Abdelkader Boucherif

First-order differential inclusions with nonlocal initial conditions

We investigate initial value problems for first-order differential inclusions with nonlocal conditions. We provide conditions on the right-hand side that are sufficient for obtaining a priori bounds on solutions. We then rely on a theorem of Bohnenblust and Karlin to prove existence of at least one solution.

Nonlinear three-point third-order boundary value problems

Abstract In this paper we consider three-point boundary value problems for third-order differential equations. We provide simple sufficient conditions in order to obtain a priori bounds on solutions. We then rely on the topological transversality theorem of Granas or on Schaefer’s theorem to prove existence of solutions.

Existence of periodic and non-periodic solutions to systems of boundary value problems for first-order differential inclusions with super-linear growth

Abstract We establish some new sufficient conditions under which periodic and non-periodic solutions exist for systems of nonlinear, first-order differential inclusions. Our new results are significant for two main reasons: they apply to differential inclusions that may have a right-hand side that grows super-linearly in the second variable and thus our ideas extend known linear-growth results; and our results also apply to systems of first-order differential inclusions, thus our work extends previous research on scalar-valued inclusions and may lead to a deeper understanding of higher-order differential inclusions. Our approach is based on novel differential inequalities and the well-known nonlinear alternative of Leray–Schauder. Some new results for ordinary differential equations with Caratheodory single-valued right-hand sides are also obtained.

Nonlocal conditions for differential inclusions in the space of functions of bounded variations

AbstractWe discuss the existence of solutions of an abstract differential inclusion, with a right-hand side of bounded variation and subject to a nonlocal initial condition of integral type. AMS Subject Classification 34A60, 34G20, 26A45, 54C65, 28B20

Existence Results for Stochastic Semilinear Differential Inclusions with Nonlocal Conditions

We discuss existence results of mild solutions for stochastic differential inclusions subject to nonlocal conditions. We provide sufficient conditions in order to obtain a priori bounds on possible solutions of a one-parameter family of problems related to the original one. We, then, rely on fixed point theorems for multivalued operators to prove our main results.

Nonlocal Conditions for Lower Semicontinuous Parabolic Inclusions

We discuss conditions for the existence of at least one solution of a discontinuous parabolic equation with lower semicontinuous right hand side and a nonlocal initial condition of integral type. Our technique is based on fixed point theorems for multivalued maps.

Existence of Solutions for Impulsive Parabolic Partial Differential Equations

We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is described by an initial-boundary value problem for a nonlinear parabolic partial differential equation subjected to impulsive effects at fixed times. Using Greens function, we convert the problem into a nonlinear integral equation. Sufficient conditions are provided that enable the application of fixed point theorems to prove existence and uniqueness of solutions.

Variational Approach to Impulsive Differential Equations with Singular Nonlinearities

We discuss the existence of periodic solutions for nonautonomous second order differential equations with singular nonlinearities. Simple sufficient conditions that enable us to obtain many distinct periodic solutions are provided. Our approach is based on a variational method.

Discontinuous Parabolic Problems with a Nonlocal Initial Condition

We study parabolic differential equations with a discontinuous nonlinearity and subjected to a nonlocal initial condition. We are concerned with the existence of solutions in the weak sense. Our technique is based on the Greens function, integral representation of solutions, the method of upper and lower solutions, and fixed point theorems for multivalued operators.

Finite Dimensional Representation of Distributed Delay Systems

It is shown that a class of distributed delay systems can be represented by a system of ordinary differential equations with arbitrary degree of accuracy.