| Summary: | In this study, we investigate the existence and multiplicity of solutions for a fractional discrete <i>p</i>−Laplacian equation on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">Z</mi></semantics></math></inline-formula>. Under suitable hypotheses on the potential function <i>V</i> and the nonlinearity <i>f</i>, with the aid of Ekeland’s variational principle, via mountain pass lemma, we obtain that this equation exists at least two nonnegative and nontrivial homoclinic solutions when the real parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is large enough.
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