Geodesic
Localization of nonsmooth lower and upper functions for periodic boundary value problems
Mathematica Bohemica, Tome 127 (2002) no. 4, pp. 531-545

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
In this paper we present conditions ensuring the existence and localization of lower and upper functions of the periodic boundary value problem $u^{\prime \prime }+k\,u=f(t,u)$, $ u(0)=u(2\,\pi )$, $u^{\prime }(0)=u^{\prime }(2\pi )$, $k\in \mathbb{R}\hspace{0.56905pt}$, $k\ne 0.$ These functions are constructed as solutions of some related generalized linear problems and can be nonsmooth in general.
In this paper we present conditions ensuring the existence and localization of lower and upper functions of the periodic boundary value problem $u^{\prime \prime }+k\,u=f(t,u)$, $ u(0)=u(2\,\pi )$, $u^{\prime }(0)=u^{\prime }(2\pi )$, $k\in \mathbb{R}\hspace{0.56905pt}$, $k\ne 0.$ These functions are constructed as solutions of some related generalized linear problems and can be nonsmooth in general.
DOI : 10.21136/MB.2002.133955
Classification : 34B15, 34C25
Keywords: second order nonlinear ordinary differential equation; periodic problem; lower and upper functions; generalized linear differential equation 
Rachůnková, Irena; Tvrdý, Milan. Localization of nonsmooth lower and upper functions for periodic boundary value problems. Mathematica Bohemica, Tome 127 (2002) no. 4, pp. 531-545. doi: 10.21136/MB.2002.133955
@article{10_21136_MB_2002_133955,
     author = {Rach\r{u}nkov\'a, Irena and Tvrd\'y, Milan},
     title = {Localization of nonsmooth lower and upper functions for periodic boundary value problems},
     journal = {Mathematica Bohemica},
     pages = {531--545},
     year = {2002},
     volume = {127},
     number = {4},
     doi = {10.21136/MB.2002.133955},
     mrnumber = {1942639},
     zbl = {1017.34013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2002.133955/}
}
TY  - JOUR
AU  - Rachůnková, Irena
AU  - Tvrdý, Milan
TI  - Localization of nonsmooth lower and upper functions for periodic boundary value problems
JO  - Mathematica Bohemica
PY  - 2002
SP  - 531
EP  - 545
VL  - 127
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2002.133955/
DO  - 10.21136/MB.2002.133955
LA  - en
ID  - 10_21136_MB_2002_133955
ER  - 
%0 Journal Article
%A Rachůnková, Irena
%A Tvrdý, Milan
%T Localization of nonsmooth lower and upper functions for periodic boundary value problems
%J Mathematica Bohemica
%D 2002
%P 531-545
%V 127
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2002.133955/
%R 10.21136/MB.2002.133955
%G en
%F 10_21136_MB_2002_133955

[1] C. De Coster, P. Habets: Lower and upper solutions in the theory of ODE boundary value problems: Classical and recent results. F. Zanolin (ed.) Non Linear Analysis and Boundary Value Problems for Ordinary Differential Equations. CISM Courses Lect. 371, Springer, Wien, 1996, pp. 1–78.

[2] R. E. Gaines, J. Mawhin: Coincidence Degree and Nonlinear Differential Equations. Lect. Notes Math. 568, Springer, Berlin, 1977. | MR

[3] P. Habets, L. Sanchez: Periodic solutions of some Liénard equations with singularities. Proc. Amer. Math. Soc. 109 (1990), 1035–1044. | MR

[4] J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J. 7 (1957), 418–449. | MR | Zbl

[5] J. Mawhin: Topological degree methods in nonlinear boundary value problems. Regional Conf. Ser. Math. 40, AMS, Rhode Island, 1979. | MR | Zbl

[6] J. Mawhin: Topological degree and boundary value problems for nonlinear differential equations. M. Furi (ed.) Topological Methods for Ordinary Differential Equations. Lect. Notes Math. 1537, Springer, Berlin, 1993, pp. 73–142. | MR | Zbl

[7] J. Mawhin, M. Willem: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences 74, Springer, Berlin, 1989. | MR

[8] P. Omari: Nonordered lower and upper solutions and solvability of the periodic problem for the Liénard and the Reyleigh equations. Rend. Ist. Mat. Univ. Trieste 20 (1988), (Fasc. supplementare), 54–64. | MR

[9] P. Omari, W. Ye: Necessary and sufficient conditions for the existence of periodic solutions of second order ordinary differential equations with singular nonlinearities. Differential Integral Equations 8 (1995), 1843–1858. | MR | Zbl

[10] I. Rachůnková: Lower and upper solutions and topological degree. J. Math. Anal. Appl. 234 (1999), 311–327. | DOI

[11] I. Rachůnková: Existence of two positive solutions of a singular nonlinear periodic boundary value problems. J. Comput. Appl. Math. 113 (2000), 27–34. | DOI

[12] I. Rachůnková, M. Tvrdý: Nonlinear systems of differential inequalities and solvability of certain nonlinear second order boundary value problems. J. Inequal. Appl. 6 (2001), 199–226. | MR

[13] I. Rachůnková, M. Tvrdý: Method of lower and upper functions and the existence of solutions to singular periodic problems for second order nonlinear differential equations. Mathematical Notes, Miskolc 1 (2000), 135–143.

[14] I. Rachůnková, M. Tvrdý: Construction of lower and upper functions and their application to regular and singular boundary value problems. Nonlinear Analysis, Theory, Methods Appl. 47 (2001), 3937–3948. | MR

[15] I. Rachůnková, M. Tvrdý, I. Vrkoč: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differential Equations 176 (2001), 445–469. | DOI | MR

[16] I. Rachůnková, M. Tvrdý, I. Vrkoč: Resonance and multiplicity in singular periodic boundary value problems. (to appear).

[17] Š. Schwabik, M. Tvrdý, O. Vejvoda: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia and D. Reidel, Praha and Dordrecht, 1979. | MR

[18] Š. Schwabik: Generalized Ordinary Differential Equations. World Scientific, Singapore, 1992. | MR | Zbl

[19] M. Tvrdý: Generalized differential equations in the space of regulated functions (Boundary value problems and controllability). Math. Bohem. 116 (1991), 225–244. | MR | Zbl

[20] M. Zhang: A relationship between the periodic and the Dirichlet BVP’s of singular differential equations. Proc. Royal Soc. Edinburgh 128A (1998), 1099–1114. | MR

Cité par Sources :