Solvability of Boundary Value Problems for a Class of Third-Order Functional Difference Equations - Liu, Yuji (Sarajevo Journal of Mathematics, 2024)
Consider the boundary value problems consisting of the functional difference equation $$ \Delta^3x(n)=f(n,x(n+2),x(n-\tau_1(n)),\dots,x(n-\tau_m(n))),\;\;n\in [0,T] $$ and the following boundary value conditions \[\begin{cases} x(0)=x(T+3)=x(1)=0,\\ x(n)=\psi(n), \;n\in [-\tau,-1],\\ x(n)=\phi(n),\;n\in [T+4,T+\delta].\end{cases} \] Sufficient conditions for the existence of at least one solution of this problem are established. We allow $f$ to be at most linear, superlinear or sublinear in the obtained results. 2000 Mathematics Subject Classification. 34B10, 34B15, 39A10