1234 BEST PROXIMITY POINT THEOREMS FOR α + F, (θ − φ)-PROXIMAL CONTRACTION ABDELKARIM KARI 1 AND MOHAMED ROSSAFI 2* Abstract. In this paper, inspired by the idea of Suzuki type α + F -proximal contrac- tion in metric spaces, we prove a new existence of best proximity point for Suzuki type α + F -proximal contraction and α + (θ - φ)-proximal contraction defined on a closed subset of a complete metric space. Our theorems extend, generalize, and improve many existing results. Keywords: proximity point, α + F -proximal contraction, α + (θ - φ)-proximal contrac- tion. 1. Introduction and preliminaries Best proximity point theorem analyses the condition under which the optimisation problem, namely, inf x∈A d(x,Tx), has a solution. The point x is called the best prox- imity (BPP (T ) of T : A → B, if d(x,Tx)= d(A,B), where {d(A,B) = inf d(x,y): x ∈ A,y ∈ B}. Note that the best proximity point reduces to a fixed point if T is a self-mapping. Sankar Raj [4] and Zhang et al. [5] defined the notion of P -property and weak P - property respectively. Hussain et al. [2] defined the concept of α + -proximal admissible for non self mapping and introduced Suzuki typeα + ψ- proximal contraction to gener- alize several best proximity results and obtained some best proximity point theorems for self-mappings. Definition 1.1. [1]. Let (A,B) be a pair of non empty subsets of a metric space (X,d). We adopt the following notations: d(A,B)= {inf d (a,b): a ∈ A,b ∈ B}; A 0 = { a ∈ A there exists b ∈ A such that d (a,b)= d (A,B)}; 11 Laboratory of Algebra, Analysis and Applications Faculty of Sciences Ben M’Sik, Hassan II Uni- versity, B.P. 7955 Casablanca, Morocco 21 email address:abdkrimkariprofes@gmail.com 32 LaSMA Laboratory Department of Mathematics Faculty of Sciences, Dhar El Mahraz University Sidi Mohamed Ben Abdellah, B. P. 1796 Fes Atlas, Morocco 42 email address:rossafimohamed@gmail.com; mohamed.rossafi@usmba.ac.ma 1 181 Abstract Keywords: 1. Introduction and preliminaries 10.20956/j.v18i2.17994