Huang and Zhang 1 reintroduced the notion of cone metric spaces and established fixed point theorems for mappings on this space. After that, many fixed point theorems have been proved in normal or nonnormal cone metric spaces by some authors see e.g. 1–26 and references contained therein . We need to recall some basic notations, definitions, and necessary results from literature. Let R be the s...

The Lipschitz function algebras were first defined in the 1960s by some mathematicians, including Schubert. Initially, the Lipschitz real-value and complex-value functions are defined and quantitative properties of these algebras are investigated. Over time these algebras have been studied and generalized by many mathematicians such as Cao, Zhang, Xu, Weaver, and others. Let  be a non-emp...

[Perov, A. I., On Cauchy problem for a system of ordinary diferential equations, (in Russian), Priblizhen. Metody Reshen. Difer. Uravn., 2 (1964), 115-134] used the concept of vector valued metric space and obtained a Banach type fixed point theorem on such a complete generalized metric space. In this article we study fixed point results for the new extensions of Banach’s contraction principle ...

Unless we say otherwise, every vector space we talk about is taken to be over C. A Banach algebra is a Banach space A that is also an algebra satisfying ‖AB‖ ≤ ‖A‖ ‖B‖ for A,B ∈ A. We say that A is unital if there is a nonzero element I ∈ A such that AI = A and IA = A for all A ∈ A, called a identity element. If X is a Banach space, let B(X) denote the set of bounded linear operators X → X, and...

In this paper we introduce the cone bounded linear mapping and demonstrate a proof to show that the cone norm is continuous. Among other things, we prove the open mapping theorem and the closed graph theorem in TVS-cone normed spaces. We also show that under some restrictions on the cone, two cone norms are equivalent if and only if the topologies induced by them are the same. In the sequel, we...

In this paper, the notion of extended cone \(b\)-metric space is introduced, established structure open ball and defined convergence a sequence. Finally, restructured Banach Kannan contraction theorems without normality condition in new setting.

In this paper, using the fixed point theory in cone metric spaces, we prove the existence of a unique solution to a first-order ordinary differential equation with periodic boundary conditions in Banach spaces admitting the existence of a lower solution.

Abstract Let ? be a proper open cone in real Banach space V . We show that the tube domain V ? i ? ? {V\oplus i\Omega} over is biholomorphic to bounded symmetric if and only normal linearly homogeneous Finsler cone, which equiva...

In this paper, we prove some properties of algebraic cone metric spaces and introduce the notion of algebraic distance in an algebraic cone metric space. As an application, we obtain some famous fixed point results in the framework of this algebraic distance.