The aim of this paper is to give several well-known results of Banach spaces possessing, the Radon-Nikodym property. It is interesting to see how this property can be described in terms of Asplund spaces, spaces admitting KKP, and rotund norm. We study nearly about an open problem [1] mentioned below.
This paper generalizes the special case of the Carlsson orthogonality in terms of the 2-HH norm in real normed linear space. Dragomir and Kikianty (2010) proved in their paper that the Pythagorean orthogonality is unique in any normed linear space, and isosceles orthogonality is unique if and only if the space is strictly convex. This paper deals with the complete proof of the uniqueness of the new orthogonality through the medium of the 2-HH norm. We also proved that the Birkhoff and Robert orthogonality via the 2-HH norm are equivalent, whenever the underlying space is a real inner-product space.
In 2014, Asadi et al.1 introduced the notion of an M−metric space which is the generalization of a partial metric space and establish Banach and Kannan fixed point theorems in M− metric space. In this paper, we give a brief survey regarding the fixed point theorem for Chatterjea contraction mapping in the framework of M−metric space. We also give some examples which support the partial answers to the question posed by Asadi et al. concerning a fixed point for Chatterjea contraction mapping.
This paper deals with a generalization of orthogonality in terms of bounded linear operators on a Banach space. The goal is to find a relation between orthogonality of images and orthogonality of elements. We prove that if the images of a bounded linear operator are orthogonal in the Pythagorean sense, then the elements are orthogonal in the sense of Birkhoff's definition. In the case of Robert's orthogonality in terms of bounded linear operators under the restriction that any element belongs to the intersection of the norm attainment set of T 1 + \lambda T 2 and T 1 -\lambda T 2 , if the images are orthogonal, then it implies that the operators are also orthogonal. Furthermore, some results in relation to the Carlsson, isosceles, and approximate Birkhoff-James orthogonality have been obtained. У цiй роботi розглядяється узагальнення ортогональностi в термiнах обмежених лiнiйних операторiв на банаховому просторi. Метою роботи є знайти спiввiдношення мiж ортогональнiстю зображень i ортогональнiстю елементiв. Доведено, що якщо образи обмеженого лiнiйного оператора ортогональнi в пiфагоровому сенсi, то елементи є ортогональний у сенсi визначення Бiркгофа. У випадку ортогональностi Роберта в термiнах обмежених лiнiйних операторiв iз умовою, що будь-який елемент належить до перетину множин де оператори T 1 + \lambda T 2 i T 1 -\lambda T 2 досягають норми, з ортогональностi образiв випливає, що оператори є також ортогональнi. Також отримано деякi результати про ортогональнiсть в сенсi Карлссона, рiвнобiчної ортогональностi та наближеної ортогональностi в сенсi Бiркгофа-Джеймса.
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