LAUSR

AbstractIn this paper, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations a0(t)Dβ2y(t)+a1(t)Dβy(t)+a2(t)y(t)=b(t),t∈I,$$a_{0}(t)D_{\beta}^{2}y(t)+a_{1}(t)D_{\beta}y(t)+a_{2}(t)y(t)=b(t),\quad t \in I, $$a0(t)≠0$a_{0}(t)\neq0$, in a neighborhood of the unique fixed point s0$s_{0}$ of the strictly increasing continuous function β, defined on an interval I⊆R$I\subseteq{\mathbb{R}}$. These equations are based on the general quantum difference operator Dβ$D_{\beta}$, which is defined by Dβf(t)=(f(β(t))−f(t))/(β(t)−t)$D_{\beta}{f(t)}= (f(\beta(t))-f(t) )/ (\beta(t)-t )$, β(t)≠t$\beta(t)\neq t$. We also construct a fundamental set of solutions for the second order linear homogeneous β-difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the Euler-Cauchy β-difference equation.