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

• Main Authors: Theodore Burton, Ioannis Purnaras
• Format: Article
• Language: English
• Published: University of Szeged 2017-06-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: progressive contractions; integral and differential equations with delay; global existence; fixed points
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=5827

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 $0=T_0<T_1<\dots<T_n=E$ with $T_i-T_{i-1} <\alpha$. It is assumed that $f$ satisfies a Lipschitz condition, but there is no growth condition on $g$. When we try for a contraction on $[0,T_1]$ the terms with $g$ add to zero and we get a unique solution $\xi_1$ on $[0,T_1]$. Then we get a complete metric space on $[0,T_2]$ with all functions equal to $\xi_1$ on $[0,T_1]$ enabling us to get a contraction. In $n$ steps we have obtained a solution on $[0,E]$. When $r(t) >0$ 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