Uncountably many solutions of a third order nonlinear difference equation with neutral delay

Volume 9, Issue 6, pp 4329--4354 http://dx.doi.org/10.22436/jnsa.009.06.74

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Authors

Zeqing Liu - Department of Mathematics, Liaoning Normal University Dalian, Liaoning 116029, People's Republic of China. Xiaoying Zhang - Department of Mathematics, Liaoning Normal University Dalian, , ., Liaoning 116029, People's Republic of China. Jeong Sheok Ume - Department of Mathematics, Changwon National University, Changwon 641-773, Korea. Shin Min Kang - Department of Mathematics and the Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Korea.

Abstract

In this paper, by using the Schauder fixed point theorem, Krasnoselskii fixed point theorem and some new techniques, we obtain the existence of uncountably many solutions for a third order nonlinear difference equation with neutral delay of the form \[\Delta(a(n; x_{a_{1n}}; x_{a_{2n}};... ; x_{a_{rn}})\Delta^2(x_n + b_nx_{n-\tau})+ \Delta h(n; x_{h_{1n}}; x_{h_{2n}}; ... ; x_{h_{kn}}) + f(n; x_{f_{1n}}; x_{f_{2n}};... ; x_{f_{kn}}) = c_n, n \geq n_0,\] The results presented improve and generalize some results in literatures. Seven examples are given to illustrate the results presented in this paper.

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Keywords

• Third order nonlinear difference equation with neutral delay
• uncountably many solutions
• Schauder fixed point theorem
• Krasnoselskii fixed point theorem.

MSC

•  39A10
•  39A20

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