# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  S. Carl, S. Heikkilä: On discontinuous implicit and explicit abstract impulsive boundary value problems. Nonlinear Anal., Theory Methods Appl. 41 (2000), 701–723. · Zbl 0985.34050  M. Federson, M. Bianconi: Linear Fredholm integral equations and the integral of Kurzweil. J. Appl. Anal. 8 (2002), 83–110. · Zbl 1043.45010  M. Federson, Š. Schwabik: Generalized ordinary differential equations approach to impulsive retarded functional differential equations. Differ. Integral Equ. 19 (2006), 1201–1234. · Zbl 1212.34251  M. Federson, P. Táboas: Impulsive retarded differential equations in Banach spaces via Bochner-Lebesgue and Henstock integrals. Nonlinear Anal., Theory Methods Appl. 50 (2002), 389–407. · Zbl 1011.34070  D. Guo, Y. J. Cho, J. Zhu: Partial Ordering Methods in Nonlinear Problems. Nova Science Publishers, Inc., New York, 2004.  S. Heikkilä, S. Kumpulainen, M. Kumpulainen: On improper integrals and differential equations in ordered Banach spaces. J. Math. Anal. Appl. 319 (2006), 579–603. · Zbl 1105.34037  S. Heikkilä, V. Lakshmikantham: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations. Marcel Dekker, Inc., New York, 1994. · Zbl 0804.34001  S. Heikkilä, S. Seikkala: On non-absolute functional Volterra integral equations and impulsive differential equations in ordered Banach spaces. Electron. J. Differ. Equ., paper No. 103 (2008), 1–11.  S. Heikkilä, G. Ye: Convergence and comparison results for Henstock-Kurzweil and McShane integrable vector-valued functions. Southeast Asian Bull. Math. 35 (2011), 407–418. · Zbl 1240.26025  J. Lu, P.-Y. Lee: On singularity of Henstock integrable functions. Real Anal. Exch. 25 (2000), 795–797. · Zbl 1015.26016  B.-R. Satco: Nonlinear Volterra integral equations in Henstock integrability setting. Electron. J. Differ. Equ., paper No. 39 (2008), 1–9. · Zbl 1169.45300  Š. Schwabik, G. Ye: Topics in Banach Space Integration. World Scientific, Hackensack, 2005. · Zbl 1088.28008  A. Sikorska-Nowak: On the existence of solutions of nonlinear integral equations in Banach spaces and Henstock-Kurzweil integrals. Ann. Pol. Math. 83 (2004), 257–267. · Zbl 1101.45006  A. Sikorska-Nowak: Existence theory for integrodifferential equations and Henstock-Kurzweil integral on Banach spaces. J. Appl. Math., Article ID31572 (2007), 1–12. · Zbl 1148.26010  A. Sikorska-Nowak: Existence of solutions of nonlinear integral equations in Banach spaces and Henstock-Kurzweil integrals. Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 47 (2007), 227–238. · Zbl 1178.45016  A. Sikorska-Nowak: Nonlinear integrodifferential equations of mixed type in Banach spaces. Int. J. Math. Math. Sci., Article ID65947 (2007), 1–14. · Zbl 1147.45009  A. Sikorska-Nowak: Nonlinear integral equations in Banach spaces and Henstock-Kurzweil-Pettis integrals. Dyn. Syst. Appl. 17 (2008), 97–107. · Zbl 1154.45011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.