Mohamed Abdalla Darwish

On initial and boundary value problems for fractional order mixed type functional differential inclusions

Abstract In this paper we prove some existence results for initial and boundary value problems for functional differential inclusions of fractional order with both retarded and advanced arguments. The Banach fixed point theorem, the nonlinear alternative of the Leray–Schauder type and the Covitz–Nadler fixed point theorem are the main tools in deriving our proofs.

EXISTENCE AND ASYMPTOTIC STABILITY OF SOLUTIONS OF A PERTURBED FRACTIONAL FUNCTIONAL-INTEGRAL EQUATION WITH LINEAR MODIFICATION OF THE ARGUMENT

We study the solvability of a perturbed quadratic functional- integral equation of fractional order with linear modication of the argu- ment. This equation is considered in the Banach space of real functions dened, bounded and continuous on an unbounded interval. Moreover, we will obtain some asymptotic characterization of solutions.

Solvability of Urysohn integral equation

In this paper we present the existence theorem of integrable solutions of Urysohn integral equation in the class of monotonic functions. The proof of that theorem is based on the fixed point theorem associated with the notion of a measure of noncompactness.

On a perturbed functional integral equation of Urysohn type

Abstract We study the existence of monotonic solutions for a perturbed functional integral equation of Urysohn type in the space of Lebesgue integrable functions on an unbounded interval. The technique associated with measures of noncompactness (in both the weak and the strong sense) and the Darbo fixed point are the main tool to prove our main result.

On integral equations of Urysohn-Volterra type

The existence of solutions to Urysohn-Volterra equation in a locally convex topological Hausdorff space has been established. The Schauder Tychonoff theorem is the main tool in our analysis.

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

Abstract In this work, we discuss the existence and Ulams 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.

Nondecreasing solutions of a quadratic Abel equation with supremum in the kernel

We prove an existence theorem for a quadratic Abel integral equation of the second kind with supremum in the kernel. The quadratic integral equation studied below contains as a special case numerous integral equations encountered in the theory of radiative transfer and in the kinetic theory of gases. We show that the singular quadratic integral equations with supremum has a monotonic solution in C[0,1]. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof.

On a perturbed quadratic fractional integral equation of Abel type

We present an existence theorem for monotonic solutions of a perturbed quadratic fractional integral equation in C[0,1]. Our equation contains the famous Chandrasekhar integral equation as a special case. The concept of the measure of noncompactness related to monotonicity, introduced by J. Banas and L. Olszowy, and a fixed point theorem due to Darbo are the main tools in carrying out our proof. Moreover, we give an example for indicating the natural realizations of our abstract result presented in this paper.

On existence and asymptotic behaviour of solutions of a fractional integral equation

We study the solvability of a quadratic integral equation of fractional order with linear modification of the argument. This equation is considered in the Banach space of real functions, defined, bounded and continuous on an unbounded interval. Moreover, we will obtain some asymptotic characterization of solutions.

Existence of fractional integral equation with hysteresis

The existence of solutions to an integral equation of fractional order with hysteresis has been established. A fixed point theorem due to Schauder is the main tool in carrying out our proof.