This paper investigates the existence of solutions for fractional differential inclusions of order with anti-periodic type integral boundary conditions by means of some standard fixed point theorems for inclusions. Our results include the cases when the multivalued map involved in the problem has convex as well as non-convex values. The paper concludes with an illustrative example.
MSC: 34A60, 34A08.
Keywords:fractional differential inclusions; anti-periodic; integral boundary conditions; fixed point theorems
The topic of fractional differential equations and inclusions has recently emerged as a popular field of research due to its extensive development and applications in several disciplines such as physics, mechanics, chemistry, engineering, etc.[1-5]. An important characteristic of a fractional-order differential operator, in contrast to its integer-order counterpart, is its nonlocal nature. This feature of fractional-order operators (equations) is regarded as one of the key factors for the popularity of the subject. As a matter of fact, the use of fractional-order operators in the mathematical modeling of several real world processes gives rise to more realistic models as these operators are capable of describing memory and hereditary properties. For some recent results on fractional differential equations, see [6-22] and the references cited therein, whereas some recent work dealing with fractional differential inclusions can be found in [23-28].
In this paper, we study a boundary value problem of fractional differential inclusions with anti-periodic type integral boundary conditions given by
where denotes the Caputo derivative of fractional order q, denotes jth derivative of with , is a multivalued map, is the family of all subsets of ℝ, are given continuous functions and ( ).
The present work is motivated by a recent paper , where the authors considered (1.1) with F as a single-valued map. The existence of solutions for problem (1.1) has been discussed for the cases when the right-hand side is convex as well as non-convex valued. The first result is based on the nonlinear alternative of Leray-Schauder type, whereas the second result is established by combining the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values. In the third result, we use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler. Though the methods used are well known, their exposition in the framework of problem (1.1) is new. We recall some preliminary facts about fractional calculus and multivalued maps in Section 2, while the main results are presented in Section 3.
2.1 Fractional calculus
Definition 2.1 Let be an -times absolutely continuous function. Then the Caputo derivative of fractional order ν for h is defined as
where denotes the integer part of the real number ν.
Definition 2.2 The Riemann-Liouville fractional integral of order ν is defined as
provided the integral exists.
Definition 2.3 A function is called a solution of problem (1.1) if there exists a function with , a.e. such that , a.e. and , .
In the sequel, the following lemma plays a pivotal role.
Lemma 2.4 ()
For a given and , the unique solution of the equation , subject to the boundary conditions of (1.1) is given by
2.2 Basic concepts of multivalued analysis
Let denote a normed space equipped with the norm . A multivalued map is
• convex (closed) valued if is convex (closed) for all ;
• bounded on bounded sets if is bounded in for all bounded sets B in , that is, ;
• upper semi-continuous (u.s.c.) on if for each , the set is a nonempty closed subset of , and if for each open set of containing , there exists an open neighborhood of such that ;
• completely continuous if is relatively compact for every bounded set B in .
Remark 2.5 If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph, that is, , , imply that .
Definition 2.6 The multivalued map has a fixed point if there is such that . The fixed point set of the map is denoted by FixG.
Definition 2.7 A multivalued map with nonempty compact convex values is said to be measurable if for any , the function
Let denote the Banach space of all continuous functions from into ℝ with the norm
Let be the Banach space of measurable functions which are Lebesgue integrable and normed by
Definition 2.8 A multivalued map is called Carathéodory if is measurable for each and is upper semicontinuous for almost all . A Carathéodory function is said to be -Carathéodory if, for each , there exists such that for all and for a.e. .
For each , we define the set of selections of F by
Let denote a nonempty closed subset of a Banach space E, and let be a multivalued operator with nonempty closed values. The map G is lower semi-continuous (l.s.c.) if the set is open for any open set B in E. Let A be a subset of . A is measurable if A belongs to the σ-algebra generated by all sets of the form , where is Lebesgue measurable in and is Borel measurable in ℝ. A subset of is decomposable if for all and measurable , the function , where stands for the characteristic function of .
Definition 2.9 Let Y be a separable metric space. A multivalued operator has the property (BC) if N is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values.
Let be a multivalued map with nonempty compact values. Define a multivalued operator associated with F as
which is called the Nemytskii operator associated with F.
Definition 2.10 Let be a multivalued function with nonempty compact values. We say F is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator ℱ is lower semi-continuous and has nonempty closed and decomposable values.
Let be a metric space induced from the normed space . Consider given by
where and . Then is a metric space and is a generalized metric space (see ), where , and .
Definition 2.11 A multivalued operator is called γ-Lipschitz if and only if there exists such that for each and is a contraction if and only if it is γ-Lipschitz with .
3 Existence results
3.1 The Carathéodory case
We recall the following lemmas to prove the existence of solutions for problem (1.1) when the multivalued map F in (1.1) is of Carathéodory type.
Lemma 3.1 (Nonlinear alternative for Kakutani maps) 
LetEbe a Banach space, letCbe a closed convex subset ofE, letUbe an open subset ofC, and . Suppose that is an upper semicontinuous compact map; here denotes the family of nonempty, compact convex subsets ofC. Then either
(i) Fhas a fixed point in , or
(ii) there is an and with .
Lemma 3.2 ()
Let be a Banach space, and let denote a family of nonempty, compact and convex subsets of . Let be an -Carathéodory multivalued map, and let Θ be a linear continuous mapping from to . Then the operator
is a closed graph operator in .
Theorem 3.3Suppose that
(H1) is Carathéodory and has nonempty compact and convex values;
(H2) there exists a continuous nondecreasing function and a function such that
(H3) there exist continuous nondecreasing functions and functions such that
(H4) there exists a constant such that
Then the boundary value problem (1.1) has at least one solution on .
Proof Define the operator by
for . We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As the first step, we show that is convex for each . This step is obvious since is convex (F has convex values), and therefore we omit the proof.
In the second step, we show that maps bounded sets (balls) into bounded sets in . For a positive number ρ, let be a bounded ball in . Then, for each , , there exists such that
Then for we have
Now we show that maps bounded sets into equicontinuous sets of . Let with and . For each , we obtain
Obviously, the right-hand side of the above inequality tends to zero independently of as . As satisfies the above three assumptions, it follows by the Ascoli-Arzelá theorem that is completely continuous.
In our next step, we show that has a closed graph. Let , and . Then we need to show that . Associated with , there exists such that for each ,
Thus it suffices to show that there exists such that for each ,
Let us consider the continuous linear operator given by
Thus, it follows by Lemma 3.2 that is a closed graph operator. Further, we have . Since , therefore, we have
for some .
Finally, we show there exists an open set with for any and all . Let and . Then there exists with such that, for , we have
and using the computations of the second step above, we have
Consequently, we have
In view of (H4), there exists M such that . Let us set
Note that the operator is upper semicontinuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we deduce that has a fixed point which is a solution of problem (1.1). This completes the proof. □
3.2 The lower semicontinuous case
This section deals with the case when F is not necessarily convex valued. Our strategy to deal with this problem is based on the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo for lower semi-continuous maps with decomposable values.
Lemma 3.4 (Bressan and Colombo )
LetYbe a separable metric space, and let be a multivalued operator satisfying the property (BC). ThenNhas a continuous selection, that is, there exists a continuous function (single-valued) such that for every .
Theorem 3.5Assume that (H2), (H3), (H4) and the following condition hold:
(H4) is a nonempty compact-valued multivalued map such that
(a) is measurable;
(b) is lower semicontinuous for each ;
then the boundary value problem (1.1) has at least one solution on .
Proof It follows from (H2) and (H4) that F is of l.s.c. type. Then from Lemma 3.4, there exists a continuous function such that for all .
Consider the problem
Observe that if is a solution of (3.1), then x is a solution to problem (1.1). In order to transform problem (3.1) into a fixed point problem, we define the operator as
It can easily be shown that is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 3.3. So, we omit it. This completes the proof. □
3.3 The Lipschitz case
Here we show the existence of solutions for problem (1.1) with a nonconvex valued right-hand side by applying a fixed point theorem for multivalued maps due to Covitz and Nadler .
Lemma 3.6 ()
Let be a complete metric space. If is a contraction, then .
Theorem 3.7Assume that the following conditions hold:
(A1) is such that is measurable for each ;
(A2) for almost all and with and for almost all ;
(A3) There exist constants , , such that
Then the boundary value problem (1.1) has at least one solution on if
Proof Observe that the set is nonempty for each by the assumption (A1), so F has a measurable selection (see Theorem III.6 ). Now we show that the operator , defined in the beginning of the proof of Theorem 3.3, satisfies the assumptions of Lemma 3.6. To show that for each , let be such that in . Then and there exists such that, for each ,
As F has compact values, we pass onto a subsequence (if necessary) to obtain that converges to v in . Thus, and for each , we have
Next we show that there exists such that
Let and . Then there exists such that, for each ,
By (H3), we have
So, there exists such that
Since the multivalued operator is measurable (Proposition III.4 ), there exists a function which is a measurable selection for U. So, and for each , we have .
For each , let us define
Analogously, interchanging the roles of x and , we obtain
Since is a contraction, it follows by Lemma 3.6 that has a fixed point x which is a solution of (1.1). This completes the proof. □
Consider the following boundary value problem of fractional differential inclusions:
where is a multivalued map given by
For , we have
with , . Here
and , , , .
Clearly, , , , , , , , with , , , and . In view of the condition
we find that . Thus, all the conditions of Theorem 3.3 are satisfied. So, there exists at least one solution of problem (3.2) on .
The authors declare that they have no competing interests.
Each of the authors, BA, SKN and AA, contributed to each part of this work equally and read and approved the final version of the manuscript.
The authors are grateful to the anonymous referees for their useful comments. The research of B. Ahmad and A. Alsaedi was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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