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Let A be a Banach algebra. An element a ϵ A has generalized Drazin inverse if there exists b ϵ A such that b = bab, ab = ba, a - a²b ϵ Aqnil. New additive results for the generalized Drazin inverse of an operator over a Banach space are presented. we extend the main results of a paper of Shakoor, Yang and Ali from 2013 and of Wang, Huang and Chen from 2017. Appling these results to 2 x 2 operator matrices we also generalize results of a paper of Deng, Cvetković-Ilić and Wei from 2010. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inertial self-adaptive projection method using Bregman distance techniques for solving pseudomonotone equilibrium problems in reflexive Banach spaces. The algorithm requires only one projection onto the feasible set without any Lipschitz-like condition on the bifunction. Using this method, a strong convergence theorem is proved under some mild conditions. Furthermore, we include numerical experiments to illustrate the behaviour of the new algorithm with respect to the Bregman function and other algorithms in the literature. [ABSTRACT FROM AUTHOR]

In this paper, starting with the geometric constants that can characterize Hilbert spaces, combined with the isosceles orthogonality of Banach spaces, the orthogonal geometric constant ΩX(α) is defined, and some theorems on the geometric properties of Banach spaces are derived. Firstly, this paper reviews the research progress of orthogonal geometric constants in recent years. Then, this paper explores the basic properties of the new geometric constants and their relationship with conventional geometric constants, and deduces the identity of ΩX(α) and X(α). Finally, according to the identities, the relationship between these the new orthogonal geometric constant and the geometric properties of Banach Spaces (such as uniformly non-squareness, smoothness, convexity, normal structure, etc.) is studied, and some necessary and sufficient conditions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study some fixed point properties of n-generalized Bregman nonspreading mappings in reflexive Banach space. We introduce a hybrid iterative scheme for finding a common solution for a countable family of equilibrium problems and fixed point problems in reflexive Banach space. Further, we give some applications and numerical example to show the importance and demonstrate the performance of our algorithm. The results in this paper extend and generalize many related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new geometric constants related to the heights of triangles: - Δ(X) and - Δ(X, I). We also propose two new geometric constants, - Δ(X) and - Δ(X), related to the midlines of equilateral triangles, and discuss the relation between the heights and midlines in equilateral triangles. We give estimates for these geometric constants in terms of other geometric parameters, and the geometric constants are used to discuss geometric properties such as uniform non-squareness, uniform normal structure, and the fixed point property. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iterative algorithm generated by a finite family of accretive operators in a q-uniformly smooth Banach space. We prove the strong convergence of the proposed iterative algorithm. The results presented in this paper are interesting extensions and improvements of known results of Qin et al. [Fixed Point Theory Appl. 2014(2014): 166], Kim and Xu [Nonlinear Anal. 61(2005), 51–60] and Benavides et al. [Math. Nachr. 248(2003), 62–71]. [ABSTRACT FROM AUTHOR]

In this paper, we present and prove new existence and uniqueness fixed point theorems for vector operators having a mixed monotone property in partially ordered product Banach spaces. Our results extend and improve existing works on T-φ-concave operators in the scalar case. As an application, we study the existence and uniqueness of positive solutions for systems of nonlinear Neumann boundary value problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of nonlinear variational type inclusions involving (H, η, φ)-monotone operators in real Banach spaces. Further, we define a proximal operator associated with an (H; η; φ)-monotone operator and show that it is single valued and Lipschitz continuous. Using proximal point operator techniques, we prove the existence and uniqueness of a solution and suggest an iterative algorithm for the system of nonlinear variational type inclusions. Furthermore, we discuss the convergence of the iterative sequences generated by the algorithms. [ABSTRACT FROM AUTHOR]

G-vector-valued sequence space frames and g-Banach frames for Banach spaces are introduced and studied in this paper. Also, the concepts of duality mapping and β-dual of a BK-space are used to define frame mapping and synthesis operator of these frames, respectively. Finally, some results regarding the existence of g-vector-valued sequence space frames and g-Banach frames are obtained. In particular, it is proved that if X is a separable Banach space and Y is a Banach space with a Schauder basis, then there exist a Y -valued sequence space Yv and a g-Banach frame for X with respect to Y and Yv. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of convergence of an iterative algorithm for a general system of variational inequalities, a nonexpansive mapping and an η-strictly pseudo-contractive mapping. Strong convergence theorems are established in the framework of real Banach spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove strong convergence theorem by a modified three step iterative process with errors for three nonexpansive mappings in the frame work of uniformly smooth Banach spaces. The main feature of this scheme is that its special cases can handle both strong convergence like Halpern type and weak convergence like Ishikawa type iteration schemes. Our result extend and generalize the result of S. H. Khan [10], Kim and Xu [12] and many other authors. [ABSTRACT FROM AUTHOR]

In this paper, we prove the generalized Hyers-Ulam stability of the following Jensen type functional equation f(x-y/n + z) + f(y-z/n + x) + f(z-x/n + y) = f(x) + f(y) + f(z) in p-Banach spaces for any fixed nonzero integer n. [ABSTRACT FROM AUTHOR]