Abstract
The aim of this paper is to investigate generalized Ulam–Hyers stability and generalized Ulam–Hyers–Rassias stability for a system of partial differential equations of first order. More precisely, we consider a system of two nonlinear equations of first order with an unknown function of two independent variables, which satisfy the corresponding compatibility condition. The study method is that of differential inequalities of the Gronwall type.
Highlights
Ulam–Hyers stability is an important problem in functional equations theory, which was studied by many authors who can be found in the monograph [1].The problem was posed by Ulam in 1940 in the following way
Generalized Ulam–Hyers–Rassias stability of system (2)
We have studied generalized Ulam–Hyers and respectively Ulam–Hyers–Rassias stability of system (2), if condition (4) is satisfied, with initial condition (3)
Summary
Ulam–Hyers stability is an important problem in functional equations theory, which was studied by many authors who can be found in the monograph [1]. The stability of differential linear equations of first order was studied in the papers [4,5]. The first authors who studied Hyers-Ulam stability of partial differential equations were Prastaro and Rassias [10]. In a set of papers, Rus [13,15], has opened a new direction of study of the Ulam stability using Gronwall type inequalities and Picard operators technique. We will study the stability of a system of partial differential equations of order one, nonlinear. We mention that those are not studied yet in literature. The compatibility condition is not verified, the system has no solution.
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