Showing papers by "Mouffak Benchohra published in 2015"

Existence and Stability Results for Nonlinear Boundary Value Problem for Implicit Differential Equations of Fractional Order

Abstract: In this paper, we establish sucient conditions for the existence and stability of solutions for a class of boundary value problem for implicit fractional dierential equations with Caputo fractional derivative. The arguments are based upon the Banach contraction principle. Two examples are included to show the applicability of our results. 2010 Mathematics Subject Classication. 26A33, 34A08.

On stability for nonlinear implicit fractional differential equations

Abstract: The purpose of this paper is to establish some types of Ulam stability: Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for a class of implicit fractional-order differential equation.

New Stability Results for Partial Fractional Differential Inclusions with Not Instantaneous Impulses

Abstract: Abstract In this work, we discuss the existence and Ulam's type stability concepts for a class of partial functional differential inclusions with not instantaneous impulses and a nonconvex valued right hand side in Banach spaces. An example is also provided to illustrate our results.

Existence results for nonlinear implicit fractional differential equations with impulse

Abstract: In this paper, we establish the existence and uniqueness of solution for a class of initial value problem for implicit fractional differential equations with Caputo fractional derivative. The arguments are based upon the Banach contraction principle, Schauder’ fixed point theorem and the nonlinear alternative of Leray-Schauder type. As applications, two examples are included to show the applicability of our results.

Integrable solutions for implicit fractional order functional differential equations with infinite delay

Abstract: In this paper we study the existence of integrable solutions for initial value problem for implicit fractional order functional differential equations with infinite delay. Our results are based on Schauder type fixed point theorem and the Banach contraction principle fixed point theorem.

Existence and Stability of Solutions for Hadamard-Stieltjes Fractional Integral Equations

Abstract: We give some existence results and Ulam stability results for a class of Hadamard-Stieltjes integral equations. We present two results: the first one is an existence result based on Schauder’s fixed point theorem and the second one is about the generalized Ulam-Hyers-Rassias stability.

Functional Differential Equations With Delay and Random Effects

Abstract: The authors study the existence of mild solutions to a functional differential equation with delay and random effects. They use a random fixed point theorem with a stochastic domain. An example is included to illustrate their results.

Existence and Ulam Stability for Partial Impulsive Discontinuous Fractional Differential Inclusions in Banach Algebras

Abstract: In this paper, we investigate some existence and Ulam’s type stability concepts of fixed point inclusions for a class of partial discontinuous fractional-order differential inclusions with impulses in Banach Algebras. Our results are obtained using weakly Picard operators theory.

Existence and Ulam Stability Results for Quadratic Integral Equations

Abstract: In this paper, we investigate some existence and Ulam's type stability concepts for functional quadratic integral equations. The proof is based on Schauder's fixed point theorem.

Global existence and stability results for partial fractional random differential equations

Abstract: Abstract We investigate some existence and stability results for the Darboux problem of partial fractional random differential equations in Banach spaces. Our existence results are based upon some fixed point theorems.

Qualitative analysis of second order functional evolution inclusions

Abstract: In this paper we investigate the existence and attractivity of mild solutions on infinite intervals to second order semilinear evolution inclusion with infinite delay in a Banach space. The proofs of the main results are based on Bohnenblust-Karlin’s fixed point theorem and the theory of evolution system. AMS (MOS) Subject Classification. 34G20, 34G25, 34K20, 34K30, 34A60.

Upper and lower solutions method for partial Hadamard fractional integral equations and inclusions

Abstract: In this paper we use the upper and lower solutions method combined with Schauder’s fixed point theorem and a fixed point theorem for condensing multivalued maps due to Martelli to investigate the existence of solutions for some classes of partial Hadamard fractional integral equations and inclusions.

Global Existence Results for Functional Differential Inclusions with Delay

Abstract: Our aim in this work is to study the existence of solutions of a functional differential inclusion with finite delay. We use the Bohnenblust–Karlin fixed-point theorem for the existence of solutions.

Partial Functional Evolution Inclusions with Infinite Delay

Abstract: In this chapter, we provide sufficient conditions for the existence of mild solutions on the semi-infinite interval \(J = \mathbb{R}_{+}\) for some classes of first order partial functional and neutral functional differential evolution inclusions with finite delay by using the recent nonlinear alternative of Frigon [114, 115] for contractive multi-valued maps in Frechet spaces [116], combined with the semigroup theory [16, 20, 168].

Second Order Functional Differential Equations with Delay

Abstract: In this chapter, we present some existence of global mild solutions for some classes of second order semi-linear functional equations with delay.

Impulsive Semi-linear Functional Differential Equations

Abstract: In this chapter, we shall prove the existence of mild solutions of first order impulsive functional equations in a separable Banach space. Our approach will be based for the existence of mild solutions, on a fixed point theorem of Burton and Kirk [88] for the sum of a contraction map and a completely continuous map.

Asymptotic behavior of fractional order Riemann-Liouville Volterra-Stieltjes integral equations

Abstract: In this paper, we present some results concerning the existence and global asymptotic stability of solutions for a functional integral equation of fractional order. We use Schauder's fixed point theorem for the existence of solutions, and we prove that all these solutions are globally asymptotically stable.

Functional Differential Inclusions with Multi-valued Jumps

Abstract: In this chapter, we are concerned by the existence of mild solutions of functional differential inclusions with delay and multi-valued jumps in a Banach space.

Partial Functional Evolution Equations with Infinite Delay

Abstract: In this chapter, we study some first order classes of partial functional, neutral functional, integro-differential, and neutral integro-differential evolution equations on a positive line \(\mathbb{R}_{+}\) with local and nonlocal conditions when the historical interval H is bounded, i.e., when the delay is finite. In the literature devoted to equations with finite delay, the phase space is much of time the space of all continuous functions on H for r > 0, endowed with the uniform norm topology. Using a recent nonlinear alternative of Leray–Schauder type for contractions in Frechet spaces due to Frigon and Granas combined with the semigroup theory, the existence and uniqueness of the mild solution will be obtained. The method we are going to use is to reduce the existence of the unique mild solution to the search for the existence of the unique fixed point of an appropriate contraction operator in a Frechet space.

Impulsive Functional Differential Inclusions with Unbounded Delay

Abstract: In this chapter, we shall establish sufficient conditions for the existence of mild, extremal mild, integral, and extremal integral solutions for some impulsive semi-linear neutral functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators.

Functional Differential Equations and Inclusions with Delay

Abstract: In this chapter, we shall prove the existence of solutions of some classes of functional differential equations and inclusions. Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the real axis \(\mathbb{R}.\) We will use some fixed point theorems combined with the semigroup theory.

Non-densely Defined Functional Differential Inclusions with Finite Delay

Abstract: In this chapter, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some non-densely defined impulsive semi-linear functional differential inclusions in separable Banach spaces with local and nonlocal conditions. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these inclusions with both multi-valued and single valued jump and the topological structure of the solutions set are considered too.

Perturbed Partial Functional Evolution Equations

Abstract: Perturbed partial functional and neutral functional evolution equations with finite and infinite delay are studied in this chapter on the semi-infinite interval \(\mathbb{R}_{+}.\)