Search

We consider an elliptic operator in a planar waveguide with a fast oscillating boundary where we impose Dirichlet, Neumann or Robin boundary conditions assuming that both the period and the amplitude of the oscillations are small. We describe the homogenized operator, establish the norm resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. It is shown that under the homogenization, the type of the boundary condition can change.

In this paper we focus our attention on a new type nonhomogeneous Sturm-Liouville systems with abstract linear operator contained in the equation. A different approach is used here for investigation such important properties as topological isomorphism and coercive solvability. Moreover we prove that the corresponding resolvent operator is compact in a suitable Hilbert space.

We consider what is perhaps the most important of all partial differential operators, theLaplace operator (Laplacian) on \(\mathbb {R}^n\).

We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory.

The main purpose of this paper is to study the sensitivity analysis for nonlinear set-valued variational equations based on -resolvent operator technique. The obtained results encompass a broad model of results.

In this paper we prove the existence of mild solutions of a general class of nonlinear evolution integrodifferential equation in Banach spaces. Based on the resolvent operator and the Schaefer fixed point theorem, a sufficient condition for the existence of general integrodifferential evolution equations is established.

We have examined the spectra of the operatorD(r,0,s,0,t)on the sequence spacesc0andc.

‎In this paper‎, ‎we investigate the essential approximate point‎ ‎spectrum and the essential defect spectrum of a $3\times3$ block‎ ‎operator matrix with unbounded entries and with domain consisting of‎ ‎vectors which satisfy certain relations between their components by‎ ‎means of the Browder resolvent set‎. ‎Furthermore‎, ‎we apply the‎ ‎obtained results to three-Group transport operators in the Banach‎ ‎space $L_{p}( [-a,a]\times[-1,1])$ where $a\gt 0$ and $p \in [1,+\infty).$‎

In this paper, we introduce a new resolvent operator and we call it relaxed resolvent operator. We prove that relaxed resolvent operator is single-valued and Lipschitz continuous and finally we appropriate the solution of a variational inclusion problem in Hilbert spaces by defining an iterative algorithm based on relaxed resolvent operator. A few concepts like Lipschitz continuity and strong monotonicity are used to prove the main result of this paper. Thus, no strong conditions are used. Some examples are constructed.

Under certain conditions on the magnetic and electric potentials, we prove that the corresponding one-dimensional magnetic Schrodinger operator on the whole axis is selfadjoint and establish that Fredholm theory is applicable to the resolvent equation of this operator.

In this paper we consider a boundary problem for a parameter-elliptic, multi-order system of differential equations defined over a bounded region in and under limited smoothness assumptions as well as under boundary conditions which include those of Dirichlet. Information is then derived concerning the asymptotic behaviour of the trace of a power of the resolvent of the Hilbert space operator, in general non-selfadjoint, induced by the boundary problem under null boundary conditions. This information will then be used in a subsequent work to derive various results pertaining to the asymptotic behaviour of the eigenvalues of this operator.

In this paper, a new resolvent algorithm, which consists of a resolvent operator technique step followed by a suitable orthogonal projection onto a moving half-space, is considered for solving a class of variational inclusions involving A-monotone mapping and H-monotone mapping in Hilbert spaces. The convergence of the iterative sequence generated by the algorithm is also proved.

In this paper, we consider and study a class of general nonlinear operator inclusion couples involving ( A , η , m )-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to ( A , η , m )-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures.

A multidimensional matrix operator of general form is considered, which need not be symmetric, with finitely many distant perturbations that are described by arbitrary abstract localized operators. We study the behavior of the resolvent of the perturbed operator as the distances between domains at which the perturbations are localized tend to infinity. An explicit formula for the resolvent of the perturbed operator is obtained; it is used to obtain a complete asymptotic expansion of the resolvent.

In this paper, a new class of over-relaxed proximal point algorithms for solving nonlinear operator equations with (A,η,m)-monotonicity framework in Hilbert spaces is introduced and studied. Further, by using the generalized resolvent operator technique associated with the (A,η,m)-monotone operators, the approximation solvability of the operator equation problems and the convergence of iterative sequences generated by the algorithm are discussed. Our results improve and generalize the corresponding results in the literature.

In this work, we present some results concerning the operators defined on various classes of exotic Banach spaces, containing in particular those studied respectively by V. Ferenczi [7, 8] and T. Gowers with B. Maurey [14, 15]. We show that, on hereditarily indecomposable or quotient hereditarily indecomposable Banach space X, the set of bounded Fredholm operators is dense in ℒ(X), this gives that the boundary of bounded Fredholm operators is nothing else but the ideal of strictly singular operators if X is hereditarily indecomposable Banach space (resp. the ideal of strictly cosingular operators if X is quotient hereditarily indecomposable Banach space). On the other hand, a comparison between sufficiently rich and exotic Banach spaces is given via some properties of the two maps spectra and Wolf essential spectra.

In this paper, we apply H(?,?)-?-cocoercive operator introduced in [2] for solving a system of generalized variational-like inclusions in q-uniformly smooth Banach spaces. By using the approach of resolvent operator associated with H(?,?)-?-cocoercive operator, an iterative algorithm for solving a system of generalized variational-like inclusions is constructed. We prove the existence of solutions of system of generalized variational-like inclusions and convergence of iterative sequences generated by the algorithm. An example through Matlab programming is constructed.

Boundary values of analytic fractional resolvent families are deduced via the approximation of fractional powers of operators, and square root reductions of fractional resolvent families are given as well.

In this paper we consider linear integro-differential equations in Banach spaces with Fredholm operators at the highest-order derivatives and convolution-type Volterra integral parts. We obtain sufficient conditions for the unique solvability (in the classical sense) of the Cauchy problem for the mentioned equations and illustrate the abstract results with pithy examples. The studies are carried out in classes of distributions in Banach spaces with the help of the theory of fundamental operator functions of degenerate integro-differential operators. We propose a universal technique for proving theorems on the form of fundamental operator functions.

In this paper, we prove the existence of solutions of fractional integrodifferential equations by using the resolvent operators and fixed point theorem. An example is given to illustrate the abstract results.

On the basis of the resolvent of a simple differential operator, a method for finding approximations to continuous functions is constructed. In this method, both the approximated function and its approximations satisfy the given integral boundary condition.

In this paper, the Resolvent–Projection algorithm for solving the variational inclusion 0∈M(x) involving an A-monotone set-valued operator M is constructed in Hilbert spaces. The convergence of the iterative sequence generated by the algorithm is proved also.

Let \( k \in C(R^+)\), A be a closed linear densely defined operator in the Banach space \(X\) and \( \{R(t)\}_{t\geq 0} \) be an exponentially bounded \(k\)-regularized resolvent operator families generated by A. In this paper, we mainly study pseudo k-resolvent and duality theory of k-regularized resolvent operator families. The conditions that pseudo k-resolvent become k-resolvent of the closed linear densely defined operator A are given. The some relations between the duality of the regularized resolvent operator families and the generator A are gotten. In addition, the corresponding results of duality of \(k\)-regularized resolvent operator families in Favard space are educed.

By using the index theory for linear bounded self-adjoint operators in a Hilbert space related to a fixed self-adjoint operator A with compact resolvent, the authors discuss the existence and multiplicity of solutions for (nonlinear) operator equations, and give some applications to some boundary value problems of first order Hamiltonian systems and second order Hamiltonian systems.

Certain estimates for the resolvent of a block-discrete Schrodinger operator with a constant diagonal perturbation are obtained. For that purpose, the resolvent is represented in terms of Chebychev polynomials of the (in general, unbounded) operator that represents a block of the perturbation. Bibliography: 12 titles.

Suppose that a pair of sequence spaces is admissible with respect to a discrete linear Volterra operator. This paper gives sufficient conditions for the same pair of spaces to be admissible with respect to the associated resolvent operator. The spaces considered include spaces of weighted bounded and weighted convergent sequences. Classes of discrete kernels are discussed for which the appropriate weight sequences are not purely exponential, but the product of an exponential and a slowly decaying sequence.

This paper investigates the admissibility of control and observation operators in UMD spaces. Necessary and/or sufficient conditions for unbounded control operators to be admissible and weakly admissible in the Salamon–Weiss sense are presented. This is illustrated by an example which shows that the UMD-property is essential. In particular, we get a direct proof of the known result of Driouich and and El-Mennaoui (Arch Math 72:56–63, 1999) on the validity of the inverse formula of the Laplace transform for C0-semigroups on UMD-spaces and in Hilbert spaces, as proved earlier by Yao (SIAM J Math Anal 26(5):1331–1341, 1995). We outline how these results can be used to prove a partial validity of the inverse Laplace transform for semigroups in general Banach spaces. In particular, we obtain the result on the inverse Laplace transform due to Hille and Philllips (Am Math Soc Transl Ser 2, 1957).

We give a very short proof of a characterization of Fredholm composition operators acting on a large class of Banach spaces of analytic functions defined on the open unit ball of the Euclidian n-complex space.

In this article we give a characterization of the convergence of projection methods which are useful for approximating the Moore-Penrose inverse of a closed densely defined operator between Hilbert spaces. We illustrate the main theorem with an example. Also a procedure for constructing the admissible sequence of projections is discussed.

In this paper, we study the behavior and sensitivity analysis of the solution set for a new system of generalized parametric multi-valued variational inclusions with (A,·)-accretive mappings in q-uniformly smoo- th Banach spaces. The present results improve and extend many known results in the literature. techniques. By using the projection method, Dafermos (2), Yen (12), Mukherjee and Verma (7), Noor (9) and Pan (10) studied the sensitivity analysis of solutions of some variational inequalities with single-valued mappings in finite-dimensional spaces or Hilbert spaces. By using the resolvent operator technique, Agarwal et al. (1), Jeong (3) stud- ied a new system of parametric generalized nonlinear mixed quasi-variational inclusions in a Hilbert space and in Lp(p ‚ 2) spaces, respectively. In 2008, using the concept and technique of resolvent operators, Lan (4) introduced and studied the behavior and sensitivity analysis of the solution set for a system of generalized parametric (A,·)-accretive variational inclusions in Banach spaces. Motivated and inspired by the research work going on this field, in this paper, we study the behavior and sensitivity analysis of the solution set for a new system of generalized parametric multi-valued variational inclusions with (A,·)-accretive mappings in q-uniformly smooth Banach spaces. The present results improve and extend many known results in the literature.

By using the Hille-Yosida theorem and the Phillips theorem in functional analysis we prove that the M/G/1 retrial queueing model with server breakdowns has a unique nonnegative time-dependent solution. Next, when the service completion rate is a constant, by studying spectral properties of the operator corresponding to the model we study asymptotic behavior of its time-dependent solution. First of all, through considering the resolvent set of the adjoint operator of the operator we obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator. In addition, we prove that zero is not an eigenvalue of the operator. Our results show that the time-dependent solution of the model strongly converges to zero.

We obtain new stability results for those properties of C0-semigroups which admit characterisation in terms of decay of resolvents of infinitesimal generators on vertical lines, e.g. analyticity, Crandall-Pazy differentiability or immediate norm continuity in the case of Hilbert spaces. As a consequence we get a generalisation of the Kato-Neuberger theorem on approximation of the identity. Finally, we present examples shedding a new light on resolvent characterisation of eventually differentiable C0-semigroups for which differentiability is stable under bounded perturbations. © 2010 Elsevier Inc. All rights reserved.

We study unbounded Hermitian operators with dense domain in Hilbert space. As is known, the obstruction for a Hermitian operator to be selfadjoint or to have selfadjoint extensions is measured by a pair of deficiency indices, and associated deficiency spaces; but in practical problems, the direct computation of these indices can be difficult. Instead, in this paper we identify additional structures that throw light on the problem. We will attack the problem of computing deficiency spaces for a single Hermitian operator with dense domain in a Hilbert space which occurs in a duality relation with a second Hermitian operator, often in the same Hilbert space.

This paper is concerned with the existence of mild solutions for partial neutral functional integrodifferential equations with infinite delay in a Banach space. The results are obtained by using the resolvent operators and Krasnoselski-Schaefer type fixed point theorem. An example is given to illustrate the results.

In this paper, a new notion of a generalized H-η-accretive operator is introduced and studied, which provides a unifying framework for the generalized m-accretive operator and the H-η-monotone operator in Banach spaces. A resolvent operator associated with the generalized H-η-accretive operator is defined, and its Lipschitz continuity is shown. As an application, the solvability for a class of variational inclusions involving the generalized H-η-accretive operators in Banach spaces is considered. By using the technique of the resolvent mapping, an iterative algorithm for solving the variational inclusion in Banach spaces is constructed. Under some suitable conditions, it is proven that the solution for the variational inclusion and the convergence of the iterative sequence generated by the algorithm exist.

In this paper, three iterations are designed to approach zeros of set-valued accretive operators in Banach spaces. The first one is the continuous Picard type iteration involving the resolvent, the second one is the approximate Picard type iteration involving the resolvent and the third one is the Halpern type iteration involving the resolvent. Some strong convergence theorems for three iterations are proved.

The main aim of this work is to use the resolvent operator technique to find the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point problems for Lipschitz mappings in Hilbert spaces. An example of applying the main result is also given. The results obtained in this work may be viewed as an extension, refinement and improvement of the previously known results.

In this paper, a new class of general nonlinear variational inclusions involving H-monotone is introduced and studied in Hilbert spaces. By applying the resolvent operator associated with H-monotone, we prove the existence and uniqueness theorems of solution for the gen- eral nonlinear variational inclusion, construct an iterative algorithm for computing approximation solution of the general nonlinear variational in- clusion and discuss the convergence of the iterative sequence generated by the algorithm. The results presented in this paper improve and extend many known results in recent literatures.

We study the existence of mild solutions for a class of neutral impulsive stochastic integro-differential equations with infinite delays. We assume that the undelayed part generates an analytic resolvent operator and transform it into an integral equation. Sufficient conditions for the existence of solutions are derived by means of the Sadovskii fixed point theorem combined with theories of analytic resolvent operators. An example is given to illustrate the theory.

In this paper the solutions of some evolution equations with fractional orders in a Banach space are considered. Conditions are given which ensure the existence of a resolvent operator for an evolution equation in a Banach space.

In this paper, we introduce a new class of accretive operators-(H(@?,@?),@h)-accretive operators, which generalize many existing monotone or accretive operators. The resolvent operator associated with an (H(@?,@?),@h)-accretive operator is defined and its Lipschitz continuity is presented. By using the new resolvent operator technique, we also introduce and study a new class of set-valued variational inclusions involving (H(@?,@?),@h)-accretive operators and construct a new algorithm for solving this class of set-valued variational inclusions. These results are new, and improve and generalize many known corresponding results.

In this paper, we introduce and study a system of nonlin- ear set-valued implicit variational inclusions (SNSIVI) with relaxed co- coercive mappings in real Banach spaces. By using resolvent operator technique for M-accretive mapping, we construct a new class of iterative algorithms for solving this class of system of set-valued implicit varia- tional inclusions. The convergence of iterative algorithms is proved in q-uniformly smooth Banach spaces. Our results generalize and improve the corresponding results of recent works.

We discuss computing the spectrum of a bounded operator and representing its resolvent operator. The results include a general convergence theorem for the polynomial convex hull of the spectrum and explicit representations for the resolvent outside. The results are formulated and proved in general Banach algebras.

In this paper we estimate the resolvent of the generator of an isometric group of operators. In particular, we establish unimprovable estimates for the integral of functions that are holomorphic in a half-plane and bounded on the whole real axis. We obtain applications of the perturbation theory for linear operators.

In this paper we introduce three kinds of resolvent families defined by purely algebraic equations, which extend the classical semigroup property and Cosine functional equation. We give their basic properties and analyticity criteria. Moreover, the relations between integrated resolvent families and resolvent families are discussed as well.

We study mixed order parameter-elliptic boundary value problems with boundary conditions of a certain structure. For such operators, we prove resolvent estimates in L p based Sobolev spaces of suitable order and the analyticity of the semigroup. Finally, we present an application of this theory to studies of the particle transport in a semi-conductor.