Best proximity point theorems for $ \alpha^{+} F, (\theta-\phi )$-proximal contraction

Mohamed Rossafi,Abdelkarim Kari

doi:10.20956/j.v18i2.17994

Abstract

In this paper, inspired by the idea of Suzuki type $ \alpha^{+} F$-proximal contraction in metric spaces, we prove a new existence of best proximity point for Suzuki type $ \alpha^{+} F$-proximal contraction and $ \alpha^{+} (\theta-\phi )$-proximal contraction defined on a closed subset of a complete metric space. Our theorems extend, generalize, and improve many existing results.

Highlights

In this paper, inspired by the idea of Suzuki type α+F -proximal contraction 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
Hussain et al [2] defined the concept of α+-proximal admissible for non self mapping and introduced Suzuki typeα+ψ- proximal contraction to generalize several best proximity results and obtained some best proximity point theorems for self-mappings
We introduce the following concept which is a α+F -proximal contraction and α+(θ, φ)-proximal contraction

Summary

By using the observations we can write

BEST PROXIMIATBYDPEOLINKTARTHIMEOKRAERMIS, MFOORHAα+MFE, (Dθ −RφO)-SPSRAOFXIIMAL CONTRACTION and. = max d(xn−1, xn)d(xn, xn+1) , d(xn−1, xn)d(xn, xn+1) , d(xn−1, xn) + d(xn, xn+1) + d(A, B) + d(A, B) − d(A, B) 2. F (d (xn, xn+1)) ≤ τ + F (d (T xn−1, T xn)) ≤ τ + F (d (T xn−1, T xn)) + α(xn−1, xn) ≤ F (M (xn−1, xn)) ≤ F (max {d(xn−1, xn), d(xn, xn+1)}). If max {d(xn−1, xn), d(xn, xn+1)} = d(xn, xn+1), F (d (T xn−1, T xn)) ≤ F (d(xn, xn+1) + τ < F (d(xn, xn+1). From the definition of the limit, there exists n0 ∈ N such that 1 d (xn, xn+1) ≤ nk , ∀n ≤ n0. (xn+1, z) for some n ∈ N, by using ( h) and definition of d∗ , we obtain the following contraction:. F (d (T xnk , T z)) + τ ≤ F (d (T xnk , T z)) + τ + α(xnk , z) ≤ F [M (xnk , z)]

Notice that

Assume that