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Infinitely many solutions for a class of discrete non-linear boundary value problems. (English) Zbl 1176.39004
The following problem is considered: $$-\Delta(\phi_p(\Delta u_{k-1})+ cx_{n-k})+ q_k\phi_p(u_k)=\lambda f(k, u_k),\quad k\in [1,N],$$ $$u_0= u_{N+1}= 0,$$ where $f: [1, N]\times\bbfR\to\bbfR$ is a continuous function, $\Delta u_{k-1}=u_k- u_{k-1}$ is the forward difference operator, $q_k\in\bbfR^+_0$ for all $k\in[1, N]$, $\phi_p(s):=|s|^{p-2}s$, $1< p< +\infty$ and $\lambda\in \bbfR^+$. Two types of results are given: the existence of either an unbounded sequence of solutions or a sequence of pairwise distinct non-zero solutions which converges to $0$, depending on whether the nonlinear term has a suitable oscillating behavior, respectively, at infinity or at zero.

39A12Discrete version of topics in analysis
39A10Additive difference equations
34B15Nonlinear boundary value problems for ODE
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