Abstract
The purpose of this work is to give sufficient conditions which guarantee the existence and the uniqueness of positive μ-pseudo almost periodic solutions for the nonlinear infinite delay integral equation . We improve the original work of [H. S. Ding, Y. Y. Chen and G. M. N’Guérékata, Existence of positive pseudo almost periodic solutions to a class of neutral integral equations, Nonlinear Analysis. Theorey, Methods and Applications 74 (2011) 7356-7364] by dropping the hypotheses of monotonicity on the functions f and h. The main results are proved by using the Hilbert’s projective metric combined with the contraction mapping principle. Our results can deal with some cases to which many results are not applicable. An example is provided to illustrate the main results of this work.
References
[1] E. Ait Dads, O. Arino, K. Ezzinbi, Existence of periodic solution for some neutral nonlinear integral equation with delay time dependent. Facta Univ. Ser. Math. Inform. 11 (1996), 79-92.Search in Google Scholar
[2] E. Ait Dads, O. Arino, K. Ezzinbi, Positive almost periodic solution for some nonlinear delay integral equation. Nonlinear Stud. 3 (1996), 85-101.Search in Google Scholar
[3] E. Ait Dads, P. Cieutat, L. Lhachimi, Positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations. Mathematical and Computer Modelling. 49 (2008), 721-739.Search in Google Scholar
[4] E. Ait Dads, K. Ezzinbi, Almost periodic solutions for some neutral nonlinear integral equation. Nonlinear Anal. 28 (1997), 1479-1489.Search in Google Scholar
[5] T.A. Burton, L. Hatvani, On the existence of periodic solutions of some nonlinear functional-diffierential equations with unbounded delay. Nonlinear Anal. 16 (1991), 389-396.Search in Google Scholar
[6] s. Busenberg, K. Cooke, Periodic solutions to delay diffierential equations arising in some models of epidemiology. Applied Nonlinear Analysis. 67-78. Academic Press (1979).10.1016/B978-0-12-434180-7.50011-5Search in Google Scholar
[7] J. Blot, P. Cieutat, K. Ezzinbi, New approach for weightad pseudo almost functions under the light of measure theory, basic results and applications. Appl. Anal. (2011), 1-34.Search in Google Scholar
[8] T. Diagana, Weighted pseudo almost periodic functions and applications. C. R.Math. Acad. Sci. Paris. 343 (2006), 643-646.Search in Google Scholar
[9] T. Diagana, Weighted pseudo almost periodic solutions to some diffierential equations. Nonlinear Anal. 68 (2008), 2250-2260.Search in Google Scholar
[10] K. Ezzinbi, M. Hachimi, Existence of positive almost periodic solutions of functional equations via Hilbert’s projective metric. Nonlinear Anal. 26 (1996), 1169-1176.Search in Google Scholar
[11] A. Fink, J. Gatica, Positive almost periodic solutions of some delay integral equations. J. Diffierential Equations. 83 (1990), 166-178.Search in Google Scholar
[12] D. Guo, V. Lakshmikantham, Positive solutions of nonlinear integral equations arising in infectious disease. J. Math. Anal. Appl. 134 (1988), 1-8.Search in Google Scholar
[13] J. Kaplan, M. Sorg, J. Yorke, Solutions of x0(t) = f (x(t), x(t − L)) have limits when f is an order relation. Nonlinear Anal. 3 (1979), 53-58.Search in Google Scholar
[14] R.W. Leggett, L.R. Williams, A fixed point theorem with application to an infectious disease model. J. Math. Anal. Appl. 76 (1980), 91-97.10.1016/0022-247X(80)90062-1Search in Google Scholar
[15] R.W. Leggett, L.R. Williams, Nonzero solutions of nonlinear integral equations modeling infectious disease. SIAM J. Math. Anal. 13 (1982), 112-121.Search in Google Scholar
[16] C. Corduneanu, Almost Periodic Functions. Wiley, New York, 1968 (Reprented, Chelsea, New York, 1989).Search in Google Scholar
[17] P. Cieutat, K. Ezzinbi, Positive pseudo almost automorphic solutions for some nonlinear infinite delay integral equations. Afr. Diaspora J. Math. 12 (2011), 19-33.Search in Google Scholar
[18] K.L. Cooke, J.L. Kaplan, A periodic treshold theorem for epidemics and population growth. Math. Biosci. 3 (1976), 87-104.10.1016/0025-5564(76)90042-0Search in Google Scholar
[19] H. S. Ding, Y. Y. Chen, G. M. N’Guérékata, Existence of positive pseudo almost periodic solutions to a class of neutral integral equations. Nonlinear Anal. 74 (2011), 7356-7364.Search in Google Scholar
[20] H. S. Ding, J. Liang, G. M. N’Guérékata, T. J. Xiao, Existence of positive almost automorphic solutions to neutral nonlinear integral equations. Nonlinear Anal. 69 (2008), 1188-1199.Search in Google Scholar
[21] R. Nussbanum, A periodicity threshold theorem for some nonlinear integral equations. SIAM J. Math. Anal. 9 (1978), 356-376.10.1137/0509024Search in Google Scholar
[22] H.L. Smith, A abstract threshold theorem for one parameter families of positive noncompact operators. Funkcial. Ekvac. 24 (1981), 141-153.Search in Google Scholar
[23] A.C. Thompson, On certain contraction mappings in a partially ordered vector space. Proc. Amer.Math. Soc. 14 (1963), 438-443.Search in Google Scholar
[24] R. Torrejón, Positive almost periodic solutions of a state dependent delay nonlinear integral equation. Nonlinear Anal. 20 (1993), 1383-1416.Search in Google Scholar
[25] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, New York, 1975.10.1007/978-1-4612-6376-0Search in Google Scholar
[26] C. Zhang, Integration of vector valued pseudo almost periodic functions. Proc. Amer. Math. Soc. 121 (1994), 167-174.Search in Google Scholar
[27] C. Zhang, Pseudo almost periodic functions of some diffierential equations. J. Math. Anal. Appl. 181 (1994), 62-676.Search in Google Scholar
[28] C. Zhang, Pseudo almost periodic functions of some diffierential equations II. J. Math. Anal. Appl. 192 (1995), 543-561.Search in Google Scholar
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