q-difference equations associated with the Rubin's q-difference operator ${\mathrm{\partial }}_q$

The aim of this paper is to prove the existence and uniqueness of solutions of the following $q$- Cauchy problem of second order linear $q$-difference problem associated with the Rubin’s $q$- difference operator ${\mathrm{\partial }}_q$ in a neighborhood of zero \begin{equation} \left{ \begin{array}{cc} q,a_0(x), \partial_q^2y(qx), +\ ,a_1(x),\partial_qy(x), + ,a_2(x)y(x) &; = ;b(x),\quad \hbox{if $y$ is odd;}\ q,a_0(x) \partial_q^2y(qx), + ,q,a_1(x)\partial_qy(qx), + ,a_2(x)y(x)&; = ;b(x),\quad \hbox{if $y$ is even,} \end{array} \right. \end{equation} with the initial conditions \begin{equation} \partial_{q}^{i-1}y(0)= b_{i};\quad b_{i} \in{$\mathbb{C}$},; i=1,2 \end{equation} where $a_i$, $i=0,1,2$, and $b$ are defined, continuous at zero and bounded on an interval $I$ containing zero such that $a_0(x)\mathrm{\ne }0$ for all $x\in I$. Then, as application of the main results, we study the second order homogenous linear $q$- difference equations as well as the $q$-Wronskian associated with the Rubin’s $q$-difference operator ${\mathrm{\partial }}_q$. Finally, we construct a fundamental set of solutions for the second order linear homogeneous $q$-difference equations in the cases when the coefficients are constants and $a_1(x)=0$ for all $x\in I$.