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1
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Isometric embedding into spaces of continuous functions

100%
Villa R.
Studia Mathematica
|
1998
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tom 129
|
nr 3
197-205
EN
We prove that some Banach spaces X have the property that every Banach space that can be isometrically embedded in X can be isometrically and linearly embedded in X. We do not know if this is a general property of Banach spaces. As a consequence we characterize for which ordinal numbers α, β there exists an isometric embedding between $C_0(α+1)$ and $C_0(β+1)$.
2
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A new Kantorovich-type theorem for Newton's method

100%
Argyros I. K.
Applicationes Mathematicae
|
1999
|
tom 26
|
nr 2
151-157
EN
A new Kantorovich-type convergence theorem for Newton's method is established for approximating a locally unique solution of an equation F(x)=0 defined on a Banach space. It is assumed that the operator F is twice Fréchet differentiable, and that F', F'' satisfy Lipschitz conditions. Our convergence condition differs from earlier ones and therefore it has theoretical and practical value.
3
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An almost nowhere Fréchet smooth norm on superreflexive spaces

100%
Matoušková E.
Studia Mathematica
|
1999
|
tom 133
|
nr 1
93-99
EN
Every separable infinite-dimensional superreflexive Banach space admits an equivalent norm which is Fréchet differentiable only on an Aronszajn null set.
4
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Approximate multi-Jensen-cubic mappings and a fixed point theorem

100%
Ramzanpour E., Bodaghi A.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2020
|
tom 19
141-154
EN
In this paper, we introduce multi-Jensen-cubic mappings and unify the system of functional equations defining the multi-Jensen-cubic mapping to a single equation. Applying a fixed point theorem, we establish the generalized Hyers-Ulam stability of multi-Jensen-cubic mappings. As a known outcome, we show that every approximate multi-Jensen-cubic mapping can be multi-Jensen-cubic.
5
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On sequential convergence in weakly compact subsets of Banach spaces

100%
Marciszewski W.
Studia Mathematica
|
1994-1995
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tom 112
|
nr 2
189-194
EN
We construct an example of a Banach space E such that every weakly compact subset of E is bisequential and E contains a weakly compact subset which cannot be embedded in a Hilbert space equipped with the weak topology. This answers a question of Nyikos.
6
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Invariant Means on Banach Spaces

100%
Łukasik R.
Annales Mathematicae Silesianae
|
2017
|
tom 31
|
nr 1
127-140
EN
In this paper we study some generalization of invariant means on Banach spaces. We give some sufficient condition for the existence of the invariant mean and some examples where we have not it.
7
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Schauder's fixed-point theorem in approximate controllability problems

88%
Babiarz A., Klamka J., Niezabitowski M.
International Journal of Applied Mathematics and Computer Science
|
2016
|
tom 26
|
nr 2
263-275
EN
The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder's fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.
8
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Asymptotic behaviour of solutions of difference equations in Banach spaces

88%
Kisiołek A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2008
|
tom 28
|
nr 1
5-13
EN
In this paper we consider the first order difference equation in a Banach space $Δx_{n} = ∑_{i=0}^∞ a^{i}_{n} f(x_{n+i})$. We show that this equation has a solution asymptotically equal to a. As an application of our result we study the difference equation $Δx_{n} = ∑_{i=0}^∞ a^i_{n}g(x_{n+i}) + ∑_{i=0}^∞ b^{i}_{n}h(x_{n+i}) + y_{n}$ and give conditions when this equation has solutions. In this note we extend the results from [8,9]. For example, in [9] the function f is a real Lipschitz function. We suppose that f has values in a Banach space and satisfies some conditions with respect to the measure of noncompactness and measure of weak noncompactness.
9
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The effect of rounding errors on a certain class of iterative methods

88%
Argyros I. K.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 3
369-375
EN
In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newton-like methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitz-type hypotheses on the mth Fréchet derivative (m ≥ 2 an integer) instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.
10
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Approximation of abstract linear integrodifferential equations

88%
Oka H., Tanaka N.
Studia Mathematica
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2000
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tom 140
|
nr 1
1-14
EN
This paper is devoted to the approximation of abstract linear integrodifferential equations by finite difference equations. The result obtained here is applied to the problem of convergence of the backward Euler type discrete scheme.
11
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Nonseparability of the quotient space cabv(∑,m;X)/L¹(m;X) for Banach spaces X without the Radon-Nikodym property

88%
Drewnowski L.
Studia Mathematica
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1993
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tom 104
|
nr 2
125-132
EN
It is shown that if (S,∑,m) is an atomless finite measure space and X is a Banach space without the Radon-Nikodym property, then the quotient space cabv(∑,m;X)/L¹(m;X) is nonseparable.
12
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Local convergence comparison between two novel sixth order methods for solving equations

88%
George S., Argyros I. K.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2019
|
tom 18
5-19
EN
The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.
13
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Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

88%
Yantir A., Kubiaczyk I., Sikorska-Nowak A.
Open Mathematics
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2015
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tom 13
|
nr 1
EN
In this paper, we present the existence result for Carathéodory type solutions for the nonlinear Sturm- Liouville boundary value problem (SLBVP) in Banach spaces on an arbitrary time scale. For this purpose, we introduce an equivalent integral operator to the SLBVP by means of Green’s function on an appropriate set. By imposing the regularity conditions expressed in terms of Kuratowski measure of noncompactness, we prove the existence of the fixed points of the equivalent integral operator. Mönch’s fixed point theorem is used to prove the main result. Finally, we also remark that it is straightforward to guarantee the existence of Carathéodory solutions for the SLBVP if Kuratowski measure of noncompactness is replaced by any axiomatic measure of noncompactness.
14
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Local convergence comparison between two novel sixth order methods for solving equations

88%
George S., Argyros I. K.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2019
|
tom 18
15
Content available remote

Vector-valued holomorphic and harmonic functions

88%
Arendt W.
Concrete Operators
|
2016
|
tom 3
|
nr 1
68-76
EN
Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.
16
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Subexponential Solutions of Linear Volterra Difference Equations

88%
Bohner M., Sultana N.
Nonautonomous Dynamical Systems
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2015
|
tom 2
|
nr 1
EN
We study the asymptotic behavior of the solutions of a scalar convolution sum-difference equation. The rate of convergence of the solution is found by determining the asymptotic behavior of the solution of the transient renewal equation.
17
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A Representation Theorem for \(\varphi\)-Bounded Variation of Functions in the Sense of Riesz

88%
Aziz W., Leiva H., Merentes N., Rzepka B.
Commentationes Mathematicae
|
2010
|
tom 50
|
nr 2
EN
In this paper we extend the well known Riesz lemma to the class of bounded \(\varphi\)-variation functions in the sense of Riesz defined on a rectangle \(I_a^b\subset \mathbb{R}^2\). This concept was introduced in [2], where the authors proved that the space \(BV_\varphi^R (I_a^b;\mathbb{R}\) of such functions is a Banach Algebra. Moreover, they characterized also the Nemytskii operator acting in this space. Thus our result creates a continuation of the paper [2].
18
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Some notes to existence and stability of the positive periodic solutions for a delayed nonlinear differential equations

75%
Dorociaková B., Olach R.
Open Mathematics
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2016
|
tom 14
|
nr 1
361-369
EN
The paper deals with the existence of positive ω-periodic solutions for a class of nonlinear delay differential equations. For example, such equations represent the model for the survival of red blood cells in an animal. The sufficient conditions for the exponential stability of positive ω-periodic solution are also considered.
19
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Existence of solutions of the dynamic Cauchy problem on infinite time scale intervals

75%
Kubiaczyk I., Sikorska-Nowak A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2009
|
tom 29
|
nr 1
113-126
EN
In the paper, we prove the existence of solutions and Carathéodory's type solutions of the dynamic Cauchy problem $x^Δ(t) = f(t,x(t))$, t ∈ T, x(0) = x₀, where T denotes an unbounded time scale (a nonempty closed subset of R and such that there exists a sequence (xₙ) in T and xₙ → ∞) and f is continuous or satisfies Carathéodory's conditions and some conditions expressed in terms of measures of noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. The results presented in the paper are new not only for Banach valued functions, but also for real-valued functions.
20
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On a quadratically convergent method using divided differences of order one under the gamma condition

75%
Argyros I., Ren H.
Open Mathematics
|
2008
|
tom 6
|
nr 2
262-271
EN
We re-examine a quadratically convergent method using divided differences of order one in order to approximate a locally unique solution of an equation in a Banach space setting [4, 5, 7]. Recently in [4, 5, 7], using Lipschitz conditions, and a Newton-Kantorovich type approach, we provided a local as well as a semilocal convergence analysis for this method which compares favorably to other methods using two function evaluations such as the Steffensen’s method [1, 3, 13]. Here, we provide an analysis of this method under the gamma condition [6, 7, 19, 20]. In particular, we also show the quadratic convergence of this method. Numerical examples further validating the theoretical results are also provided.
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