On the existence of nontrivial solutions of a system of nonlinear partial differential equations
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Date
2023
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University 20 august 1955-Skikda
Abstract
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.