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Fixed point theorems using diameters of level sets and the notion of R.G.I. mapping

Akkouchi M.

Demonstratio Mathematica

2002

Vol. 35, nr 3

621-628

In this paper, we generalize and improve a recent result established by W. A. Kirk and L. M. Saliga (see [3], Theorem 4.3, p. 149) by using a result of W. Walter (see [4]). Indeed, We prove that the conclusions of Kirk-Saliga theorem are not only satisfied but, moreover, they are equivalent for a wide class of contractive gauge functions. Our main result contains also a new equivalent conclusion (see Property (4) in Theorem 2.2 below). As a consequence, we recapture (and improve the results of) a theorem proved by M. Angrisani and M. Clavelli in [2]. Theorem 2.2 completes and improves the main result of[1].