Global existence of solutions of integral equations with delay: progressive contractions

Abstract

In the theory of progressive contractions an equation such as \[ x(t) = L(t)+\int^t_0 A(t-s)[ f(s,x(s)) + g(s,x(s-r(s))]ds, \] with initial function $\omega$ with $\omega (0) =L(0)$ defined by $ t\leq 0 \implies x(t) =\omega (t)$ is studied on an interval $[0,E]$ with $r(t) \geq \alpha >0$. The interval $[0,E]$ is divided into parts by $0=T_00$ on $[0,\infty)$ we obtain a unique solution on that interval as follows. As we let $E= 1,2,\dots$ we obtain a sequence of solutions on $[0,n]$ which we extend to $[0,\infty)$ by a horizontal line, thereby obtaining functions converging uniformly on compact sets to a solution on $[0,\infty)$. Lemma 2.1 extends progressive contractions to delay equations

Keywords

• progressive contractions
• integral and differential equations with delay
• global existence
• fixed points