Positive kernels, fixed points, and integral equations

Burton, Theodore Allen and Purnaras, Ioannis K.: Positive kernels, fixed points, and integral equations. Electronic journal of qualitative theory of differential equations 44. pp. 1-21. (2018)

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Abstract

There is substantial literature going back to 1965 showing boundedness properties of solutions of the integro-differential equation x 0 (t) = − Z t 0 A(t − s)h(s, x(s))ds when A is a positive kernel and h is a continuous function using Z T 0 h(t, x(t)) Z t 0 A(t − s)h(s, x(s))dsdt ≥ 0. In that study there emerges the pair: Integro-differential equation and Supremum norm. In this paper we study qualitative properties of solutions of integral equations using the same inequality and obtain results on L p solutions. That is, there occurs the pair: Integral equations and L p norm. The paper also offers many examples showing how to use the L p idea effectively.

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