kernel, and the K i .i D 1; 2; / are the iterated kernels of K.
The fractional integral is in fact a kind of convolution of functions. However, the kernel of the fractional integral Now we suppose that Then we have Lemma 3.4. Let the functions u, p and q be continuous and nonnegative on OE0; b. If the inequality (4) holds, then , is the resolvent kernel, and the k i .i D 1; 2; / are the iterated kernels defined by (5).
Proof. From (4), we have for all t 2 OE0; b. By induction, we have for all n 2 N, for all t 2 OE0; b. Now we prove that for all n 2 N and 0 Ä s < t Ä b. In fact, Using the ratio test it is easily seen that P 1 iDOE1=˛C1 kpk 1 b˛ i .i˛/ is convergent. Hence P 1 i DOE1=˛C1 k i .t; s/ converges absolutely and uniformly for 0 Ä s < t Ä b. This, combined with (9), implies that R t 0 R.t; s/p.s/ds D R t 0 P 1 iD1 k i .t; s/p.s/ds uniformly for t 2 OE0; b. The convergence of At last, letting n ! 1 in (7), we get the desired inequality (6).  Next we discuss the comparison property of the fractional integral. It is well-known that if an integrable function f is nonnegative on OE0; b, then R t 2 t 1 f .s/ds 0 for any t 1 ; t 2 2 OE0; b with t 1 < t 2 . Equivalently, the function So for 0 <˛< 1 2 , 2˛ 1 < 0, and the function F . / is strictly decreasing for t > 0.
Fortunately, we have the following relatively weak result. Proof. We first suppose that f 2 C 1 OE0; b is nonnegative and nondecreasing. Then f 0 .t / 0 for all t 2 OE0; b. It follows that For the general case that f 2 C OE0; b, we choose a sequence of nonnegative and nondecreasing functions ff n g 2 C 1 OE0; b such that lim n f n D f in C OE0; b. Then each F n D I0 f n is nondecreasing, and the limit F D lim n F n is therefore nondecreasing.

Existence results
In this section, we study the existence of solutions to the weighted functional differential equations (1)- (2). We begin with the definition of solutions to these equations. for some constant C . See also [5,Theorem 2.23].
For the forthcoming analysis, we need the following hypothesis.
(H1) f W .0; b B ! X is continuous. (H2) There exist a nonnegative Á 2 L p .0; b with p > 1=˛and a continuously non-decreasing function Then .W; kzk W / becomes a Banach space. Define an operator P W W ! W by .t s/˛ 1 f .s; e s C e z s /ds C .0/t˛ 1 ; t > 0: We will prove by the Schauder's fixed point theorem that P has at least a fixed point z, and hence z C e is a solution to (1)- (2). First note that the continuity of P can be derived by .H 2/ and the Lebesgue dominated convergence theorem. We now show that P maps bounded subsets in W into bounded subsets. Let B r D fz 2 W I kzk W Ä rg. Then, for any z 2 B r and t 2 .0; b, we have where M b D sup 0ÄsÄb M.s/. It follows from (H2) and Holder's inequality that .t s/ .˛ 1/q ds/ 1=q kÁk p C k .0/k: where kÁk p D . R b 0 jÁ.s/j p ds/ 1=p and 1=p C 1=q D 1; .˛ 1/q > 1. Therefore, kP zk W Ä l for every z 2 B r , which implies that P maps bounded subsets into bounded subsets in W .
Next we prove that e PB is equicontinuous for every bounded subsets B W . Let z 2 B r and t 1 ; t 2 2 .0; b with t 1 < t 2 , then we have kf .s; e s C e z s /kds C .˛ 1/qC1/ 1=q and r 2 D OE.˛ 1/qC1=q > 0. It follows that kt 1 2 .P z/.t 2 / t 1 1 .P z/.t 1 /k ! 0 as t 2 t 1 ! 0, and the convergence is independent of z 2 B r , which implies that the set e PB r is equicontinuous.
Now we have to verify that there exists a closed convex bounded subset B r 0 , such that PB r 0 B r 0 . We derive from inequlity (12), i.e., lim r!C1 .r/ r Define B r 0 D fz 2 W I kzk W Ä r 0 g. Then B r 0 is closed, convex and bounded in W . Then, for every z 2 B r 0 and t 2 .0; b, similar to the proof of P maps bounded subsets in W into bounded subset, we have It then follows that kP zk W Ä r 0 for all z 2 Br 0 , and hence PB r 0 B r 0 . Now we prove that there exists a compact subset M B r 0 such that PM M . We first construct a series of sets fM n g B r 0 by M 0 D B r 0 ; M 1 D convPM 0 ; M nC1 D convPM n ; n D 1; 2; : From the above proof it is easy to see that M nC1 M n for n D 1; 2; and each f M n is equi-continuous. Further, from Lemma 2.5, 2.6 and 2.7 we can derive thať If z 2 W is a fixed point of P , then y D z C e is a solution of (1)-(2). Let K.b/ D supfK.t /I t 2 .0; bg, where K.:/ is the function that appeared in Definition 2.4. Let N D OEb.2LK b b 1 ˛= .1 C˛// 1=˛ , and h i D i b=N . Then 0 < h 1 < h 2 < ::: < h N D b and for i D 1; 2; :::; N: We first focus on the interval .0; h 1 . Let W 1 D fz W . 1; h 1 ! X I zj .0;h 1 2 C 1 ˛. .0; h 1 I X /; zj 0 D 0g and define kzk W 1 D kz 0 k B C supfjt 1 ˛z .t /jI 0 < t < h 1 g D supfjt 1 ˛z .t /jI 0 < t < h 1 g for z 2 W 1 .Then .W 1 ; kzk W 1 / is a Banach space. Define the operator P 1 W W 1 ! W 1 by For z; z 2 W 1 and t 2 .0; h 1 , we have we have and hence From (20) and the Banach contraction principle we know that there exists a unique z 2 W 1 satisfying for t 2 .0; h 1 , which is the unique solution to the integral equation (13)  .t s/˛ 1 f .s; e s C e z s /ds C .0/t˛ 1 (24) Since the function z is uniquely defined on .0; h 1 , the second integral can be considered as a known function. Using the same arguments as above, we can obtain that there exists a unique function z 2 W 2 satisfying z.t / D 1 .˛/

An example
In this section, we discuss an example to illustrate our results. For any real constant > 0 we set In [2], the authors discussed an example similar to the problem (26)-(27) without weight. They have to suppose that '.0/ D 0 since otherwise the solution y may be unbounded at the right neighbourhood of 0. Here we do not need this restriction. Furthermore, we don't have any restriction on the constant c.