In this paper we introduce the concept of cone metric spaces with Banach algebras, replacing Banach spaces by Banach algebras as the underlying spaces of cone metric spaces. With this modification, we shall prove some fixed point theorems of generalized Lipschitz mappings with weaker conditions on generalized Lipschitz constants. An example shows that our main results concerning the fixed point...

In the theory of Banach spaces a rather small class of spaces has always played a central role (actually even before the formulation of the general theory). This class —the class of classical Banach spaces— contains the Lp (p) spaces (p a measure, 1 < p < °°) and the C(K) spaces (K compact Hausdorff) and some related spaces. These spaces are very important in various applications of Banach spac...

A twisted sum Z of two Banach spaces X and Y is defined (see [11]) through a short exact sequence: 0 −→ X −→ Z −→ Y −→ 0. These short exact sequences in the category of (quasi-)Banach spaces are considered naturally in the investigation of three space properties (a property P in the category of quasi-Banach spaces is called a three space property if for every short exact sequence as above, Z ha...

In this paper, we give characterizations of retro Banach frames in Banach spaces. The notion of almost exact retro Banach frame is defined and a characterization of retro Banach frame has been given. Also results exhibiting relationship between frames, almost exact retro Banach frames and Riesz bases has been proved. Finally, we give some perturbation results of retro Banach frames and an almos...

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for e...

Equilibrium problems have many uses in optimization theory and convex analysis and which is why different methods are presented for solving equilibrium problems in different spaces, such as Hilbert spaces and Banach spaces. The purpose of this paper is to provide a method for obtaining a solution to the equilibrium problem in Banach spaces. In fact, we consider a hybrid proximal point algorithm...

We characterize those classes C of separable Banach spaces admitting a separable universal space Y (that is, a space Y containing, up to isomorphism, all members of C) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class NU of non-universal separable Banach spaces is strongly bounded. This settles in the aff...

Following results of Bourgain and Gorelik we show that the spaces lp, 1 < p < ∞, as well as some related spaces have the following uniqueness property: If X is a Banach space uniformly homeomorphic to one of these spaces then it is linearly isomorphic to the same space. We also prove that if a C(K) space is uniformly homeomorphic to c0, then it is isomorphic to c0. We show also that there are B...

A projectional skeleton in a Banach space is a σ-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the ...