Best proximity and best periodic points for proximal nonunique contractions

Taking into account nonunique fixed point and periodic point results of Ciric (Institut Mathematique 17, 52–58, 1974) and best proximity point results in the literature, we introduce the concept of proximal nonunique contraction for nonself mappings. Then, we obtain some nonunique best proximity point theorems. Further, we describe the concept of best periodic point of nonself mappings and we present some nonunique best periodic point results. We also provide many examples to show that our new concepts are meaningful and to support our theorems. Additionally, we present a new notion for the nonself mapping H called property $$P_{b}$$ , and then we provide some results including sufficient conditions to guarantee that H has the property $$P_{b}$$ .