Nonlinear second order periodic boundary value problems with Caratheodory functions. (English) Zbl 0662.34022

Nieto, Juan J.

Appl. Anal. 34, No. 1-2, 111-128 (1989).

The authors consider the periodic boundary value problem \(-u''=f(t,u)\), \(u(0)=u(2\pi)\), \(u'(0)=u'(2\pi)\) when f satisfies Carathéodory conditions. It is shown that a generalized upper and lower solution method is still valid and develop monotone iterative technique for finding minimal and maximal solutions. Moreover, the authors prove that the set of solutions between the (generalized) lower and upper solutions is a compact and convex set provided that f is decreasing in u for fixed t.

Reviewer: J.J.Nieto

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34C25	Periodic solutions to ordinary differential equations

Keywords:

minimal solution; lower solutions; Caratheodory conditions; lower solution method; monotone iterative technique; maximal solutions; upper solutions

References:

[1]	DOI: 10.1007/BF01320867 · Zbl 0368.35032 · doi:10.1007/BF01320867
[2]	Brezis H., Analyse fonctionnelle, theorie et applications (1983)
[3]	Dugundji J., Pacific J. Math 1 pp 353– (1951)
[4]	DOI: 10.1017/S000497270001011X · Zbl 0592.34028 · doi:10.1017/S000497270001011X
[5]	Gupta C. P., J. Math. Anal. Appl 34 (1986)
[6]	DOI: 10.1016/0362-546X(82)90097-9 · Zbl 0494.34029 · doi:10.1016/0362-546X(82)90097-9
[7]	DOI: 10.1080/00036818408839488 · Zbl 0532.34030 · doi:10.1080/00036818408839488
[8]	Ladde G. S., Monotone iterative techniques for nonlinear differential equations (1986)
[9]	Lakshmikantham V., Periodic boundary value problems of first and second order differential equations · Zbl 0712.34058
[10]	DOI: 10.1016/0022-247X(83)90102-6 · Zbl 0525.34031 · doi:10.1016/0022-247X(83)90102-6
[11]	DOI: 10.1016/0362-546X(83)90088-3 · Zbl 0524.34023 · doi:10.1016/0362-546X(83)90088-3
[12]	Mawhin J., Acad. Roy. Relg. Bull. Cl. Sci 7 (1983)
[13]	Nkashama M. N., J, Math. Anal. Appl 7 (1983)
[14]	DOI: 10.1016/0096-3003(85)90026-8 · Zbl 0596.34008 · doi:10.1016/0096-3003(85)90026-8
[15]	DOI: 10.1016/0022-0396(85)90122-6 · Zbl 0537.35049 · doi:10.1016/0022-0396(85)90122-6
[16]	Nieto J. J., Differential Equations: Qualitative Theory 47 pp 799– (1987)
[17]	DOI: 10.1016/0022-247X(87)90213-7 · Zbl 0648.34019 · doi:10.1016/0022-247X(87)90213-7
[18]	DOI: 10.1016/0022-247X(87)90213-7 · Zbl 0648.34019 · doi:10.1016/0022-247X(87)90213-7
[19]	Nieto J. J., Lecture Notes Pure Appl. Math 109 pp 381– (1987)