The Existence Of Solutions For Two Types Of Quadratic Integral Equations

Integral equation is an important branch of modern mathematics, many problems of mathematics, natural science and engineering technology can be summed up to the problems in integral equation. Quadratic integral equations appear in applications of the real world, the research results have been applied to different fields.This paper mainly studies the existence and uniqueness of two kinds of quadratic integral equations. By studying the characters in the space of functions with tempered moduli of continuity and Ln-H¨older function space, the existence of solution is converted into the fixed point of operator. After discussing the condition of relatively compactness in the space, we obtain the existence of solutions to quadratic integral equation of Fredholm type in the space of functions with tempered moduli of continuity and Ln-H¨older function space. By studying the characters in H¨older function space and equivalent conditions of relatively compactness in the space and using Schauder fixed point theorem,we obtain the existence and uniqueness of solutions involving Erd?elyi-Kober integral equation in H¨older function space. After finishing the work above and using the basic properties of Mittag-Leffer function and the fixed point theorems, we study the existence of solution to the cauchy problem for nonlinear fractional differential equations in continuous function space.