Periodic singular problem with quasilinear differential operator

Rachůnková, Irena ; Tvrdý, Milan

Mathematica Bohemica, Tome 131 (2006) no. 3, pp. 321-336

Abstract (VO)

We study the singular periodic boundary value problem of the form \[ \left(\phi (u^{\prime })\right)^{\prime }+h(u)u^{\prime }=g(u)+e(t),\quad u(0)=u(T),\quad u^{\prime }(0)=u^{\prime }(T), \] where $\phi \:\mathbb{R}\rightarrow \mathbb{R}$ is an increasing and odd homeomorphism such that $\phi (\mathbb{R})=\mathbb{R},$ $h\in C[0,\infty ),$ $e\in L_1J$ and $g\in C(0,\infty )$ can have a space singularity at $x=0,$ i.e. $\limsup _{x\rightarrow 0+}|g(x)|=\infty $ may hold. We prove new existence results both for the case of an attractive singularity, when $\liminf _{x\rightarrow 0+}g(x)=-\infty ,$ and for the case of a strong repulsive singularity, when $\lim _{x\rightarrow 0+}\int _x^1g(\xi )\hspace{0.56905pt}\text{d}\xi =\infty .$ In the latter case we assume that $\phi (y)=\phi _p(y)=|y|^{p-2}y,$ $p>1,$ is the well-known $p$-Laplacian. Our results extend and complete those obtained recently by Jebelean and Mawhin and by Liu Bing.

Détail

DOI : 10.21136/MB.2006.134137

Classification : 34B15, 34B16, 34B18, 34C25
Keywords: singular periodic boundary value problem; positive solution; $\phi $-Laplacian; $p$-Laplacian; attractive singularity; repulsive singularity; strong singularity; lower function; upper function

Bibliographie

[1] D. Bonheure, C. De Coster: Forced singular oscillators and the method of lower and upper functions. Topol. Methods Nonlinear Anal. 22 (2003), 297–317. | DOI | MR

[2] D. Bonheure, C. Fabry, D. Smets: Periodic solutions of forced isochronous oscillators. Discrete Contin. Dyn. Syst. 8 (2002), 907–930. | DOI | MR

[3] A. Cabada, R. Pouso: Existence result for the problem $\left(\phi (u^{\prime })\right)^{\prime }=f(t,u,u^{\prime })$ with periodic and Neumann boundary conditions. Nonlinear Anal., Theory Methods Appl. 30 (1997), 1733–1742. | MR

[4] M. del Pino, R. Manásevich, A. Montero: $T$-periodic solutions for some second order differential equations with singularities. Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), 231–243. | MR

[5] M. del Pino, R. Manásevich, A. Murúa: Existence and multiplicity of solutions with prescribed period for a second order quasilinear O.D.E. Nonlinear Anal., Theory Methods Appl. 18 (1992), 79–92. | DOI | MR

[6] L. Derwidué: Systèmes différentiels non linéaires ayant des solutions périodiques. Acad. R. Belgique, Bull. Cl. Sci. 49 (1963), 11–32.

[7] P. Drábek: Solvability and Bifurcation of Nonlinear Equations. Research Notes in Math. Vol. 264, Pitman, Boston, 1992. | MR

[8] R. Faure: Solutions périodiques d’equations différentielles et mèthode de Leray-Schauder (Cas des vibrations forceés). Ann. Inst. Fourier (Grenoble) 14 (1964), 195–204. | DOI | MR

[9] A. Fonda: Periodic solution of scalar second order differential equations with a singularity. Acad R. Belgique, Mém. Cl. Sci. 4 (1993), 1–39. | MR

[10] N. Forbat, A. Huaux: Détermination approachée et stabilité locale de la solution périodique d’une equation différentielle non linéaire. Mém. Public. Soc. Sci. Arts Letters Hainaut 76 (1962), 3–13. | MR

[11] W. G. Ge, J. Mawhin: Positive solutions to boundary value problems for second order ordinary differential equations with singular nonlinearities. Result. Math. 34 (1998), 108–119. | DOI | MR

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

[13] A. Huaux: Sur l’existence d’une solution périodique de l’equation différentielle non linéaire $u^{\prime \prime }+0.2\,u^{\prime }+u/(1-u)=0.5\cos \omega t$. Acad. R. Belgique, Bull. Cl. Sci. 48 (1962), 494–504. | MR

[14] P. Jebelean, J. Mawhin: Periodic solutions of singular nonlinear perturbations of the ordinary $p$-Laplacian. Adv. Nonlinear Stud. 2 (2002), 299–312. | MR

[15] P. Jebelean, J. Mawhin: Periodic solutions of forced dissipative $p$-Liénard equations with singularities. Vietnam J. Math. 32 (2004), 97–103. | MR

[16] D. Q. Jiang, J. Y. Wang: A generalized periodic boundary value problem for the one-dimensional $p$-Laplacian. Ann. Polon. Math. 65 (1997), 265–270. | DOI | MR

[17] A. C. Lazer, S. Solimini: On periodic solutions of nonlinear differential equations with singularities. Proc. Amer. Math. Soc. 99 (1987), 109–114. | DOI | MR

[18] Liu Bing: Periodic solutions of dissipative dynamical systems with singular potential and $p$-Laplacian. Ann. Pol. Math. 79 (2002), 109–120. | DOI | MR | Zbl

[19] J. Mawhin: Topological degree and boundary value problems for nonlinear differential equations. Topological Methods for Ordinary Differential Equations, M. Furi, P. Zecca (eds.), Lecture Notes in Mathematics, Vol. 1537, Springer, Berlin, 1993, pp. 74–142. | MR | Zbl

[20] R. Manásevich, J. Mawhin: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differential Equations 145 (1998), 367–393. | DOI | MR

[21] J. Mawhin: Periodic Solutions of Systems with $p$-Laplacian-like Operators. Nonlinear Analysis and its Applications to Differential Equations, M. R. Grossinho, M. Ramos, C. Rebelo, L. Sanchez (eds.), Progress in Nonlinear Differerential Equations and their Applications, Vol. 43, Birkhäuser, Boston, 2001, pp.  37–63. | MR | Zbl

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

[23] I. Rachůnková, M. Tvrdý: Second-order periodic problem with $\phi $-Laplacian and impulses. Nonlinear Anal., Theory Methods Appl. 63 (2005), e257–e266. | DOI

[24] I. Rachůnková, M. Tvrdý: Periodic problems with $\phi $-Laplacian involving non-ordered lower and upper functions. Fixed Point Theory 6 (2005), 99–112. | MR

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

[26] I. Rachůnková, M. Tvrdý, I. Vrkoč: Resonance and multiplicity in periodic BVPs with singularity. Math. Bohem. 128 (2003), 45–70. | MR

[27] S. Staněk: Periodic boundary value problem for second order functional differential equations. Math. Notes (Miskolc) 1 (2000), 63–81. | DOI | MR

[28] P. J. Torres: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differential Equations 190 (2003), 643–662. | DOI | MR | Zbl

[29] P. Yan: Nonresonance for one-dimensional $p$-Laplacian with regular restoring. J. Math. Anal. Appl. 285 (2003), 141–154. | DOI | MR | Zbl

[30] M. R. Zhang: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203 (1996), 254–269. | DOI | MR | Zbl

[31] M. R. Zhang: Nonuniform nonresonance at the first eigenvalue of the $p$-Laplacian. Nonlinear Anal., Theory Methods Appl. 29 (1997), 41–51. | DOI | MR | Zbl

[32] M. R. Zhang: A relationship between the periodic and the Dirichlet BVPs of singular differential equations. Proc. Roy. Soc. Edinburgh Sect. A 128A (1998), 1099–1114. | MR | Zbl