Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-17T00:33:52.904Z Has data issue: false hasContentIssue false
  • English
  • Français

Three Fixed Point Theorems: Periodic Solutions of a Volterra Type Integral Equation with Infinite Heredity

Published online by Cambridge University Press:  20 November 2018

Muhammad N. Islam
Muhammad N. Islam*
Affiliation:
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316 USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's and Schaefer's fixed point theorems are employed in the analysis. The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions. We employ Liapunov's direct method to obtain such an a priori bound. In the process, we compare these theorems in terms of assumptions and outcomes.

Keywords

45D0545J05Volterra integral equationperiodic solutionsLiapunov's methodKrasnosel'skii's fixed point theoremSchaefer's fixed point theorem
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 1 , 01 March 2013 , pp. 80 - 91
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Adivar, M. and Raffoul, Y., Existence results for periodic solutions of integro-dynamic equations on time scales. Ann. Mat. Pura. Appl. 188 (2009), no. 4, 543–559. http://dx.doi.org/10.1007/s10231-008-0088-z Google Scholar
[2] Burton, T. A., Integral equations, implicit functions and fixed points. Proc. Amer. Math. Soc. 124 (1996), no. 8, 2383–2390. http://dx.doi.org/10.1090/S0002-9939-96-03533-2 Google Scholar
[3] Burton, T. A., Liapunov Functionals for Integral Equations, Trafford Publishing, 2008.Google Scholar
[4] Burton, T. A. and Colleen Kirk, A fixed point theorem of Krasnoselskii Schaefer type. Math. Nachr. 189 (1998), 23–31. http://dx.doi.org/10.1002/mana.19981890103 Google Scholar
[5] Burton, T. A., Eloe, P. W., and Islam, M. N., Periodic solutions of linear integro-differential equations. Math. Nachr. 147 (1990), 175–184. http://dx.doi.org/10.1002/mana.19901470120 Google Scholar
[6] Burton, T. A., Eloe, P. W., and Islam, M. N., Nonlinear integrodifferential equations and a priori bounds on periodic solutions. Ann. Mat. Pura. Appl. 161 (1992), 271–283. http://dx.doi.org/10.1007/BF01759641 Google Scholar
[7] Burton, T. A. and Zhang, B., Periodic solutions of abstract differential equations with infinite delay. J. Differential Equations 90 (1991), no. 2, 357–396. http://dx.doi.org/10.1016/0022-0396(91)90153-Z Google Scholar
[8] Corduneanu, C., Integral Equations and Applications. Cambridge University Press, Cambridge, 1991.Google Scholar
[9] Driver, R. D., Existence and continuous dependence of solutions of a neutral functional-differential equation. Arch. Rational Mech. Anal. 19 (1965), 149–166. http://dx.doi.org/10.1007/BF00282279 Google Scholar
[10] Eloe, P. W. and Henderson, J., Nonlinear boundary value problems and a priori bounds on solutions. SIAM J. Math. Anal. 15 (1984), no. 4, 642–647. http://dx.doi.org/10.1137/0515049 Google Scholar
[11] Eloe, P. W. and Islam, M. N., Periodic solutions of nonlinear intrgral equations with infinite memory. Appl. Anal. 28 (1988), no. 2, 79–93. http://dx.doi.org/10.1080/00036818808839451 Google Scholar
[12] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics. Mathematics and its Applications 74. Kluwer, Dordrecht, 1992.Google Scholar
[13] Gopalsamy, K. and Zhang, B. G., On a neutral delay logistic equation. Dynam. Stability Systems 2 (1988), no. 3-4, 183–195.Google Scholar
[14] Granas, A., Sur la méthode de continuité de Poincaré. C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 17, A983–A985.Google Scholar
[15] Granas, A., Guenther, R. B., and Lee, J. W., Nonlinear boundary value problems for some classes of ordinary differential equations. Rocky Mountain J. Math. 10 (1980), no. 1, 35–58. http://dx.doi.org/10.1216/RMJ-1980-10-1-35 Google Scholar
[16] Gripenberg, G., Londen, S. O., and Staffans, O., Volterra Integral and Functional Equations. Encyclopedia of Mathematics and its Applications 34. Cambridge University Press, Cambridge, 1990.Google Scholar
[17] Hale, J. K., and Lunel, S. M. V., Introduction to Functional-Differential Equations. Applied Mathematical Sciences 99. Springer-Verlag, New York, 1993.Google Scholar
[18] Islam, M. N., Periodic solutions of Volterra type integral equations with finite delay. Comm. Appl. Anal. 15 (2011), 57–68.Google Scholar
[19] Kaufmann, E., Kosmatov, N., and Raffoul, Y., The connection between boundedness and periodicity in nonlinear functional neutral dynamic equations on a time scale. Nonlinear Dyn. Syst. Theory 9 (2009), no. 1, 89–98.Google Scholar
[20] Kuang, Y., Delay Differential Equations with Applications to Population Dynamics. Mathematics in Science and Engineering, 191. Academic Press, Boston, MA, 1993.Google Scholar
[21] Kuang, Y., Global Stability in one or two species neutral delay population models. Canad. Appl. Math. Quart. 1 (1993), 23–45.Google Scholar
[22] Liu, Y. and Li, Z., Krasnoselskii type fixed point theorems and applications. Proc. Amer. Math. Soc. 136 (2007), no. 4, 1213–1220.Google Scholar
[23] Miller, R. K., Nolinear Volterra Integral Equations. W. A. Benjamin, Menlo Park, CA, 1971.Google Scholar
[24] Smart, D. R., Fixed Point Theorems. Cambridge Tracts in Mathematics 66. Cambridge University Press, London, 1980.Google Scholar