Periodic solutions for nonlinear systems with p-Laplacian-like operators

Let us consider the so-called one-dimensional p-Laplacian operator (,p(u$))$, where p>1 and ,p : R R is given by ,p(s)=|s| p&2 s for s{0 and ,p(0)=0. Various separated two-point boundary value problems containing this operator have received a lot of attention with respect to existence and multiplicity of solutions. See, for example, [3, 9, 11, 13, 21, 22, 28, 31, 33], and the references therein. Periodic boundary conditions have been considered in [12, 14]. The case of separated two-point boundary conditions when ,p is replaced by a one-dimensional possibly not homogeneous operator ,, has been dealt with in a series of papers, cf. [2, 8, 15, 16, 17, 18, 19]. Our aim in this paper is to study existence of periodic solutions to some system cases involving the fairly general vector-valued operator ,. Thus we will consider the boundary value problem

[1]  Jean Mawhin,et al.  The Leray-Schauder degree of S1-equivariant operators associated to autonomous neutral equations in spaces of periodic functions , 1991 .

[2]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[3]  J. Mawhin.  Topological degree and boundary value problems for nonlinear differential equations , 1993 .

[4]  P. Ubilla,et al.  Multiplicity of solutions for a class of nonlinear second-order equations , 1997 .

[5]  L. Bobisud.  Steady-State Turbulent Flow with Reaction , 1991 .

[6]  F. Zanolin,et al.  On a pseudo Fuccik spectrum for strongly nonlinear second-order ODEs and an existence result , 1994 .

[7]  F. Zanolin,et al.  Strongly Nonlinear Second-Order ODEs with Rapidly Growing Terms , 1996 .

[8]  F. Zanolin,et al.  Strongly nonlinear second-order ODEs with unilateral conditions , 1993, Differential and Integral Equations.

[9]  Seth F. Oppenheimer,et al.  Existence and Uniqueness Results for Some Nonlinear Boundary Value Problems , 1996 .

[10]  K. Deimling.  Nonlinear functional analysis , 1985 .

[11]  Manuel del Pino,et al.  A homotopic deformation along p of a Leray-Schauder degree result and existence for (¦u′¦p − 2u′)′ + ƒ(t, u) = 0, u(0) = u(T) = 0, p > 1 , 1989 .

[12]  Zongming Guo.  Boundary value problems of a class of quasilinear ordinary differential equations , 1993, Differential and Integral Equations.

[13]  G. Metzen,et al.  The existence of solutions to a class of semilinear differential equations , 1995, Differential and Integral Equations.

[14]  Raúl Manásevich,et al.  Time-mappings and multiplicity of solutions for the one-dimensional p -Laplacian , 1993 .

[15]  Meirong Zhang,et al.  Nonuniform nonresonance at the first eigenvalue of the p -Laplacian , 1997 .

[16]  Pavel Drábek,et al.  Generalization of Fredholm alternative for nonlinear differential operators , 1986 .

[17]  Wenjie Gao,et al.  Boundary value problems for general second order equations and similarity solutions to the Rayleigh problem , 1995 .

[18]  P. Hartman.  Ordinary Differential Equations , 1965 .

[19]  Jean Mawhin,et al.  Continuation theorems and periodic solutions of ordinary differential equations , 1995 .

[20]  A. Cañada,et al.  Periodic solutions of nonlinear vector ordinary differential equations of higher order at resonance , 1983 .

[21]  F. Browder.  Nonlinear functional analysis , 1970 .

[22]  C. Coster.  Pairs of positive solutions for the one-dimensional p -Laplacian , 1994 .

[23]  Jean Mawhin,et al.  Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces , 1972 .

[24]  G. Villari.  Soluzioni periodiche di una classe di equazioni differenziali del terz’ ordine quasi lineari , 1966 .

[25]  Manuel del Pino,et al.  Existence and multiplicity of solutions with prescribed period for a second order quasilinear O.D.E. , 1992 .

[26]  James R. Ward,et al.  Asymptotic conditions for periodic solutions of ordinary differential equations , 1981 .

[27]  N. Rouche,et al.  Ordinary differential equations : stability and periodic solutions , 1980 .

[28]  J. Mawhin,et al.  Continuation theorems for periodic perturbations of autonomous systems , 1992 .

[29]  C. Fabry,et al.  Periodic solutions of second order differential equations with a p-Laplacian and asymmetric nonlinearities , 1992 .

[30]  A. Cañada,et al.  Solvability of some operator equations and periodic solutions of nonlinear functional differential equations , 1983 .

[31]  R. Ortega,et al.  Existence theorems for equations in normed spaces and boundary value problems for nonlinear vector ordinary differential equations , 1984, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[32]  F. Zanolin,et al.  A Fredholm like Result for Strongly Nonlinear Second Order ODE′s , 1994 .

[33]  Donal O'Regan,et al.  Some general existence principles and results for f y ′ =qf t,y,y ′ ,0 , 1993 .

[34]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .