Tripled best proximity point in complete metric spaces

Abstract In this paper, we introduce a new type of contraction to seek the existence of tripled best proximity point results. Here, using the new contraction and P-property, we generalize and extend results of W. Shatanawi and A. Pitea and prove the existence and uniqueness of some tripled best proximity point results. Examples are also given to support our results.


Introduction
Fixed point theory has become the focus of many researchers and that is due the fact that it has many applications in di erent elds, such as physics, engineering, computer sciences, ..., etc,... However, sometimes maps do not have a xed point so the best we can do is to get the minimum "distance" of a input and its output, which it turns out to be very interesting and it has many applications such a point is called best proximity point. Introduction of coupled xed point by Guo and Lakshmikantham [1] in the year 1987 leads to the introduction of tripled xed point by Vasile Berinde and Marine Borcut [2]. After this we had seen many coupled and tripled xed point results on di erent spaces and under di erent contractions. B. Samet [3] proved some best proximity points theorems endowed with P-property. In [4] W. Shantanawi et. al. proved best proximity point and coupled best proximity point theorems. For more results on best proximity point and its application, readers can see research papers [5][6][7][8][9][10][11] and references therein.
W. Shantanawi et. al. [4] motivated us to introduce tripled best proximity point. In this paper, we proved some tripled best proximity point theorems and examples are also given.
Let A and B be any two nonempty subsets of a metric space (X, d). De ne

Let θ be a continuous function in Θ and ϕ be a comparison function satisfying
for all x, y, z, u, v, w ∈ X. Then (u, u, u) is the unique tripled best proximity point of F.

Similarly, d(v, F(v, u, v)) = d(A, B) and d(w, F(w, v, u)) = d(A, B).
Thus, (u, v, w) is a tripled best proximity point of F. Now, we show that u = v = w. Lastly, from (c) and using (3.1), we have (3.17) To prove the uniqueness, let t be another tripled best proximity point. Now, This completes the proof.
Then by following Theorem 3.1, we get that (u, u, u) is the unique tripled best proximity point.
Taking A = B in Theorem 3.1, we can get a triple xed point which is given below:   -1,1) is the unique tripled best proximity point but not of the form (u, u, u). This is because (3.1) is not satis ed. Therefore, by Theorem 3.1, we cannot get the results.

Conclusion
In closing, we would like to bring to the readers' attention that our results were proven in metric spaces. So, we can prove these results in partial metric spaces, metric like spaces, or M-metric spaces.