Abstract
In this paper, we investigate the set of solutions for nonlinear Volterra type integral equations in Banach spaces in the weak sense and under Henstock–Kurzweil–Pettis integrability. Moreover, a fixed point result is presented for weakly sequentially continuous mappings defined on the function space C(K, X), where K is compact Hausdorff and X is a Banach space. The main condition is expressed in terms of axiomatic measure of weak noncompactness.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Agarwal R., O’Regan D., Sikorska-Nowak A.: The set of solutions of integrodifferential equations and the Henstock–Kurzweil–Pettis integral in Banach spaces. Bull. Aust. Math. Soc. 78, 507–522 (2008)
Angosto C., Cascales B.: Measures of weak noncompactness in Banach spaces. Topol. Appl. 156, 1412–1421 (2009)
Arino O., Gautier S., Penot J.P.: A fixed point theorem for sequentially continuous mapping with application to ordinary differential equations. Funct. Ekvac. 27, 273–279 (1984)
Banaś J., Rivero J.: On measure of weak noncompactness. Ann. Math. Pura Appl. 151, 213–224 (1988)
Banaś J., Martinón A.: Measures of weak noncompactness in Banach sequence spaces. Portugal. Math. 59(2), 131–138 (1995)
Cichoń M.: Weak solutions of differential equations in Banach spaces. Discret Math. Differ. Incl. 15, 5–14 (1995)
Cichoń M., Kubiaczyk I., Sikorska A.: Henstock–Kurzweil and Henstock–Kurzweil–Pettis integrals and some existence theorems. Proc. ISCM Herlany 2000, 53–56 (1999)
Cichoń M., Kubiaczyk I., Sikorska A.: The Henstock–Kurzweil–Pettis integrals and existence theorems for the Cauchy problem. Czech. Math. J. 54(129), 279–289 (2004)
Cichoń M.: On solutions of differential equations in Banach spaces. Nonlinear Anal. Theory. Methods Appl. 60(4), 651–667 (2005)
DeBlasi F.S.: On a property of the unit sphere in Banach space. Bull. Math. Soc. Sci. Math. R.S. Roum. 21, 259–262 (1977)
Di Piazza L.: Kurzweil–Henstock type integration on Banach spaces. Real Anal. Exch. 29(2), 543–555 (2003)
Diestel, J.; Uhl, J.J.: Vector measures., Math. Surv., 15 (1977)
Dobrakov I.: On representation of linear operators on C 0(T, X). Czech. Math. J. 20, 13–30 (1971)
Edwards, R.E.: Functional Analysis, Theory and Applications., Holt, Reinhart and Winston, New York, (1965)
Gamez J.L., Mendoza J.: On Denjoy-Dunford and Denjoy-Pettis integrals. Stud. Math. 130, 115–133 (1998)
Garcia-Falset J.: Existence of fixed points and measures of weak noncompactness. Nonlinear Anal. Theory Methods Appl. 71, 2625–2633 (2009)
Gordon R.A.: Rienmann integration in Banach spaces. Rocky Mt. J. Math. 21(3), 923–949 (1991)
Gordon, R.A.: The Integrals of Lebesgue, Denjoy, Perron and Henstock. Grad. Stud. Math., 4 AMS, Providence, 1994
Yu L., Barbot J.-P., Boutat D., Benmerzouk D.: Observability forms for switched systems with zeno phenomenon or high switching frequency. Autom. Control, IEEE Trans. 56(2), 436–441 (2011)
Kaliaj S.B., Tato A.D., Gumeni F.D.: Controlled convergence theorems for Henstock–Kurzweil–Pettis integral on m-dimensional compact intervals. Czech. Math. J. 62(1), 243–255 (2012)
Kryczka A., Prus S., Szczepanik M.: Measure of weak noncompactness and real interpolation of operators. Bull. Aust. Math. Soc. 62, 389–401 (2000)
Kryczka A., Prus S.: Measure of weak noncompactness under complex interpolation. Stud. Math. 147, 89–102 (2001)
O’Regan D.: Weak solutions of ordinary differential equations in Banach spaces. Appl. Math. Lett. 12, 101–105 (1999)
Satco B.: A Kolmós-type theorem for the set-valued Henstock–Kurzweil–Pettis integrals and applications. Czech. Math. J. 56(131), 1029–1047 (2006)
Satco B.: Volterra integral inclusions via Henstock–Kurzweil–Pettis integral. Discuss. Math. Differ. Incl. Control Optim. 26, 87–101 (2006)
Sikorska-Nowak A.: Nonlinear integral equations in Banach spaces and Henstock–Kurzweil–Pettis integrals. Dyn. Syst. Appl. 17, 97–108 (2008)
Sikorska-Nowak A.: Integrodifferential equations on time scales with Henstock–Kurzweil–Pettis delta integrals. Abstr. Appl. Anal. Vol. 2010, 17 (2010). doi:10.1155/2010/836347
Sikorska-Nowak A.: The set of solutions of Volterra and Urysohn integral equations in Banach spaces. Rocky Mt. J. Math. 40(4), 1313–1331 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Ben Amar, A. On an integral equation under Henstock–Kurzweil–Pettis integrability. Arab. J. Math. 4, 91–99 (2015). https://doi.org/10.1007/s40065-014-0125-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40065-014-0125-2