Fixed point results for single and multi-valued three-points contractions
Mohamed Jleli, Evgeniy Petrov, Bessem Samet
TL;DR
This work addresses the existence of fixed points for three-point contractions in spaces equipped with three metrics by introducing single-valued and multivalued contraction frameworks governed by a $Φ$-function. For the single-valued case, a Picard-type iteration under the contractive inequality $d_1(Fx,Fy)+d_2(Fy,Fz)+d_3(Fz,Fx)\leq φ(d_1(x,y)+d_2(y,z)+d_3(z,x))$ with $φ\inΦ$ yields a fixed point, with at most two fixed points, in a complete $ (M,d_1)$ setting. The multivalued extension uses the Hausdorff metric $H$ and diameter $\mathcal{D}$ to prove the existence of a fixed point via a constructive sequence and continuity of $F$ into $(CB(M),H)$ for $λ\in(0,1)$. Together, the results generalize Banach/Nadler-type fixed points to triangle-perimeter contractions across triple-metric spaces and their multivalued analogs, enriching the fixed-point theory with new contraction notions and concrete examples.
Abstract
In this paper, we are concerned with the study of the existence of fixed points for single and multi-valued three-points contractions. Namely, we first introduce a new class of single-valued mappings defined on a metric space equipped with three metrics. A fixed point theorem is established for such mappings. The obtained result recovers that established recently by the second author [J. Fixed Point Theory Appl. 25 (2023) 74] for the class of single-valued mappings contracting perimeters of triangles. We next extend our study by introducing the class of multivalued three points contractions. A fixed point theorem, which is a multi-valued version of that obtained in the above reference, is established. Some examples showing the validity of our obtained results are provided.