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