Periodic solutions of equations with oscillating nonlinearities

Krasnoselʹskiǐ, A. M.; Mawhin, Jean

doi:10.1016/s0895-7177(00)00216-8

Cited by 20 publications

(15 citation statements)

References 4 publications

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“…where n ∈ N, b is continuous and 2π-periodic, and f is continuous and unbounded in general. Theorems 4 and 5 below generalize some results from [11] where bounded non-linearities f are considered. Set…”

Section: Consider the Equationmentioning

confidence: 65%

On Resonant Differential Equations with Unbounded Non-Linearities

2002

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We present a method to study asymptotically linear degenerate problems with sublinear unbounded non-linearities. The method is based on the uniform convergence to zero of projections of non-linearity increments onto some finite-dimensional spaces. Such convergence was used for the analysis of resonant equations with bounded non-linearities by many authors. The unboundedness of nonlinear terms complicates essentially the analysis of most problems: existence results, approximate methods, systems with parameters, stability, dissipativity, etc. In this paper we present statements on projection convergence for unbounded non-linearities and apply them to various resonant asymptotically linear problems: existence of forced periodic oscillations and unbounded sequences of such oscillations, existence of unbounded solutions, sharp analysis of integral equations with simple degeneration of the linear part (a scalar twopoint boundary value problem is considered as an example), existence of non-trivial cycles for higher order autonomous ordinary differential equations, and Hopf bifurcations at infinity.

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Section: Consider the Equationmentioning

confidence: 65%

On Resonant Differential Equations with Unbounded Non-Linearities

2002

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“…x 0 ψ(x)dx. They proved that each solution with large initial condition is unbounded if Other conditions for the existence of bounded and unbounded solutions are described in [1,2,6,8,15,16] and their references.…”

Section: Thus If Limmentioning

confidence: 99%

A new method for the boundedness of semilinear Duffing equations at resonance

Wang¹,

Wang²,

Piao³

2016

DCDS-A

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We introduce a new method for the boundedness problem of semilinear Duffing equations at resonance. In particular, it can be used to study a class of semilinear equations at resonance without the polynomial-like growth condition. As an application, we prove the boundedness of all the solutions for the equationẍ + n 2 x + g(x) + ψ(x) = p(t) under the Lazer-Leach condition on g and p, where n ∈ N + , p(t) and ψ(x) are periodic and g(x) is bounded.

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“…The boundedness of all the solutions, and the existence of periodic or unbounded solutions for (1.1) have been widely studied in the literature. We refer the readers to the papers [1,6,16,21,[26][27][28][29]31,34,36,37,39]. All the results mentioned above require g(x) to be a bounded function with asymptotic limits or generalized limits.…”

Section: Introductionmentioning

confidence: 99%

“…generally speaking, one can expect that (1.1) has an unbounded sequence of periodic solutions [21] (or subharmonic solutions [16]). If g, p are of class C 7 and lim inf h→∞ Λ(h), lim sup h→∞ Λ(h) are finite, we proved in [33] that all the solutions of (1.1) are bounded under some asymptotic assumptions regarding the behavior of the functions g(x) and Γ(h).…”

Section: Introductionmentioning

confidence: 99%

Periodic and unbounded motions in asymmetric oscillators at resonance

2015

Journal of Mathematical Analysis and Applications

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We consider the existence of periodic and unbounded solutions for the asymmetric oscillatorwhere x + = max{x, 0}, x − = max{−x, 0}, a and b are two positive constants, p(t) is a 2π-periodic smooth function and g(x) satisfies lim |x|→+∞ x −1 g(x) = 0. We have proved previously that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions Φ p (θ) and Λ(h). In this paper, we consider some critical cases and obtain some new sufficient conditions for the existence of 2π-periodic and unbounded solutions. In particular, the obtained periodic solutions are new and cannot be deduced from Landesman-Lazer's and Fabry-Fonda's existence conditions. In contrast with many existing results, the function g(x) may be unbounded or oscillatory without asymptotic limits.

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Periodic solutions of equations with oscillating nonlinearities

Cited by 20 publications

References 4 publications

On Resonant Differential Equations with Unbounded Non-Linearities

On Resonant Differential Equations with Unbounded Non-Linearities

A new method for the boundedness of semilinear Duffing equations at resonance

Periodic and unbounded motions in asymmetric oscillators at resonance

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