Fixed-point Theorem and the Nishida-Nirenberg Method in Solving Certain Nonlinear Singular Partial Differential Equations

Abstract

In their 2012 work, Lope, Roque, and Tahara considered singular nonlinear partial differential equations of the form tut = F(t; x; u; ux), where the function F is assumed to be continuous in t and holomorphic in the other variables. They have shown that under some growth conditions on the coefficients of the partial Taylor expansion of F as t 0, the equation has a unique solution u(t; x) with the same growth order as that of F(t; x; 0; 0). Koike considered systems of partial differential equations using the Banach fixed point theorem and the iterative method of Nishida and Nirenberg. In this paper, we prove the result obtained by Lope and others using the method of Koike, thereby avoiding the repetitive step of differentiating a recursive equation with respect to x as was done by the aforementioned authors.

Keywords

• Singular partial differential equations
• nonlinear
• fixed point