×

zbMATH — the first resource for mathematics

A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems. (English) Zbl 0719.34039
The authors study the nonlinear second order periodic boundary value problem \((1)\quad -u''=f(t,u),\quad u(0)=u(2\pi),\quad u'(0)=u'(2\pi)\) in the Sobolev space \(W^{2,1}(I)\), \(I=[0,2\pi]\), where f is a Carathéodory function. They generalize the monotone iterative method to prove the existence of periodic solutions of (1) when the lower and upper solutions \(\alpha\), \(\beta\) of (1) do not necessarily satisfy \((2)\quad \alpha '(0)\geq \alpha '(2\pi),\quad \beta '(0)\leq \beta '(2\pi)\) which were required in previous works. A new approach to the monotone iterative technique by considering the monotone iterates as orbits of a (discrete) dynamical system is presented. They also prove that the set of solutions of (1) between the lower and upper solutions is a compact and convex set provided that f is decreasing in u for fixed t.
Reviewer: M.Shahin (Dallas)

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Hale, J, Theory of functional differential equations, (1977), Springer-Verlag New York
[2] Heikkila, S; Lakshmikantam, V; Leela, S, Applications of monotone techniques to differential equations with discontinuous right hand side, Differential integral equations, 1, 287-297, (1988) · Zbl 0723.34005
[3] Ladde, G.S; Lakshmikantam, V; Vatsala, A.S, Monotone iterative techniques for nonlinear differential equations, (1985), Pitman Boston, MA · Zbl 0658.35003
[4] Lakshmikantam, V, Periodic boundary value problems of first and second order differential equations, J. appl. math. simulation, 2, 131-138, (1989) · Zbl 0712.34058
[5] Lakshmikantam, V; Leela, S, Remarks on first and second order periodic boundary-value problems, Nonlinear anal., 8, 281-287, (1984) · Zbl 0532.34029
[6] Nieto, J.J, Nonlinear second order periodic boundary value problems, J. math. anal. appl., 130, 22-29, (1988) · Zbl 0678.34022
[7] \scJ. J. Nieto, Nonlinear second order periodic boundary value problem with Carathéodory functions, Appl. Anal., to appear.
[8] Nieto, J.J; Rao, V.S.H, Periodic solutions of second order nonlinear differential equations, Acta math. hungar., 48, 59-66, (1986) · Zbl 0615.34036
[9] Temam, R, Infinite-dimensional dynamical systems in mechanics and physics, (1988), Springer-Verlag New York · Zbl 0662.35001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.