Existence of at Least One Homoclinic Solution for a Nonlinear Second-Order Difference Equation

2019 ◽  
Vol 20 (3-4) ◽  
pp. 433-439 ◽  
Author(s):  
Martin Bohner ◽  
Giuseppe Caristi ◽  
Shapour Heidarkhani ◽  
Shahin Moradi
Keyword(s):  
Difference Equation ◽  
Variational Methods ◽  
Sufficient Conditions ◽  
Homoclinic Solution ◽  
Second Order ◽  
Technical Approach ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractThis paper presents sufficient conditions for the existence of at least one homoclinic solution for a nonlinear second-order difference equation with p-Laplacian. Our technical approach is based on variational methods. An example is offered to demonstrate the applicability of our main results.

scholarly journals The Asymptotic Behavior of Solutions of Second order Difference Equations With Damping Term

2018 ◽  
Vol 14 (2) ◽  
pp. 7806-7811
Author(s):  
Jai Kumar S ◽  
K. Alagesan
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Asymptotic Behavior Of Solutions ◽  
Damping Term ◽  
Order Difference ◽  
Keywords And Phrases ◽  
Second Order Difference Equation ◽  
Order Difference Equation

  The author presents some sufficient conditions for second order difference equation with damping term of the form                                                                             ^(an ^(xn + cxn-k)) + pn^xn + qnf(xn+1-l) = 0 An example is given to illustrate the main results. 2010 AMS Subject Classification: 39A11 Keywords and Phrases: Second order, difference equation, damping term.

scholarly journals Necessary and sufficient conditions for oscillation of second order neutral difference equations

2003 ◽  
Vol 34 (2) ◽  
pp. 137-146 ◽  
Author(s):  
E. Thandapani ◽  
K. Mahalingam
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Real Sequence ◽  
Order Difference ◽  
Neutral Difference Equations ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Necessary And Sufficient

Consider the second order difference equation of the form$\Delta^2(y\n-py_{n-1-k})+q_nf(y_{n-\ell})=0,\quad n=1,2,3,\ldots  \hskip 1.9cm\hbox{(E)}$where $ \{q_n\}$ is a nonnegative real sequence, $ f:{\Bbb R}\rightarrow {\Bbb R}$ is continuous such that $ uf(u)>0$ for $ u\not= 0$, $ 0\le p

scholarly journals Oscillation Criteria of Solution for a Second Order Difference Equation with Forced Term

2010 ◽  
Vol 2010 ◽  
pp. 1-6
Author(s):  
Chen Huiqin ◽  
Jin Zhen
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Order Difference ◽  
Oscillation Criteria ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Forced Term

We will consider oscillation criteria for the second order difference equation with forced termΔ(anΔ(xn+λxn−τ))+qnxn−σ=rn(n≥0). We establish sufficient conditions which guarantee that every solution is oscillatory or eventually positive solutions converge to zero.

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 Boundedness of solutions and stability of certain second-order difference equation with quadratic term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Emin Bešo ◽  
Senada Kalabušić ◽  
Naida Mujić ◽  
Esmir Pilav
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Second Order ◽  
Quadratic Term ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractWe consider the second-order rational difference equation $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$xn+1=γ+δxnxn−12, where γ, δ are positive real numbers and the initial conditions $x_{-1}$x−1 and $x_{0}$x0 are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

scholarly journals The global behavior of a quadratic difference equation

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6203-6210
Author(s):  
Vahidin Hadziabdic ◽  
Midhat Mehuljic ◽  
Jasmin Bektesevic ◽  
Naida Mujic
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Julia Set ◽  
Second Order ◽  
Global Behavior ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

In this paper we will present the Julia set and the global behavior of a quadratic second order difference equation of type xn+1 = axnxn-1 + ax2n-1 + bxn-1 where a > 0 and 0 ? b < 1 with non-negative initial conditions.

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