On Best Proximity Point Results for Some Type of Mappings

Abstract

In this paper, we give new conditions for existence and uniqueness of a best proximity point for Geraghty- and Caristi-type mappings. The presented results are most valuable generalizations of the Geraghty and Caristi fixed point theorems.

On Fixed Point Results in Controlled Metric Spaces

“…The authors in [11] manifested that a Banach contraction is a specific case of F-contractions, while there are many F -contractions which need not be a Banach contraction. For more details, we refer the readers to ( [12][13][14][15][16][17][18][19][20][21][22]).…”

Some Results on Iterative Proximal Convergence and Chebyshev Center

“…The geometrical property, that is, the proximal normal structure, is the sufficient condition for the existence of the best proximity [1]. For details about the best proximity point, one can see research papers in [1,2,5,[15][16][17][18][19]. We now prove the following result, which shows that the above condition can be dropped if a reflexive Banach space satisfies Opial's condition.…”

On interpolative F-contractions with shrink map

“…Altun and Tasdemir [9] presented the study of best proximity points using interpolative proximal contraction inequalities. Along with the aforementioned studies, many other interesting studies on fixed point theory are available in [10][11][12][13][14][15][16][17][18][19][20][21][22][23]; they help readers to verify the existence of fixed points for self-mappings and best proximity points for nonself mappings. Jleli et al [23] introduced the concept of E-fixed point (also called φ-fixed point), which states that, for maps V : K → K and E : K → [0, ∞), a point k ∈ K is called E-fixed point of V : K → K if V (k) = k and E(k) = 0, and proved the existence of such points by using a single inequality involving both maps V and E. It is important to note that Jleli et al [23] used the lower semicontinuity of E. This use of the lower semicontinuity of E by Jleli et al [23] arises the question whether the condition of lower semicontinuity of E can be left and some other technique be adopted.…”