the discrete eigenvalue problem of fractional order 0.5 and 0.75

‎in this paper‎, ‎we investigate the eigenvalues of the following problem‎‎egin{equation*}‎‎egin{cases}‎‎left(^{}_{5}abla_k^{alpha}left( ^{}_{k}abla_0^{alpha}ight)+{^{}_{k}abla}_0^{alpha}left( ^{}_5abla_{k}^{alpha}ight)ight)u(k)=lambda u(k)‎, ‎quad k in [1,4]_{mathbb{n}_{0}}‎, ‎‎‎u(0)=0= u(5)‎,‎end{cases}‎‎end{equation*}‎‎where $alpha=0.5‎, ‎0.75$‎, ‎$^{}_kabla_{0}^{0.75}$ and $^{}_{5}abla_k^{0.75}$ are the left and right nabla discrete fractional difference‎, ‎respectively‎, ‎and $lambda>0$ is a parameter‎. ‎by‎‎using left and right nabla discrete fractional difference definitions‎, ‎we obtain the matrix structure of the problem and by using maple software‎ ‎the eigenvalues provide.