Oscillatory Behaviors of Solutions of Riccati'S \(\alpha\)-Difference Equation

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Maria Susai Manuel
Chandrasekar V
Britto Antony Xavier G

Abstract

In this paper, by introducing $\alpha$-difference equation with the definition of generalized $\alpha$-difference operator, we discuss the oscillatory and nonoscillatory behaviours of the solutions of the Riccati's generalized $\alpha$-difference equation

\[{\label{trans.01}}\Delta_{\alpha(\ell)}\big(p(k)\Delta_{\alpha(\ell)} u(k)\big)+\alpha r(k) u(k+\ell) = 0, \ k\in[0, \infty), \ \alpha > 0\]

where the real valued functions $p > 0$ and $r$ are defined on $[0,\infty)$ and $\Delta_{\alpha(\ell)} u(k) = u(k+\ell) - \alpha u(k)$.

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