Estimates of majorizing sequences in the Newton-Kantorovich method: A further improvement

Let be an operator, with X and Y Banach spaces, and f^′ be Hölder continuous with exponent θ. The convergence of the sequence of Newton-Kantorovich approximations

     

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function and an absolutely summing operator u from X to another Banach space Y such that the composition is not Bochner integrable; in particular, h is not McShane integrable.     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.     

On the Cauchy-Rassias stability of a generalized additive functional equation

Let X and Y be Banach spaces and f:X→Y an odd mapping. We solve the following generalized additive functional equation

     

Operators on the space of vector-valued totally measurable functions

Let L(X,Y) stand for the space of all bounded linear operators between real Banach spaces X and Y, and let Σ be a σ-algebra of sets. A bounded linear operator T from the Banach space B(Σ,X) of X-valued Σ-totally measurable functions to Y is said to be σ-smooth if ‖T(fn)Y_‖→0 whenever a sequence of scalar functions (‖fn(⋅)X_‖) is order convergent to 0 in B(Σ). It is shown that a bounded linear operator is σ-smooth if and only if its representing measure is variationally semi-regular, i.e., as An↓∅ (here stands for the semivariation of m on A∈Σ). As an application, we show that the space Lσs(B(Σ,X),Y) of all σ-smooth operators from B(Σ,X) to Y provided with the strong operator topology is sequentially complete. We derive a Banach-Steinhaus type theorem for σ-smooth operators from B(Σ,X) to Y. Moreover, we characterize countable additivity of measures in terms of continuity of the corresponding operators .     

On controlled extensions of functions

For any numerical function we give sufficient conditions for resolving the controlled extension problem for a closed subset A of a normal space X. Namely, if the functions , and satisfy the equality E(f(a),g(a))=h(a), for every a∈A, then we are interested to find the extensions f? and ? of f and g, respectively, such that , for every x∈X. We generalize earlier results concerning E(u,v)=u·v by using the techniques of selections of paraconvex-valued LSC mappings and soft single-valued mappings.     

Continuous dependence results for inhomogeneous ill-posed problems in Banach space

We apply semigroup theory and other operator-theoretic methods to prove Hölder-continuous dependence on modeling for the inhomogeneous ill-posed Cauchy problem in Banach space. The inhomogeneous ill-posed Cauchy problem is given by , u(0)=χ, 0?t<T; where −A is the infinitesimal generator of a holomorphic semigroup on a Banach space X, χ∈X, and . For a suitable function f, the approximate problem is given by , v(0)=χ. Under certain stabilizing conditions, we prove that for a related norm, where and M are computable constants independent of β, 0<β<1, and ω(t) is a harmonic function. These results extend earlier work of Ames and Hughes on the homogeneous ill-posed problem.     

Periodic solutions for a class of differential inclusions in general Banach spaces

Let X be a real Banach space, an m-accretive operator and a multi-function which is 2π-periodic with respect to its first argument, has nonempty, closed, convex and weakly compact values and is strongly-weakly upper semicontinuous. In this paper we prove the existence of at least one solution for the problem

     

Mean inequalities in certain Banach function spaces

Boundedness of the Hardy operator and its adjoint is characterized between Banach function spaces Xq and Lp. By applying a limiting procedure, corresponding boundedness of the geometric mean operator is also derived.     

On the asymptotic stability of equilibrium

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a positive definite function V which has infinitesimal upper bound such that is negative definite. In this paper we prove that if is bounded then the condition that is negative definite can be weakened and replaced by that and is negative definite.     

Generalized vector valued almost periodic and ergodic distributions

Schwartz's almost periodic distributions are generalized to the case of Banach space valued distributions , and furthermore for a given arbitrary class A to for φ∈ test functions D(R,C)}. It is shown that this extension process is iteration complete, i.e. . Moreover the T from are characterized in various ways, also tempered distributions with P={X-valued functions of polynomial growth} are shown. Under suitable assumptions , , where for all h>0}, is defined with the corresponding extension of Mh. With an extension of the indefinite integral from to D^′(R,X) a distribution analogue to the Bohl-Bohr-Amerio-Kadets theorem on the almost periodicity of bounded indefinite integrals of almost periodic functions is obtained, also for almost automorphic, Levitan almost periodic and recurrent functions, similar for a result of Levitan concerning ergodic indefinite integrals. For many of the above results a new (Δ)-condition is needed, we show that it holds for most of the A needed in applications. Also an application to the study of asymptotic behavior of distribution solutions of neutral integro-differential-difference systems is given.     

Notes on ordered reproducing Hilbert spaces over the complex plane

Let f, g be entire functions. If there exist M₁,M₂>0 such that |f(z)|?M₁|g(z)| whenever |z|>M₂ we say that f?g. Let X be a reproducing Hilbert space with an orthogonal basis . We say that X is an ordered reproducing Hilbert space (or X is ordered) if f?g and g∈X imply f∈X. In this note, we show that if then X is ordered; if then X is not ordered. In the case , there are examples to show that X can be of order or opposite.     

Additive Jordan isomorphisms of nest algebras on normed spaces

Let X be a real or complex Banach space. Let and be two nest algebras on X. Suppose that φ is an additive bijective mapping from onto such that φ(A²)=φ(A)² for every . Then φ is either a ring isomorphism or a ring anti-isomorphism. Moreover, if X is a real space or an infinite dimensional complex space, then there exists a continuous (conjugate) linear bijective mapping T such that either φ(A)=TAT⁻¹ for every or φ(A)=TA∗T−1 for every .     

Bounded tightness for locally convex spaces and spaces C(X)

We show that metrizability and bounded tightness are actually equivalent for a large class of locally convex spaces including (LF)-spaces, (DF)-spaces, the space of distributions D′(Ω), etc. A consequence of this fact is that for the bounded tightness for the weak topology of X is equivalent to the following one: X is linearly homeomorphic to a subspace of . This nicely supplements very recent results of Cascales and Raja. Moreover, we show that a metric space X is separable if the space C_p(X) has bounded tightness.     

Convergence in mean of some random Fourier series

For a symmetric stable process X(t,ω) with index α∈(1,2], f∈Lp[0,2π], p?α, and , we establish that the random Fourier-Stieltjes (RFS) series converges in the mean to the stochastic integral , where fβ is the fractional integral of order β of the function f for . Further it is proved that the RFS series is Abel summable to . Also we define fractional derivative of the sum of order β for an, An(ω) as above and . We have shown that the formal fractional derivative of the series of order β exists in the sense of mean.     

A hypercyclicity criterion with applications

A continuous linear operator on a topological vector space X is called hypercyclic if there is x∈X such that the orbit {Tnx}_n?0 is dense in X. We establish a criterion for hypercyclicity, and study some applications. In particular, we establish hypercyclic left-multipliers on the space L(X,Y) of continuous linear operators between X and Y, provided with the topology of uniform convergence on bounded sets, for some spaces X,Y of holomorphic functions.     

Analytic discs in the boundary and compactness of Hankel operators with essentially bounded symbols

We derive conditions for compactness of Hankel operators () with bounded, holomorphic symbols f for a large class of convex and bounded domains Ω with Ω⊆Dk.     

Resolvents of operators and partial functional differential equations with nonautonomous past

To a backward evolution family on a Banach space X we associate an abstract differential operator G through the integral equation on a Banach space of X-valued functions on . We compute the resolvent of the restriction of this operator to a smaller domain to obtain a generator. We then apply the results to prove existence, exponential stability and exponential dichotomy of solutions to partial functional equations with nonautonomous past as discussed in [S. Brendle, R. Nagel, Dist. Contin. Dynam. Systems 8 (2002) 953-966]. Our main tools are spectral mapping theorems for evolution semigroups and hyperbolicity criteria.     

Reflection about k-jets and holomorphic extension of CR mappings

Let be a CR mapping between real analytic generic submanifolds M, M₁ of and , respectively. According to Webster's theory (Proc. Amer. Math. Soc. 86 (1982) 236-240) and its further developments, f has holomorphic extension to a full neighborhood of M in when the following requirements are fulfilled: f extends to a wedge W continuous up to M; f is of class C^k; (where denotes the complex tangent bundle); M₁ is “k-nondegenerate.” We deal here with the case where is strictly smaller than but is still real analytic in suitable sense. We show that a suitably refined condition of k-nondegeneracy still entails holomorphic extension of f.