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Positive solutions of discrete $$(n,p)$$ boundary value problems. (English) Zbl 0893.39001
Consider the boundary value problem $\Delta^ny +\lambda F(k,y, \Delta y, \dots, \Delta^{n-2} y)= \lambda G(k,y,\Delta y, \dots, \Delta^{n-1}y), \quad k=n-1, \dots, N,\tag{1}$ $\Delta^i y(0)=0, \quad 0\leq i\leq n-2, \quad \Delta^p y(N+n-p) =0, \tag{2}$ where $$n\geq 2$$, $$\lambda >0$$ and $$p$$ is a fixed integer $$(0\leq p\leq n-1)$$. The author obtains conditions on $$\lambda$$ for which the problem (1), (2) has a positive solution, i.e. she determines the set of eigenvalues.