EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.

Hàjek-Rèniy Type Inequality for Banach Space Valued Martingales and Its Applications

In this paper,we establish the Hàjek-Rèniy type inequality for Banach space valued martingales generalizing the recent results of Tómcs and L'ibor [1].Then p-uniformly smoothable Banach space is characterized in terms of the Hàjek-Rèniy type inequality for Banach space valued martingales.Those results generalize the recent results of Gan Shixin [2].

Gap Functions and Error Bounds for Set-Valued Vector Quasi Variational Inequality Problems

One of the classical approaches in the analysis of a variational inequality problem is to transform it into an equivalent optimization problem via the notion of gap function. The gap functions are useful tools in deriving the error bounds which provide an estimated distance between a specific point and the exact solution of variational inequality problem. In this paper, we follow a similar approach for set-valued vector quasi variational inequality problems and define the gap functions based on scalarization scheme as well as the one with no scalar parameter. The error bounds results are obtained under fixed point symmetric and locally α-Holder assumptions on the set-valued map describing the domain of solution space of a set-valued vector quasi variational inequality problem.

Davis's Inequalities for Two-Parameter Banach Space Valued Strong Martingales and Geometrical Properties of Banach Spaces

In this paper we investigated theL 1 norm inequalities of theP square and the maximal functions of two-parameterB-valued strong martingales, which can be applied to characterizep-smoothness andq-convexity of Banach spaces.

Auxiliary principle and three-step iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities

An auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities is extended. The existence and uniqueness of the solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities are proved, a novel and innovative three-step iterative algorithm to compute approximate solution is constructed, and the existence of the solution of the generalized set-valued strongly nonlinear mixed variational-like inequality is shown using the auxiliary principle iterative sequences generated by the algorithm technique. The convergence of three-step is also proved.

On the Inequalities for Traces and Singular Values of the Quaternion Matrices

In this paper we derive some inequalities for traces and singular values of the quaternion matrices,extend and improve some of the corresponding results appeared in other papers we know.

SOME INEQUALITIES OF GENERALIZED SINGULAR VALUES OF MATRICES

@1 Definition 1 Let A=(αij)∈Cn×n,B=(bij)∈Cn×n,is nonsingular.The generalizedsingular values of A（relative to B）are following determinate nonnegative real numberswhen ||·||2 denotes the Euclid vector norm,〈n〉={1,2,…,n}.Definition 2 Let A,B∈Cn×n,if there exist λ∈C and x∈Cn\{0}。

Singular Value Inequalities for Compact Normal Operators

We give singular value inequality to compact normal operators, which states that if is compact normal operator on a complex separable Hilbert space, where is the cartesian decomposition of , then Moreover, we give inequality which asserts that if?is compact normal operator, then .Several inequalities will be proved.

More Results on Singular Value Inequalities for Compact Operators

The well-known arithmetic-geometric mean inequality for singular values, according to Bhatia and Kittaneh, says that if and are compact operators on a complex separable Hilbert space, then Hirzallah has proved that if are compact operators, then We give inequality which is equivalent to and more general than the above inequalities, which states that if are compact operators,

DIFFERENTIAL INEQUALITIES OF OPERATOR-VALUED ANALYTIC FUNCTIONS

Some differential inequalities and integral inequalities for operator-valued an- alytic functions are investigated.

Projected subgradient method for non-Lipschitz set-valued mixed variational inequalities

A projected subgradient method for solving a class of set-valued mixed variational inequalities （SMVIs） is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven that the sequence generated by the method can strongly converge to the unique solution to the problem in the Hilbert spaces.

Two-Parameter Vector-Valued Martingales and Geometrical Properties of Banach Spaces

We obtained a number of inequalities and laws of large numbers for two parameter vector valued martingales.In the other direction we characterized p smoothness and q convexity of Banach spaces by using these inequalities and laws of large numbers for two parameter vector valued martingales.

A Class of Strongly Nonlinear Singular Perturbed Boundary Value Problems

In this paper, a class of strongly nonlinear singular perturbed boundary value problems are coasidered by the theory of differential inequalities and the correction of boundary layer, under which the existence of solution is proved and the uniformly valid asymptotic expansions is obtained as well.

THE SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER SEMILINEAR ELLIPTIC EQUATIONS

In this paper a singular perturbation of boundary value problem for elliptic partial differential equations of higher order is considered by using the differential inequalities. The uniformly valid asymptotic expansion in entire region is obtained.

Initial-boundary value problem of nonlinear hyperbolic system for conservation laws with delta-shock waves

For a nonlinear hyperbolic system of conservation laws, the initial-boundary value problem is concerned with the boundary conditions. A boundary entropy condition is derived based on Dubois F and Le Floch P＇s results by taking a suitable entropy-flux pair （Journal of Differential Equations, 1988, 71（1）： 93-122）. The solutions of the initial-boundary value problem for the system are constructively obtained, in which initial-boundary data are in piecewise constant states. The delta-shock waves appear in their solutions.

THE NONLINEAR BOUNDARY VALUE PROBLEM FOR A CLASS OF INTEGRO-DIFFERENTIAL SYSTEM

In this paper, using the theory of differential inequalities, we study the nonlinear boundary value problem for a class of integro-differential system. Under appropriate assumptions, the existence of solution is proved and the uniformly valid asymptotic expansions for arbitrary n-th order approximation and the estimation of remainder term are obtained simply and conveniently.

Singular Perturbations for a Class of Fourth-order Nonlinear Boundary Value Problem

This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p（y＇＇（0）,y＇＇＇（0））=0,q（y＇＇（1）,y＇＇＇（1））=0 where f：[0,1]×R3→R is continuous,p,q：R2→R are continuous.Under certain conditions,by introducing an appropriate stretching transformation and constructing boundary layer corrective terms,an asymptotic expansion for the solution of the problem is obtained.And then the uniformly validity of solution is proved by using the differential inequalities.

OPERATOR-VALUED FOURIER MULTIPLIER THEOREMS ON TRIEBEL SPACES

The authors establish operator-valued Fourier multiplier theorems on Triebel spaces on R^N, where the required smoothness of the multiplier functions depends on the dimension N and the indices of the Triebel spaces. This is used to give a sufficient condition of the maximal regularity in the sense of Triebel spaces for vector-valued Cauchy problems with Dirichlet boundary conditions.

CONTINUITY OF l^2-VALUED TWO-PARAMETER ORNSTEIN UHLENBECK PROCESSES

Let {Y (t);t= (t 1,t 2)≥0}={X k(t 1,t 2);t 1≥0,t 2≥0} ∞ k=1 be a sequence of two parameter Ornstein Uhlenbeck processes (OUP 2) with coefficients α k>0,β k>0 . A Fernique type inequality is established and the sufficient condition for a.s. l 2 continuity of Y(·) is studied by means of the inequality.

Bohr Inequality for Multiple Operators

An absolute value equation is established for linear combinations of two operators.When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in literature [3] and its equivalent deformation are obtained.Based on the equivalent deformation of the operator equation and using the properties of conjugate number as well as the operator, an absolute value identity of multiple operators is given by means of mathematical induction. As Corollaries, Bohr inequalities are extended to multiple operators and some related inequalities are reduced to, such as inequalities in [2]and [3].