scholarly journals Global structure and one‐sign solutions for second‐order Sturm–Liouville difference equation with sign‐changing weight

2021 ◽  
Author(s):  
Fumei Ye
Keyword(s):  
Difference Equation ◽  
Global Structure ◽  
Second Order ◽  
Sturm Liouville

scholarly journals Global structure and one-sign solutions for second-order Sturm-Liouville difference equation with sign-changing weight

2020 ◽  
Author(s):  
fumei ye
Keyword(s):  
Difference Equation ◽  
Liouville Problem ◽  
Global Structure ◽  
Second Order ◽  
Sturm Liouville

This paper is devoted to study the discrete Sturm-Liouville problem $$ \left\{\begin{array}{ll} -\Delta(p(k)\Delta u(k-1))+q(k)u(k)=\lambda m(k)u(k)+f_1(k,u(k),\lambda)+f_2(k,u(k),\lambda),\ \ k\in[1,T]_Z,\\[2ex] a_0u(0)+b_0\Delta u(0)=0,\ a_1u(T)+b_1\Delta u(T)=0, \end{array}\right. $$ where $\lambda\in\mathbb{R}$ is a parameter, $f_1, f_2\in C([1,T]_Z\times\mathbb{R}^2, \mathbb{R})$, $f_1$ is not differentiable at the origin and infinity. Under some suitable assumptions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcate from intervals of the line of trivial solutions or from infinity, respectively.

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.

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