## Positive solutions to a generalized second-order difference equation with summation boundary value problem.(English)Zbl 1244.39008

Summary: By using Krasnoselskii’s fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem $$\Delta^2 u(t - 1) + a(t)f(u(t)) = 0, t \in \{1, 2, \dots, T\}, u(0) = \beta \sum^\eta_{s=1} u(s), u(T + 1) = \alpha \sum^\eta_{s=1} u(s)$$, where $$f$$ is continuous, $$T \geq 3$$ is a fixed positive integer, $$\eta \in \{1, 2, \dots, T - 1\}, 0 < \alpha < (2T + 2)/\eta(\eta + 1), 0 < \beta < (2T + 2 - \alpha \eta(\eta + 1))/\eta(2T - \eta + 1)$$, and $$\Delta u(t - 1) = u(t) - u(t - 1)$$. We show the existence of at least one positive solution if $$f$$ is either superlinear or sublinear.

### MSC:

 39A12 Discrete version of topics in analysis
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### References:

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