Finding the Fixed Points Inside Large Mapping Sets: Integral Equations

Authors : Theodore A Burton, Ioannis K Purnaras

Pages : 41-47

Doi:10.31197/atnaa.379110

View : 9 | Download : 8

Publication Date : 2017-09-30

Article Type : Research Paper

Abstract :Let xfinsert ignore into journalissuearticles values(t,x); > 0 for x 6= 0 and let Ainsert ignore into journalissuearticles values(t−s); satisfy some classical properties yielding a nice resolvent. Using repeated application of a fixed point mapping and induction we develop an asymptotic formula showing that solutions of the Caputo equation cDqxinsert ignore into journalissuearticles values(t); = −finsert ignore into journalissuearticles values(t,xinsert ignore into journalissuearticles values(t););, 0 < q < 1, xinsert ignore into journalissuearticles values(0); ∈<, xinsert ignore into journalissuearticles values(0); 6= 0, and more generally of the integral equation xinsert ignore into journalissuearticles values(t); = xinsert ignore into journalissuearticles values(0);−Zt 0 Ainsert ignore into journalissuearticles values(t−s);finsert ignore into journalissuearticles values(s,xinsert ignore into journalissuearticles values(s););ds,xinsert ignore into journalissuearticles values(0); 6= 0, all satisfy xinsert ignore into journalissuearticles values(t); → 0 as t →∞.
Keywords : compact maps, repeated mappings, integral equations, fixed points, limit sets