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We study the existence of solutions to the boundary-value problem $$displaylines{ Delta(phi_p(Delta u(k-1)))=f(k,u(k),Delta u(k)),quad kin mathbb{T}_{[1, N]}, cr Delta u(0)=A, quad u(N+1)=B, }$$ with barrier strips conditions, where N is a fixed natural number greater than 1, $phi_p(s)=|s|^{p-2}s$, p greater than 1.