Operators In Banach Spaces Research Articles

Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces

We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.

English

We propose a Halpern-type forward-backward splitting with inertial extrapolation step for finding a zero of the sum of accretive operators in Banach spaces. Strong convergence of the sequence of iterates generated by the method proposed is obtained under mild assumptions. We give some numerical results in compressed sensing to validate the theoretical analysis results. Our result is one of the lew available inertial-type methods for zeros of the sum of accretive operators in Banach spaces.

Strong Convergence of New Algorithm for Monotone Operator in Banach Spaces

Let E be a uniformly convex and uniformly smooth real Banach space with dual The convergence of a new iterative algorithm for approximating zero points of monotone (not strongly monotone) and bounded mapping is established and analyzed. The algorithm is proved to converge strongly to which is not involving the resolvent operator. Moreover, applications to solutions of variational inequality problems, convex minimization problems and a cluster of semi-pseudo mappings are included. Numerical results are reported to illustrate the behavior of the algorithm with different sequences of stepsizes and also to compare it with other algorithms.

Well-Posedness of Fractional Degenerate Differential Equations in Banach Spaces

Abstract We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces $\begin{array}{} B_{p,q}^s \end{array}$ (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 < β < α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and $\begin{array}{} B_{p,q}^s \end{array}$-well-posedness of above equation.

Bounded reflexivity of kernels of elementary operators

Kernels of elementary operators of length two give rise to many interesting classes of operators. In this paper we study bounded reflexivity of these kernels. For general Banach spaces, a sufficient condition is given for the kernels to be boundedly reflexive, in particular this assures that the commutant of any Banach space operator is boundedly reflexive. We also show that the bounded reflexive cover of the set of all Toeplitz matrices in Mm×n is the entire Mm×n. For finite-dimensional Banach spaces, using Kronecker canonical form for matrix pencils, we obtain some necessary and sufficient conditions for the kernels to be boundedly reflexive. These conditions enable us to easily determine bounded reflexivity by simply computing rank or nullity of matrices.

Newton–Kantorovich regularization method for nonlinear ill-posed equations involving $$m-$$accretive operators in Banach spaces

In this paper, we study nonlinear ill-posed problems involving $$m-$$accretive mappings in Banach spaces. We consider Newton–Kantorovich regularization method for the implementation of Lavrentiev regularization method. Using general Holder type source condition we obtain an error estimate. We also use the adaptive parameter choice strategy proposed by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005) for choosing the regularization parameter.

System of fuzzy fractional differential equations in generalized metric space

In this paper, we study the existence of integral solutions of fuzzy fractional differential systems with nonlocal conditions under Caputo generalized Hukuhara derivatives. These models are considered in the framework of completegeneralized metric spaces in the sense of Perov. The novel feature of our approach is the combination of the convergentmatrix technique with Schauder fixed point principle of vector valued operators in semilinear Banach spaces. Somecomputational examples are represented to demonstrate our theoretical results.

Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

In this article, we study a modified viscosity splitting method combined with inertial extrapolation for accretive operators in Banach spaces and then establish a strong convergence theorem for such iterations under some suitable assumptions on the sequences of parameters. As an application, we extend our main results to solve the convex minimization problem. Moreover, the numerical experiments are presented to support the feasibility and efficiency of the proposed method.

On the convergence of operator splitting for the Rosenau–Burgers equation

We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.

INTEGRO-DIFFERENTIAL EQUATIONS EMBODYING POWERS OF A DIFFERENTIAL OPERATOR

We establish solvability and correctness criteria for two Fredholm type linear integro-differential operators B2, B4 encompassing up to second and fourth powers, respectively, of a differential operator A�with aknown inverse I = A�1. We also derive explicit solution formulae to corresponding initial and boundary value problems by using the inverse of the differential operator. The approach is based on the theory of the extensions of linear operators in Banach spaces. Three example problems for ordinary and partial integro-differential operators are solved.

Iterations for systems of variational inequalities, fixed points of pseudocontractions and zeros of accretive operators

In this paper, we introduce implicit composite three-step Mann iterations for finding a common solution of a general system of variational inequalities, a fixed point problem of a countable family of pseudocontractive mappings and a zero problem of an accretive operator in Banach spaces. Strong convergence of the suggested iterations are given.

Two new algorithms for finding a common zero of accretive operators in Banach spaces

Two new algorithms for finding a common zero of accretive operators in Banach spaces

Mild solution for fractional integro-differential equations with non-instantaneous impulses through sectorial operator in Banach space

Mild solution for fractional integro-differential equations with non-instantaneous impulses through sectorial operator in Banach space

Sylvester equations and polynomial separation of spectra

Sylvester equations $AX-XB=C$ have unique solutions for all $C$ when the spectra of $A$ and $B$ are disjoint. Here $A$ and $B$ are bounded operators in Banach spaces. We discuss the existence of polynomials $p$ such that the spectra of $p(A)$ and $p(B)$ are well separated, either inside and outside of a circle or separated into different half planes. Much of the discussion is based on the following inclusion sets for the spectrum: $V_p(T)=\{\lambda \in \mathbb C : |p(\lambda)| \le \|p(T)\| \}$ where $T$ is a bounded operator. We also give an explicit series expansion for the solution in terms of $p(M)$, where $M=\begin{pmatrix} A&C &B\end{pmatrix}$, in the case where the spectra of $A$ and $B$ lie in different components of $V_p(M)$ .

Strong convergence theorems for $\delta$-inverse strongly accretive operators in Banach spaces

Strong convergence theorems for $\delta$-inverse strongly accretive operators in Banach spaces

Sphere and projective space of a C *-algebra with a faithful state

Abstract Let 𝒜 be a unital C *-algebra with a faithful state ϕ. We study the geometry of the unit sphere 𝕊 ϕ = {x ∈ 𝒜 : ϕ(x * x) = 1} and the projective space ℙ ϕ = 𝕊 ϕ /𝕋. These spaces are shown to be smooth manifolds and homogeneous spaces of the group 𝒰 ϕ (𝒜) of isomorphisms acting in 𝒜 which preserve the inner product induced by ϕ , which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in ℙ ϕ , and prove the existence of minimal geodesics, both with given initial data, and given endpoints.

Isolated eigenvalues, poles and compact perturbations of Banach space operators

Given a Banach space operator $A$, the isolated eigenvalues $E(A)$ and the poles $\Pi(A)$ (resp., eigenvalues $E^a(A)$ which are isolated points of the approximate point spectrum and the left ploles $\Pi^a(A)$) of the spectrum of $A$ satisfy $\Pi(A)\subseteq E(A)$ (resp., $\Pi^a(A)\subseteq E^a(A)$), and the reverse inclusion holds if and only if $E(A)$ (resp., $E^a(A)$) has empty intersection with the B-Weyl spectrum (resp., upper B-Weyl spectrum) of $A$. Evidently $\Pi(A)\subseteq E^a(A)$, but no such inclusion exists for $E(A)$ and $\Pi^a(A)$. The study of identities $E(A)=\Pi^a(A)$ and $E^a(A)=\Pi(A)$, and their stability under perturbation by commuting Riesz operators, has been of some interest in the recent past. This paper studies the stability of these identities under perturbation by (non-commuting) compact operators. Examples of analytic Toeplitz operators and operators satisfying the abstract shift condition are considered.

Pseudospectra in a non-Archimedean Banach space and essential pseudospectra in Eω

In this work, we introduce and study the pseudospectra and the essential pseudospectra of linear operators in a non-Archimedean Banach space and in the non-Archimedean Hilbert space E?, respectively. In particular, we characterize these pseudospectra. Furthermore, inspired by T. Diagana and F. Ramaroson [12], we establish a relationship between the essential pseudospectrum of a closed linear operator and the essential pseudospectrum of this closed linear operator perturbed by completely continuous operator in the non-Archimedean Hilbert space E?.

Norm-Attainability and Range-Kernel Orthogonality of Elementary Operators

Various aspects of elementary operators have been characterized by many mathematicians. In this paper, we consider norm-attainability and orthogonality of these operators in Banach spaces. Characterizations and generalizations of norm-attainability and orthogonality are given in details. We first give necessary and sufficient conditions for norm-attainability of Hilbert space operators then we give results on orthogonality of the range and the kernel of elementary operators when they are implemented by norm-attainable operators in Banach spaces.

(h, k)-Dichotomy and Lyapunov Type Norms

Abstract The paper considers a general concept of nonuniform (h, k)-dichotomy for evolution operators in Banach spaces. Two characterizations of this concept in terms of some families of norms compatible with the dichotomy projectors are given.