A recent study of the Darbaugh problem's stability in partial differential equations using fixed-point theory

Abstract

The stability analysis of solutions to nonlinear PDEs involves a fundamental and serious problem in applied mathematics and physics. The paper aims for the first time ever to provide a stability analysis for a class of nonlinear initial-boundary value problems called Darbaugh problems. To achieve this goal, we convert the PDE into a fixed-point equation via a nonlinear solution operator so that we can investigate several kinds of stability interested in Hyers-Ulam and Lagrange stability using the fixed-point theories of Banach Contraction Principle, Krasnoselskii’s Theorem and Schaefer’s Theorem, by using a variety of suitable function spaces, especially Banach algebras such as C(X), and Sobolev spaces to make use of their topological and multiplicative properties. New stability theorems are proved. One of them states that if the nonlinearity satisfies a Lipschitz-type condition where the constant can be taken sufficiently small relative to the norm of the linear solution operator then the problem has uniform Hyers-Ulam stability. Another one states that if the operator is decomposable into contractive and compact parts then we get Lagrange stability. Finally, uniform a priori bound gives bounded-input-bounded-output stability. In conclusion, the fixed-point method gives rise to explicit conditions for stability that can be verified algebraically. This provides a unified and robust alternative to the classical energy methods and adds analytical techniques to our toolbox of methods for the analysis of nonlinear PDEs.