Operators In Banach Spaces Research Articles

Generalized Inverse Operator for an Integrodifferential Operator in the Banach Space

By using the theory of generalized inversion of operators in Banach spaces, we establish conditions for the generalized invertibility of integrodifferential operators with degenerate kernels in Banach spaces. We obtain formulas for the construction of bounded projectors and the bounded generalized inverse operator for an integrodifferential operator with degenerate kernel in a Banach space. The results of investigations are illustrated by an example.

Ill-posed problems associated with sectorial operators in Banach space

Ill-posed problems of the form u′(t)=Au+h, 0<t<T, u(0)=φ are studied in Banach space under two related scenarios, both of which consider a well-posed logarithmic approximation. In the first scenario, −A generates a bounded holomorphic semigroup of angle θ=π2, and so A is sectorial of angle 0. In the second, where θ may be relaxed to any value in (0,π2], we impose an additional assumption that e−A is sectorial. Regularization results for the original ill-posed problem are then established in the linear case where h=h(t), and subsequently extended to nonlinear problems where h is replaced by h(t,u(t)). We compare the two cases in terms of regularization convergence rates and apply our results to backward heat equations, both linear and nonlinear in Lp(Rn), 1<p<∞.

A new iterative method based on the modi&amp;#64257;ed proximal-point algorithm for &amp;#64257;nding a common null point of an in&amp;#64257;nite family of accretive operators in Banach spaces

In this paper, we introduce and study a new iterative method for finding a common null point of an infinite family of accretive operators with a strongly accretive and Lipschitzian operator, by using the proximal-point algorithm. And also we prove that the common null point is a unique solution of variational inequality without imposing any compactness-type condition on either the operators or the space considered. Finally, some applications of the main results to equilibrium problems and fixed point problems with an infinite family of pseudocontractive mappings are given. The main result is a generalization and improvement of numerous well-known results in the available literature.

On the FPV Property, Monotone Operator Structure and the Monotone Polar of Representable Monotone Sets

We study the pointwise partial ordering of representative functions for a monotone operator and in particular we focus on the bigger conjugate representative functions that represent a fixed initial (non-maximal) monotone operator. The first problem considered is that of constructing a new representative function from a given member of this class when wanting to add an additional monotonically related point. This study allows us to prove that all bigger conjugate representable monotone sets are monotonically closed. This result sheds light on the structure of the domains for maximal monotone operators and enables us to study the sum theorem for FPV operators in Banach spaces which posses a dual space that has a strictly convex renorm.

On generalized Bregman nonspreading mappings and zero points of maximal monotone operator in a reflexive Banach space

The purpose of this paper is to investigate the existence and approximation of fixed points of generalized Bregman nonspreading mapping which are also the zero points of maximal monotone operator in a real reflexive Banach space. Without assuming the ‘AKTT’ condition, we prove a strong convergence theorem for approximating a common element in the set of fixed points of system of generalized Bregman nonspreading mappings and the set of zero points of system of maximal monotone operators in a real reflexive Banach space. Our results improve and generalized many recent results in the literature.

Compositions and convex combinations of Bregman weakly relatively nonexpansive operators in reflexive Banach spaces

In this paper, we investigate the right and left Bregman weakly relatively nonexpansive operators in reflexive Banach spaces. We first introduce the notions of convex hull demi-closedness principle, strictly strongly convex hull demi-closedness principle, strongly demi-closedness principle, and composition demi-closedness principle of a family of nonlinear mappings using another new notions of convex asymptotic fixed points, and composition asymptotic fixed points for such mappings in a Banach space E. We analyze, in particular, compositions and convex combinations of Bregman weakly relatively nonexpansive operators, and prove the convergence of the Mann iterative method for operators of these types. Finally, we use our results to approximate common zeros of maximal monotone mappings and solutions to convex feasibility problems. To support our results, we include nontrivial examples in the paper. Therefore, our results improve and generalize many known results in the current literature.

Iterative Approximations for a Class of Generalized Nonexpansive Operators in Banach Spaces

In this article, we prove some weak and strong convergence theorems for mappings satisfying condition (E) using the AK iterative scheme in the setting of Banach spaces. We offer a new example of mapping with condition (E) in support of our main result. Our results extend and improve many well-known corresponding results of the current literature.

Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach space

Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach space

Solving nonlinear non-local problems using positive square-root operators

A non-constructive existence theory for certain operator equations L u = D u , using the substitution u = B 1 2 ξ with B = L −1 , is developed, where L is a linear operator (in a suitable Banach space) and D is a homogeneous nonlinear operator such that Dλu = λ α D u for all λ ≥ 0 and some α ∈ R , α ≠ ~1. This theory is based on the positive-operator approach of Krasnosel’skii. The method has the advantage of being able to tackle the nonlinear right-hand side D in cases where conventional operator techniques fail. By placing the requirement that the operator B must have a positive square root, it is possible to avoid the usual regularity condition on either the mapping D or its Fréchet derivative. The technique can be applied in the case of elliptic PDE problems, and we show the existence of solitary waves for a generalization of Benjamin’s fluid dynamics problem.

On Norm-Attainable Operators in Banach Spaces

Norm-attainable operators have been studied over a period of time with nice results obtained particularly in Hilbert spaces. In this work, we consider the Banach space setting by characterizing nonpower operators onHand elementary operators. Lastly, we characterize a new notion of norm-attainability for power operators in general Banach spaces.

Iterative algorithm for computing fixed points of demicontractive and zeros points of multivalued accretive operators in certain Banach spaces with application

In this paper, an iterative algorithm for finding a common point of the set of common zero of an infinite family of multivalued accretive operators and the set of fixed points of a demicontractive operator is constructed and studied in certain Banach spaces having a weakly continuous duality map. Under suitable control conditions, strong convergence of the sequence generated by proposed algorithm to a common point of the two sets is established. Moreover, application to convex minimization problems involving an infinite family of lower semi-continuous and convex functions are included.The main theorems develop and complement the recent results announced by researchers in this area.

Property $(w)$ of upper triangular operator Matrices

Let $M_C=\begin{pmatrix} A &amp; C \\ 0 &amp; B \\ \end{pmatrix}\in\LB(\x,\y)$ be be an upper triangulate Banach spaceoperator. The relationship between the spectra of $M_C$ and $M_0,$ and theirvarious distinguished parts, has been studied by a large number of authors inthe recent past. This paper brings forth the important role played by SVEP,the {\it single-valued extension property,} in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type $M_0$ satisfies property $(w)$ $\Leftrightarrow$ $M_C$ satisfies property $(w)$ to hold. Moreover, we explore certain conditions on $T\in\LB(\hh)$ and $S\in\LB(\K)$ so that the direct sum $T\oplus S$ obeys property $(w)$, where $\hh$ and $\K$ are Hilbert spaces.

Two Projection Algorithms for Solving the Split Common Fixed Point Problem

We study the split common fixed point problem for Bregman relatively nonexpansive operators in real reflexive Banach spaces. Using Bregman distances, we propose two new projection algorithms for solving this problem.

A Fixed-Point Theorem for Ordered Contraction-Type Decreasing Operators in Banach Space with Lattice Structure

In this work, we mainly improve the results in Amini-Harandi and Emami (2010). By introducing a new kind of ordered contraction-type decreasing operator in Banach space, we obtain a unique fixed point by using the iterative algorithm. An example is also presented to illustrate the theorem.

Weighted composition operators in functional Banach spaces: an axiomatic approach

We work with very general Banach spaces of analytic functions in the disk or other domains which satisfy a minimum number of natural axioms. Among the preliminary results, we discuss some implications of the basic axioms and identify all functional Banach spaces in which every bounded analytic function is a pointwise multiplier. Next, we characterize (in various ways) the weighted composition operators among the bounded operators on such spaces, thus generalizing some well-known results on multiplication or composition operators. We also characterize the invertible weighted composition operators on the disk and on general Banach spaces of analytic functions on bounded domains under different sets of axioms whose connections we discuss by providing appropriate examples This generalizes and complements various recent results by Gunatillake, Bourdon, and Hyvärinen–Lindström–Nieminen–Saukko.

Regular factorizations and group inverses of linear operators in Banach spaces

The main topic of this paper is the group invertibility and expressions of group inverses of linear operators in Banach spaces. We first give some properties of regular factorizations and then derive some new characterizations of the group invertibility of linear operators. Based on these results, we use any one of generalized inverses to give some concise expressions of the group inverse.

Abstract Formulation of the Miura Transform

Miura transform is known as the transformation between Korweg de-Vries equation and modified Korweg de-Vries equation. Its formal similarity to the Cole-Hopf transform has been noticed. This fact sheds light on the logarithmic type transformations as an origin of a certain kind of nonlinearity in the soliton equations. In this article, based on the logarithmic representation of operators in infinite-dimensional Banach spaces, a structure common to both Miura and Cole-Hopf transforms is discussed. In conclusion, the Miura transform is generalized as the transform in abstract Banach spaces, and it is applied to the higher order abstract evolution equations.

On the convergence of iterates of convolution operators in Banach spaces

Let $G$ be a locally compact abelian group and let $M(G)$ be the measure algebra of $G$. A measure $\mu \in M(G)$ is said to be power bounded if $\sup _{n\geq 0}\lVert \mu ^{n} \rVert _{1}&lt;\infty $. Let $\mathbf {T} = \{ T_{g}:g\in G\}$ be a bounded and continuous representation of $G$ on a Banach space $X$. For any $\mu \in M(G)$, there is a bounded linear operator on $X$ associated with µ, denoted by $\mathbf {T}_{\mu }$, which integrates $T_{g}$ with respect to µ. In this paper, we study norm and almost everywhere behavior of the sequences $\{ \mathbf {T}_{\mu }^{n}x\}$ $(x\in X)$ in the case when µ is power bounded. Some related problems are also discussed.

A Viewpoint to Measure of Non-Compactness of Operators in Banach Spaces

This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection $$\mathcal{C}(X)$$ (consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication) is a normed semigroup, and the mapping J from $$\mathcal{C}(X)$$ onto $$\mathcal{F}(\Omega)$$ is a fully order-preserving positively linear surjective isometry, where Ω is the closed unit ball of X* and $$\mathcal{F}(\Omega)$$ the collection of all continuous and w*-lower semicontinuous sublinear functions on X* but restricted to Ω. Furthermore, both $$E_\mathcal{C}=\overline{J\mathcal{C}-J\mathcal{C}}$$ and $$E_\mathcal{K}=\overline{J\mathcal{K}-J\mathcal{K}}$$ are Banach lattices and $$E_\mathcal{K}$$ is a lattice ideal of $$E_\mathcal{C}$$ . The quotient space $$E_\mathcal{C}/E_\mathcal{K}$$ is an abstract M space, hence, order isometric to a sublattice of C(K) for some compact Haudorspace K, and $$(FQJ)\mathcal{C}$$ which is a closed cone is contained in the positive cone of C(K), where $$Q:E_\mathcal{C}\rightarrow{E_\mathcal{C}/E_\mathcal{K}}$$ is the quotient mapping and $$F:E_\mathcal{C}/E_\mathcal{K}\rightarrow{C(K)}$$ is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given: Let BX be the closed unit ball of a Banach space X, then $$\mu(T)=\mu(T(B_X))=\parallel(FQJ)\overline{T(B_X)}\parallel_{\mathcal{C}(\mathcal{K})}, \forall{T}\in{B(X)}.$$

New Approximation Techniques for Solving Variational Inclusions Problem via SP Iterative Algorithm with Mixed Errors for Accretive Lipschitzian Operators

Using different convergence techniques and under the lack of parametrical restrictions, the convergence and stability results of SP iterative algorithm with mixed errors for accretive Lipschitzian operators in Banach spaces are established. We propose numerical examples to verify effectiveness of new convergence techniques and to show that SP iterative algorithm with mixed errors converges more effectively than the Mann, Ishikawa and Noor iterative algorithms with mixed errors. Moreover, new iterative approximation of solution for variational inclusion problem in Banach spaces is investigated by using SP iterative algorithm with mixed errors for accretive Lipschitzian operators. Our results are improvement and generalization of results of Kim[15], Gu[10], Gu and Lu[11], Chugh and Kumar[7] and many others in the literature.