Cauchy's theorem in Banach spaces.

On Peano’s Theorem in Banach Spaces

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 → ...

Kreps-yan Theorem for Banach Ideal Spaces

Let C be a closed convex cone in a Banach ideal space X on a measurable space with a σ-finite measure. We prove that conditions C ∩ X+ = {0} and C ⊃ −X+ imply the existence of a strictly positive continuous functional on X , whose restriction to C is non-positive. Let (Ω,F ) be a measurable space, which is complete with respect to a measure (that is, a countably-additive function) μ : F 7→ [0,∞...

On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings

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...

Strong convergence theorem for finite family of m-accretive operators in Banach spaces

The purpose of this paper is to propose a compositeiterative scheme for approximating a common solution for a finitefamily of m-accretive operators in a strictly convex Banach spacehaving a uniformly Gateaux differentiable norm. As a consequence,the strong convergence of the scheme for a common fixed point ofa finite family of pseudocontractive mappings is also obtained.

A Mean Ergodic Theorem For Asymptotically Quasi-Nonexpansive Affine Mappings in Banach Spaces Satisfying Opial's Condition