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On Diviccaro, Fisher and Sessa open questions

Ljubomir B. Ćirić — 1993

Archivum Mathematicum

Let K be a closed convex subset of a complete convex metric space X and T , I : K K two compatible mappings satisfying following contraction definition: T x , T y ) ( I x , I y ) + ( 1 - a ) max { I x . T x ) , I y , T y ) } for all x , y in K , where 0 < a < 1 / 2 p - 1 and p 1 . If I is continuous and I ( K ) contains [ T ( K ) ] , then T and I have a unique common fixed point in K and at this point T is continuous. This result gives affirmative answers to open questions set forth by Diviccaro, Fisher and Sessa in connection with necessarity of hypotheses of linearity and non-expansivity of I in their Theorem [3]...

A new class of nonexpansive type mappings and fixed points

Ljubomir B. Ćirić — 1999

Czechoslovak Mathematical Journal

In this paper a new class of self-mappings on metric spaces, which satisfy the nonexpensive type condition (3) below is introduced and investigated. The main result is that such mappings have a unique fixed point. Also, a remetrization theorem, which is converse to Banach contraction principle is given.

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