Existence theorems for second-order discrete boundary value problems. (English) Zbl 1113.39019

The authors consider the following second order difference boundary value problem:
\[ \Delta (p_n\,(\Delta x_{n-1})^\delta)+q_n\, x_n^\delta=f(n,x_n), \; n \in \{1, \dots,k\}; \quad \Delta\, x_0=A, \quad x_{k+1}=B. \]
Here \(\delta >0\), \(\{p_n\}\) and \(\{q_n\}\) are real sequences, \(p_n \neq 0\) for all \(n \in \{1,\dots,k+1\}\) and \(A\) and \(B\) are two given constants.
Under different suitable assumptions on functions \(f\), \(p_n\) and \(q_n\), the authors prove some existence results of at least one (or at least two) solution of the considered problem. The proofs follow from the Linking Theorem and the Mountain Pass Lemma in the critical point theory.


39A12 Discrete version of topics in analysis
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


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