Existence and Numerical Solutions of Nonlinear Quadratic Integral Equations Defined on Unbounded Intervals

Abstract

The first goal of this article is to discuss the existence of solutions of nonlinear quadratic integral equations. These equations are considered in the Banach space L ^p (ℝ₊). The arguments used in the existence proofs are based on Schauder's and Darbo's fixed point theorems. In particular, to apply Schauder's fixed point theorem based method, a special care is devoted to the proof of the L ^p -compactness of the operators associated with our nonlinear quadratic integral equations. The second goal of this work is to study a numerical method for solving nonlinear Volterra integral equations of a fairly general type. Finally, we provide the reader with some examples that illustrate the different results of this work.

Keywords:

• Darbo's fixed-point theorem
• Measure of non compactness
• Nonlinear quadratic integral equations
• Numerical solution of quadratic-Volterra equation
• Schauder's fixed-point theorem

AMS Subject Classification:

• Primary 45G10, 47H30
• Secondary 65R20