Mathématiques

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On the existence of nontrivial solutions of a system of nonlinear partial differential equations
(University 20 august 1955-Skikda, 2023) Abada, Asma; Slimani, K
The purpose of this work is to study some nonlinear fractional differential problems with boundary conditions in a bounded domain in order to generalize some results from the classical differential problems to the fractional case, which has a diversity of applications in various fields, manly physics, engineering, mathematical biology, signal processing, and image processing. In this study, under some suitable conditions on the nonlinearities, we apply the Leray-Schauder degree and the Schauder fixed point theorem to establish the existence of solutions; also, the Banach principle of contraction and the absurd reasoning are applied to prove the uniqueness of solutions. The first problem is a coupled semilinear fractional Laplacian system in a fractional Sobolev space, and the second problem is a semilinear equation involving the distributional Riesz fractional gradient in a Bessel potential space. The third problem is a nonlocal nonlinear equation related to the distributional Riesz fractional derivative in a Bessel-potential space. Finally, the fourth problem is a nonlinear problem involving left and right Riemann-Liouville fractional derivatives in a new fractional space of Sobolev type, the study of this problem is illustrated with an example to affirm the validity of methods used.