Real Banach Space Research Articles

On Removability Properties of $$\psi $$ ψ -Uniform Domains in Banach Spaces

Suppose that E is a real Banach space with dimension at least 2. The main aim of this paper is to show that a domain D in E is a \(\psi \)-uniform domain if and only if \(D\backslash P\) is a \(\psi _1\)-uniform domain, and D is a uniform domain if and only if \(D\backslash P\) also is a uniform domain, whenever P is a countable subset of D satisfying a quasihyperbolic separation condition. This condition requires that the quasihyperbolic distance with respect to D between each pair of distinct points in P has a lower bound greater than or equal to \(\frac{1}{2}\).

Strong Convergence Theorem for Multiple Sets Split Feasibility Problems in Banach Spaces

ABSTRACTThe purpose of this article is to study the iterative approximation of solution to multiple sets split feasibility problems in p-uniformly convex real Banach spaces that are also uniformly smooth. We propose an iterative algorithm for solving multiple sets split feasibility problems and prove a strong convergence theorem of the sequence generated by our algorithm under some appropriate conditions in p-uniformly convex real Banach spaces that are also uniformly smooth.

Variable quasi-Bregman monotone sequences

We introduce a notion of variable quasi-Bregman monotone sequence which unifies the notion of variable metric quasi-Fejer monotone sequences and that of Bregman monotone sequences. The results are applied to analyze the asymptotic behavior of proximal iterations based on variable Bregman distance and of algorithms for solving convex feasibility problems in reflexive real Banach spaces.

Fundamental aspects of vector-valued Banach limits

This paper is divided into four parts. In the first we study the existence of vector- valued Banach limits and show that a real Banach space with a monotone Schauder basis admits vector- valued Banach limits if and only if it is -complemented in its bidual. In the second we prove two vector- valued versions of Lorentz' intrinsic characterization of almost convergence. In the third we show that the unit sphere in the space of all continuous linear operators from to which are invariant under the shift operator on cannot be obtained via compositions of surjective linear isometries with vector- valued Banach limits. In the final part we show that if enjoys the Krein–Milman property, then the set of vector- valued Banach limits is a face of the unit ball in the space of all continuous linear operators from to which are invariant under the shift operator on .

Complex Structures on Banach spaces with a subsymmetric basis

We prove that a real Banach space E with a subsymmetric basis and isomorphic to the sequence space E[E] has a unique complex structure. We also show that if a real Banach space X has the property P, then all its complex structures also satisfies P, when P is any of the following properties: bounded approximation property, G.L-l.u.st, being injective and being complemented in a dual space.

On liskovets approach for nonlinear ill-posed problems under arbitrarily perturbative operators

The purpose of this paper is to present a modification of Liskovets’s approach of solution of nonlinear ill-posed problems involving monotone operators in real reflexive Banach space under arbitrarily perturbative operators. The aspect of convergence of Tikhonov regularization is considered in combination with finite-dimensional approximations of the space.

Orthogonality of bounded linear operators

In this paper we explore the connection between strict convexity of a real normed linear space X and orthogonality of operators in the sense of Birkhoff–James in K(X), the space of all compact linear operators on X. We prove that a real reflexive Banach space X is strictly convex iff for any T,A∈K(X), T⊥BA⇒T⊥SBA or Ax=0 for some x∈SX with ‖Tx‖=‖T‖. We prove that if H is an infinite dimensional real Hilbert space and T∈K(H), then for all A∈B(H), A⊥BT⇒T⊥BA if and only if T is the zero operator. We also prove that for a real Hilbert space H, T⊥BA⇒A⊥BT for all A∈B(H) if and only if T is the zero operator.

A strong convergence theorem for finding a common fixed point of a finite family of Bregman nonexpansive mappings in Banach spaces which solves a generalized mixed equilibrium problem

In this paper, we study an iterative method for finding a common fixed point of a finite family of Bregman strongly nonexpansive mappings in real reflexive Banach spaces. Moreover, we prove a strong convergence theorem for finding common fixed points which solve a generalized mixed equilibrium problem. As a special case, we solve mixed variational inequality problems.

Extended new generalized mixed vector variational-like inequalities in reflexive Banach spaces

In this paper, we introduce and study an extended new generalized mixed vector variational-like inequality (ENGMVVLI) problem in real reflexive Banach spaces. This problem is more general than many existing equilibrium problems and variational inequalities available in the literature. We prove the existence results of (ENGMVVLI) by utilizing the KKM lemma and Nadlar’s results with relaxed semi- α-monotone mappings. Also, by using the Brouwer’s fixed point theorem, we prove the existence of solutions for (ENGMVVLI) without using the monotonicity concept. Finally, we give an example to justify the validity of our results.

Approximate and Near Weak Invariance for Nonautonomous Differential Inclusions

Let X be a real Banach space and I a nonempty interval. Let K:IźX$K:I\rightsquigarrow X$ be a multi-function with the graph K$\mathcal {K} $. We give here a characterization for K$\mathcal {K} $ to be approximate/near weakly invariant with respect to the differential inclusion xź(t)źF(t,x(t))$x^{\prime }(t)\in F(t, x(t))$ by means of an appropriate tangency concept and Lipschitz conditions on F. The tangency concept introduced in this paper extends in a natural way the quasi-tangency concept introduced by Cârjăź et al. (Trans Amer Math Soc. [2009];361:343---90) (see also Cârjăź et al. ([2007])). Viability, invariance and applications. Amsterdam: Elsevier Science B V) in the case when F is independent of t. As an application, we give some results concerning the set of solutions for the differential inclusion xź(t)źF(t,x(t))$x^{\prime }(t)\in F(t,x(t))$.

Harmonic Analysis of Nonquasianalytic Operators in a Real Banach Space

In a real Banach space, we consider linear bounded invertible operators such that the norms of their powers are nonquasianalytic. We obtain the condition of the existence of a nontrivial invariant subspace and the condition of decomposability (in the sense of Foia¸s) of such operators. Bibliography: 14 titles.

ON C-DELTA INTEGRAL OF BANACH SPACE VALUED FUNCTIONS ON TIME SCALES

Abstract. In this paper we introduce the Banach-valued C-deltaintegral on time scales and investigate some properties of these in-tegrals. 1. Introduction and preliminariesThe calculus on time scales was introduced for the rst time in 1988by Hilger[2] to unify the theory of di erence equations and the theoryof di erential equations. In 2012, Gwang Sik Eun, Ju Han Yoon, YoungKuk Kim and Byung Moo Kim introduced the C- integral on time scalesand inrestigated some properties of the integral. In this paper, we studythe Banach-valued C-delta integral on time scales. We prove that theC-integral and the strong C-integral are equivalent if and only if theBanach space is nite dimensional and F is the inde nite strong C-integral on time scales if and only if the C-variational VFis absolutelycontinuous on time scales.Throughout this paper, Xis a real Banach space with norm kkandits dual X. I denote the family of all subintervals of [a;b] T . A timescale Tis a nonempty closed subset of real number R with the subspacetopology inherited from the standard topology of R. For t2Twe de nethe forward jump operator ˙(t) = inffs2T : s>tgwhere inf ˚=supT, while the backward jump operator ˆ(t) = supfs2T : s t, we say that tis right-scattered, whileif ˆ(t) <t, we say that tis left-scattered. If ˙(t) = t, we say that tisright-dense, while if ˆ(t) = t, we say that tis left-dense. The forwardgraininess function (t) of t2Tis de ned by (t) = ˙(t) t, whlie the

SOME STRONG CONVERGENCE RESULTS OF RANDOM ITERATIVE ALGORITHMS WITH ERRORS IN BANACH SPACES

In this paper, we study the strong convergence and stability of a new two step random iterative scheme with errors for accretive Lipschitzian mapping in real Banach spaces. The new iterative scheme is more acceptable because of much better convergence rate and less restrictions on parameters as compared to random Ishikawa iterative scheme with errors. We support our analytic proofs by providing numerical examples. Applications of random iterative schemes with errors to variational inequality are also given. Our results improve and establish random generalization of results obtained by Chang [4], Zhang [31] and many others.

SOME OBSERVATIONS ON THE NUMERICAL INDEX AND THE POLYNOMIAL NUMERICAL INDEX

In this paper, we study both the numerical index and the polynomial numerical index. First, we give a sufficient condition for a Banach space X to have lushness. Second, we study the relation between the renormings of a Banach space and the k-order polynomial numerical index. This shows that every real Banach spaces of dimension greater that 1 can be renormed to have 2-order polynomial numerical index for any .

Iterative algorithms for infinite accretive mappings and applications to p-Laplacian-like differential systems

Some new iterative algorithms with errors for approximating common zero point of an infinite family of m-accretive mappings in a real Banach space are presented. A path convergence theorem and some new weak and strong convergence theorems are proved by means of some new techniques, which extend the corresponding works by some authors. As applications, an infinite p-Laplacian-like differential system is investigated, from which we construct an infinite family of m-accretive mappings and discuss the connections between the equilibrium solution of the differential systems and the zero point of the m-accretive mappings.

$${{C^m}}$$ C m Positive Eigenvectors for Linear Operators Arising in the Computation of Hausdorff Dimension

We consider a broad class of linear Perron–Frobenius operators $${\Lambda:X \rightarrow X}$$ , where $${X}$$ is a real Banach space of $${C^m}$$ functions. We prove the existence of a strictly positive $${C^m}$$ eigenvector $${v}$$ with eigenvalue $${r=r(\Lambda) =}$$ the spectral radius of $${\Lambda}$$ . We prove (see Theorem 6.5 in Sect. 6 of this paper) that $${r(\Lambda)}$$ is an algebraically simple eigenvalue and that, if $${\sigma(\Lambda)}$$ denotes the spectrum of the complexification of $${\Lambda,\sigma(\Lambda) \backslash \{r(\Lambda)\} \subseteq \{\zeta \in \mathbb{C} \big| |\zeta| \le r_*\}}$$ , where $${r_* < r(\Lambda)}$$ . Furthermore, if $${u \in X}$$ is any strictly positive function, $${(\frac 1r \Lambda)^k(u) \rightarrow s_u v}$$ as $${k \rightarrow \infty}$$ , where $${s_u > 0}$$ and convergence is in the norm topology on $${X}$$ . In applications to the computation of Hausdorff dimension, one is given a parametrized family $${\Lambda_s,s > s_*}$$ , of such operators and one wants to determine the (unique) value $${s_0}$$ such that $${r(\Lambda_{s_0})=1}$$ . In another paper (Falk and Nussbaum in C $${^{\rm m}}$$ Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension, submitted) we prove that explicit estimates on the partial derivatives of the positive eigenvector $${v_s}$$ of $${\Lambda_s}$$ can be obtained and that this information can be used to give rigorous, sharp upper and lower bounds for $${s_0}$$ .

Strong convergence results for nonlinear mappings in real Banach spaces

Let X be a real Banach space, K be a nonempty closed convex subset of X, T : K → K be a nearly uniformly L-Lipschitzian mapping with sequence {an}. Let kn ⊂ [1, ∞) and En be sequences with limn→∞ kn = 1, limn→∞ En = 0 and F(T) = {ρ ∈ K : T ρ = ρ} 6= ∅. Let {αn}n≥0 be real sequence in [0, 1] satisfying the following conditions: (i)P n≥0 αn = ∞ (ii) limn→∞ αn = 0. For arbitrary x0 ∈ K, let {xn}n≥0 be iteratively defined by xn+1 = (1 − αn)xn + αnT nxn, n ≥ 0. If there exists a strictly increasing function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0 such that &lt; T nx − T nρ, j(x − ρ) &gt;≤ knkx − ρk 2 − Φ(kx − ρk) + En for all x ∈ K, then, {xn}n≥0 converges strongly to ρ ∈ F(T). It is also proved that the sequence of iteration {xn} defined by xn+1 = (1 − bn − dn)xn + bnT nxn + dnwn, n ≥ 0, where {wn}n≥0 is a bounded sequence in K and {bn}n≥0, {dn}n≥0 are sequences in [0,1] satisfying appropriate conditions, converges strongly to a fixed point of T.

A path convergence theorem and construction of fixed points for nonexpansive mappings in certain Banach spaces

In this paper, we introduce a new iterative process to approximate fixed points of nonexpansive maps in real Banach spaces having weakly continuous duality map and establish strong convergence theorems for the proposed iterative process. There is no compactness assumption on K or on T. Our results improve important recent results.

Stability of a Quartic Functional Equation in Restricted Domains

LetXbe a real normed space andYa Banach space andf:X→Y. We prove the Ulam-Hyers stability theorem for the quartic functional equationf(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y)=0in restricted domains. As a consequence we consider a measure zero stability problem of the above inequality whenf:R→Y.

Iterative Approximations for Zeros of Sum of Accretive Operators in Banach Spaces

We study the approximation of zero for sum of accretive operators using a modified Mann type forward-backward splitting algorithm and obtain strong convergence of the sequence generated by our scheme to the zero of sum of accretive operators in uniformly convex real Banach spaces which are also uniformly smooth. Our result is new and complements many recent and important results in this direction in the literature.