×

Monotone method for a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. (English) Zbl 1157.45005

The authors study the existence of solutions to a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. For this purpose, a monotone iterative technique and lower and upper solutions are used. Furthermore, under wide monotonicity conditions and using noncompactness measure conditions, they obtain the existence of extremal solutions and a unique solution between lower and upper solutions. At the end of the paper, the theoretical results are applied to an illustrative example.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45G15 Systems of nonlinear integral equations
45L05 Theoretical approximation of solutions to integral equations
45J05 Integro-ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific: World Scientific Singapore · Zbl 0837.34003
[2] 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
[3] 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
[4] Liu, Y., Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations, J. Math. Anal. Appl., 327, 435-452 (2007) · Zbl 1119.34062
[5] Nieto, J. J.; Rodríguez-Lóez, R., Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, J. Math. Anal. Appl., 318, 593-610 (2006) · Zbl 1101.34051
[6] Nieto, J. J., Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Anal., 51, 1223-1232 (2002) · Zbl 1015.34010
[7] Ahmad, B.; Sivasundaram, S., The monotone iterative technique for impulsive hybrid set valued integro-differential equations, Nonlinear Anal., 65, 2260-2276 (2006) · Zbl 1111.45006
[8] Rachunková, I.; Tvrdý, M., Non-ordered lower and upper functions in second order impulsive periodic problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 12, 397-415 (2005) · Zbl 1086.34026
[9] Yang, X. X.; Shen, J. H., Nonlinear boundary value problems for first order impulsive functional differential equations, Appl. Math. Comput., 189, 2, 1943-1952 (2007) · Zbl 1125.65074
[10] Zhang, W.; Fan, M., Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Math. Comput. Model., 39, 479-493 (2004) · Zbl 1065.92066
[11] 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
[12] 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
[13] Zhang, X.; Shuai, Z.; Wang, K., Optimal impulsive harvesting policy for single population, Nonlinear Anal. RWA, 4, 639-651 (2003) · Zbl 1011.92052
[14] d’Onofrio, A., A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D: Nonlinear Phenom., 208, 220-235 (2005) · Zbl 1087.34028
[15] 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)
[16] Choisy, M.; Guegan, J. F.; Rohani, P., Dynamics of infectious diseases and pulse vaccination: Teasing apart the embedded resonance effects, Physica D: Nonlinear Phenom., 22, 26-35 (2006) · Zbl 1110.34031
[17] 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
[18] H. Zhang, L. Chen, J.J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Anal. RWA, in press, (doi:10.1016/j.nonrwa.2007.05.004); H. Zhang, L. Chen, J.J. Nieto, A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Anal. RWA, in press, (doi:10.1016/j.nonrwa.2007.05.004)
[19] Gao, S.; Teng, Z.; Nieto, J. J.; Torres, A., Analysis of an SIR epidemic model with pulse vaccination and distributed time delay, J. Biomedicine Biotechnol., 2007 (2007), Art. ID 64870, 10 pp
[20] 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
[21] Akhmet, M. U.; Turan, M., The differential equation on time scales through impulsive differential equations, Nonlinear Anal., 65, 2043-2060 (2006) · Zbl 1110.34006
[22] Carl, S.; Heikkilä, S., On discontinuous implicit and explicit abstract impulsive boundary value problems, Nonlinear Anal., 41, 701-723 (2000) · Zbl 0985.34050
[23] Chen, L. J.; Sun, J. T., Nonlinear boundary problem of first order impulsive integro-differential equations, J. Comput. Appl. Math., 202, 2, 392-401 (2007) · Zbl 1122.45007
[24] Guo, D. J., Initial value problems for nonlinear second order impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl., 200, 1-13 (1996) · Zbl 0851.45012
[25] 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
[26] 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
[27] 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
[28] Liu, Z. B.; Liu, L. S.; Wu, Y. H.; Zhao, J., Initial value problems in infinite interval of first order nonlinear impulsive integro-differential equations in Banach spaces, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 14, 13-26 (2007) · Zbl 1116.45008
[29] Liu, L. S.; Wu, C. X.; Guo, F., A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach spaces, J. Math. Anal. Appl., 275, 369-385 (2002) · Zbl 1014.45007
[30] Lu, H. Q., Extremal solutions of nonlinear first order impulsive integro-differential equations in Banach spaces, Indian J. Pure Appl. Math., 30, 11, 1181-1197 (1999) · Zbl 0943.45004
[31] Mitropolskiy, Yu. A.; Iovane, G.; Borysenko, S. D., About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their applications, Nonlinear Anal., 66, 2140-2165 (2007) · Zbl 1119.26026
[32] Nieto, J. J.; Rodríguez-López, R., Hybrid metric dynamical systems with impulses, Nonlinear Anal., 64, 368-380 (2006) · Zbl 1094.34007
[33] Nieto, J. J.; Rodríguez-López, R., New comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl., 328, 1343-1368 (2007) · Zbl 1113.45007
[34] Sun, J. L.; Ma, Y. H., Initial value problems for seconder order mixed monotone type of impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl., 247, 506-516 (2000) · Zbl 0964.34008
[35] 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. Model., 40, 509-518 (2004) · Zbl 1112.34052
[36] Yang, X. X.; Shen, J. H., Periodic boundary value problems for second-order impulsive integro-differential equations, J. Comput. Appl. Math., 209, 176-186 (2007) · Zbl 1155.45007
[37] Zeng, G. Z.; Chen, L. S.; Sun, L. H., Existence of periodic solution of order one of planar impulsive autonomous system, J. Comput. Appl. Math., 186, 2, 466-481 (2006) · Zbl 1088.34040
[38] Deimling, K., Nonlinear Functional Analysis (1985), Springer Verlag: Springer Verlag Berlin · Zbl 0559.47040
[39] Guo, D. J.; Lakshmikantham, V.; Liu, X. Z., Nonlinear Integral Equations in Abstract Spaces (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht
[40] 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
[41] Fang, Y. P.; Huang, N. J., Iterative algorithm for a system of variational inclusions involving \(H\)-accretive operators in Banach spaces, Acta Math. Hungar., 108, 3, 183-195 (2005) · Zbl 1094.49007
[42] Du, Y., Fixed points of increasing operators in ordered Banach spaces and applications, Appl. Anal., 38, 1-20 (1990) · Zbl 0671.47054
[43] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman: Pitman Boston, Mass, USA · Zbl 0658.35003
[44] Nieto, J. J.; Jiang, Y.; Jurang, Y., Monotone iterative method for functional-differential equations, Nonlinear Anal., 32, 741-747 (1998) · Zbl 0937.34053
[45] Vatsala, A. S.; Yang, J., Monotone iterative technique for semilinear elliptic systems, Bound. Value Probl., 2005, 93-106 (2005) · Zbl 1143.65388
[46] Drici, Z.; McRae, F. A.; Vasundhara Devi, J., Monotone iterative technique for periodic boundary value problems with causal operators, Nonlinear Anal., 64, 1271-1277 (2006) · Zbl 1208.34103
[47] West, I. H.; Vatsala, A. S., Generalized monotone iterative method for initial value problems, Appl. Math. Lett., 17, 1231-1237 (2004) · Zbl 1112.34304
[48] Jiang, D.; Nieto, J. J.; Zuo, W., On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations, J. Math. Anal. Appl., 289, 691-699 (2004) · Zbl 1134.34322
[49] Nieto, J. J.; Rodríguez-Lóez, R., Monotone method for first-order functional differential equations, Comput. Math. Appl., 52, 471-484 (2006) · Zbl 1140.34406
[50] Ahmad, B.; Nieto, J. J., The monotone iterative technique for three-point second-order integrodifferential boundary value problems with \(p\)-Laplacian, Bound. Value Probl., 2007 (2007), Art. ID 57481, 9pp · Zbl 1149.65098
[51] He, Z. M.; He, X. M., Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions, Comput. Math. Appl., 48, 73-84 (2004) · Zbl 1070.65136
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.