×

Advanced differential equations with nonlinear boundary conditions. (English) Zbl 1092.34032

Consider the nonlinear boundary value problem \[ x'(t)=f(t, c(t),x( \alpha(t))), \quad 0= g(x(0),x(T))\tag{*} \] under the assumption that \(f,\alpha\) and \(g\) are continuous functions with \(b\leq \alpha(t)\leq T\). By means of the method of monotone iteration based on lower and upper solutions for (*), the author derives sufficient conditions for (*) to have an extremal solution or a unique solution. Linear advanced differential inequalities are discussed, too.

MSC:

34K10 Boundary value problems for functional-differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Franco, D.; Nieto, J. J., First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions, Nonlinear Anal., 42, 163-173 (2000) · Zbl 0966.34025
[2] Franco, D.; Nieto, J. J.; O’Regan, D., Existence of nonnegative solutions for resonant periodic boundary value problems with impulses, Nonlinear Stud., 9, 1-10 (2002)
[3] Jankowski, T., Monotone iterative technique for differential equations with nonlinear boundary conditions, Nonlinear Stud., 8, 381-388 (2001) · Zbl 1091.34507
[4] 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
[5] Jankowski, T., Monotone method for second-order delayed differential equations with boundary value conditions, Appl. Math. Comput., 149, 589-598 (2004) · Zbl 1039.34058
[6] 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
[7] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman: Pitman Boston · Zbl 0658.35003
[8] Liz, E.; Nieto, J. J., Periodic boundary value problems for a class of functional differential equations, J. Math. Anal. Appl., 200, 680-686 (1996) · Zbl 0855.34080
[9] 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
[10] 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
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.