In this paper, we introduce new classes of proximal multi-valued contractions in a metric space and nonexpansive mappings Banach show the existence best proximity points for both classes. Further, mappings, prove point theorem on starshape sets. As consequence, also obtain some fixed theorems. Finally, give examples to illustrate our main results.

In this paper, we first introduce a new class of the pointwise cyclic-noncyclic proximal contraction pairs. Then consider coincidence quasi-best proximity point problem for class. Finally, study points weak Kannan We an example to indicate validity main result.

In this work, we prove the existence of best proximity point results for ⊥-contraction (orthogonal-contraction) mappings on an O-complete metric space (orthogonal-complete space). Subsequently, these are employed to establish common result. Finally, provide suitable examples demonstrate validity our results.