Shangquan Bu

OPERATOR-VALUED FOURIER MULTIPLIERS ON PERIODIC BESOV SPACES AND APPLICATIONS

Let 1 p, q ∞, s ∈ R and let X be a Banach space. We show that the analogue of Marcinkiewicz’s Fourier multiplier theorem on Lp(T) holds for the Besov space Bs p,q(T;X) if and only if 1 < p < ∞ and X is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann’s result (Math. Nachr. 186 (1997), 5–56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.

Approximation of Jensen measures by image measures under holomorphic functions and applications

We show that Jensen measures defined on C n or more generally on a complex Banach space X can be approximated by the image of Lebesgue measure on the torus under X-valued polynomials defined on C. We give similar characterizations for Jensen measures in terms of analytic martingales and Hardy martingales. The results are applied to approximate plurisubharmonic martingales by Hardy martingales, which enables us to give a characterization of the analytic Radon-Nikodym property of Banach spaces in terms of convergence of plurisubharmonic martingales, thus solving a problem of G. A. Edgar

Tools for maximal regularity

Let A be the generator of an analytic C0-semigroup on a Banach space X .W e associate a closed operator A1 with A dened on Rad(X) and show that when X is a UMD-space, the Cauchy problem associated with A has maximal regularity if and only if the operator A1 generates an analytic C0-semigroup on Rad(X). This allows us to exploit known results on analytic C0-semigroups to study maximal regularity. Our results show that R-boundedness is a local property for semigroups: an analytic C0-semigroup T of negative type is R-bounded if and only if it is R-bounded at z = 0. As applications, we give a perturbation result for positive semigroups. Finally, we show the following: when X is a UMD-space, T is an analytic C0-semigroup of negative type, then for every f2 L p (R+;X), the mild solution of the corresponding inhomogeneous Cauchy problem with initial value 0 belongs toW ;p (R+;X) for every 0 << 1.

Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces

Abstract In this paper, we study a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of variational inequalities for two inverse strongly accretive mappings and the set of common fixed points of an infinite family of nonexpansive mappings in a real 2-uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the corresponding results announced by many others.

Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces

In this paper, we introduce two iterative algorithms for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of finite variational inequalities for inverse strongly monotone mappings and the set of common fixed points of an asymptotically k-strictly pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems by using the proposed algorithms under some suitable conditions. Our results improve and extend the corresponding results announced by many others.

Functional calculus, variational methods and Liapunov's theorem

Abstract. Given the generator −A of a holomorphic semigroup on a Hilbert space H, we show that A is associated with a closed form if and only if

Hybrid algorithm for generalized mixed equilibrium problems and variational inequality problems and fixed point problems

A+w\in BIP(H)

Modified extragradient methods for variational inequality problems and fixed point problems for an infinite family of nonexpansive mappings in Banach spaces

for some

Approximation of common fixed points of a countable family of continuous pseudocontractions in a uniformly smooth Banach space

w\in\Bbb{R}

Fourier Series in Banach spaces and Maximal Regularity

. Under this condition we also show that Liapunovs classical theorem is true, in the linear as well as the semilinear case.