Properties of Chmielinski-orthogonality using Kadets-Klee property

The aim of this paper is to study new results of an approximate orthogonality of Birkhoff-James techniques in real Banach space , namely Chiemelinski orthogonality (even there is no ambiguity between the concepts symbolized by orthogonality) and provide some new geometric characterizations which is considered as the basis of our main definitions. Also, we explore relation between two different types of orthogonalities. First of them orthogonality in a real Banach space and the other orthogonality in the space of bounded linear operator . We obtain a complete characterizations of these two orthogonalities in some types of Banach spaces such as strictly convex space, smooth space and reflexive space. The study is designed to give different results about the concept symmetry of Chmielinski-orthogonality for a compact linear operator on a reflexive, strictly convex Banach space having Kadets-Klee property by exploring a new type of a generalized some results with Birkhoff James orthogonality in the space of bounded linear operators. We also exhibit a smooth compact linear operator with a spectral value that is defined on a reflexive, strictly convex Banach space having Kadets-Klee property either having zero nullity or not -right-symmetric.

Strong Convergence of an Inertial Iterative Algorithm for Generalized Mixed Variational-like Inequality Problem and Bregman Relatively Nonexpansive Mapping in Reflexive Banach Space

In this paper, we consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems and also prove some properties for the solution set of generalized mixed variational-like inequality problem. Furthermore, we introduce and study an inertial hybrid iterative method for solving the generalized mixed variational-like inequality problem involving Bregman relatively nonexpansive mapping in Banach space. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational examples to emphasize the efficiency and relevancy of the main result.

Existence of common fixed points of non-linear semigroups of Enriched Kannan contractive mappings

In this article, we consider the non-linear semigroup of \textit{enriched Kannan} contractive mapping and prove the existence of common fixed point on a non-empty closed convex subset $\mathcal C$ of a real Banach space $\mathscr X$, having uniformly normal structure.

Multiobjective Convex Optimization in Real Banach Space

In this paper, we consider convex multiobjective optimization problems with equality and inequality constraints in real Banach space. We establish saddle point necessary and sufficient Pareto optimality conditions for considered problems under some constraint qualifications. These results are motivated by the symmetric results obtained in the recent article by Cobos Sánchez et al. in 2021 on Pareto optimality for multiobjective optimization problems of continuous linear operators. The discussions in this paper are also related to second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones due to Mishra and Wang in 2005. Further, we establish Karush–Kuhn–Tucker optimality conditions using saddle point optimality conditions for the differentiable cases and present some examples to illustrate our results. The study in this article can also be seen and extended as symmetric results of necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds by Ruiz-Garzón et al. in 2019.

Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization

In this paper we investigate quasi equilibrium problems in a real Banach space under the assumption of Brezis pseudomonotonicity of the function involved. To establish existence results under weak coercivity conditions we replace the quasi equilibrium problem with a sequence of penalized usual equilibrium problems. To deal with the non compact framework, we apply a regularized version of the penalty method. The particular case of set-valued quasi variational inequalities is also considered.

Siegel domains over Finsler symmetric cones

Let Ω be a proper open cone in a real Banach space V. We show that the tube domain V ⊕ i ⁢ Ω {V\oplus i\Omega} over Ω is biholomorphic to a bounded symmetric domain if and only if Ω is a normal linearly homogeneous Finsler symmetric cone, which is equivalent to the condition that V is a unital JB-algebra in an equivalent norm and Ω is the interior of { v 2 : v ∈ V } {\{v^{2}:v\in V\}} .

REVISITING THE RECTANGULAR CONSTANT IN BANACH SPACES

Let X be a real Banach space. The rectangular constant $\mu (X)$ and some generalisations of it, $\mu _p(X)$ for $p \geq 1$ , were introduced by Gastinel and Joly around half a century ago. In this paper we make precise some characterisations of inner product spaces by using $\mu _p(X)$ , correcting some statements appearing in the literature, and extend to $\mu _p(X)$ some characterisations of uniformly nonsquare spaces, known only for $\mu (X)$ . We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning $\mu (l_p)$ and $\mu _p(l_p)$ .

Remarks on a recent paper titled: “On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces”

In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition $0<\kappa < \frac{1}{\sqrt{2}}$ 0 < κ < 1 2 , as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is $E=l_{p}$ E = l p , $2\leq p<\infty $ 2 ≤ p < ∞ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition $0<\kappa \leq \frac{1}{\sqrt{2}}$ 0 < κ ≤ 1 2 , then E is necessarily $l_{2}$ l 2 , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.

An algorithm for approximating a common solution of some nonlinear problems in Banach spaces with an application

In this paper, we construct a new Halpern-type subgradient extragradient iterative algorithm. The sequence generated by this algorithm converges strongly to a common solution of a variational inequality, an equilibrium problem, and a J-fixed point of a continuous J-pseudo-contractive map in a uniformly smooth and two-uniformly convex real Banach space. Also, the theorem is applied to approximate a common solution of a variational inequality, an equilibrium problem, and a convex minimization problem. Moreover, a numerical example is given to illustrate the implementability of our algorithm. Finally, the theorem proved complements, improves, and unifies some related recent results in the literature.

STRONG CONVERGENCE THEOREM FOR UNIFORMLY L-LIPSCHITZIAN MAPPING OF GREGUS TYPE IN BANACH SPACES

In this paper, we introduced a new mapping called Uniformly L-Lipschitzian mapping of Gregus type, and used the Mann iterative scheme to approximate the fixed point. A Strong convergence result for the sequence generated by the scheme is shown in real Banach space. Our result generalized and unifybmany recent results in this area of research. In addition, using Java(jdk1.8.0_101), we give a numericalbexample to support our claim.