صندلی اداری

QUALITATIVE PROPERTIES OF SOLUTIONS OF RICCATI'S α-DIFFERENCE EQUATIONS

M. Maria Susai Manuel, V. Chandrasekar, G. Britto Antony Xavier

Abstract


In this paper, by introducing $\alpha$-difference equation with the definition of generalized $\alpha$-difference operator, we discuss the general properties and boundedness behaviour of solutins of the generalized Riccati's $\alpha$-difference equation

\begin{equation}{\label{ricc.01}}

p(k)u(k+\ell)+\alpha^2 p(k-\ell)u(k-\ell)=\alpha q(k)u(k), k\in[\ell, \infty),

\end{equation}

where the real valued functions $p$ and $q$ are defined on $[\ell,\infty)$ and $p(k)>0$ for all $k\in[\ell,\infty)$. Equation (\ref{ricc.01}) equivalently can be written as

\begin{equation}{\label{ricc.02}}

-\Delta_{\alpha(\ell)}\Big(p(k-\ell)\Delta_{\alpha(\ell)}u(k-\ell)\Big)+\alpha f(k)u(k)=0, k\in[\ell,\infty),

\end{equation}

where $f(k)=q(k)-p(k)-p(k-\ell)$.


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