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Periodic solutions for periodic second-order differential equations with variable potentials

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Published/Copyright: November 6, 2018
Journal of Applied Analysis
From the journal Journal of Applied Analysis Volume 24 Issue 2

Abstract

We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials -(p⁢x′)′⁢(t)-r⁢(t)⁢p⁢(t)⁢x′⁢(t)+q⁢(t)⁢x⁢(t)=f⁢(t,x⁢(t)), where the functions p⁢(t)>0, q⁢(t), r⁢(t) and f⁢(t,x) are 𝒞2 and T-periodic in the variable t.

Keywords: Periodic orbit; third-order differential equation; quadratic system; averaging theory
MSC 2010: 37G15; 37C80; 37C30

Funding statement: The first author is partially supported by a MICINN/FEDER grant number MTM2009-03437, by an AGAUR grant number 2009SGR-410, by an ICREA Academia, two FP7+PEOPLE+2012+IRSES numbers 316338 and 318999, and FEDER-UNAB10-4E-378.

A Appendix: Averaging theory of first order

In this appendix, we recall one of the basic results from the averaging theory that we need for proving the main results of this paper.

We consider the problem of the bifurcation of T-periodic solutions from differential systems of the form

(A.1)𝐱′=F0⁢(t,𝐱)+ε⁢F1⁢(t,𝐱)+ε2⁢F2⁢(t,𝐱,ε)

with ε=0 to ε≠0 sufficiently small. Here the functions F0,F1:ℝ×Ω→ℝn and F2:ℝ×Ω×(-ε0,ε0)→ℝn are 𝒞2 functions, T-periodic in the first variable, and Ω is an open subset of ℝn. The main assumption is that the unperturbed system

(A.2)𝐱′=F0⁢(t,𝐱)

has a submanifold of dimension n of periodic solutions. A solution of this problem is given using the averaging theory.

Let 𝐱⁢(t,𝐳,ε) be the solution of system (A.2) such that 𝐱⁢(0,𝐳,ε)=𝐳. We write the linearization of the unperturbed system along the periodic solution 𝐱⁢(t,𝐳,0) as

(A.3)𝐲′=D𝐱⁢F0⁢(t,𝐱⁢(t,𝐳,0))⁢𝐲,

where 𝐲 is an n×n matrix. In what follows, we denote by M𝐳⁢(t) some fundamental matrix of the linear differential system (A.3).

We assume that there exists an open set V with Cl⁢(V)⊂Ω such that, for each 𝐳∈Cl⁢(V), 𝐱⁢(t,𝐳,0) is T-periodic. The set Cl⁢(V) is isochronous for the system (A.1); i.e., it is a set formed only by periodic orbits, all of them having the same period. Then an answer to the problem of the bifurcation of T-periodic solutions from the periodic solutions 𝐱⁢(t,𝐳,0) contained in Cl⁢(V) is given in the following result.

Theorem 2 (Perturbations of an isochronous set).

We assume that there exists an open and bounded set V with Cl⁢(V)⊂Ω such that, for each z∈Cl⁢(V), the solution x⁢(t,z,0) is T-periodic. Then we consider the function F:Cl⁢(V)→Rn,

(A.4)ℱ⁢(𝐳)=∫0TM𝐳-1⁢(t)⁢F1⁢(t,𝐱⁢(t,𝐳,0))⁢d⁢t.

If there exists α∈V with F⁢(α)=0 and

(A.5)det⁡((d⁢ℱ/d⁢𝐳)⁢(𝐳*))≠0,

then there exists a T-periodic solution x⁢(t,ε) of system (A.1) such that, when ε→0, we have x⁢(0,ε)→α.

Theorem 2 goes back to Malkin [7] and Roseau [8]; for a shorter and easier proof, see [4].

Acknowledgements

We thank the reviewer for his/her comments.

References

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Received: 2017-10-12
Revised: 2018-02-02
Accepted: 2018-02-05
Published Online: 2018-11-06
Published in Print: 2018-12-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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  2. Quantitative approximation by shift invariant multivariate sublinear-Choquet operators
  3. Periodic solutions for periodic second-order differential equations with variable potentials
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https://doi.org/10.1515/jaa-2018-0013
Keywords for this article
Periodic orbit; third-order differential equation; quadratic system; averaging theory

Articles in the same Issue

  1. Frontmatter
  2. Quantitative approximation by shift invariant multivariate sublinear-Choquet operators
  3. Periodic solutions for periodic second-order differential equations with variable potentials
  4. Oscillatory behavior of higher order nonlinear homogeneous neutral delay dynamic equations with positive and negative coefficients
  5. Non-Archimedean hyperstability of Cauchy–Jensen functional equations on a restricted domain
  6. Koebe domains for the class of typically real functions that are convex in two directions
  7. Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case
  8. Composite relaxed resolvent operator and Yosida approximation operator for solving a system of Yosida inclusions
  9. Multiple values and unicity problem of meromorphic mappings sharing different families of moving hyperplanes
  10. Generalization of different type integral inequalities for generalized (𝑠, 𝑚)-preinvex Godunova–Levin functions
  11. On the multiobjective control problem
  12. Fuzzy 𝛿-𝐼-continuity in mixed fuzzy ideal topological spaces
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