System of Non-Linear Volterra Integral Equations in a Direct-Sum of Hilbert Spaces

Abstract

We use the contraction mapping theorem to present the existence and uniqueness of solutions in a short time to a system of non-linear Volterra integral equations in a certain type of direct-sum H[a; b] of a Hilbert space V[a; b]. We extend the local existence and uniqueness of solutions to the global existence and uniqueness of solutions to the proposed problem. Because the kernel function is a transcendental function in H[a; b] on the interval [a; b], the results are novel and very important in numerical approximation.

Keywords

• system of non-linear integral equations
• Reproducing kernel Hilbert spaces
• Fixed point theorem