Solvability of three point boundary value problems for second order differential equations with deviating arguments. (English) Zbl 1154.34367

Summary: The monotone iterative technique is used to boundary problems for second order ordinary differential equations with deviating arguments \[ x''(t)=f(t,x(t),x(\alpha(t))),\quad t\in[0,T], \]
\[ x(0)=0, x(T)=rx(\gamma), \] where the numbers \(\gamma,T,r\) are fixed with \(0<\gamma<T\). The cases of positive and negative \(r\) are considered separately. Corresponding results are formulated when the problem has extremal solutions or weakly coupled extremal quasi-solutions.


34K10 Boundary value problems for functional-differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
Full Text: DOI


[1] Bellen, A., Monotone methods for periodic solutions of second order scalar functional differential equations, Numer. Math., 42, 15-30 (1983) · Zbl 0536.65065
[2] Cabada, A.; Habets, P.; Lois, S., Monotone method for the Neumann problem with lower and upper solutions in the reverse order, Appl. Math. Comput., 117, 1-14 (2001) · Zbl 1031.34021
[3] Cherpion, M.; DeCoster, C.; Habets, P., A constructive monotone iterative method for second-order BVP in the presence of lower and upper solutions, Appl. Math. Comput., 123, 75-91 (2001) · Zbl 1024.65063
[4] Eloe, P. W.; Zhang, Y., A quadratic monotone iteration scheme for two-point boundary value problems for ordinary differential equations, Nonlinear Anal., 33, 443-453 (1998) · Zbl 0939.34019
[5] Jankowski, T., Existence of solutions for functional antiperiodic boundary value problems, Math. Pannon., 12, 201-215 (2001) · Zbl 0981.34056
[6] Jankowski, T., Functional differential equations of second order, Bull. Belg. Math. Soc., 10, 291-298 (2003) · Zbl 1045.34037
[7] Jankowski, T., Existence of solutions of boundary value problems for differential equations with delayed arguments, J. Comput. Appl. Math., 156, 239-252 (2003) · Zbl 1048.34107
[8] Jankowski, T., Monotone method for second order delayed differential equations with boundary value conditions, Appl. Math. Comput., 149, 589-598 (2004) · Zbl 1039.34058
[9] Jankowski, T., Advanced differential equations with nonlinear boundary conditions, J. Math. Anal. Appl., 304, 490-503 (2005) · Zbl 1092.34032
[11] Jiang, D.; Wei, J., Monotone method for first- and second-order periodic boundary value problems and periodic solutions of functional differential equations, Nonlinear Anal., 50, 885-898 (2002) · Zbl 1014.34049
[12] Jiang, D.; Fan, M.; Wan, A., A monotone method for constructing extremal solutions to second order periodic boundary value problems, J. Comput. Appl. Math., 136, 189-197 (2001) · Zbl 0993.34011
[13] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman: Pitman Boston · Zbl 0658.35003
[14] Lakshmikantham, V., Periodic boundary value problems of first and second order differential equations, J. Appl. Math. Sim., 2, 131-138 (1989) · Zbl 0712.34058
[15] Lakshmikantham, V.; Vatsala, A. S., Generalized Quasilinearization for Nonlinear Problems (1998), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0997.34501
[16] 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
[17] Nieto, J. J.; Rodríguez-López, R., Remarks on periodic boundary value problems for functional differential equations, J. Comput. Appl. Math., 158, 339-353 (2003) · Zbl 1036.65058
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