Boundedness in Nonlinear Asymmetric Oscillations

Abstract In this paper, the boundedness of all solutions of the nonlinear equation ( ϕ p ( x ′))′+( p -1)[ αϕ p ( x + )− βϕ p ( x − )]+ f ( x )+ g ( x )= e ( t ) is discussed, where e ( t ) ∈C 7 is 2 π p -periodic, f,g are bounded C 6 functions, ϕ p ( u )=∣u∣ p −2 u , p ⩾2, α,β are positive constants, x + =max{ x ,0}, x − =max{− x ,0}.

[1]  Bin Liu,et al.  Boundedness in Asymmetric Oscillations , 1999 .

[2]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[3]  Ongren Peking.  NONLINEAR OSCILLATIONS AT A POINT OF RESONANCE , 1982 .

[4]  Bin Liu,et al.  Boundedness in Nonlinear Oscillations at Resonance , 1999 .

[5]  E. Zehnder,et al.  Boundedness of solutions via the twist-theorem , 1987 .

[6]  Rafael Ortega,et al.  Asymmetric Oscillators and Twist Mappings , 1996 .

[7]  Mark Levi,et al.  Quasiperiodic motions in superquadratic time-periodic potentials , 1991 .

[8]  Rafael Ortega,et al.  Boundedness in a Piecewise Linear Oscillator and a Variant of the Small Twist Theorem , 1999 .

[9]  J. Littlewood.  Some problems in real and complex analysis , 1968 .

[10]  G. R. Morris.  A case of boundedness in Littlewood's problem on oscillatory differential equations , 1976, Bulletin of the Australian Mathematical Society.

[11]  J. L. Massera.  The existence of periodic solutions of systems of differential equations , 1950 .

[12]  Rafael Ortega,et al.  Roots of Unity and Unbounded Motions of an Asymmetric Oscillator , 1998 .

[13]  Christian Fabry,et al.  Oscillations of a forced asymmetric oscillator at resonance , 2000 .

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

[15]  Pavel Drábek,et al.  The Fredholm Alternative at the First Eigenvalue for the One Dimensionalp-Laplacian , 1999 .