Title:
|
An existence and multiplicity result for a periodic boundary value problem (English) |
Author:
|
Rudolf, Boris |
Language:
|
English |
Journal:
|
Mathematica Bohemica |
ISSN:
|
0862-7959 (print) |
ISSN:
|
2464-7136 (online) |
Volume:
|
133 |
Issue:
|
1 |
Year:
|
2008 |
Pages:
|
41-61 |
Summary lang:
|
English |
. |
Category:
|
math |
. |
Summary:
|
A periodic boundary value problem for nonlinear differential equation of the second order is studied. Nagumo condition is not assumed on a part of nonlinearity. Existence and multiplicity results are proved using the method of lower and upper solutions. Results are applied to the generalized Liénard oscillator. (English) |
Keyword:
|
periodic boundary value problem |
Keyword:
|
multiplicity result |
Keyword:
|
method of lower and upper solutions |
Keyword:
|
Liénard oscillator |
MSC:
|
34B15 |
MSC:
|
34C25 |
idZBL:
|
Zbl 1199.34065 |
idMR:
|
MR2400150 |
DOI:
|
10.21136/MB.2008.133946 |
. |
Date available:
|
2009-09-24T22:34:21Z |
Last updated:
|
2020-07-29 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133946 |
. |
Reference:
|
[1] Bihari I.: A generalization of a lemma of Bellman and its applications to uniqueness problems of differential equations.Acta Math. Sci. Hung. 7 (1956), 71–94. MR 0079154, 10.1007/BF02022967 |
Reference:
|
[2] Fabry C., Mawhin J., Nkashama M. N.: A multiplicity result for periodic solutions of forced nonlinear boundary value problems.Bull. London Math. Soc. 18 (1986), 173–186. MR 0818822, 10.1112/blms/18.2.173 |
Reference:
|
[3] Gaines R., Mawhin J.: Coincidence Degree and Nonlinear Differential Equations.Lect. Notes Math. 568 Springer, Berlin, 1977. MR 0637067, 10.1007/BFb0089537 |
Reference:
|
[4] Gossez J., Omari P.: Periodic solutions of a second order ordinary differential equation: A necessary and sufficient condition for nonresonance.J. Differ. Equations 94 (1991), 67–82. MR 1133541, 10.1016/0022-0396(91)90103-G |
Reference:
|
[5] Habets P., Omari P.: Existence and localization of solutions of second order elliptic boundary value problem using lower and upper solutions in the reversed order.Topol. Methods Nonlinear Anal. 8 (1996), 25–56. MR 1485756, 10.12775/TMNA.1996.020 |
Reference:
|
[6] Korman Ph.: Remarks on Nagumo’s condition.Portugaliae Mathematica 55 (1998), 1–9. Zbl 0894.34015, MR 1612323 |
Reference:
|
[7] Mawhin J.: Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces.J. Differ. Equations 12 (1972), 610–636. Zbl 0244.47049, MR 0328703, 10.1016/0022-0396(72)90028-9 |
Reference:
|
[8] Mawhin J.: Points fixes, points critiques et problèmes aux limites.Sémin. Math. Sup. no. 92, Presses Univ. Montréal, Montréal, 1985. Zbl 0561.34001, MR 0789982 |
Reference:
|
[9] Mawhin J., Willem M.: Multiple solutions of the periodic boundary value problem for some forced pendulum type equations.J. Differ. Equations 52 (1984), 264–287. MR 0741271, 10.1016/0022-0396(84)90180-3 |
Reference:
|
[10] Nieto J. J.: Nonlinear second-order periodic boundary value problems.J. Math. Anal. Appl. 130 (1988), 22–29. Zbl 0678.34022, MR 0926825, 10.1016/0022-247X(88)90383-6 |
Reference:
|
[11] Omari P.: Non-ordered lower and upper solutions and solvability of the periodic problem for the Liénard and the Raleigh equations.Rend. Inst. Mat. Univ. Trieste 20 (1988), 54–64. |
Reference:
|
[12] Rachůnková I.: Multiplicity results for four-point boundary value problems.Nonlinear Anal., Theory Methods Appl. 18 (1992), 497–505. 10.1016/0362-546X(92)90016-8 |
Reference:
|
[13] Rachůnková I.: On the existence of two solutions of the periodic problem for the ordinary second order differential equation.Nonlinear Anal., Theory Methods Appl. 22 (1994), 1315–1322. MR 1280199, 10.1016/0362-546X(94)90113-9 |
Reference:
|
[14] Rachůnková I.: Upper and lower solutions and topological degree.J. Math. Anal. Appl. 234 (1999), 311–327. 10.1006/jmaa.1999.6375 |
Reference:
|
[15] Rudolf B., Kubáček Z.: Remarks on J. J. Nieto’s Paper: Nonlinear second-order periodic boundary value problems.J. Math. Anal. Appl. 146 (1990), 203–206. MR 1041210, 10.1016/0022-247X(90)90341-C |
Reference:
|
[16] Rudolf B.: A multiplicity result for a periodic boundary value problem.Nonlinear Anal., Theory Methods Appl. 28 (1997), 137–144. Zbl 0859.34016, MR 1416037, 10.1016/0362-546X(95)00144-K |
Reference:
|
[17] Rudolf B.: Method of lower and upper solutions for a generalized boundary value problem.Archivum Mathematicum (Brno) 36 (2000), 595–602. Zbl 1090.34520, MR 1822829 |
Reference:
|
[18] Scorza Dragoni G.: Il problema dei valori ai limiti studiato il grande per gli integrali di una equazione differenziale del secondo ordine.Giorn. Mat. Battagliani, III. Ser. 69 (1931), 77–112. |
Reference:
|
[19] Šeda V.: On some nonlinear boundary value problems for ordinary differential equations.Archivum Mathematicum (Brno) 25 (1989), 207–222. MR 1188065 |
Reference:
|
[20] Thompson H. B.: Second order ordinary differential equations with fully nonlinear two-point boundary conditions.Pacific J. Math. 172 (1996), 255–297. Zbl 0862.34015, MR 1379297, 10.2140/pjm.1996.172.255 |
. |