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

Islam, Muhammad N.

Canadian mathematical bulletin, Tome 56 (2013) no. 1, pp. 80-91

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-123-5

Mots-clés : 45D05, 45J05, Volterra integral equation, periodic solutions, Liapunov's method, Krasnosel'skii's fixedpoint theorem, Schaefer's fixed point theorem

Bibliographie

[1] [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. Google Scholar | DOI

[2] [2] Burton, T. A., Integral equations, implicit functions and fixed points. Proc. Amer. Math. Soc. 124 (1996), no. 8, 2383–2390. Google Scholar | DOI

[3] [3] Burton, T. A., Liapunov Functionals for Integral Equations, Trafford Publishing, 2008. Google Scholar

[4] [4] Burton, T. A. and Colleen Kirk, A fixed point theorem of Krasnoselskii Schaefer type. Math. Nachr. 189 (1998), 23–31. Google Scholar | DOI

[5] [5] Burton, T. A., Eloe, P. W., and Islam, M. N., Periodic solutions of linear integro-differential equations. Math. Nachr. 147 (1990), 175–184. Google Scholar | DOI

[6] [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. Google Scholar | DOI

[7] [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. Google Scholar | DOI

[8] [8] Corduneanu, C., Integral Equations and Applications. Cambridge University Press, Cambridge, 1991. Google Scholar

[9] [9] Driver, R. D., Existence and continuous dependence of solutions of a neutral functional-differential equation. Arch. Rational Mech. Anal. 19 (1965), 149–166. Google Scholar | DOI

[10] [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. Google Scholar | DOI

[11] [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. Google Scholar | DOI

[12] [12] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics. Mathematics and its Applications 74. Kluwer, Dordrecht, 1992. Google Scholar

[13] [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] [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] [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. Google Scholar | DOI

[16] [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] [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] [18] Islam, M. N., Periodic solutions of Volterra type integral equations with finite delay. Comm. Appl. Anal. 15 (2011), 57–68. Google Scholar

[19] [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] [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] [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] [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] [23] Miller, R. K., Nolinear Volterra Integral Equations. W. A. Benjamin, Menlo Park, CA, 1971. Google Scholar

[24] [24] Smart, D. R., Fixed Point Theorems. Cambridge Tracts in Mathematics 66. Cambridge University Press, London, 1980. Google Scholar