Differentiation with respect to parameters of solutions of nonlocal boundary value problems for difference equations

Abstract

For the $n$-th order difference equation, ${\Delta }^{n}u=f\left(t,u,\Delta u,\dots ,{\Delta }^{n-1}u,\lambda \right)$, the solution of the boundary value problem satisfying ${\Delta }^{i-1}u\left(t_0\right)=A_i,1\le i\le n-1$, and $u\left(t_1\right)-{\sum }_{j=1}^m{a}_{j}u\left({\tau }_j\right)=A_n$, where $t_0,{\tau }_1,\dots ,{\tau }_m,t_1\in \mathbb{Z}$, $t_0<\cdots <t_0+n-1<{\tau }_1<\cdots <{\tau }_m<t_1$, and $a_1,\dots ,a_m,A_1,\dots ,A_n\in ℝ$, is differentiated with respect to the parameter $\lambda$.