Dominated inessential operators

We consider the following three closed algebraic ideals of operators on a Banach lattice: compact, strictly singular, and inessential operators. Suppose that 0?A?B and B is compact or strictly singular. We show that, under certain assumptions, A (or some power of A) is inessential.     

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {A_n}_(n∈N) be a family of monotone and Lipschitz continuos mappings of C into E~*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for{A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.     

On similarity of quasinilpotent operators

Bounded linear operators on separable Banach spaces algebraically similar to the classical Volterra operator V acting on C[0,1] are characterized. From this characterization it follows that V does not determine the topology of C[0,1], which answers a question raised by Armando Villena. A sufficient condition for an injective bounded linear operator on a Banach space to determine its topology is obtained. From this condition it follows, for instance, that the Volterra operator acting on the Hardy space Hp of the unit disk determines the topology of Hp for any p∈[1,∞].     

On uniqueness of fractional powers of multi-valued linear operators and the incomplete Cauchy problem

We obtain uniqueness of additive families {A _t } _t>0 of fractional powers of a multi-valued sectorial linear operator A ?? A ₁ in a Banach space, satisfying a certain kind of continuity with respect to the exponent and a spectral property, from uniqueness of the solutions of the second-order incomplete Cauchy problem associated with A. We show the close relationship between the multiplicativity and the uniqueness of fractional powers.     

Least squares and bounded variation regularization with nondifferentiable functionals

Let A be an operator from a real Banach space into a real Hilbert space. In this paper we study least squares regularization methods for the ill-posed operator equation A(u) = f using nonlinear nondifferentiable penalty functionals. We introduce a notion of distributional approximation, and use constructs of distributional approximations to establish convergence and stability of approximations of bounded variation solutions of the operator equation. We also show that the results provide a framework for a rigorous analysis of numerical methods based on Euler-Lagrange equations to solve the minimization problem. This justifies many of the numerical implementation schemes of bounded variation minimization that have been recently proposed.     

Asymptotic behavior of trajectories for some nonautonomous,almost periodic processes

In this paper, asymptotics are studied for some almost periodic processes on a complete metric space (X, d): (1) It is shown that any precompact positive trajectory of a contractive periodic process is asymptotically almost periodic as t → +∞. This property does not hold for general almost periodic contractive processes. (2) A compactness result is obtained for weakly almost periodic complete trajectories of some (possibly nonlinear) processes in a uniformly convex Banach space. (3) The existence of almost periodic trajectories is studied for “affine” processes in a uniformly convex Banach space. These results are applicable to some evolution equations of the form $\text{du}\text{dt}\text{+ A (t) u(t) ? f(t)}$, where $\text{?(t)}$ is almost periodic: R → V uniformly convex Banach space and A(t) is a periodic, time-dependent, m-accretive operator in V.     

Pullback attractors for a non-autonomous incompressible non-Newtonian fluid

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional (2D) bounded domains. We first prove the existence of pullback attractors AV in space V (has H²-regularity, see notation in Section 2) and AH in space H (has L²-regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing AV=AH, which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.     

Hyper-Tauberian algebras and weak amenability of Figà-Talamanca-Herz algebras

We study certain commutative regular semisimple Banach algebras which we call hyper-Tauberian algebras. We first show that they form a subclass of weakly amenable Tauberian algebras. Then we investigate the basic and hereditary properties of them. Moreover, we show that if A is a hyper-Tauberian algebra, then the linear space of bounded derivations from A into any Banach A-bimodule is reflexive. We apply these results to the Figà-Talamanca-Herz algebra A_p(G) of a locally compact group G for p∈(1,∞). We show that A_p(G) is hyper-Tauberian if the principal component of G is abelian. Finally, by considering the quantization of these results, we show that for any locally compact group G, A_p(G), equipped with an appropriate operator space structure, is a quantized hyper-Tauberian algebra. This, in particular, implies that A_p(G) is operator weakly amenable.     

On the perturbation of the group generalized inverse for a class of bounded operators in Banach spaces

Given a bounded operator A on a Banach space X with Drazin inverse AD and index r, we study the class of group invertible bounded operators B such that I+AD(B−A) is invertible and R(B)∩N(Ar)={0}. We show that they can be written with respect to the decomposition X=R(Ar)⊕N(Ar) as a matrix operator, , where B₁ and are invertible. Several characterizations of the perturbed operators are established, extending matrix results. We analyze the perturbation of the Drazin inverse and we provide explicit upper bounds of ‖B^?−AD‖ and ‖BB^?−ADA‖. We obtain a result on the continuity of the group inverse for operators on Banach spaces.     

Symmetric family of Fredholm operators of indices zero, stability of essential spectra and application to transport operators

In this paper, we prove that, if the product A=A₁?An is a Fredholm operator where the ascent and descent of A are finite, then Aj is a Fredholm operator of index zero for all j, 1?j?n, where A₁,…,An be a symmetric family of bounded operators. Next, we investigate a useful stability result for the Rako?evi?/Schmoeger essential spectra. Moreover, we show that some components of the Fredholm domains of bounded linear operators on a Banach space remain invariant under additive perturbations belonging to broad classes of operators A such as γ(Am)<1 where γ(⋅) is a measure of noncompactness. We also discuss the impact of these results on the behavior of the Rako?evi?/Schmoeger essential spectra. Further, we apply these latter results to investigate the Rako?evi?/Schmoeger essential spectra for singular neutron transport equations in bounded geometries.     

On abstract Volterra equations with kernels having a positive resolvent

We consider the nonlinear abstract Volterra equation of convolution type: V $$u(t) + b * Au(t) = u_0 + b * g(t), t \geqq 0,$$ whereA ism-accretive in Banach spaceX, b is a given real kernel,u ₀ andg are given. Boundedness and asymptotic properties of the solutions are established under the assumption that the kernel satisfies certain natural positivity conditions.     

Narrow orthogonally additive operators in lattice-normed spaces

We consider a new class of narrow orthogonally additive operators in lattice-normed spaces and prove the narrowness of every C-compact norm-laterally-continuous orthogonally additive operator from a Banach–Kantorovich space V into a Banach space Y. Furthermore, every dominated Urysohn operator from V into a Banach sequence lattice Y is also narrow. We establish that the order narrowness of a dominated Urysohn operator from a Banach–Kantorovich space V into a Banach space with mixed norm W implies the order narrowness of the least dominant of the operator.     

Best proximity points of non-self mappings

Let A,B be nonempty subsets of a Banach space X and let T:A→B be a non-self mapping. Under appropriate conditions, we study the existence of solutions for the minimization problem min _x∈A ∥x?Tx∥.     

Topological characterisation of weakly compact operators

Let X be a Banach space. Then there is a locally convex topology for X, the “Right topology,” such that a linear map T, from X into a Banach space Y, is weakly compact, precisely when T is a continuous map from X, equipped with the “Right” topology, into Y equipped with the norm topology. When T is only sequentially continuous with respect to the Right topology, it is said to be pseudo weakly compact. This notion is related to Pelczynski's Property (V).     

Summability and vector amarts

We show that an operator is absolutely summing if and only if it maps amarts into uniform amarts, from which we can deduce a theorem of A. Bellow and another of Edgar-Sucheston. We also show that the absolute value of a Banach lattice valued potential is a potential if and only if the lattice is an A-M space from which we deduce that the L¹-bounded amarts form a Ricsz space if and only if the space is finite dimensional.     

Linear functionals on nonlinear spaces and applications to problems from viscoplasticity theory

A classical result in the theory of monotone operators states that if C is a reflexive Banach space, and an operator A: C→C^* is monotone, semicontinuous and coercive, then A is surjective. In this paper, we define the ‘dual space’ C^* of a convex, usually not linear, subset C of some Banach space X (in general, we will have C^*⊃X^*) and prove an analogous result. Then, we give an application to problems from viscoplasticity theory, where the natural space to look for solutions is not linear. Copyright © 2007 John Wiley & Sons, Ltd.     

An abstract Volterra integral equation in a reflexive Banach space

This paper discusses the existence, uniqueness, and asymptotic behavior of solutions to the equation u(t) + ∝₀^ta(t ? s) Au(s) ds = f(t), where A is a maximal monotone operator mapping the reflexive Banach space V into its dual V′.     

Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces

This paper is concerned with almost automorphy of the solutions to a nonautonomous semilinear evolution equation u^′(t)=A(t)u(t)+f(t,u(t)) in a Banach space with a Stepanov-like almost automorphic nonlinear term. We establish a composition theorem for Stepanov-like almost automorphic functions. Furthermore, we obtain some existence and uniqueness theorems for almost automorphic solutions to the nonautonomous evolution equation, by means of the evolution family and the exponential dichotomy. Some results in this paper are new even if A(t) is time independent.     

Invariance for quasi-dissipative systems in Banach spaces

In a separable Banach space E, we study the invariance of a closed set K under the action of the evolution equation associated with a maximal dissipative linear operator A perturbed by a quasi-dissipative continuous term B. Using the distance to the closed set, we give a general necessary and sufficient condition for the invariance of K. Then, we apply our result to several examples of partial differential equations in Banach and Hilbert spaces.     

Semilinear differential inclusions via weak topologies

The paper deals with the multivalued initial value problem x^′∈A(t,x)x+F(t,x) for a.a. t∈[a,b], x(a)=x₀ in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x and the investigation includes the case when both A and F have a superlinear growth in x. We prove the existence of local and global classical solutions in the Sobolev space W1,p([a,b],E) with 1<p<∞. Introducing a suitably defined Lyapunov-like function, we are able to investigate the topological structure of the solution set. Our main tool is a continuation principle in Frechét spaces and we prove the required pushing condition in two different ways. Some examples complete the discussion.