On a periodic problem for super-linear second-order ODEs
-
Jiří Šremr
Abstract
The present paper concerns the periodic problem
where p, f : [0, ω] → ℝ are Lebesgue integrable functions and q : [0, ω] × ℝ → ℝ is a Carathéodory function. We assume that the anti-maximum principle holds for the corresponding linear problem and provide sufficient conditions guaranteeing the existence and uniqueness of a positive solution to the given non-linear problem. The general results obtained are applied to the non-autonomous Duffing type equation with a super-linear power non-linearity.
Funding statement: The research has been supported by the internal grant FSI-S-20-6187 of FME BUT
Communicated by Jozef Džurina
References
[1] Barletta, G.—D’aguí, G.—Papageorgiou, N. S.: Nonlinear nonhomogeneous periodic problems, NoDEA Nonlinear Differential Equations Appl. 23(2) (2016), Art. No. 18.10.1007/s00030-016-0356-3Search in Google Scholar
[2] Bibikov, Yu. N.—Savel’eva, A. G.: Periodic perturbations of a nonlinear oscillator, Differ. Equ. 52(4) (2016), 405–412.10.1134/S0012266116040017Search in Google Scholar
[3] Cabada, A.—Cid, J. Ä.—López-Somoza, L.: Maximum Principles for the Hill’s Equation, Academic Press, London, 2018.Search in Google Scholar
[4] Fonda, A.: Playing Around Resonance. An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations. Birkhäuser Advanced Texts Basler Lehrbücher, Birkhäuser/Springer, Cham, 2016.10.1007/978-3-319-47090-0Search in Google Scholar
[5] Gidoni, P.: Existence of a periodic solution for superlinear second order ODEs, J. Differential Equations 354 (2023), 401–417.10.1016/j.jde.2022.11.054Search in Google Scholar
[6] Habets, P.—De Coster, C.: Two-Point Boundary Value Problems: Lower and Upper Solutions. Mathematics in Science and Engineering, Vol. 205, Elsevier B.V., Amsterdam, 2006.Search in Google Scholar
[7] Hartman, P.: Ordinary Differential Equations, John Wiley, New York, 1964.Search in Google Scholar
[8] Lasota, A.—Opial, Z.: Sur les solutions péridoques des équations différentielles ordinaires, Ann. Polon. Math. 16 (1964), 69–94.10.4064/ap-16-1-69-94Search in Google Scholar
[9] Lomtatidze, A.: On periodic boundary value problem for second order ordinary differential equations, Commun. Contemp. Math. 22(6) (2020), Art. ID 1950049.10.1142/S0219199719500494Search in Google Scholar
[10] Lomtatidze, A.: Theorems on differential inequalities and periodic boundary value problem for second-order ordinary differential equations, Mem. Differential Equations Math. Phys. 67 (2016), 1–129.10.1016/B978-0-12-803652-5.00001-6Search in Google Scholar
[11] Lomtatidze, A.—Šremr, J.: On periodic solutions to second-order Duffing type equations, Nonlinear Anal. Real World Appl. 40 (2018), 215–242.10.1016/j.nonrwa.2017.09.001Search in Google Scholar
[12] Lomtatidze, A.—Šremr, J.: On a periodic problem for second-order Duffing type equations, preprint No. 73-2015, Institute of Mathematics, Czech Academy of Sciences, 2015.Search in Google Scholar
[13] Nowakowski, A.—Plebaniak, R.: Fixed point theorems and periodic problems for nonlinear Hill’s equation, NoDEA Nonlinear Differential Equations Appl. 30(2) (2023), Art. No. 16.10.1007/s00030-022-00825-9Search in Google Scholar
[14] Qian, D.—Torres, P. J.—WANG, P.: Periodic solutions of second order equations via rotation numbers, J. Differential Equations 266(8) (2019), 4746–4768.10.1016/j.jde.2018.10.010Search in Google Scholar
[15] ŠRemr, J.: On a structure of the set of positive solutions to second-order equations with a super-linear non-linearity, Georgian Math. J. 29(1) (2022), 139–152.10.1515/gmj-2021-2117Search in Google Scholar
[16] Šremr, J.: Positive solutions to the forced non-autonomous Duffing equations, Georgian Math. J. 29(2) (2022), 295–307.10.1515/gmj-2021-2118Search in Google Scholar
[17] Torres, P. J.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations 190(2) (2003), 643–662.10.1016/S0022-0396(02)00152-3Search in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Algebraic structures formalizing the logic of effect algebras incorporating time dimension
- Walks on tiled boards
- Composition on FLew-algebras
- On high power sums of a hybrid arithmetic function
- On indices of quintic number fields defined by x5 + ax + b
- Note on fundamental system of solutions to the differential equations (D2 − 2Dα + α2 ± β2) y = 0
- Several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals
- A new Approach of Generalized Fractional Integrals in Multiplicative Calculus and Related Hermite–Hadamard-Type Inequalities with Applications
- On a periodic problem for super-linear second-order ODEs
- Existence and Uniqueness of Solutions for Fractional Dynamic Equations with Impulse Effects
- Periodic Solutions for Conformable Non-autonomous Non-instantaneous Impulsive Differential Equations
- Self referred equations with an integral boundary condition
- Approximation by matrix means of double Vilenkin-Fourier series
- Maps on self-adjoint operators preserving some relations related to commutativity
- Chains in the Rudin-Frolík order
- Relative versions of first and second countability in hyperspaces
- Proportion estimation in multistage pair ranked set sampling
- The Marshall-Olkin Extended Gamma Lindley distribution: Properties, characterization and inference
Articles in the same Issue
- Algebraic structures formalizing the logic of effect algebras incorporating time dimension
- Walks on tiled boards
- Composition on FLew-algebras
- On high power sums of a hybrid arithmetic function
- On indices of quintic number fields defined by x5 + ax + b
- Note on fundamental system of solutions to the differential equations (D2 − 2Dα + α2 ± β2) y = 0
- Several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals
- A new Approach of Generalized Fractional Integrals in Multiplicative Calculus and Related Hermite–Hadamard-Type Inequalities with Applications
- On a periodic problem for super-linear second-order ODEs
- Existence and Uniqueness of Solutions for Fractional Dynamic Equations with Impulse Effects
- Periodic Solutions for Conformable Non-autonomous Non-instantaneous Impulsive Differential Equations
- Self referred equations with an integral boundary condition
- Approximation by matrix means of double Vilenkin-Fourier series
- Maps on self-adjoint operators preserving some relations related to commutativity
- Chains in the Rudin-Frolík order
- Relative versions of first and second countability in hyperspaces
- Proportion estimation in multistage pair ranked set sampling
- The Marshall-Olkin Extended Gamma Lindley distribution: Properties, characterization and inference
