Nonlocal fractional functional differential equations with measure of noncompactness in Banach space

In this paper, we are concerned with the following fractional functional differential equations with nonlocal initial conditions in Banach space $$\begin{aligned} \hbox {D}^{\alpha }x(t)=Ax(t)+f(t,x(t),x_{t}),\ \ t\in [0, T], \ \ x(0)=\phi +g(x). \end{aligned}$$Dαx(t)=Ax(t)+f(t,x(t),xt),t∈[0,T],x(0)=ϕ+g(x).By virtue of the theory of measure of noncompactness associated with Darbo’s fixed point theorem, upon making some suitable assumptions, some existence results of mild solutions are obtained. Moreover the results obtained are utilized to study the existence of solutions to fractional parabolic equations as an illustrative example to show the practical usefulness of the analytical results.


Introduction
In this paper, we are concerned with the nonlocal initial value problem D a xðtÞ ¼ AxðtÞ þ f ðt; xðtÞ; x t Þ; t 2 ½0; T; where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators T(t) in a separable Banach space X, f : ½0; T Â X Â C ! X; g : L p ð½0; T; XÞ ! X; are given X-valued functions. The fractional derivative is understood in the Riemann-Liouville sense. The aim of this paper is to study the existence of mild solutions for the fractional functional differential Eq. (1.1) in a separable Banach space. The technique used here is the measure of noncompactness associated with Darbo's fixed point theorem.
The fractional derivative is understood in the Riemann-Liouville sense. The origin of fractional calculus goes back to Newton and Leibnitz in the seventieth century. One observes that fractional order can be very complex in viewpoint of mathematics and they have recently proved to be valuable in various fields of science and engineering. In fact, one can find numerous applications in electrochemistry, electromagnetism, viscoelasticity, biology and hydrogeology. For example space-fractional diffusion equations have been used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium [1,2] or to model activator-inhibitor dynamics with anomalous diffusion [3]. For details, see [4][5][6][7] and the references therein.
Differential equations of fractional order have appeared in many branches of physics and technical sciences [8,9]. It has seen considerable development in the last decade, see  and the references therein. Recently, the existence and uniqueness problem for various fractional differential equations were considered by Ahmad [10], Bhaskarc [11], Lakshmikantham and Leela [12] et al. The nonlocal Cauchy problem was considered by Anguraj, Karthikeyan and N'Guérékata [13], and the importance of nonlocal initial conditions in different fields has been discussed in [6,7] and the references therein.
The nonlocal problem (1.1) was motivated by physical problems. Indeed, the nonlocal initial condition xð0Þ ¼ / þ gðxÞ can be applied in physics with better effect than the classical initial condition xð0Þ ¼ /. For this reason, the problem (1.1) has gotten considerable attention in recent years, see [30][31][32] and the references therein. See also [33][34][35] and the references therein for recent generalizations of problem (1.1) to various kinds of differential equations.
To the best of our knowledge, the existence of mild solutions for the fractional functional differential Eq. (1.1) with nonlocal initial conditions using the theory of measure of noncompactness is a subject that has not been treated in the literature. Our purpose in this paper is to establish some results concerning the existence of mild solutions for equations that can be modeled in the form (1.1) by virtue of the theory of measure of noncompactness associated with Darbo's fixed point theorem. Upon making some appropriate assumptions, some sufficient conditions for the existence of mild solutions for the fractional functional differential Eq. (1.1) are given. It is worthwhile mentioning that the cases of T(t) or f compact and of f Lipschitz are special cases of our conditions. Also we hope that the concept of measure of noncompactness considered here may be a stimulant for further investigations concerning solutions of fractional differential equations of other types.
The rest of this paper is organized as follows. In ''Notations, definitions and auxiliary facts'' section, we give some notations, definitions and auxiliary facts. ''Main results'' section contains the main results of this paper with two existence theorems. An example is given to illustrate our results in ''Applications'' section.

Notations, definitions and auxiliary facts
Let ðX; k Á kÞ be a real separable Banach space. Denote by Cð½0; T; XÞ the space of X-valued continuous functions on [0,T] and by L p ð½0; T; XÞ the space of X-valued measurable functions on ½0; T with Z T 0 kxðtÞk p dt\1; provided with norm Let r be a given positive real number, if x : ½Àr; T ! X, define x t 2 Cð½Àr; 0; XÞ by x t ðhÞ ¼ xðt þ hÞ; for À r h 0; and denote kxk C ¼ sup n kxðtÞk : t 2 ½Àr; 0 o ; for x t 2 Cð½Àr; 0; XÞ: We need some basic definitions and properties of the fractional calculus theory which are used further in this paper.
Definition 2.1 [36] The fractional integral of order a [ 0 with the lower limit t 0 for a function f is defined as provided the right-hand side is pointwise defined on ½t 0 ; 1Þ, where f is an abstract continuous function and CðaÞ is the Gamma function [36].  In this paper, we denote v by the Hausdorff's measure of noncompactness of X and denote v p by the Hausdorff's measure of noncompactness of L p ð½0; T; XÞ. To discuss the problem in this paper, we need the following lemmas.
where B and convB mean the closure and convex hull of B, respectively; for any bounded subset B DðQÞ, where Z is a Banach space; CÞ means Hausdorff distance between B and C in Y; (9) If fW n g 1 n¼1 is a decreasing sequence of bounded closed nonempty subsets of Y and lim n!1 for any bounded closed subset C W.
In 1955, Darbo [38] proved the fixed point property for a-set contraction (i.e., aðSðAÞÞ kaðAÞ with k 2 ½0; 1) on a closed, bounded and convex subset of Banach spaces. Since then many interesting works have appeared. For example, in 1972, Sadovskii [39] proved the fixed point property for condensing functions (i.e., aðSðAÞÞ\aðAÞ with aðAÞ 6 ¼ 0) on closed, bounded and convex subset of Banach spaces. It should be noted that any a-set contraction is a condensing function, but the converse is not true (see [40]). In 2007, Hajji and Hanebaly [41] proved the existence of a common fixed point for commuting mappings satisfying where a is the measure of noncompactness on a closed, bounded and convex subset X of a locally convex space X, T i and S are continuous functions from X to X with T i , and in addition, are affine or linear. Furthermore, for every i 2 I, T i are equal to the identity function, moreover the obtain in particular Darbo's (see [38]) as well as Sadovskii's (see [39]) fixed point theorems, which are used to study the existence of solutions of one equation. Recently, Hajji [42] present common fixed point theorems for commuting operators which generalize Darbo's and Sadovskii's fixed point theorems, furthermore, as examples and applications, they study the existence of common solutions of equations in Banach spaces using measure of noncompactness. Our purpose in this paper is to establish some results concerning the existence of mild solutions for equations that can be modeled in the form (1.1) by virtue of the theory of measure of noncompactness associated with Darbo's fixed point theorem.

Lemma 2.2 ([37], Darbo-Sadovskii) If W Y is bounded closed and convex, the continuous map
We call B & Lð½0; T; XÞ uniformly integrable if there exists g 2 Lð½0; T; R þ Þ such that kuðsÞk gðsÞ; for all u 2 B and a.e. s 2 ½0; T: ds: where is equicontinuous for all bounded set B in X and t [ 0. It is known that the analytic semigroup is equicontinuous.
The following lemma is obvious.
Proof Note that, 1 CðaÞ Estimating the terms on the right-hand side of (2.1) yields It follows from the assumption of gðsÞ that I 0 ! 0 as h ! 0. Using H€ older inequality, one obtains II 0 ! 0 as h ! 0 and e ! 0.
For II, one has ðt À sÞ aÀ1 gðsÞds ¼I 00 þ II 00 ; where ðt À sÞ aÀ1 gðsÞds: Using the assumption that T(t) is equicontinuous in X, integrating with s ! gðsÞ 2 Lð½0; T; R þ Þ, one sees that I 00 ! 0 as h ! 0. From the assumption of gðsÞ and H€ older inequality, it is easy to see that II 00 ! 0 as h ! 0 and e ! 0. Therefore, the family of functions
Define F : L p ð½Àr; T; XÞ ! L p ð½Àr; T; XÞ by Math Sci (2015) 9:59-69 63 Thus, one has Thus, we conclude that Fx exists. Second, we show that there is a k 2 N such that FðB k Þ B k .
Suppose contrary that for each k 2 N there is x k 2 B k and t k 2 ½Àr; T such that If t k 2 ½Àr; 0, then and if t k 2 ½0; T, one has Divided by k on both sides of (3.2), one has which contradicts the hypotheses (H 4 ). Therefore, there is a k 2 N such that FðB k Þ B k . From now on, we will restrict F on such B k . Third, we will verify that F is a v C -contraction. To this end, from the hypothesises (H 2 ) (1) and (3), one can prove that F is continuous by the continuity of g and of the operator f. The hypothesis (H 1 ) and Lemma 2.5 imply that FB k & Cð½0; T; XÞ is bounded and equicontinuous on [0, T], so is convðFB k Þ. As X is separable, from Lemma 2.  If X is a Hilbert space, and / is a proper, convex and lower semicontinuous function from X into ðÀ1; þ1Þ, then its subdifferential oU is m-accretive. Let A ¼ oU then A generates an equicontinuous nonlinear contraction semigroup (cf. [45,46]). From above we can get the following existence result.
Corollary 3.1 If X is a separable Hilbert space, the hypotheses (H 2 )-(H 4 ) are true, and A ¼ oU with / is proper, convex and lower semicontinuous from X into ðÀ1; þ1Þ. Then the nonlocal Eq. (3.1) has at least one integral solution provided that Let us now formulate an existence result when g is uniformly bounded.