Title | PERIODIC SOLUTIONS OF VOLTERRA TYPE INTEGRAL EQUATIONS WITH FINITE DELAY |
Publication Type | Journal Article |
Year of Publication | 2011 |
Authors | ISLAM, MUHAMMADN |
Secondary Title | Communications in Applied Analysis |
Volume | 15 |
Issue | 1 |
Start Page | 57 |
Pagination | 68 |
Date Published | 01/2011 |
Type of Work | scientific: mathematics |
ISSN | 1083–2564 |
AMS | 45D05, 45J05 |
Abstract | In this paper, we study the existence of periodic solutions of Volterra type integral equations with finite delay. These equations often arise from delay differential equations. 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.
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URL | http://www.acadsol.eu/en/articles/15/1/5.pdf |
Short Title | PERIODIC SOLUTIONS OF INTEGRAL EQUATIONS |
Refereed Designation | Refereed |
Full Text | REFERENCES[1] M. Adivar and Y. Raffoul, Existence results for periodic solutions of integro-dynamic equations on time scales, Ann. Mat. Pura. Appl., 188(2009), 543-559.
[2] T. A. Burton, Integral equations, implicit functions and fixed points, Proc. Amer. Math. Soc., 124(1996), 2383-2390 [3] T. A. Burton, Liapunov Functionals for Integral Equations, Trafford Publishing, Victoria, Canda, 2008. [4] T. A. Burton and C. Kirk, A fixed point theorem of Krasnoselskii Schaefer type, Math. Nachr., 189 (1998), 23-31. [5] T. A. Burton, P. W. Eloe, and M. N. Islam, Periodic solutions of linear integrodifferential equations, Math. Nachr., 147 (1990), 175-184. [6] T. A. Burton, P. W. Eloe, and M. N. Islam, Nonlinear integrodifferential equations and a priori bounds on periodic solutions, Ann. Mat. Pura. Appl., CLXI (1992), 271-283. [7] T. A. Burton, and Bo Zhang, Periodic solutions of abstract differential equations with infinite delay, J. Differential Equations, 90 (1991), 357-396. [8] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991. [9] P. W. Eloe, and J. Henderson, Nonlinear boundary value problems and a priori bounds on solutions, SIAM J. Math. Anal., 15 (1984), 642-647. [10] P. W. Eloe, and M. N. Islam, Periodic solutions of nonlinear integral equations with infinite memory, Applicable Analysis, 28 (1988), 79-93. [11] A. Granas, Sur la methode de continuit ́e de Poincar ́e, C. R. Acad. Sci., Paris, 282 (1976), 983-985. [12] A. Granas, R. B. Guenther, and J. W. Lee, Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mountain J. Math., 10 (1980), 35-58. [13] G. Gripenberg, S. O. Londen, and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990.
[14] J. K. Hale, and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [15] M. N. Islam, Three fixed point theorems; periodic solutions of a Volterra type integral equation with infinite heredity, Canadian Mathematical Bulletin, to appear. [16] E. Kaufmann, N. Kosmatov, and Y. Raffoul, The connection between boundedness and periodicity in nonlinear functional neutral dynamic equations on a time scale, Nonlinear Dyn. Syst. Theory, 9(2009), 89-98. [17] Y. Liu and Z. Li, Krasnoselskii type fixed point theorems and applications, Proc. Amer. Math. Soc., 136(2008), 1213-1220. [18] R. K. Miller, Nolinear Volterra Integral Equations, Benjamin, New York, 1971. [19] D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1980. |