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Boundary value problems for a class of impulsive functional equations. (English) Zbl 1142.34362
Summary: This paper is related to the existence and approximation of solutions for impulsive functional differential equations with periodic boundary conditions. We study the existence and approximation of extremal solutions to different types of functional differential equations with impulses at fixed times, by the use of the monotone method. Some of the options included in this formulation are differential equations with maximum and integro-differential equations. In this paper, we also prove that the Lipschitzian character of the function which introduces the functional dependence in a differential equation is not a necessary condition for the development of the monotone iterative technique to obtain a solution and to approximate the extremal solutions to the equation in a given functional interval. The corresponding results are established for the impulsive case. The general formulation includes several types of functional dependence (delay equations, equations with maxima, integro-differential equations). Finally, we consider the case of functional dependence which is given by nonincreasing and bounded functions.
MSC:
34K13Periodic solutions of functional differential equations
34K45Functional-differential equations with impulses
References:
[1]Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[2]Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations, (1995) · Zbl 0837.34003
[3]Franco, D.; Nieto, J. J.: Maximum principles for periodic impulsive first order problems, J. comput. Appl. math. 88, 149-159 (1998) · Zbl 0898.34010 · doi:10.1016/S0377-0427(97)00212-4
[4]Wan, Z.; Chen, Y.; Chen, J.: Remarks on the periodic boundary value problems for first-order differential equations, Comput. math. Appl. 37, No. 8, 49-55 (1999) · Zbl 0936.34013 · doi:10.1016/S0898-1221(99)00100-5
[5]Nieto, J. J.; Rodríguez-López, R.: Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions, Comput. math. Appl. 40, 433-442 (2000) · Zbl 0958.34055 · doi:10.1016/S0898-1221(00)00171-1
[6]Nieto, J. J.; Rodríguez-López, R.: Remarks on periodic boundary value problems for functional differential equations, J. comput. Math. appl. 158, 339-353 (2003) · Zbl 1036.65058 · doi:10.1016/S0377-0427(03)00452-7
[7]Nieto, J. J.; Rodríguez-López, R.: Monotone method for first-order functional differential equations, Comput. math. Appl. 52, 471-484 (2006) · Zbl 1140.34406 · doi:10.1016/j.camwa.2006.01.012
[8]He, Z.; Yu, J.: Periodic boundary value problem for first-order impulsive functional differential equations, J. comput. Appl. math. 138, 205-217 (2002) · Zbl 1004.34052 · doi:10.1016/S0377-0427(01)00381-8
[9]Nieto, J. J.; Rodríguez-López, R.: Periodic boundary value problems for non-Lipschitzian impulsive functional differential equations, J. math. Anal. appl. 318, 593-610 (2006) · Zbl 1101.34051 · doi:10.1016/j.jmaa.2005.06.014
[10]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 · doi:10.1016/j.jmaa.2006.06.029
[11]Nieto, J. J.; Rodríguez-López, R.: Hybrid metric dynamical systems with impulses, Nonlinear anal. 64, 368-380 (2006) · Zbl 1094.34007 · doi:10.1016/j.na.2005.05.068
[12]Nieto, J. J.; Rodríguez-López, R.: Comparison results and approximation of extremal solutions for second-order functional differential equations, J. nonlinear funct. Anal. differential equations 1, No. 1, 67-102 (2007) · Zbl 1163.34043
[13]Chen, L.; Sun, J.: Nonlinear boundary value problem of first order impulsive functional differential equations, J. math. Anal. appl. 318, 726-741 (2006) · Zbl 1102.34052 · doi:10.1016/j.jmaa.2005.08.012
[14]Jankowski, T.; Nieto, J. J.: Boundary value problems for first order impulsive ordinary differential equations with delay arguments, Indian J. Pure. appl. Math. 38, 203-211 (2007)
[15]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 · doi:10.1016/j.jmaa.2005.04.005
[16]Gallo, A.; Piccirillo, A. M.: About new analogies of Gronwall–Bellman–bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems, Nonlinear anal. 67, 1550-1559 (2007) · Zbl 1124.26013 · doi:10.1016/j.na.2006.07.038
[17]J.J. Nieto, R. Rodríguez-López, Comparison results and approximation of solutions for impulsive functional differential equations, DCDIS Ser. A: Math. Anal. (in press) · Zbl 1153.34048
[18]Franco, D.; Liz, E.; Nieto, J. J.; Rogovchenko, Y.: A contribution to the study of functional differential equations with impulses, Math. nachr. 218, 49-60 (2000) · Zbl 0966.34073 · doi:10.1002/1522-2616(200010)218:1<49::AID-MANA49>3.0.CO;2-6
[19]Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations, (1985)
[20]Smart, D. R.: Fixed point theorems, (1974)