A NASC for equicontinuous maps for integral equations
Document Type
Article
Publication Date
1-1-2017
Abstract
We offer necessary and sufficient conditions for a mapping of the form (P ?)(t) = p(t) ? ?0t C(t, s)g(s, ?(s))ds to send sets of bounded continuous functions on [0, ?) into equicontinuous sets. When that equicontinuity holds then one may study the problem of obtaining a bounded solution of the integral equation by means of a Schauder-type fixed point theorem. When the mapped sets are equicontinuous then we use Schaefer’s fixed point theorem to show that we can obtain a bounded positive solution provided that we know that the resolvent kernel, R(t, s), of C is non-negative and that p(t) ? ?0t R(t, s)p(s)ds is bounded and positive, while g(t, x) does not grow too fast near x = 0. The known literature shows that there are wide classes of important problems from applied mathematics and fractional equations for which these conditions hold. For those classes, the problem of obtaining a positive solution is largely solved when equicontinuity, characterized by our theorem, holds.
Recommended Citation
Burton, T. A. and Zhang, Bo, "A NASC for equicontinuous maps for integral equations" (2017). College of Health, Science, and Technology. 737.
https://digitalcommons.uncfsu.edu/college_health_science_technology/737