Equations containing locally Henstock-Kurzweil integrable functions. (English) Zbl 1274.45017

Heikkilä, Seppo; Ye, Guoju

Appl. Math., Praha 57, No. 6, 569-580 (2012).

The authors consider a functional integral equation of Volterra type \[ u(t)=h(t,u)+{}^{K}\int _a^t g(s,u(s),u)\, d\,s, \quad t\in [a,b), \] where \(-\infty <a<b\leq \infty \), \(X\) is an ordered Banach space, \({h:[a,b)\times L^1_{\text{loc}}([a,b),X)\to X}\), \(g\:[a,b)\times X\times L^1_{loc}([a,b),X)\to X\) and the integral is understood in the sense of Henstock-Kurzweil. The results are formulated for locally Henstock-Kurzweil integrable functions \(g\) and \(h\), thus they need not be continuous.
The existence of least and greatest solutions for the given equation is proved using two different approaches: a fixed point theorem in ordered Banach space and monotone techniques. Comparison results for such solutions are presented as well. Furthermore, an application to functional impulsive Cauchy problem and some illustrating examples (including examples of weakly sequentially complete Banach spaces possessing normal order cones) are given.

Reviewer: Giselle Antunes Monteiro (Praha)

MSC:

45N05	Abstract integral equations, integral equations in abstract spaces
26A39	Denjoy and Perron integrals, other special integrals
34A37	Ordinary differential equations with impulses
28B15	Set functions, measures and integrals with values in ordered spaces
46B40	Ordered normed spaces
47H07	Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces

Keywords:

Volterra integral equation; Henstock-Kurzweil integral; ordered Banach space; fixed point; Cauchy problem; functional integral equation

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