scholarly journals Homoclinic solutions of 2nth-order difference equations containing both advance and retardation

2016 ◽  
Vol 14 (1) ◽  
pp. 520-530 ◽  
Author(s):  
Yuhua Long ◽  
Yuanbiao Zhang ◽  
Haiping Shi
Keyword(s):  
Difference Equation ◽  
Critical Point ◽  
Difference Equations ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Nonlinear Difference Equation ◽  
Mountain Pass Lemma ◽  
Order Difference ◽  
Existence And Multiplicity ◽  
New Criteria

AbstractBy using the critical point method, some new criteria are obtained for the existence and multiplicity of homoclinic solutions to a 2nth-order nonlinear difference equation. The proof is based on the Mountain Pass Lemma in combination with periodic approximations. Our results extend and improve some known ones.

scholarly journals Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations

2018 ◽  
Vol 16 (1) ◽  
pp. 1435-1444 ◽  
Author(s):  
Peng Mei ◽  
Zhan Zhou
Keyword(s):  
Difference Equation ◽  
Critical Point ◽  
Critical Point Theory ◽  
Point Theory ◽  
Nonlinear Difference Equation ◽  
Subharmonic Solutions ◽  
Existence And Multiplicity ◽  
Explicit Criteria

AbstractWe consider a 2nth-order nonlinear difference equation containing both many advances and retardations with p-Laplacian. Using the critical point theory, we obtain some new explicit criteria for the existence and multiplicity of periodic and subharmonic solutions. Our results generalize and improve some known related ones.

scholarly journals Boundary Value Problems for Fourth Order Nonlinearp-Laplacian Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Qinqin Zhang
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Mountain Pass ◽  
Mountain Pass Lemma

We consider the boundary value problem for a fourth order nonlinearp-Laplacian difference equation containing both advance and retardation. By using Mountain pass lemma and some established inequalities, sufficient conditions of the existence of solutions of the boundary value problem are obtained. And an illustrative example is given in the last part of the paper.

scholarly journals Sequences of positive homoclinic solutions to difference equations with variable exponent

2020 ◽  
Vol 70 (2) ◽  
pp. 417-430
Author(s):  
Robert Stegliński ◽  
Magdalena Nockowska-Rosiak
Keyword(s):  
Variational Principle ◽  
Difference Equation ◽  
Difference Equations ◽  
Critical Point Theory ◽  
Variable Exponent ◽  
Point Theory ◽  
Homoclinic Solutions ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

Abstract We study the existence of infinitely many positive homoclinic solutions to a second-order difference equation on integers with pk-Laplacian. To achieve our goal we use the critical point theory and the general variational principle of Ricceri.

scholarly journals Homoclinic Solutions for a Higher Order ϕc-Laplacian Difference Equation Containing Both Advance and Retardation

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Peng Mei ◽  
Zhan Zhou
Keyword(s):  
Difference Equation ◽  
Critical Point ◽  
Critical Point Theory ◽  
Point Theory ◽  
Higher Order ◽  
Homoclinic Solutions ◽  
Nonlinear Difference Equation ◽  
Mixed Nonlinearities

We consider a 2mth-order nonlinear difference equation containing both advance and retardation with ϕc-Laplacian. Using the critical point theory, some new and concrete criteria for the existence of homoclinic solutions with mixed nonlinearities are obtained.

Oscillation Criteria for Fourth Order Nonlinear Difference Equations

2007 ◽  
Vol 14 (2) ◽  
pp. 203-222
Author(s):  
Ravi P. Agarwal ◽  
Said R. Grace ◽  
Elvan Akin-Bohner
Keyword(s):  
Difference Equation ◽  
Fourth Order ◽  
Difference Equations ◽  
Nonlinear Difference Equations ◽  
Order Difference ◽  
Oscillation Criteria ◽  
Order Difference Equation ◽  
New Criteria

Abstract Some new criteria for the oscillation of the fourth order difference equation Δ2 (𝑎(𝑛)(Δ2𝑥(𝑛)) α ) + 𝑞(𝑛)𝑓(𝑥(𝑛 + 1)) = 0, where α is the ratio of two positive odd integers are established.

scholarly journals Positive strongly decreasing solutions of Emden-Fowler type second-order difference equations with regularly varying coefficients

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2751-2770
Author(s):  
Aleksandra Kapesic ◽  
Jelena Manojlovic
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Order Difference ◽  
Varying Coefficients ◽  
Regularly Varying ◽  
Necessary And Sufficient

Positive decreasing solutions of the nonlinear difference equation ?(pn|?xn|?-1?xn)=qn|xn+1|?-1xn+1, n ? 1, ? > ? > 0, are studied under the assumption that p; q are regularly varying sequences. Necessary and sufficient conditions are established for the existence of regularly varying strongly decreasing solutions and it is shown that the asymptotic behavior of all such solutions is governed by a unique formula.

scholarly journals A classification scheme for nonoscilatory solutions of a higher order neutral nonlinear difference equation

1999 ◽  
Vol 67 (1) ◽  
pp. 122-142 ◽  
Author(s):  
Li Wan-Tong ◽  
Sui Sun Cheng ◽  
Guang Zhang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Classification Scheme ◽  
Neutral Type ◽  
Higher Order ◽  
Nonlinear Difference Equation ◽  
Nonoscillatory Solutions ◽  
Order Difference ◽  
Asymptotic Behaviors ◽  
Existence Criteria

AbstractNonoscillatory solutions of a nonlinear neutral type higher order difference equations are classified by means of their asymptotic behaviors. Existence criteria are then provided for justification of such classficiation.

scholarly journals Nontrivial Periodic Solutions to Some Semilinear Sixth-Order Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yuhua Long
Keyword(s):  
Periodic Solutions ◽  
Difference Equations ◽  
Index Theory ◽  
Sixth Order ◽  
Variational Technique ◽  
Order Difference ◽  
Existence And Multiplicity ◽  
Nontrivial Periodic Solutions ◽  
New Criteria

We establish some new criteria to guarantee nonexistence, existence, and multiplicity of nontrivial periodic solutions of some semilinear sixth-order difference equations by using minmax method,Z2index theory, and variational technique. Our results only make some assumptions on the periodT, which are very easy to verify and rather relaxed.

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 Multiple solutions of fourth-order difference equations with different boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yuhua Long ◽  
Shaohong Wang ◽  
Jiali Chen
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Difference Equations ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Dirichlet Boundary Conditions ◽  
Invariant Sets ◽  
Mountain Pass Lemma ◽  
Order Difference

Abstract In the present paper, a class of fourth-order nonlinear difference equations with Dirichlet boundary conditions or periodic boundary conditions are considered. Based on the invariant sets of descending flow in combination with the mountain pass lemma, we establish a series of sufficient conditions on the existence of multiple solutions for these boundary value problems. In addition, some examples are provided to demonstrate the applicability of our results.

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