Fixed points for multivalued convex contractions on Nadler sense types in a geodesic metric space (Q2310967)

Summary: In [Pac. J. Math. 30, 475--488 (1969; Zbl 0187.45002)], based on the concept of the Hausdorff metric, \textit{S. B. Nadler jun.} introduced the notion of multivalued contractions. He demonstrated that, in a complete metric space, a multivalued contraction possesses a fixed point. Later on, Nadler's fixed point theorem was generalized by many authors in different ways. Using a method given by \textit{M. Angrisani} and \textit{M. Clavelli} [Ann. Mat. Pura Appl. (4) 170, 1--12 (1996; Zbl 0870.54046)] and \textit{S. Mureşan} [``The compactness of the fixed points set for multivalued mappings'', in: Proceedings of the 27th annual congress of American Romanian academy of arts and sciences (ARA), Volume II, Oradea, Romania, 29 May--2 June 2002. Montréal, QC.: Polytechnic International Press. 764--765 (2002)], we prove in this paper that, for a class of convex multivalued left A-contractions in the sense of Nadler and the right A-contractions with a convex metric, the fixed points set is non-empty and compact. In this paper we present the fixed point theorems for convex multivalued left A-contractions in the sense of Nadler and right A-contractions on the geodesic metric space. Our results are particular cases of some general theorems, to the multivalued left A-contractions in the sense of Nadler and right A-contractions, and particular cases of the results given by \textit{I. A. Rus} [Principles and applications of fixed point theory. Cluj-Napoca, Romania: Dacia Publishing House (1979); Fixed Point Theory 9, No. 2, 541--559 (2008; Zbl 1172.54030)], \textit{S. B. Nadler jun.} [Pac. J. Math. 30, 475--488 (1969; Zbl 0187.45002)], \textit{S. Mureşan} [loc.\,cit.; Fixed Point Theory 5, No. 1, 87--95 (2004; Zbl 1064.47054)], \textit{A. Bucur} et al. [ibid. 10, No. 1, 19--34 (2009; Zbl 1194.54056)], \textit{I.-R. Petre} and \textit{M. Bota} [Publ. Math. 83, No. 1--2, 139--159 (2013; Zbl 1289.54145)], and are applicable in many fields, such as economy, management, society, biology, ecology, etc.