On a nonlinear Diophantine equation with powers of three consecutive $k$--Lucas Numbers

Let $(L_n^{(k)}{)}_{n\ge 2-k}$ be the sequence of $k$–generalized Lucas numbers for some fixed integer $k\ge 2$ whose first $k$ terms are $0,\dots ,0,2,1$ and each term afterwards is the sum of the preceding $k$ terms. In this paper, we completely solve the nonlinear Diophantine equation ${\left(L_{n+1}^{(k)}\right)}^x+{\left(L_n^{(k)}\right)}^x-{\left(L_{n-1}^{(k)}\right)}^x=L_m^{(k)}$, in nonnegative integers $n$, $m$, $k$, $x$, with $k\ge 2$.