On fractional differential equations and inclusions with nonlocal and average-valued (integral) boundary conditions

This paper is concerned with the existence of solutions for boundary value problems of fractional differential equations and inclusions supplemented with nonlocal and average-valued (integral) boundary conditions. The existence results for the single-valued case (equations) are obtained by means of fixed point theorems due to O’Regan and Sadovski, whereas the existence of solutions for the multivalued case (inclusions) is established via nonlinear alternative for contractive maps. The obtained results are well illustrated by examples.


Introduction
The study of fractional differential equations has recently attracted the attention of many researchers and modelers. The interest in the subject owes to its widespread applications in a variety of applied sciences and engineering disciplines such as biological sciences, ecology, aerodynamics, control theory, viscoelasticity, electro-dynamics of complex medium, electron-analytical chemistry, environmental issues, et cetera. The recent trend in the mathematical modeling of several phenomena indicates the popularity of fractional calculus modeling tools due to the nonlocal characteristic of fractional-order differential and integral operators, which are capable of tracing the past history of many materials and processes; see, for instance, [-] and the references therein.
Differential inclusions, regarded as a generalization of differential equations and inequalities, have very important and interesting applications in optimal control theory and stochastic processes []. In fact, the tools of differential inclusions facilitate the investigation of dynamical systems having velocities not uniquely determined by the state of the system.
Boundary value problems of fractional-order differential equations and inclusions supplemented with several kinds of conditions such as classical, nonlocal, multipoint, periodic/antiperiodic, fractional-order, and integral boundary conditions have extensively been investigated by many researchers. In particular, the study of nonlocal boundary value problems finds interesting applications in physical and chemical processes, where the classical initial/boundary conditions fail to describe some peculiar phenomena occurring inside the domain. On the other hand, integral boundary conditions help to formulate computational fluid dynamics (blood flow) problems in a better way as such conditions allow one to describe the cross-section of vessels in a more realistic arbitrary manner instead of always assuming circular type cross-section []. Also, ill-posed parabolic backward problems in time partial differential equations can be regularized with the aid of integral boundary conditions; see, for example, mathematical models for bacterial selfregularization []. For details and examples, we refer the reader to a variety of results [-]. In a recent article [], the authors studied a boundary value problem of fractional differential equations with nonlocal and average-type integral boundary conditions given by where c D α and c D β denote the Caputo fractional derivatives of orders α and β with n - < α < n (n ≥ ) and  < β < , and f : [, ] × R × R → R and h : C([, ], R) → R are continuous functions. Applying the Leray-Schauder nonlinear alternative, Krasnoselskii's fixed point theorem and Banach's fixed point theorem together with Hölder inequality, some existence results for problem (.) were obtained.
The objective of the present paper is to continue the study initiated in [] and provide a variety in the existence criteria for solutions of the problem at hand. Precisely, we establish two more existence results for problem (.), which are based on fixed point theorems due to O'Regan and Sadovski (Section ). Then we switch onto investigating the multivalued analogue of (.) is a multivalued map, and P(R) is the family of all nonempty subsets of R. In Section , we discuss the existence of solutions for problem (.) by means of the nonlinear alternative for contractive maps.

Preliminaries
In this section, we present some basic definitions on fractional calculus and an auxiliary lemma [, ].
Definition . The fractional integral of order r with the lower limit zero for a function f is defined as provided that the right-hand side is pointwise defined on [, ∞), where is the gamma function defined by (r) = ∞  t r- e -t dt.
Definition . The Riemann-Liouville fractional derivative of order r > , n - < r < n, n ∈ N , of a function f is defined as provided that the function f has absolutely continuous derivatives up to order (n -).
Definition . The Caputo derivative of order r of a function f : We need the following known lemma [].

Lemma . Let y ∈ AC[, ] and x ∈ AC n [, ]. Then the linear problem
is equivalent to the fractional integral equation

Existence results for a single-valued problem
In view of Lemma ., we introduce the operator F : X → X by which can be expressed as where F , : X → X are given by For computational convenience, we set the notations Our first existence result relies on a fixed point theorem of O'Regan [].

is continuous and completely continuous, and F
for all x, y ∈Ū). Then, either (C) F has a fixed point u ∈Ū; or (C) there exist a point u ∈ ∂U and λ ∈ (, ) with u = λF(u), whereŪ and ∂U, respectively, represent the closure and boundary of U.
In the proof of the next result, we use the notations Theorem . Assume that: , R) and nondecreasing functions ψ i :

Then the boundary value problem (.) has at least one solution on [, ].
Proof By assumption (A  ), there exists a number r  >  such that We shall show that the operators F  and F  defined by (.) and (.), respectively, satisfy all the conditions of Lemma .. Step Consequently, we have Hence, the operator F  is equicontinuous. Thus, it follows by the Arzelá-Ascoli theorem that which proves the continuity of F  . This completes the proof of Step .
Step . The operator F  :¯ r  → X is contractive. This is a consequence of (A  ). Indeed, for x, y ∈ X, we have which, by taking the supremum for t ∈ [, ], yields Consequently, we get which, in view of (A  ), implies that F  is a contraction.
Step . The set F(¯ r  ) is bounded. Assumptions (A  ) and (A  ) imply that Hence, for any x ∈¯ r  . This, together with the boundedness of the set F  (¯ r  ), implies that the set F(¯ r  ) is bounded.
Step . Finally, we will show that case (C) in Lemma . does not hold. On the contrary, suppose that (C) holds. Then, we have that there exist λ ∈ (, ) and x ∈ ∂ r  such that x = λFx. So, we have x X = r  and Further, we have that Thus, we have which yields a contradiction, Thus, the operators F  and F  satisfy all the conditions of Lemma .. Hence, the operator F has at least one fixed point x ∈¯ r  , which is the solution of problem (.). This completes the proof.
Our second existence result is based on Sadovskii's fixed point theorem. Let us first recall some auxiliary material before proceeding further. Proof Let B r = {x ∈ X : x ≤ r} be a closed bounded and convex subset of X, where r will be fixed later. We define the map F : B r → X by where F  and F  are defined by (.) and (.), respectively. Notice that problem (.) is equivalent to the fixed point problem F(x) = x. Step , where i , i = , , , are defined by (.) and (.). As in Theorem ., Step , we can prove that and, as in Step , we can get

Consequently,
Step . F  is compact. This was proved in Theorem ., Step .
Step . F  is continuous and γ -contractive.
To show the continuity of F  for t ∈ [, ], let us consider a sequence x n converging to x. Then, as in Step  of the proof of Theorem ., we can show that which implies that F  is continuous. Also, F  is γ -contractive by Theorem . (Step ) with γ = ( + η +  (-β) ) < .
Step . F is condensing. Since T  is a continuous γ -contraction and T  is compact, by Lemma ., F : From the previous four steps we conclude by Lemma . that the map F has a fixed point, which, in turn, implies that problem (.) has a solution.
Example . Consider the following boundary value problem: where h(x) =   x(/), η = /, ξ = /, and a and b are suitably chosen real numbers.
Observe that With the given values, we find that  .,  .,  .. Further, < ( + η + / (β)) - holds since ( + η + / (β)) - . >   = , and (A  ) is satisfied for  < a < / and for any finite real value of b since  is relatively compact for every B ∈ P b (X). If a multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, that is, x n → x * , y n → y * , y n ∈ G(x n ) imply y * ∈ G(x * ). A map G has a fixed point if there is x ∈ X such that x ∈ G(x). The fixed point set of a multivalued operator G will be denoted by Fix G. A multivalued map G : [; ] → P cl (R) is said to be measurable if for every y ∈ R, the function t → d(y, G(t)) = inf{|y -z| : z ∈ G(t)} is measurable.
a.e. on [, ] and for all x , y ≤ a and for a.e. t ∈ [, ].
For each y ∈ C([, ], R), define the set of selections of F by The following lemma will be used in the sequel.
To prove our main result in this section, we use the following form of the nonlinear alternative for contractive maps ([], Corollary .).
Theorem . Let X be a Banach space, and D a bounded neighborhood of  ∈ X. Let Z  : X → P cp,c (X) and Z  :D → P cp,c (X) be two multivalued operators satisfying (a) Z  is contraction, and (b) Z  is u.s.c. and compact. Then, if G = Z  + Z  , then either (i) G has a fixed point inD, or (ii) there are a point u ∈ ∂D and λ ∈ (, ) with u ∈ λG(u).
Theorem . Assume that (A  ) and (A  ) hold. In addition, we suppose that: Proof To transform problem (.) into a fixed point problem, we introduce the operator N : X → P(X) as follows: Now, we define the operator A  : X → X by and the multivalued operator A  : X → P(X) by Observe that N = A  + A  . We shall show that the operators A  and A  satisfy all the conditions of Theorem . on [, ]. Also, we establish that the operators A  and A  are such that A  , A  : B r → P cp,c (X), where B r = {x ∈ X : x X ≤ r} is a bounded set in C([, ], R). First, we prove that A  is compact-valued on B r . Note that the operator A  is equivalent to the composition L • S F , where L is the continuous linear operator from L  ([, ], R) into X defined by Suppose that x ∈ B r is arbitrary and let {v n } be a sequence in S F,x . Then, by the definition of S F,x , we have v n (t) ∈ F(t, x(t), c D β x(t)) for almost all t ∈ [, ]. Since F(t, x(t), c D β x(t)) is compact for all t ∈ J, there is a convergent subsequence of {v n (t)} (we denote it by {v n (t)} again) that converges in measure to some v(t) ∈ S F,x for almost all t ∈ J. On the other hand, L is continuous, so L(v n )(t) → L(v)(t) pointwise on [, ].
In order to show that the convergence is uniform, we have to show that {L(v n )} is an equicontinuous sequence. Let t  , t  ∈ [, ] with t  < t  . Then, we have supplemented with the boundary conditions (.), where Clearly, where p(t) = / √  + t, ψ  (|x|) = |x|, ψ  (|x|) = π . Using the data of Example . and condition (H  ), we find that M > M  .. Hence, the hypothesis of Theorem . is satisfied, which implies that the fractional differential inclusion (.) together with (.) has a solution on [, ].