Multiple solutions of fourth-order functional difference equation with periodic boundary conditions

2020 ◽  
Vol 104 ◽  
pp. 106292 ◽  
Author(s):  
Shaohong Wang ◽  
Yuhua Long
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Multiple Solutions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Functional Difference ◽  
Functional Difference Equation

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

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.

scholarly journals Solvability of a fourth order boundary value problem with periodic boundary conditions

1988 ◽  
Vol 11 (2) ◽  
pp. 275-284
Author(s):  
Chaitan P. Gupta
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Elastic Beam ◽  
Periodic Boundary Conditions ◽  
Uniqueness Of Solutions

Fourth order boundary value problems arise in the study of the equilibrium of an elastaic beam under an external load. The author earlier investigated the existence and uniqueness of the solutions of the nonlinear analogues of fourth order boundary value problems that arise in the equilibrium of an elastic beam depending on how the ends of the beam are supported. This paper concerns the existence and uniqueness of solutions of the fourth order boundary value problems with periodic boundary conditions.

scholarly journals Solvability of a fourth-order boundary value problem with periodic boundary conditions II

1991 ◽  
Vol 14 (1) ◽  
pp. 127-137 ◽  
Author(s):  
Chaitan P. Gupta
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Periodic Boundary ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Periodic Boundary Conditions ◽  
Special Case

Letf:[0,1]×R4→Rbe a function satisfying Caratheodory's conditions ande(x)∈L1[0,1]. This paper is concerned with the solvability of the fourth-order fully quasilinear boundary value problemd4udx4+f(x,u(x),u′(x),u″(x),u‴(x))=e(x),   0<x<1, withu(0)−u(1)=u′(0)−u′(1)=u″(0)-u″(1)=u‴(0)-u‴(1)=0. This problem was studied earlier by the author in the special case whenfwas of the formf(x,u(x)), i.e., independent ofu′(x),u″(x),u‴(x). It turns out that the earlier methods do not apply in this general case. The conditions need to be related to both of the linear eigenvalue problemsd4udx4=λ4uandd4udx4=−λ2d2udx2with periodic boundary conditions.

scholarly journals Existence of One-Signed Solutions of Discrete Second-Order Periodic Boundary Value Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Ruyun Ma ◽  
Yanqiong Lu ◽  
Tianlan Chen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Nonlinear Difference Equation ◽  
Periodic Boundary Value Problems ◽  
The Fixed Point Theorem

We prove the existence of one-signed periodic solutions of second-order nonlinear difference equation on a finite discrete segment with periodic boundary conditions by combining some properties of Green's function with the fixed-point theorem in cones.

Molecular Dynamics Using Non-Variational Polarizable Force Fields: Theory, Periodic Boundary Conditions Implementation and Application to the Bond Capacity Model

2019 ◽  
Author(s):  
Pier Paolo Poier ◽  
Louis Lagardere ◽  
Jean-Philip Piquemal ◽  
Frank Jensen
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Force Fields ◽  
Periodic Boundary Conditions ◽  
Computational Effort ◽  
Polarizable Force Fields ◽  
Energy Models ◽  
Capacity Model ◽  
Energy Gradient ◽  
Polarization Models

<div> <div> <div> <p>We extend the framework for polarizable force fields to include the case where the electrostatic multipoles are not determined by a variational minimization of the electrostatic energy. Such models formally require that the polarization response is calculated for all possible geometrical perturbations in order to obtain the energy gradient required for performing molecular dynamics simulations. </p><div> <div> <div> <p>By making use of a Lagrange formalism, however, this computational demanding task can be re- placed by solving a single equation similar to that for determining the electrostatic variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p><div><div><div> </div> </div> </div> <p> </p><div> <div> <div> <p>variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p> </div> </div> </div> </div> </div> </div> </div> </div> </div>

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