# Article

Full entry | PDF   (0.3 MB)
Keywords:
second order nonlinear ordinary differential equation; periodic problem; lower and upper functions; generalized linear differential equation
Summary:
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.
References:
[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 0637067
[3] P. Habets, L. Sanchez: Periodic solutions of some Liénard equations with singularities. Proc. Amer. Math. Soc. 109 (1990), 1035–1044. MR 1009991
[4] J. Kurzweil: Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J. 7 (1957), 418–449. MR 0111875 | Zbl 0090.30002
[5] J. Mawhin: Topological degree methods in nonlinear boundary value problems. Regional Conf. Ser. Math. 40, AMS, Rhode Island, 1979. MR 0525202 | Zbl 0414.34025
[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 1226930 | Zbl 0798.34025
[7] J. Mawhin, M. Willem: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences 74, Springer, Berlin, 1989. MR 0982267
[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 1113247
[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 1347985 | Zbl 0831.34048
[10] I. Rachůnková: Lower and upper solutions and topological degree. J. Math. Anal. Appl. 234 (1999), 311–327.
[11] I. Rachůnková: Existence of two positive solutions of a singular nonlinear periodic boundary value problems. J. Comput. Appl. Math. 113 (2000), 27–34.
[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.
[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 1972337
[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.
[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 0542283
[18] Š. Schwabik: Generalized Ordinary Differential Equations. World Scientific, Singapore, 1992. MR 1200241 | Zbl 0781.34003
[19] M. Tvrdý: Generalized differential equations in the space of regulated functions (Boundary value problems and controllability). Math. Bohem. 116 (1991), 225–244. MR 1126445 | Zbl 0744.34021
[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 1642144

Partner of