Volume 4, Issue 3
On Two-Point Boundary Value Problems for Second-Order Difference Equation

Huijuan Li, Gaofeng Du & Cunyan Yue

J. Nonl. Mod. Anal., 4 (2022), pp. 605-614.

Published online: 2022-06

[An open-access article; the PDF is free to any online user.]

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  • Abstract

In this paper, we aim to investigate the difference equation $$∆^2 y(t − 1) + |y(t)| = 0, t ∈ [1, T]_{\mathbb{Z}}$$ with different boundary conditions $y(0) = 0$ or $∆y(0) = 0$ and $y(T + 1) = B$ or $∆y(T) = B,$ where $T ≥ 1$ is an integer and $B ∈\mathbb{R}.$ We will show that how the values of $T$ and $B$ influence the existence and uniqueness of the solutions to the about problem. In details, for the different problems, the $TB$-plane explicitly divided into different parts according to the number of solutions to the above problems. These parts of $TB$-plane for the value of $T$ and $B$ guarantee the uniqueness, the existence and the nonexistence of solutions respectively.

  • AMS Subject Headings

39A27, 39A05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JNMA-4-605, author = {Li , HuijuanDu , Gaofeng and Yue , Cunyan}, title = {On Two-Point Boundary Value Problems for Second-Order Difference Equation}, journal = {Journal of Nonlinear Modeling and Analysis}, year = {2022}, volume = {4}, number = {3}, pages = {605--614}, abstract = {

In this paper, we aim to investigate the difference equation $$∆^2 y(t − 1) + |y(t)| = 0, t ∈ [1, T]_{\mathbb{Z}}$$ with different boundary conditions $y(0) = 0$ or $∆y(0) = 0$ and $y(T + 1) = B$ or $∆y(T) = B,$ where $T ≥ 1$ is an integer and $B ∈\mathbb{R}.$ We will show that how the values of $T$ and $B$ influence the existence and uniqueness of the solutions to the about problem. In details, for the different problems, the $TB$-plane explicitly divided into different parts according to the number of solutions to the above problems. These parts of $TB$-plane for the value of $T$ and $B$ guarantee the uniqueness, the existence and the nonexistence of solutions respectively.

}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2022.605}, url = {http://global-sci.org/intro/article_detail/jnma/20727.html} }
TY - JOUR T1 - On Two-Point Boundary Value Problems for Second-Order Difference Equation AU - Li , Huijuan AU - Du , Gaofeng AU - Yue , Cunyan JO - Journal of Nonlinear Modeling and Analysis VL - 3 SP - 605 EP - 614 PY - 2022 DA - 2022/06 SN - 4 DO - http://doi.org/10.12150/jnma.2022.605 UR - https://global-sci.org/intro/article_detail/jnma/20727.html KW - Second-order difference equation, Different boundary conditions, Boundary value problems. AB -

In this paper, we aim to investigate the difference equation $$∆^2 y(t − 1) + |y(t)| = 0, t ∈ [1, T]_{\mathbb{Z}}$$ with different boundary conditions $y(0) = 0$ or $∆y(0) = 0$ and $y(T + 1) = B$ or $∆y(T) = B,$ where $T ≥ 1$ is an integer and $B ∈\mathbb{R}.$ We will show that how the values of $T$ and $B$ influence the existence and uniqueness of the solutions to the about problem. In details, for the different problems, the $TB$-plane explicitly divided into different parts according to the number of solutions to the above problems. These parts of $TB$-plane for the value of $T$ and $B$ guarantee the uniqueness, the existence and the nonexistence of solutions respectively.

Huijuan Li, Gaofeng Du & Cunyan Yue. (2022). On Two-Point Boundary Value Problems for Second-Order Difference Equation. Journal of Nonlinear Modeling and Analysis. 4 (3). 605-614. doi:10.12150/jnma.2022.605
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