Fractional functional differential inclusions with finite delay
Introduction
This paper deals with the existence of solutions, for initial value problems (IVP for short), for fractional differential inclusions with infinite delay where is the standard Riemann-Liouville fractional derivative, is a multivalued map with compact values, is the family of all nonempty subsets of , , and .
For any function defined on and any , we denote by the element of defined by Here represents the history of the state from time up to the present time .
Differential equations with fractional order have been recently proved to be valuable tools in the modeling of many physical phenomena [10], [20], [21], [36], [37], [3], [9], [18], [31], [32], [41], [43].
The arbitrary (fractional) order integral (Riemann–Liouville) operator is a singular integral operator, and for details on the arbitrary fractional order differential operator, see the references [14], [15], [17]. There has been a significant development in fractional differential equations in recent years; see the monographs of Miller and Ross [40], Samko et al. [45] and the papers of Diethelm et al. [10], [12], [13], El-Sayed [16], Mainardi [36], Momani and Hadid [38], Momai et al. [39], Podlubny et al. [44], and Yu and Gao [48], [49].
Very recently, some basic theory for initial value problems of fractional differential equations involving the Riemann–Liouville differential operator was discussed by Benchohra et al. [5] and Lakshmikantham and Vastala [33], [34], [35].
El-Sayed and Ibrahim [17] and Benchohra et al. [6] initiated the study of fractional multivalued differential inclusions.
In the case where , existence results for fractional boundary value problems and a relaxation theorem were studied by Ouahab [42].
Our goal in this paper is to complement and extend some of the above results by giving some existence results and a relaxation theorem for Problem (1) and (2). Also,we prove that the set of solutions is compact.
Section snippets
Preliminaries
Throughout this paper we will use the following notations: , , , .
Let be a metric space induced from the normed space . Consider given by where . Then is a metric space and is a generalized metric space; see [30]. Definition 2.1 A multivalued operator is called -Lipschitz if
Filippov’s theorem
Now, we present a Filippov-type result for the problem (1) and (2). Let , and be a solution of the problem where . Theorem 3.1 Assume the following conditions: is a function satisfying: For all is measurable; The map is integrable where is defined in problem(3)–(4).
There exists a function such that
Relaxation theorem
In this section, we examine to what extent the convexification of the right-hand side of the inclusion introduces new solutions. More precisely, we want to find out if the solutions of the nonconvex problem are dense in those of the convex one. Such a result is known in the literature as a Relaxation theorem and has important implications in optimal control theory. It is well-known that in order to have optimal state-control pairs, the system has to satisfy certain convexity requirements. If
Existence results
In this section, we present the existence result for the problem (1), (2). Our considerations are based on the following fixed point theorem for contractive multivalued operators given by Covitz and Nadler [8] (see also Deimling, [11] Theorem 11.1). Lemma 5.1 Let be a complete metric space. If is a contraction, then .
- (A1)
; is measurable, for each .
- (A2)
There exists a function
Some properties of solution sets
We shall examine the properties of the map defined by, where
Lemma 6.1 Assume that all the conditions ofTheorem 5.2are satisfied. Then the set-valued map is Lipschitz, with Lipschitz constant equal to Proof It is clear that is a closed set in . We show that Let and . Since is a
References (49)
- et al.
Positive solutions for boundary value problem of nonlinear fractional differential equations
J. Math. Anal. Appl.
(2005) - et al.
Existence results for fractional order functional differential equations with infinite delay
J. Math. Anal. Appl.
(2008) - et al.
Existence and uniqueness for a nonlinear fractional differential equation
J. Math. Anal. Appl.
(1996) - et al.
Analysis of fractional differential equations
J. Math. Anal. Appl.
(2002) Nonlinear functional differential equations of arbitrary orders
Nonlinear Anal.
(1998)- et al.
Multivalued fractional differential equations
Appl. Math. Comput.
(1995) Modification of the application of a contraction mapping method on a class of fractional differential equation
Appl. Math. Comput.
(2003)- et al.
Damping description involving fractional operators
Mech. Syst. Signal Process.
(1991) - et al.
A fractional calculus approach of self-similar protein dynamics
Biophys. J.
(1995) - et al.
Integral conditional expectations, and martingales of multivalued functions
J. Multivariate Anal.
(1977)
Existence of fractional differential equations
J. Math. Anal. Appl.
Differential Inclusions
Set-Valued Analysis
Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces
Appl. Anal.
Existence results for fractional functional differential inclusions with infinite delay and application to control theory
Fract. Calc. Appl. Anal.
Multivalued contraction mappings in generalized metric spaces
Israel J. Math.
On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity
Multivalued Differential Equations
Numerical solution of fractional order differential equations by extrapolation
Numer. Algorithms
Fractional order evolution equations
J. Fract. Calc.
Fractional order diffusion-wave equations
Internat. J. Theoret. Phys.
Fixed point results for multivalued contractions on gauge spaces, set valued mappings with applications in nonlinear analysis
Topological Fixed Point Theory of Multivalued Mappings
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