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On the existence of solutions for nonlinear first-order implicit impulsive integro-differential equations. (English) Zbl 1170.45008

The authors obtain new existence theorems for the solutions of a new class of initial value problems pertaining to nonlinear first-order implicit impulsive integro-differential equations in Banach spaces under some weaker conditions by employing the Mönch fixed-point theorem.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
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[1] Agarwal, R. P.; O’regan, D., A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem, Appl. Math. Comput., 161, 433-439 (2005) · Zbl 1070.34042
[2] Akhmet, M. U.; Kirane, M.; Tleubergenova, M. A.; Weber, G. W., Control and optimal response problems for quasilinear impulsive integrodifferential equations, European. J. Oper. Res., 169, 3, 1128-1147 (2006) · Zbl 1259.49037
[3] Akhmet, M. U.; Tleubergenova, M. A.; Yilmaz, O., Asymptotic behavior of linear impulsive integro-differential equations, Comput. Math. Appl., 56, 4, 1071-1081 (2008) · Zbl 1155.45301
[4] Akhmet, M. U.; Turan, M., The differential equation on time scales through impulsive differential equations, Nonlinear Anal., 65, 2043-2060 (2006) · Zbl 1110.34006
[5] Carl, S.; Heikkilä, S., On discontinuous implicit and explicit abstract impulsive boundary value problems, Nonlinear Anal., 41, 701-723 (2000) · Zbl 0985.34050
[6] Franco, D.; Nieto, J. J., Remarks on a periodic boundary value problem for second order impulsive integro-differential equations, (International Conference on Differential Equations, vol. 1, 2 (Berlin 1999) (2000), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 500-502 · Zbl 0965.45010
[7] Guo, D. J., Multiple positive solutions for first order nonlinear impulsive integro-differential equations in Banach spaces, Appl. Math. Comput., 143, 233-249 (2003) · Zbl 1030.45009
[8] Heikkilä, S.; Kumpulainen, M.; Seikkala, S., Uniqueness and existence results for implicit impulsive differential equations, Nonlinear Anal., 42, 13-26 (2000) · Zbl 0969.34012
[9] Huang, N. J.; Lan, H. Y., Nonlinear first-order implicit impulsive differential equations in Banach spaces, Indian J. Pure Appl. Math., 35, 10, 1201-1214 (2004) · Zbl 1078.34041
[10] Lan, H. Y.; Huang, N. J.; Kim, J. K., First order nonlinear implicit impulsive integro-differential equations in Banach spaces, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 13, 803-813 (2006) · Zbl 1136.34325
[11] Li, Y. X.; Liu, Z., Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces, Nonlinear Anal., 66, 83-92 (2007) · Zbl 1109.34005
[12] Lu, H. Q.; Liu, L. S., The existence theorems of whole solutions of nonlinear impulsive Volterra type integral equations in Banach spaces and applications, Acta Math. Scientia, 20, 1, 101-108 (2000), (in Chinese) · Zbl 0962.45009
[13] Z.G. Luo, J.J. Nieto, New results for the periodic boundary value problem for impulsive integro-differential equations, Nonlinear Anal. TMA, in press (corrected proof, available online 18 March 2008, doi:10.1016/j.na.2008.03.004); Z.G. Luo, J.J. Nieto, New results for the periodic boundary value problem for impulsive integro-differential equations, Nonlinear Anal. TMA, in press (corrected proof, available online 18 March 2008, doi:10.1016/j.na.2008.03.004)
[14] Sakthivel, R.; Mahmudov, N. I.; Nieto, J. J.; Kim, J. H., On controllability of nonlinear impulsive integrodifferential systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15, 3, 333-343 (2008) · Zbl 1147.93010
[15] Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific: World Scientific Singapore · Zbl 0837.34003
[16] Wei, W.; Hou, S. H.; Teo, K. L., On a class of strongly nonlinear impulsive differential equation with time delay, Nonlinear Dyn. Syst. Theory, 6, 281-293 (2006) · Zbl 1114.47067
[17] Xu, L. G.; Xu, D. Y., Exponential stability of nonlinear impulsive neutral integro-differential equations, Nonlinear Anal., 69, 9, 2910-2923 (2008) · Zbl 1155.34354
[18] Zhang, J. Q., Solutions of second impulsive integro-differential equations in Banach spaces, Acta Math. Scientia, 19, 5, 13-17 (1999), (in Chinese)
[19] Zhang, X. Y.; Yu., Z. X., Solutions of nonlinear impulsive integro-differential equations of mixed type, J. Qufu Normal Univ., 27, 2, 13-17 (2001), (in Chinese) · Zbl 0993.45014
[20] Zhou, Q. H., Global exponential stability for a class of impulsive integro-differential equation, Internat J. Bifur. Chaos Appl. Sci. Engrg., 18, 3, 735-743 (2008) · Zbl 1147.34357
[21] Borysenko, S.; Iovane, G., About some new integral inequalities of Wendroff type for discontinuous functions, Nonlinear Anal. TMA, 66, 2190-2203 (2007) · Zbl 1135.26012
[22] Choisy, M.; Guegan, J. F.; Rohani, P., Dynamics of infectious diseases and pulse vaccination: Teasing apart the embedded resonance effects, Physica D, 22, 26-35 (2006) · Zbl 1110.34031
[23] Gao, S.; Chen, L.; Nieto, J. J.; Torres, A., Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24, 6037-6045 (2006)
[24] Guo, D. J., Existence of positive solutions for \(n\) th-order nonlinear impulsive singular integro-differential equations in Banach spaces, Nonlinear Anal. TMA, 68, 9, 2727-2740 (2008) · Zbl 1140.45017
[25] Iovane, G., Some new integral inequalities of Bellman-Bihari type with delay for discontinuous functions, Nonlinear Anal. TMA, 66, 498-508 (2007) · Zbl 1118.26022
[26] Li, W.; Huo, H., Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics, J. Comput. Appl. Math, 174, 227-238 (2005) · Zbl 1070.34089
[27] Liu, L. S.; Liu, Z. B.; Wu, Y. H., Infinite boundary value problems for \(n\) th-order nonlinear impulsive integro-differential equations in Banach spaces, Nonlinear Anal. TMA, 67, 9, 2670-2679 (2007) · Zbl 1124.45008
[28] Li, J.; Nieto, J. J.; Shen, J., Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl., 325, 226-236 (2007) · Zbl 1110.34019
[29] Mitropolskiy, Yu. A.; Iovane, G.; Borysenko, S. D., About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their applications, Nonlinear Anal. TMA, 66, 2140-2165 (2007) · Zbl 1119.26026
[30] Nieto, J. J.; Oregan, D., Variational approach to impulsive differential equations, Nonlinear Anal. RWA, 10, 2, 680-690 (2009) · Zbl 1167.34318
[31] Nieto, J. J.; Rodríguez-López, R., New comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl., 328, 2, 1343-1368 (2007) · Zbl 1113.45007
[32] Nieto, J. J.; Rodríguez-López, R., Boundary value problems for a class of impulsive functional equations, Comput. Math. Appl., 55, 12, 2715-2731 (2008) · Zbl 1142.34362
[33] d’onofrio, A., On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. Math. Lett., 18, 729-732 (2005) · Zbl 1064.92041
[34] Tang, S.; Chen, L., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol., 44, 185-199 (2002) · Zbl 0990.92033
[35] Wang, W.; Wang, H.; Li, Z., The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy, Chaos Solitons Fractals, 32, 1772-1785 (2007) · Zbl 1195.92066
[36] Wang, W. X.; Zhang, L. L.; Liang, Z. D., Initial value problems for nonlinear impulsive integro-differential equations in Banach space, J. Math. Anal. Appl., 320, 2, 510-527 (2006) · Zbl 1097.45011
[37] Yan, J.; Zhao, A.; Nieto, J. J., Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems, Math. Comput. Modelling, 40, 509-518 (2004) · Zbl 1112.34052
[38] Zavalishchin, S. T.; Sesekin, A. N., Dynamic Impulse Systems: Theory and Applications (1997), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0880.46031
[39] Zhang, H.; Chen, L.; Nieto, J. J., A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Anal. RWA, 9, 4, 1714-1726 (2008) · Zbl 1154.34394
[40] Zhang, W.; Fan, M., Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Math. Comput. Modelling, 39, 479-493 (2004) · Zbl 1065.92066
[41] Zhang, X. G.; Liu, L. S., Initial value problems for nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. TMA, 64, 11, 2562-2574 (2006) · Zbl 1093.45006
[42] Zhang, H. J.; Liu, L. S.; Wu, Y. H., Positive solutions for \(n\) th-order nonlinear impulsive singular integro-differential equations on infinite intervals in Banach spaces, Nonlinear Anal. TMA, 70, 2, 772-787 (2009) · Zbl 1158.45005
[43] Lan, H. Y., Monotone method for a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces, J. Comput. Appl. Math, 222, 2, 531-543 (2008) · Zbl 1157.45005
[44] H.Y. Lan, Existence and uniqueness results for nonlinear first-order implicit impulsive integro-differential equations with monotone conditions, Dyn. Contin. Discrete Impuls. Syst. Ser. A (accepted on 2007-09-13, the possible publication in Feb 2010, http://www.monotone.uwaterloo.ca/ journal); H.Y. Lan, Existence and uniqueness results for nonlinear first-order implicit impulsive integro-differential equations with monotone conditions, Dyn. Contin. Discrete Impuls. Syst. Ser. A (accepted on 2007-09-13, the possible publication in Feb 2010, http://www.monotone.uwaterloo.ca/ journal)
[45] Guo, D. J.; Lakshmikantham, V.; Liu, X. Z., Nonlinear Integral Equations in Abstract Spaces (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht
[46] Deimling, K., Nonilnear Functional Analysis (1985), Springer Verlag: Springer Verlag Berlin
[47] Hein, H. P., On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7, 1351-1371 (1983) · Zbl 0528.47046
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