Abstract
Abstract In this paper we define a new class of multivalued generalized contractions on cone metric spaces. Then, by using a necessary new technique, we prove two common fixed point theorems for a pair of those mappings on complete cone metric spaces over solid, not necessarily normal cone. Our main theorems are generalizations of the theorem of Wardowski (Appl. Math. Lett. 24:275-278, 2011) and many existing theorems in the literature. By using our main theorems, we can obtained some important corollaries which are generalizations of the well-known metric fixed point theorems to setting of cone metric spaces over a solid non-normal cone. MSC:47H10, 54H25.
Highlights
Introduction and preliminariesThere exist many generalizations of the concept of metric spaces in the literature
Huang and Zhang [ ] reintroduced cone metric spaces and defined the convergence via interior points of the cone which determines an order on E. They considered and proved several fixed point theorems only in cone metric spaces over a normal cone, their approach enables the investigation of cone metric spaces over a cone which is not necessarily normal
In the present paper we will introduce the concept of a generalized multivalued contraction on cone metric spaces and using a new technique of proof, we prove two common fixed point theorems for a pair of those multivalued mappings on cone metric spaces over solid non-normal cones
Summary
It is easy to show that the generalized multivalued contraction defined in Definition . Is an example of the cone generalized multivalued contraction defined in Definition. If each right hand side of (C ), (C ), (C ), and (C ) with x = x and y = x is θ in E, d(x , x ) = θ and from the property (d ) of the metric d it follows x = x This and x ∈ Sx imply x ∈ Sx. Further, d(x , u) = d(x , u) = θ for each fixed u ∈ Tx implies x = u ∈ Tx = Tx. x ∈ Tx. x ∈ Tx In this case x is a common fixed point of S and T and proof is done.
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