Fixed point and endpoint theorems of multivalued mappings in convex $ b $-metric spaces with an application

Author:

Ji Dong1,Yu Yao1,Li Chaobo2

Affiliation:

1. Department of Mathematics, Harbin University, Harbin 150086, China

2. Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, China

Abstract

<abstract><p>In this paper, we investigated several new fixed points theorems for multivalued mappings in the framework $ b $-metric spaces. We first generalized $ S $-iterative schemes for multivalued mappings to above spaces by means of a convex structure and then we developed the existence of fixed points and approximate endpoints of the multivalued contraction mappings using iteration techniques. Furthermore, we introduced the modified $ S $-iteration process for approximating a common endpoint of a multivalued $ \alpha_{s} $-nonexpansive mapping and a multivalued mapping satisfying conditon $ (E^{'}) $. We also showed that this new iteration process converges faster than the $ S $-iteration process in the sense of Berinde. Some convergence results for this iterative procedure to a common endpoint under some certain additional hypotheses were proved. As an application, we applied the $ S $-iteration process in finding the solution to a class of nonlinear quadratic integral equations.</p></abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Reference39 articles.

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2. S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5–11.

3. M. Gromov, Metric structure for riemannian and non-riemannian spaces, In: Progr. Math., Birkhauser, Boston, 152 (1984).

4. J. Markin, A fixed point theorem for set valued mappings, Bull. Amer. Math. Soc., 74 (1968), 639–640. https://doi.org/10.1090/S0002-9904-1968-11971-8

5. S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475–488. https://doi.org/10.2140/pjm.1969.30.475

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