Hussein A. H. Salem

On the existence of continuous solutions for a singular system of non-linear fractional differential equations

Abstract In the this paper, we establish the existence of continuous solutions to some non-linear fractional integral and differential equations. Quadratic system of fractional integral equations cases will be considered. Our analysis rely on Krasnoselskii’s fixed point theorem on a cone.

Fractional order boundary value problem with integral boundary conditions involving Pettis integral

In this article, we investigate the existence of Pseudo solutions for some fractional order boundary value problem with integral boundary conditions in the Banach space of continuous function equipped with its weak topology. The class of such problems constitute a very interesting and important class of problems. They include two, three, multi-point and nonlocal boundary-value problems as special cases. In our investigation, the right hand side of the above problem is assumed to be Pettis integrable function. To encompass the full scope of this article, we give an example illustrating the main result.

On the fractional calculus in abstract spaces and their applications to the Dirichlet-type problem of fractional order

Abstract In the following pages, based on the linear functional over a Banach space E and on the definition of fractional integrals of real-valued functions, we define the fractional Pettis-integrals of E -valued functions and the corresponding fractional derivatives. Also, we show that the well-known properties of fractional calculus over the domains of the Lebesgue integrable also hold in the Pettis space. To encompass the full scope of the paper, we apply this abstract result to investigate the existence of Pseudo-solutions to the following fractional-order boundary value problem { D α x ( t ) + λ a ( t ) f ( t , x ( t ) ) = 0 , t ∈ [ 0 , 1 ] , α ∈ ( n − 1 , n ] , n ≥ 2 , x ( 1 ) + ∫ 0 1 u ( τ ) x ( τ ) d τ = l , x ( k ) ( 0 ) = 0 , k = 0 , 1 , … , n − 2 , in the Banach space C [ I , E ] under Pettis integrability assumptions imposed on f . Our results extend all previous results of the same type in the Bochner integrability setting and in the Pettis integrability one. Here, λ ∈ R , u ∈ L p , a ∈ L q and l ∈ E .

Global monotonic solutions of multi term fractional differential equations

Abstract Based on the Leray–Schauder principle, a fixed point theorem is established to study the existence of a global monotonic solution for some multi term differential equations of fractional type. Some existence result for the inclusion problem is proved.

On the quadratic integral equations and their applications

The class of quadratic integral equations contains, as a special case, numerous integral equations encountered in the theory of radiative transfer, the queuing theory, the kinetic theory of gases and the theory of neutron transport. As a pursuit of this, in the following pages, sufficient conditions are given for the existence of positive continuous solutions to some possibly singular quadratic integral equations. Meanwhile, we prove the existence of maximal and minimal solutions of our problems. The method used here depends on both Schauder and Schauder-Tychonoff fixed point principles. Unlike all previous contributions of the same type, no assumptions in terms of the measure of noncompactness were imposed on the nonlinearity of the given functions. As far as we know, the approach presented in this paper, in particular, the discussion of the existence of maximal and minimal solutions to the quadratic integral equations was never applied in the field of the quadratic integral equations and so is new.

Multi-term fractional differential equation in reflexive Banach space

In this paper, we investigate the solvability of a multi-term fractional differential equation in a reflexive Banach space relative to weak topology. A particular case of our main result is treated.

On Solutions of Fractional Order Boundary Value Problems with Integral Boundary Conditions in Banach Spaces

The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.

On the Theory of Fractional Calculus in the Pettis-Function Spaces

Throughout this paper, we outline some aspects of fractional calculus in Banach spaces. Some examples are demonstrated. In our investigations, the integrals and the derivatives are understood as Pettis integrals and the corresponding derivatives. Our results here extended all previous contributions in this context and therefore are new. To encompass the full scope of our paper, we show that a weakly continuous solution of a fractional order integral equation, which is modeled off some fractional order boundary value problem (where the derivatives are taken in the usual definition of the Caputo fractional weak derivative), may not solve the problem.