PERIODIC SYSTEMS WITH TIME DEPENDENT MAXIMAL MONOTONE OPERATORS

We consider a first order periodic system in R^(N),involving a time dependent maximal monotone operator which need not have a full domain and a multivalued perturbation.We prove the existence theorems for both the convex and nonconvex problems.We also show the existence of extremal periodic solutions and provide a strong relaxation theorem.Finally,we provide an application to nonlinear periodic control systems.

Iterative convergence theorems for maximal monotone operators and relatively nonexpansive mappings

In this paper, some iterative schemes for approximating the common element of the set of zero points of maximal monotone operators and the set of fixed points of relatively nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Some strong convergence theorems are obtained, to extend the previous work.

New iterative schemes for strongly relatively nonexpansive mappings and maximal monotone operators

In this paper, some new iterative schemes for approximating the common element of the set of fixed points of strongly relatively nonexpansive mappings and the set of zero points of maximal monotone operators in a real uniformly smooth and uniformly convex Banach space are proposed. Some weak convergence theorems are obtained, which extend and complement some previous work.

MODIFIED APPROXIMATE PROXIMAL POINT ALGORITHMS FOR FINDING ROOTS OF MAXIMAL MONOTONE OPERATORS

In order to find roots of maximal monotone operators, this paper introduces and studies the modified approximate proximal point algorithm with an error sequence {e k} such that || ek || \leqslant hk || xk - [(x)\tilde]k ||\left\| { e^k } \right\| \leqslant \eta _k \left\| { x^k - \tilde x^k } \right\| with ?k = 0￥ ( hk - 1 ) < + ￥\sum\limits_{k = 0}^\infty {\left( {\eta _k - 1} \right)} and infk \geqslant 0 hk = m\geqslant 1\mathop {\inf }\limits_{k \geqslant 0} \eta _k = \mu \geqslant 1 . Here, the restrictions on {η k} are very different from the ones on {η k}, given by He et al (Science in China Ser. A, 2002, 32 (11): 1026–1032.) that supk \geqslant 0 hk = v < 1\mathop {\sup }\limits_{k \geqslant 0} \eta _k = v . Moreover, the characteristic conditions of the convergence of the modified approximate proximal point algorithm are presented by virtue of the new technique very different from the ones given by He et al.

AN ITERATIVE ALGORITHM FOR MAXIMAL MONOTONE MULTIVALUED OPERATOR EQUATIONS

A proximal iterative algorithm for the mulitivalue operator equation 0 ∈ T(x) is presented, where T is a maximal monotone operator. It is an improvement of the proximal point algorithm as well know. The convergence of the algorithm is discussed and all example is given.

Anti-periodic solutions to a class of second-order evolution equations

In this paper we discuss the anti-periodic problem for a class of abstractnonlinear second-order evolution equations associated with maximal monotone operators in Hilbertspaces and give some new assumptions on operators. We establish the existence and uniqueness ofanti-periodic solutions, which improve andgeneralize the results that have been obtained. Finally weillustrate the abstract theory by discussing a simple example of an anti-periodic problem fornonlinear partial differential equations.

EXISTENCE OF SOLUTIONS OF A FAMILY OF NONLINEAR BOUNDARY VALUE PROBLEMS IN L^2-SPACES

By using the perturbation results of sums of ranges of accretive mappings of Calvert and Gupta(1978),the abstract results on the existence of solutions of a family of nonlinear boundary value problems in L2(Ω) are studied.The equation discussed in this paper and the methods used here are extension and complement to the corresponding results of Wei Li and He Zhen's previous papers.Especially,some new techniques are used in this paper.

Generalized mixed equilibrium problem in Banach spaces

This paper uses a hybrid algorithm to find a common element of the set of solutions to a generalized mixed equilibrium problem, the set of solutions to variational inequality problems, and the set of common fixed points for a finite family of quasi-C- nonexpansive mappings in a uniformly smooth and strictly convex Banach space. As applications, we utilize our results to study the optimization problem. It shows that our results improve and extend the corresponding results announced by many others recently.

RESEARCH ON THE EXISTENCE OF SOLUTION OF EQUATION INVOLVING p-LAPLACIAN OPERATOR

By using the perturbation theories on sums of ranges for nonlinear accretive mappings of Calvert and Gupta （1978）,the abstract result on the existence of a solution u ∈ L^p （Ω） to nonlinear equations involving p-Laplacian operator △p, where 2N/N＋1〈p〈＋∞ and N （≥ 1 ） denotes the dimension of R^N,is studied. The equation discussed and the methods shown in the paper are continuation and complement to the corresponding results of Li and Zhen＇s previous papers. To obtain the result ,some new techniques are used.

EXISTENCE AND UNIQUENESS OF NON-TRIVIAL SOLUTION OF PARABOLIC p-LAPLACIAN-LIKE DIFFERENTIAL EQUATION WITH MIXED BOUNDARIES

One parabolic p-Laplacian-like differential equation with mixed boundaries is au/at in the corresponding studies is replaced by a（au/at）, studied in this paper, where the item au/at which makes it more general. The sufficient condition of the existence and uniqueness of non-trivial solution in L2（O, T; L2 （Ω）） is presented by employing the techniques of splitting the boundary problems into operator equation. Compared to the corresponding work, the restrictions imposed on the equation are weaken and the proof technique is simplified. It can be regarded as the extension and complement of the previous work.

CONVERGENCE OF INVARIANT MEASURES FOR MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

This article is concerned with the weak convergence of invariant measures asso- ciated with multivalued stochastic differential equations in the finite dimensional space.

Iterative Convergence Theorems for Maximal Monotone Operators and Nonexpansive Mappings and Their Applications

In this paper, two iterative schemes for approximating common element of the set of zero points of maximal monotone operators and the set of fixed points of a kind of generalized nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Two strong convergence theorems are obtained and their applications on finding the minimizer of a kind of convex functional are discussed, which extend some previous work.

Projection Scheme for Zero Points of Maximal Monotone Operators in Banach Spaces

A new projection scheme with errors for zero points of maximal monotone operators is introduced and is proved to be strongly convergent to zero points of maximal monotone operators in Banach space by using the techniques of Lyapunov functional and generalized projection operator, etc.

Existence and Uniqueness of Renormalized Solution of Nonlinear Degenerated Elliptic Problems

In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion β（u）-div（α（x, Du）＋F（u）） ∈ f in fΩ, where f ∈ L1 （Ω）. A vector field a（.,.） is a Carath6odory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general L1-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.

MULTIVALUED STOCHASTIC MCKEAN-VLASOV EQUATION

In this work, under global Lipschitz conditions, we prove the existence and uniqueness of strong solutions for multivalued stochastic McK ean-Vlasov equation. Moreover, under continuous and linear growth assumptions, we also obtain the existence of weak solutions.

THE EXISTENCE OF SOLUTIONS OF NONLINEAR BOUNDARY VALUE PROBLEMS INVOLVING THE p-LAPLACIAN OPERATOR IN L^s-SPACES

By using the perturbation theories on sums of ranges of nonlinear accretive mappings of Calvert and Gupta, we study the abstract results on the existence of a solution u ∈ L^s （Ω） of nonlinear boundary value problems involving the p-Laplacian operator, where 2≤ s〈＋∞, and 2N/N＋1 〈 p ≤ 2 for N（≥ 1） which denotes the dimension of R^N. To obtain the result, some new techniques are used in this paper. The equation discussed in this paper and our methods here are extension and complement to the corresponding results of L. Wei and Z. He.

Existence of Solutions to Nonlinear Neumann Boundary Value Problems with p-Laplacian Operator and Iterative Construction

By using some results of pseudo-monotone operator, we discuss the existence and uniqueness of the solution of one kind nonlinear Neumann boundary value problems involving the p-Laplacian operator. We also construct an iterative scheme converging strongly to this solution.

Existence of Solutions for Nonlinear Neumann Boundary Value Problems

Using perturbation theories on sums of ranges of nonlinear accretive mappings of Calvert and Gupta, we present the abstract results on the existence of solutions of one kind nonlinear Neumann boundary value problems related to p-Laplacian operator. The equation discussed in this paper and the method used here extend and complement some of the previous work.

Existence of anti-periodic solutions for hemivariational inequalities

J.Y. Park and T. G. Ha [Nonlinear Anal., 2008, 68： 747-767; 2009, 7h 3203-3217] investigated the existence of anti-periodic solutions for hemivariational inequalities with a pseudomonotone operator. In this note, we point out that the methods used there are not suitable for the proof of the existence of anti-periodic solutions for hemivariational inequalities and we shall give a straightforward approach to handle these problems. The main tools in our study are the maximal monotone property of the derivative operator with anti- periodic conditions and the surjectivity result for L-pseudomonotone operators.

Multivalued Stochastic Differential Equations with Non-Lipschitz Coeffcients

The existence and uniqueness of solutions to the multivalued stochastic differential equations with non-Lipschitz coefficients are proved, and bicontinuous modifications of the solutions are obtained.