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

Yang, Yitao; Meng, Fanwei

J. Appl. Math. Comput. 40, No. 1-2, 319-340 (2012).

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

Keywords:

difference equation; positive solution; \(p\)-Laplacian; fixed point theorem; discrete four-point boundary value problem

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