Nontrivial solutions for a class of fractional difference boundary value problems and fixed-point problems

Abstract

In this work, we use the variant fountain theorem to study the existence of nontrivial solutions for the superquadratic fractional difference boundary value problem:

　　　　　　　　 \begin{array}{l}\{\begin{array}{l}{}_T{\mathrm{\Delta }}_{t-1}^{\nu }\left(t{\mathrm{\Delta }}_{\nu -1}^{\nu }x\left(t\right)\right)=f\left(x\right(t+\nu -1\left)\right),t\in [0,T{]}_{{\mathbb{N}}_0},\hfill \\ x(\nu -2)={\left[{}_t{\mathrm{\Delta }}_{\nu -1}^{\nu }x\left(t\right)\right]}_{t=T}=0.\hfill \end{array}\hfill \end{array}

The existence of nontrivial solutions is obtained in the case of super quadratic growth of the nonlinear term f by change of fountain theorem.

Key words： Fractional difference; boundary value problem; fountain theorem; nontrivial solution

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