## Eigenvalue problem for finite difference equations with $$p$$-Laplacian.(English)Zbl 1295.39006

Summary: In this paper, we consider a discrete four-point boundary value problem $\Delta\bigl(\phi_p\bigl(\Delta u(k-1)\bigr)\bigr)+\lambda e(k)f\bigl(u(k)\bigr)=0,\quad k\in N(1,T),$ subject to boundary conditions $\Delta u(0)-\alpha u(l_1)=0,\quad \Delta u(T)+\beta u(l_2)=0,$ by a simple application of a fixed-point theorem. If $$e(k)$$, $$f(u(k))$$ are nonnegative, the solutions of the above problem may not be nonnegative, this is the main difficulty for us to study positive solution of this problem. In this paper, we give restrictive conditions $$\alpha l_1\leq 1$$, $$\beta(T+1-l_2)\leq 1$$ to guarantee the solutions of this problem are nonnegative, if it has, under the conditions $$e(k)$$, $$f(u(k))$$ are nonnegative. We first construct a new operator equation which is equivalent to the problem and provide sufficient conditions for the nonexistence and existence of at least one or two positive solutions. In doing so, the usual restrictions $$f_0=\lim_{u\to 0^+}\frac{f(u)}{\phi_p(u)}$$ and $$f_\infty=\lim_{u\to\infty}\frac{f(u)}{\phi_p(u)}$$ exist are removed.

### MSC:

 39A12 Discrete version of topics in analysis 39A10 Additive difference equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 39A22 Growth, boundedness, comparison of solutions to difference equations
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