scholarly journals Existence and Nonexistence of Solutions for Fourth-Order Nonlinear Difference Boundary Value Problems via Variational Methods

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Xia Liu ◽  
Tao Zhou ◽  
Haiping Shi
Keyword(s):  
Variational Methods ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Nonlinear Difference Equation ◽  
Nontrivial Solutions ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions

This paper is concerned with boundary value problems for a fourth-order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence ofndistinct pairs of nontrivial solutions, and nonexistence of solutions. Some examples are provided to show the effectiveness of the main results.

Existence and nonexistence results for a fourth-order discrete neumann boundary value problem

2014 ◽  
Vol 51 (2) ◽  
pp. 186-200
Author(s):  
Xia Liu ◽  
Yuanbiao Zhang ◽  
Haiping Shi
Keyword(s):  
Boundary Value Problem ◽  
Fourth Order ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Point Theory ◽  
Nonlinear Difference Equation ◽  
Neumann Boundary Value Problem ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions

In this paper, a fourth-order nonlinear difference equation is considered. By making use of the critical point theory, we establish various sets of sufficient conditions for the existence and nonexistence of solutions for Neumann boundary value problem and give some new results. Results obtained generalize and complement the existing ones.

scholarly journals Boundary value problems of a discrete generalized beam equation via variational methods

2018 ◽  
Vol 16 (1) ◽  
pp. 1412-1422
Author(s):  
Xia Liu ◽  
Tao Zhou ◽  
Haiping Shi
Keyword(s):  
Variational Methods ◽  
Boundary Value Problems ◽  
Critical Point ◽  
Critical Point Theory ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Beam Equation

AbstractThe authors explore the boundary value problems of a discrete generalized beam equation. Using the critical point theory, some sufficient conditions for the existence of the solutions are obtained. Several consequences of the main results are also presented. Examples are given to illustrate the theorems.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

Existence of Two Solutions for a Second-Order Discrete Boundary Value Problem

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Pasquale Candito ◽  
Giovanni Molica Bisci
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Second Order ◽  
Nontrivial Solutions ◽  
Discrete Boundary Value Problem

AbstractThe existence of two nontrivial solutions for a class of nonlinear second-order discrete boundary value problems is established. The approach adopted is based on variational methods.

scholarly journals Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with ϕc-Laplacian

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1839 ◽  
Author(s):  
Yanshan Chen ◽  
Zhan Zhou
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Curvature Operator ◽  
Nontrivial Solutions ◽  
Three Solutions ◽  
The Maximum Principle ◽  
Mean Curvature Operator ◽  
The Mean

In this paper, based on critical point theory, we mainly focus on the multiplicity of nontrivial solutions for a nonlinear discrete Dirichlet boundary value problem involving the mean curvature operator. Without imposing the symmetry or oscillating behavior at infinity on the nonlinear term f, we respectively obtain the sufficient conditions for the existence of at least three non-trivial solutions and the existence of at least two non-trivial solutions under different assumptions on f. In addition, by using the maximum principle, we also deduce the existence of at least three positive solutions from our conclusion. As far as we know, our results are supplements to some well-known ones.

scholarly journals Positive solutions of fourth-order superlinear singular boundary value problems

2002 ◽  
Vol 66 (1) ◽  
pp. 95-104 ◽  
Author(s):  
Guoliang Shi ◽  
Shaozhu Chen
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
Closed Interval ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Singular Boundary ◽  
Point Boundary ◽  
Singular Boundary Value Problems ◽  
Necessary And Sufficient

This paper investigates fourth-order superlinear singular two-point boundary value problems and obtains necessary and sufficient conditions for existence of C2 or C3 positive solutions on the closed interval.

Existence and Multiple Solutions for Higher Order Difference Dirichlet Boundary Value Problems

2018 ◽  
Vol 19 (5) ◽  
pp. 539-544
Author(s):  
Lianwu Yang
Keyword(s):  
Boundary Value Problems ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Point Theory ◽  
Higher Order ◽  
Nonlinear Difference Equation ◽  
Linking Theorem ◽  
Order Difference ◽  
Existence And Multiplicity

AbstractIn this paper, a higher order nonlinear difference equation is considered. By using the critical point theory, we obtain the existence and multiplicity for solutions of difference Dirichlet boundary value problems and give some new results. The proof is based on the variational methods and linking theorem.

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 Variational Approach to Impulsive Problems: A Survey of Recent Results

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Fang-fang Liao ◽  
Juntao Sun
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Periodic Solutions ◽  
Variational Approach ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Homoclinic Solutions ◽  
Nontrivial Solutions ◽  
Impulsive Problems

We present a survey on the existence of nontrivial solutions to impulsive differential equations by using variational methods, including solutions to boundary value problems, periodic solutions, and homoclinic solutions.

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