On the existence of periodic boundary conditions for nonlinear second order differential equations

SynopsisThis paper studies a linked system of second order ordinary differential equationswhere xx ∈ [ar, br] and the coefficients qrars are continuous, real valued and periodic of period (br − ar), 1 ≤ r,s ≤ k. We assume the definiteness condition det{ars(xr)} > 0 and 2k possible multiparameter eigenvalue problems are then formulated according as periodic or semi-periodic boundary conditions are imposed on each of the equations of (*). The main result describes the interlacing of the 2k possible sets of eigentuples thus extending to the multiparameter case the well known theorem concerning 1-parameter periodic equation.

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Stability regions for multi-parameter systems of periodic second order ordinary differential equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050001711x ◽

1979 ◽

Vol 84 (3-4) ◽

pp. 249-257 ◽

Cited By ~ 2

Author(s):

Patrick J. Browne ◽

B. D. Sleeman

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Ordinary Differential Equations ◽

Periodic Boundary ◽

Periodic Boundary Conditions ◽

Second Order ◽

Parameter System ◽

Stability Regions ◽

The Stability ◽

Image Position

SynopsisThis paper studies the stability regions associated with the multi-parameter systemwhere the functions qr(xr), ars(xr) are periodic and the system is subjected to periodic or semi-periodic boundary conditions.

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Periodic solutions for a class of conservative Liénard-type equations

4open ◽

10.1051/fopen/2019003 ◽

2019 ◽

Vol 2 ◽

pp. 5

Author(s):

Emily R. Korfanty ◽

Ankai Liu ◽

Wenying Feng

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Periodic Solutions ◽

Periodic Boundary ◽

Coincidence Degree ◽

Degree Theory ◽

Periodic Boundary Conditions ◽

Operator Equations ◽

Second Order Differential Equations ◽

Semilinear Operator

In this paper, we study solvability of a class of second-order differential equations in a conservative Liénard form subject to periodic boundary conditions. Results on existence of non-trivial T-periodic solutions or positive T-periodic solutions are obtained respectively. Applications of the theorems are shown by examples. The results are proved by applying the coincidence degree theory for semilinear operator equations.

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Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas ◽

10.1007/s13398-021-01080-w ◽

2021 ◽

Vol 115 (3) ◽

Author(s):

Robert Stegliński

Keyword(s):

Difference Equation ◽

Boundary Conditions ◽

Difference Equations ◽

Periodic Boundary ◽

Periodic Boundary Conditions ◽

Second Order ◽

Linear Difference Equations ◽

Order Difference ◽

Second Order Difference Equation ◽

Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

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