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Existence results for fractional neutral integro-differential equations with state-dependent delay. (English) Zbl 1228.45014
Summary: We study the existence of mild solutions for a class of abstract fractional neutral integro-differential equations with state-dependent delay. The results are obtained by using the Leray-Schauder alternative fixed point theorem. An example is provided to illustrate the main results.

##### MSC:
 45K05 Integro-partial differential equations 47H10 Fixed-point theorems
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##### References:
 [1] Bartha, M., Periodic solutions for differential equations with state-dependent delay and positive feedback, Nonlinear anal. TMA, 53, 6, 839-857, (2003) · Zbl 1028.34062 [2] Cao, Y.; Fan, J.; Gard, T.C., The effects of state-dependent time delay on a stage-structured population growth model, Nonlinear anal. TMA, 19, 2, 95-105, (1992) · Zbl 0777.92014 [3] Cuevas, C.; N’Guérékata, G.; Rabelo, M., Mild solutions for impulsive neutral functional differential equations with state-dependent delay, Semigroup forum (80), 375-390, (2010) · Zbl 1197.34153 [4] Domoshnitsky, A.; Drakhlin, M.; Litsyn, E., On equations with delay depending on solution, Nonlinear anal. TMA, 49, 5, 689-701, (2002) · Zbl 1012.34066 [5] Chen, F.; Sun, D.; Shi, J., Periodicity in a food-limited population model with toxicants and state dependent delays, J. math. anal. appl., 288, 1, 136-146, (2003) · Zbl 1087.34045 [6] Hartung, F., Linearized stability in periodic functional differential equations with state-dependent delays, J. comput. appl. math., 174, 2, 201-211, (2005) · Zbl 1077.34074 [7] Hartung, F.; Herdman, T.; Turi, J., Parameter identification in classes of neutral differential equations with state-dependent delays, Nonlinear anal. TMA, 39, 3, 305-325, (2000) · Zbl 0955.34067 [8] Kuang, Y.; Smith, H., Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonlinear anal. TMA, 19, 9, 855-872, (1992) · Zbl 0774.34054 [9] Torrejón, R., Positive almost periodic solutions of a state-dependent delay nonlinear integral equation, Nonlinear anal. TMA, 20, 12, 1383-1416, (1993) · Zbl 0787.45003 [10] dos Santos, J.P.C., On state-dependent delay partial neutral functional integro-differential equations, Appl. math. comput., 216, 1637-1644, (2010) · Zbl 1196.45013 [11] dos Santos, J.P.C., Existence results for a partial neutral integro-differential equation with state-dependent delay, Electron. J. qual. theory differ. equ., 29, 1-12, (2010) · Zbl 1208.45009 [12] dos Santos, J.P.C.; Cuevas, C.; de Andrade, B., Existence results for a fractional equation with state-dependent delay, Adv. difference equ., 2011, 1-15, (2011), Article ID 642013 · Zbl 1216.45003 [13] Hernández, E.; Ladeira, L.; Prokopczyk, A., A note on state dependent partial functional differential equations with unbounded delay, Nonlinear anal. RWA, 7, 4, 510-519, (2006) · Zbl 1109.34060 [14] Hernández, E.; McKibben, M., On state-dependent delay partial neutral functional differential equations, Appl. math. comput., 186, 1, 294-301, (2007) · Zbl 1119.35106 [15] Hernández, E.; McKibben, M.; Henríquez, H., Existence results for partial neutral functional differential equations with state-dependent delay, Math. comput. modelling (49), 1260-1267, (2009) · Zbl 1165.34420 [16] Cascaval, R.C.; Eckstein, E.C.; Frota, C.L.; Goldstein, J.A., Fractional telegraph equations, J. math. anal. appl., 276, 145-159, (2002) · Zbl 1038.35142 [17] Eidelman, S.D.; Kochubei, A.N., Cauchy problem for fractional diffusion equations, J. differential equations, 199, 211-255, (2004) · Zbl 1129.35427 [18] J.A.T. Machado, M.F. Silva, R.S. Barbosa, I.S. Jesus, C.M. Reis, M.G. Marcos, A.F. Galhano, Some applications of fractional calculus in engineering, Math. Problems Eng., doi:10.1155/2010/639801. · Zbl 1191.26004 [19] Ahn, V.V.; McVinish, R., Fractional differential equations driven by levy noise, J. appl. stoch. anal., 16, 2, 97-119, (2003) · Zbl 1042.60034 [20] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (), 223-276 [21] Hilfer, H., Applications of fractional calculus in physics, (2000), World Scientific Publ. Co. Singapore · Zbl 0998.26002 [22] Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002 [23] Agarwal, R.P.; de Andrade, B.; Cuevas, C., On type of periodicity and ergodicity to a class of fractional order differential equations, Adv. difference equ., 2010, 1-25, (2010), Article ID 179750 · Zbl 1194.34007 [24] Agarwal, R.P.; de Andrade, B.; Cuevas, C., Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear anal. RWA (11), 3532-3554, (2010) · Zbl 1248.34004 [25] R.P. Agarwal, J.P.C. dos Santos, C. Cuevas, Analytic Resolvent Operator and Existence Results for Fractional Integro-differential Equations (submitted for publication). · Zbl 1330.45008 [26] Agarwal, R.P.; Benchohra, M.; Hammani, S., A survey on existence results of nonlinear fractional differential equations and inclusions, Acta appl. math., 109, 3, 973-1033, (2010) · Zbl 1198.26004 [27] Agarwal, R.P.; Belmekki, M.; Benchohra, M., A survey on semilinear differential equations and inclusions involving riemann – liouville fractional derivative, Adv. difference equ., (2009), 47 pages, Article ID 981728 · Zbl 1182.34103 [28] Agarwal, R.P.; Lakshmikantham, V.; Nieto, J.J., On the concept of solution for fractional differential equations with uncertainly, Nonlinear anal., 72, 6, 2859-2862, (2010) · Zbl 1188.34005 [29] Agarwal, R.P.; Zhou, Y.; He, Y., Existence of fractional neutral functional differential equations, Comput. math. appl., 59, 1095-1100, (2010) · Zbl 1189.34152 [30] Benchohra, M.; Henderson, J.; Ntouyas, S.K.; Ouahab, A., Existence results for fractional order functional differential equations with infinite delay, J. math. anal. appl., 338, 1340-1350, (2008) · Zbl 1209.34096 [31] Cuevas, C.; de Souza, J.C., $$S$$-asymptotically $$\omega$$-periodic solutions of semilinear fractional integro-differential equations, Appl. math. lett., 22, 865-870, (2009) · Zbl 1176.47035 [32] Cuevas, C.; de Souza, J.C., Existence of $$S$$-asymptotically $$\omega$$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear anal. (72), 3-4, 1683-1689, (2010) · Zbl 1197.47063 [33] Cuevas, C.; Rabelo, M.; Soto, H., Pseudo almost automorphic solutions to a class of semilinear fractional differential equations, Commun. appl. nonlinear anal. (17), 33-48, (2010) [34] dos Santos, J.P.C.; Cuevas, C., Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. math. lett. (23), 960-965, (2010) · Zbl 1198.45014 [35] Lakshmikantham, V., Theory of fractional differential equations, Nonlinear anal., 60, 10, 3337-3343, (2008) · Zbl 1162.34344 [36] Lakshmikantham, V.; Devi, J.V., Theory of fractional differential equations in Banach spaces, Eur. J. pure appl. math., 1, 38-45, (2008) · Zbl 1146.34042 [37] Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for $$p$$-type fractional neutral differential equations, Nonlinear anal. TMA, 71, 2724-2733, (2009) · Zbl 1175.34082 [38] Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal. TMA, 71, 3249-3256, (2009) · Zbl 1177.34084 [39] Zhou, Y.; Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal. RWA, 11, 4465-4475, (2010) · Zbl 1260.34017 [40] E. Bazhlekova, Fractional evolution equations in Banach spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. [41] Ph. Clément, G. Gripenberg, S.-O. Londen, Regularity properties of solutions of fractional evolutions equations, Helsinki University of Technology Institute of Mathematics Research Reports, A413. · Zbl 1010.47046 [42] Cuesta, E., Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete cont. dyn. syst., Supplement, 277-285, (2007) · Zbl 1163.45306 [43] Diethelm, K., The analysis of fractional differential equations, (2010), Springer-Verlag Berlin [44] Hino, Y.; Murakami, S.; Naito, T., () [45] Granas, A.; Dugundji, J., Fixed point theory, (2003), Springer-Verlag New York · Zbl 1025.47002 [46] Martin, R., Nonlinear operators and differential equations in Banach spaces, (1987), Robert E. Krieger Publ. Co. Florida
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