A best proximity point theorem for relatively nonexpansive mappings in the absence of the proximal normal structure property. (English) Zbl 07884569
Summary: The well-known W. A. Kirk [Am. Math. Mon. 72, 1004–1006 (1965; Zbl 0141.32402)] fixed point theorem for nonexpansive mapping relies on the geometric notion called normal structure property. D. Göhde [Math. Nachr. 28, 45–55 (1964; Zbl 0139.31402)] provided sufficient conditions for the existence of a fixed point of a nonexpansive mapping without using normal structure property. In [A. A. Eldred et al., Stud. Math. 171, No. 3, 283–293 (2005; Zbl 1078.47013)], W. A. Kirk et al. introduced a notion called relatively nonexpansive mapping and provided sufficient conditions for the existence of best proximity points for such mappings using the proximal normal structure property. The main result of this manuscript provides the existence of best proximity points of a relatively nonexpansive mapping without using the proximal normal structure property. Also, our main result extends Göhde’s fixed point theorem in best proximity point setting. An example is given to illustrate our main result.
MSC:
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H10 | Fixed-point theorems |
Keywords:
Göhde’s fixed point theorem; cyclic contraction; relatively nonexpansive mappings; fixed points; best proximity pointsReferences:
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