Existence, Uniqueness, and Approximation for Solutions of a Functional-integral Equation in $L^p$ Spaces

ABSTRACT

In this work we consider the general functional-integral equation:

$$y(t)=f(t,{\int }_0^{1}k(t,s)g(s,y(s))ds),t\in [0,1],$$

and give conditions that guarantee existence and uniqueness of solution in $L^p([0,1])$, with 1 $1p\infty$.We use Banach Fixed Point Theorem and employ the successive approximation method and Chebyshev quadrature for approximating the values of integrals. Finally, to illustrate the results of this work, we provide some numerical examples.

Key words
functional-integral equations; L^p spaces; existence; uniqueness; successive approximation