in this paper, a vector version of the intermediate value theorem is established. the main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{on fixed point theorems for monotone increasing vector valued mappings via scalarizing}, positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

As a cornerstone of functional analysis, Hahn–Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications o...

We show that if X is a separable Banach space (or more generally a Banach with an infinite-dimensional separable quotient) then there is a continuous mapping f : X → X such that the autonomous differential equation x′ = f(x) has no solution at any point. In order to put our results into context, let us start by formulating the classical theorem of Peano. Theorem 1. (Peano) Let X = R, f : R×X → ...

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael’s Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse o...

1. The norm || • || in a Banach algebra A is said to be minimal [l ] if, given any other norm || •||1 in A (with respect to which A need not be complete), the condition ||a||iá||a|| for each oG^4 implies that ||a[|i = ||a||. We shall say that || •|| is absolutely minimal if, given any other norm ||-||i whatever in A, then ||a||iè||a|| for each aEA. An absolutely minimal norm is of course minima...

Let C be a nonempty closed convex subset of a smooth Banach space E and let A be an accretive operator of C into E. We first introduce the problem of finding a point u ∈ C such that 〈Au, J(v− u)〉 ≥ 0 for all v ∈ C, where J is the duality mapping of E. Next we study a weak convergence theorem for accretive operators in Banach spaces. This theorem extends the result by Gol’shteı̆n and Tret’yakov i...

In this paper, we first introduce a class of nonlinear mappings called generic generalized nonspreading which contains the class of generalized nonspreading mappings in a Banach space and then prove an attractive point theorem for such mappings in a Banach space. Furthermore, we prove a mean convergence theorem of Baillon’s type and a weak convergence theorem of Mann’s type for such nonlinear m...

The generalized Roper-Suffridge extension operator in Banach spaces is introduced. We prove that this operator preserves the starlikeness on some domains in Banach spaces and does not preserve convexity in some cases. Furthermore, the growth theorem and covering theorem of the corresponding mappings are given. Some results of Roper and Suffridge and Graham et al. in Cn are extended to Banach sp...

Let A and B be two infinite sets of non-negative integers. Similar to Kneser’s Theorem (Theorem 1.1 bellow) we characterize the structure of A + B when the upper Banach density of A + B is less than the sum of the upper Banach density of A and the upper Banach density of B.