In this paper, we introduce the concept of $\alpha$-partial weakly Zamfirescu mappings and give some approximate fixed point results for this mapping in $\alpha$-complete metric spaces. We also give some approximate fixed point results in $\alpha$-complete metric space endowed with an arbitrary binary relation and approximate fixed point results in $\alpha$-complete metric space endowed with graph. As application, we give homotopy results for $\alpha$-partial weakly Zamfirescu mapping.