Monotone generalized variational inequalities and generalized complementarity problems

Some existence results for generalized variational inequalities and generalized complementarity problems involving quasimonotone and pseudomonotone set-valued mappings in reflexive Banach spaces are proved. In particular, some known results for nonlinear variational inequalities and complementarity problems in finite-dimensional and infinite-dimensional Hilbert spaces are generalized to quasimonotone and pseudomonotone set-valued mappings and reflexive Banach spaces. Application to a class of generalized nonlinear complementarity problems studied as mathematical models for mechanical problems is given.The research of the first author was supported by the National Natural Science Foundation of P. R. China and by the Ethel Raybould Fellowship, University of Queensland, St. Lucia, Brisbane, Australia.     

On generalized variational inequalities

We obtain a new version of the minimax inequality of Ky Fan. As an application, an existence result for the generalized variational inequality problem with set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces is given. Also, some existence results for the generalized variational inequality problem for quasimonotone and pseudomonotone mappings are obtained. Dedicated to the memory of T. Rapcsák.     

Existence results for evolution hemivariational inequalities of second order

In this paper, we consider evolution hemivariational inequalities of second order with a time-dependent pseudomonotone operator and nonmonotone multivalued perturbations. We present the existence of solutions for such inequality. The proof profits from a result on the surjectivity of operators of pseudomonotone type. We discuss some examples which indicate the practical importance of our theoretical findings.     

FIXED POINT THEOREMS OF GENERALIZED CONTRACTIVE TYPE MAPPINGS(Ⅱ)

In this paper the author obtains several new fixed point theorems for generalized contractive type mappings by means of Kwapisz's contractive gauge function and then proves that lots of contractive type mappings are topologically equivalent to Banach contraction with given contractive constant C∈[0, 1).     

Stable pseudomonotone variational inequality in reflexive Banach spaces

Stability of a generalized variational inequality with either the mapping or the set perturbed is discussed in reflexive Banach spaces, provided that the mappings are pseudomonotone in the sense of Karamardian. As a byproduct, generalized variational inequality having nonempty and bounded set is proved to be equivalent to the strictly feasibility.     

拟-A-Proper映射的拓扑度

拓扑度理论为研究非线性方程多解问题提供了强有力的工具。迄今拓扑度方法已成为非线性泛函分析中最基本的方法之一(关于非线性泛函分析的发展近况请见田方增[1])。 本文提出很广泛的拟-A-proper映射类,用A-proper映射([5])逼近拟-A-proper映射的方法(类似[2])建立了后者的拓扑度;证明出两类映射拓扑度的性质几乎全同;讨论     

Existence of antiperiodic solutions for hemivariational inequalities

In this paper we consider the problem of existence of antiperiodic solutions for first-order and second-order hemivariational inequalities with a pseudomonotone operator. We first give the surjectivity result and then prove a existence of antiperiodic solutions for hemivariational inequalities with the surjectivity result.     

Existence of solutions and periodic solutions for nonlinear evolution inclusions

In this paper we consider nonlinear-dependent systems with multivalued perturbations in the framework of an evolution triple of spaces. First we prove a surjectivity result for generalized pseudomonotone operators and then we establish two existence theorems: the first for a periodic problem and the second for a Cauchy problem. As applications we work out in detail a periodic nonlinear parabolic partial differential equation and an optimal control problem for a system driven by a nonlinear parabolic equation.     

Common fixed point theorems of Gregus type for weakly compatible mappings satisfying generalized contractive conditions

We prove a common fixed point theorem of Gregus type for four mappings satisfying a generalized contractive condition in metric spaces using the concept of weak compatibility which generalizes theorems of [I. Altun, D. Turkoglu, B.E. Rhoades, Fixed points of weakly compatible mappings satisfying a general contractive condition of integral type, Fixed Point Theory Appl. 2007 (2007), article ID 17301; A. Djoudi, L. Nisse, Gregus type fixed points for weakly compatible mappings, Bull. Belg. Math. Soc. 10 (2003) 369-378; A. Djoudi, A. Aliouche, Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type, J. Math. Anal. Appl. 329 (1) (2007) 31-45; P. Vijayaraju, B.E. Rhoades, R. Mohanraj, A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type, Int. J. Math. Math. Sci. 15 (2005) 2359-2364; X. Zhang, Common fixed point theorems for some new generalized contractive type mappings, J. Math. Anal. Appl. 333 (2) (2007) 780-786]. We prove also a common fixed point theorem which generalizes Theorem 3.5 of [H.K. Pathak, M.S. Khan, T. Rakesh, A common fixed point theorem and its application to nonlinear integral equations, Comput. Math. Appl. 53 (2007) 961-971] and common fixed point theorems of Gregus type using a strict generalized contractive condition, a property (E.A) and a common property (E.A).     

Generalization of Ulam stability problem for Euler-Lagrange quadratic mappings

In 1968 S.M. Ulam proposed the problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?” In 1978 P.M. Gruber proposed the Ulam type problem: “Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this object by objects, satisfying the property exactly?” In this paper we solve the generalized Ulam stability problem for non-linear Euler-Lagrange quadratic mappings satisfying approximately a mean equation and an Euler-Lagrange type functional equations in quasi-Banach spaces and p-Banach spaces.     

MULTIVALUED DIFFERENTIAL EQUATIONS IN BANACH SPACES AND THEIR APPLICATIONS

IIntroductionConsider the initial value problemDtL干Au＋1ugi．U＜t＜I,oh山)”w,where A is of of。s(S＋)and F is a multivalued msp．If F is singl。vlued Hiranoti goteistence results for Fbeing compact．Ahmed and Xiang[21 extended Hirano’s results forFsatls州ngthat＜Fu。,u。一U＞－＋ 0 lfunconverges wewyto h;While Liul3]e枕ended theabOV6 th6lltlollsd T6SllltS to COf正erpolldlllg ClAsS Of yatlatlollal ill6qttallt16s ld lfittodllC6d 8 hO16gmerdC0nd让ND th幼劝m比a bro劝0 d%sdpenurb砒…     

Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebe?s canonical slit domains

This paper presents a boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region G onto the second, third and fourth categories of Koebe?s canonical slit domains. The method can be also used for calculating the conformal mappings of simply and doubly connected regions. The method is an extension of the author?s method for the first category of Koebe?s canonical slit domains (see [M.M.S. Nasser, Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. Sci. Comput. 31 (2009) 1695-1715]). Three numerical examples are presented to illustrate the performance of the proposed method.     

On the Tikhonov regularization of affine pseudomonotone mappings

The pseudomonotonicity of affine mappings on polyhedral convex sets is characterized in the one-dimensional case and in a higher-dimensional setting. The obtained results allow us to investigate the pseudomonotonicity of the regularized mappings (in the sense of Tikhonov regularization). Among other things, it is shown that there exists a pseudomonotone affine variational inequality problem VI( $K,F$ ) with a nonempty solution set for which the regularized problem VI( $K,F_\varepsilon $ ) is not pseudomonotone for every $\varepsilon \in (0,\frac{1}{2})$ . In addition, we prove that the feasibility of a pseudomonotone linear complementarity problem implies the solution uniqueness of the regularized problem.     

On the Ulam stability of mixed type mappings on restricted domains

In 1941 D.H. Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 D.G. Bourgin was the second author to treat the Ulam problem for additive mappings. In 1982-1998 we established the Hyers-Ulam stability for the Ulam problem of linear and nonlinear mappings. In 1983 F. Skof was the first author to solve the Ulam problem for additive mappings on a restricted domain. In 1998 S.M. Jung investigated the Hyers-Ulam stability of additive and quadratic mappings on restricted domains. In this paper we improve the bounds and thus the results obtained by S.M. Jung, in 1998. Besides we establish the Ulam stability of mixed type mappings on restricted domains. Finally, we apply our recent results to the asymptotic behavior of functional equations of different types.     

Generalized Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces

In 1940 S.M. Ulam proposed the famous Ulam stability problem. In 1941 D.H. Hyers solved the well-known Ulam stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. The first author of this paper investigated the Hyers-Ulam stability of Cauchy and Jensen type additive mappings. In this paper we generalize results obtained for Jensen type mappings and establish new theorems about the Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces.     

A NOTE ON GENERALIZED DEGREE FOR GENERALIZED GRADIENT MAPPINGS

In this paper the author studies the generalized degree, for Clarke''s generalized gradient mappings which are multivalued A-proper in Banach spaces. The results of H. Amann on degree theory for gradient mappings which are compact vector fields in Hilbert spaces are extended.     

A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact

In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the Kelvin–Voigt viscoelastic law, include the thermal effects and consider the general nonmonotone and multivalued subdifferential boundary conditions. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is proved by using a surjectivity result for operators of pseudomonotone type. The uniqueness is obtained for a large class of operators of subdifferential type satisfying a relaxed monotonicity condition.     

The coupling of boundary integral and finite element methods for nonmonotone nonlinear problems ∗

In this paper, we apply the coupling of the boundary integral and finite element methods to study the weak solvability of certain nonmonotone nonlinear exterior boundary value problems. In order to convert the original exterior problem into an equivalent nonlocal boundary value problem on a finite region, we employ two different approaches based on the use of one and two integral equations on the coupling boundary. Existence of a solution for the associated weak formulation, and convergence properties of the corresponding Galerkin approximations are deduced from fundamental results in nonlinear functional analysis. Indeed, the main arguments of our proofs are based on a surjectivity theorem for mappings of type (S) and on the Fredholm alternative for nonlinear A-proper mappings.     

Existence of solutions for hyperbolic hemivariational inequalities

In this paper we study a hyperbolic hemivariational inequality with a nonlinear, pseudomonotone operator depending on the derivative of an unknown function and a linear, monotone operator depending on an unknown function. Using the surjectivity result for L-pseudomonotone operators, an existence result for such inequalities is proved.     

Existence of anti-periodic solutions for hemivariational inequalities

J. Y. Park and T. G. Ha [Nonlinear Anal., 2008, 68: 747-767; 2009, 71: 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 antiperiodic conditions and the surjectivity result for L-pseudomonotone operators.