Existence of positive periodic solutions for Liénard equations with an indefinite singularity of attractive type

Lu, Shiping; Yu, Xingchen

doi:10.1186/s13661-018-1020-0

Cited by 7 publications

(5 citation statements)

References 29 publications

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“…where b(t) is a piecewise-constant sign-changing function. Motivated by the two works [6,22], the mathematicians paid their attention to the singular equations containing both attractive and repulsive singularities simultaneously or indefinite singularity (see [8,10,21,24,[35][36][37]47,52]). Thereinto, Hakl and Zamora [24], Godoy and Zamora [21] generalized equation (1.5) in a wider application.…”

Section: Introductionmentioning

confidence: 99%

“…al. [35][36][37] by means of coincidence degree theory generalized indefinite singularity to Liénard equation.…”

Section: Introductionmentioning

confidence: 99%

“…Thirdly, our results can be applied to both strong singularity and weak singularity. In fact, following Lazer and Solimini's work, there are many good papers appearing in this field by different kinds of methods such as the fixed point theorems [8,9,27], lower and upper solution [1,22,23], Leray-Schauder alternative principle [12,28,32] and coincidence degree theory [5,29,37], and so on. Different from above methods, in this paper, we use the theory of the fixed point in cones to study the periodic solution.…”

Section: Introductionmentioning

confidence: 99%

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Second-order differential equation with indefinite and repulsive singularities

Cui¹,

Xia²

2022

Preprint

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This paper concerns a second-order differential equation with indefinite and repulsive singularities. It is the first time to study differential equation containing both indefinite and repulsive singularities simultaneously. A set of sufficient conditions are obtained for the existence of positive periodic solutions. The theoretical underpinnings of this paper are the positivity of Green's function and fixed point theorem in cones. Our results improve and extend the results in previous literatures. Finally, three examples and their numerical simulations (phase diagrams and time diagrams of periodic solutions) are given to show the effectiveness of our conclusions.

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Section: Introductionmentioning

confidence: 99%

“…al. [35][36][37] by means of coincidence degree theory generalized indefinite singularity to Liénard equation.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Second-order differential equation with indefinite and repulsive singularities

Cui¹,

Xia²

2022

Preprint

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“…As we all know, due to seasonal fluctuations in the environment and hereditary factors, time delays have been introduced into the biological system (see [11,14,21,[27][28][29][30][31][32][33][34][35][36]). Zhao et al [8] further considered a discrete Lotka-Volterra competition system with infinite delays and single feedback control variable as follows:…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Discrete Allelopathic Phytoplankton Model with Infinite Delays and Feedback Controls

Zhao

Qin

Chen

2020

Discrete Dynamics in Nature and Society

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A discrete allelopathic phytoplankton model with infinite delays and feedback controls is studied in this paper. By applying the discrete comparison theorem, a set of sufficient conditions which guarantees the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the extinction of the system are obtained. Our results extend and supplement some known results and show that the feedback controls and toxic substances play a crucial role on the permanence and extinction of the system.

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“…For more works about superlinear/sublinear problems with a weight function having an indefinite sign, see e.g. [21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Positive periodic solution for indefinite singular Liénard equation with p-Laplacian

Zhou

2019

Adv Differ Equ

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The efficient conditions guaranteeing the existence of positive T-periodic solution to the p-Laplacian-Liénard equation(R + stands for positive real numbers) with a singularity at x = 0, g(x) is continuous on (0; +∞), μ is a constant with μ > 0, the signs of α 1 and α 2 are allowed to change. The approach is based on the continuation theorem for p-Laplacian-like nonlinear systems obtained by Manásevich and Mawhin in (J. Differ. Equ. 145:367-393, 1998). MSC: 34B15

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Existence of positive periodic solutions for Liénard equations with an indefinite singularity of attractive type

Cited by 7 publications

References 29 publications

Second-order differential equation with indefinite and repulsive singularities

Second-order differential equation with indefinite and repulsive singularities

Dynamics of a Discrete Allelopathic Phytoplankton Model with Infinite Delays and Feedback Controls

Positive periodic solution for indefinite singular Liénard equation with p-Laplacian

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