Optimal solutions to variational inequalities on Banach lattices

We study the existence of maximum and minimum solutions to generalized variational inequalities on Banach lattices. The main tools of analysis are the variational characterization of the generalized metric projection operator and order-theoretic fixed point theory.     

ψ-序压缩算子的不动点定理

2005年,张宪在Banach空间中通过其中的锥所定义的半序引进了序压缩算子,证明了几个相应的定理．但是在一般的度量空间中,能否定义序压缩算子,能否得到类似的结论呢？本文在度量空间X中,通过X上的泛函ψ-所定义的半序,引进了ψ--序压缩算子,并且得到了相应的不动点定理．     

Isotonicity of the metric projection with applications to variational inequalities and fixed point theory in Banach spaces

In this paper, we discuss isotonicity characterizations of the metric projection operator, including its necessary and sufficient conditions for isotonicity onto sublattices in Banach spaces. Then, we demonstrate their applications to variational inequalities and fixed point theory in Banach spaces. Our work generalizes many existing results obtained in earlier work.     

Projection algorithms for the system of mixed variational inequalities in Banach spaces

In this paper, the system of mixed variational inequalities is introduced and considered in Banach spaces, which includes some known systems of variational inequalities and the classical variational inequalities as special cases. Using the projection operator technique, we suggest some iterative algorithms for solving the system of mixed variational inequalities and prove the convergence of the proposed iterative methods under suitable conditions. Our theorems generalize some known results shown recently.     

The existence of zeros of set-valued mappings in reflexive Banach spaces

In this paper, we establish the equivalence between the existence of zeros for set-valued mappings and the solvability of variational inequality, under some conditions involving the generalized projection operator. Basing on this result, we obtain some existence theorems of zeros for quasimonotone set-valued mappings. As application, we derive several fixed point theorems for generalized inward mappings in Hilbert spaces.     

Variational conclusions of set-valued bifunctions on convex subsets of Banach spaces with applications

In this paper, variational conclusions of set-valued bifunctions on convex subsets of Banach spaces are investigated and several new results are obtained. The results are applied to the fixed point theory and variational inequalities. We obtain two fixed point theorems and existence theorems of solutions of variational inequalities.     

Error Estimate of Data Dependence for Discontinuous Operators by New Iteration Process with Convergence Analysis

Abstract

In this paper, we introduce a new discontinuous operator and investigate the existence and uniqueness of fixed points for the operators in complete metric spaces. We also provide rate of convergence and data dependency of S-iterative scheme for a fixed point of the discontinuous operators in Banach spaces. Moreover, we prove the estimation Collage theorems and compare error estimate between data dependency and Collage theorems. Numerical examples are provided to support our results.     

Fixed point theory for generalized contractions in cone metric spaces

In this paper, we prove some fixed point theorems for generalized contractions in cone metric spaces. Our theorems extend some results of Suzuki (2008) [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc Amer Math Soc 136(5) (2008), 1861-1869] and Kikkawa and Suzuki (2008) [M. Kikkawa and T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal 69(9) (2008), 2942-2949].     

The general iterative methods for nonexpansive mappings in Banach spaces

In this paper, we introduce a general iterative approximation method for finding a common fixed point of a countable family of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The main result extends various results existing in the current literature.     

A note on cone metric fixed point theory and its equivalence

The main aim of this paper is to investigate the equivalence of vectorial versions of fixed point theorems in generalized cone metric spaces and scalar versions of fixed point theorems in (general) metric spaces (in usual sense). We show that the Banach contraction principles in general metric spaces and in TVS-cone metric spaces are equivalent. Our theorems also extend some results in Huang and Zhang (2007) [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476], Rezapour and Hamlbarani (2008) [Sh. Rezapour, R. Hamlbarani, Some notes on the paper Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 345 (2008) 719-724] and others.     

Non-zero solutions for a class of generalized variational inequalities in reflexive Banach spaces

In this paper, we study the existence of nonzero solutions for a class of generalized variational inequalities involving set-contractive mappings by using the fixed point index approach in reflexive Banach spaces. Under some suitable assumptions, we show some new existence theorems of nonzero solutions for this class of generalized variational inequalities in reflexive Banach spaces.     

Controlled fuzzy metric spaces and some related fixed point results

In this paper, we introduce a new extension in the subject of fuzzy metric, called controlled fuzzy metric space. This notion is a generalization of fuzzy b‐metric space. Also, we prove a Banach‐type fixed point theorem and a new fixed point theorem for some self‐mappings satisfying fuzzy ψ ‐contraction condition that is more general than existing theorems. Furthermore, we establish some examples about our main results.     

对带误差项的Banach空间中完全广义集值拟变分包含的摄动迭代算法

研究了一类新的实Banach空间中的广义集值拟变分包含:f∈N(x，y) M(z，v) W(g(u)-h(w)，u)，它包含了近几年许多作者所作的变分包含同题．在买Banach空间中，利用极大增生算子的性质，建立了Banach空间中的广义集值拟变分包含和不动点问题间的等价性．利用这种等价性，建立了一些摄动迭代算法，并证明了近似解序列强收敛于精确解．本文的算法和结果改进和一般化了最近许多文章中相应的算法和结果。     

Some common fixed point theorems for Banach operator pairs with applications in best approximation

The ordered pair (T,I) of two self-maps of a metric space (X,d) is called a Banach operator pair if the set F(I) of fixed points of I is T-invariant i.e. T(F(I))⊆F(I). Some common fixed point theorems for a Banach operator pair and the existence of common fixed points of best approximation are presented in this paper. The results prove, generalize and extend some results of Al-Thagafi [M.A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996) 318-323], Carbone [A. Carbone, Applications of fixed point theorems, Jnanabha 19 (1989) 149-155], Chen and Li [J. Chen, Z. Li, Common fixed points for Banach operator pairs in best approximations, J. Math. Anal. Appl. 336 (2007) 1466-1475], Habiniak [L. Habiniak, Fixed point theorems and invariant approximation, J. Approx. Theory 56 (1989) 241-244], Jungck and Sessa [G. Jungck, S. Sessa, Fixed point theorems in best approximation theory, Math. Japon. 42 (1995) 249-252], Sahab, Khan and Sessa [S.A. Sahab, M.S. Khan, S. Sessa, A result in best approximation theory, J. Approx. Theory 55 (1988) 349-351], Shahzad [N. Shahzad, Invariant approximations and R-subweakly commuting maps, J. Math. Anal. Appl. 257 (2001) 39-45] and of few others.     

半序概率度量空间中压缩算子的不动点定理

半序方法是研究非线性算子方程问题的主要方法之一.在概率度量空间中引入半序,并且利用半序方法研究了非线性算子的不动点问题,推广了度量空间中序压缩算子的不动点定理,获得若干新的结果.     

Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings

In this paper, the famous Banach contraction principle and Caristi's fixed point theorem are generalized to the case of multi-valued mappings. Our results are extensions of the well-known Nadler's fixed point theorem [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-487], as well as of some Caristi type theorems for multi-valued operators, see [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188; J.P. Aubin, Optima and Equilibria. An Introduction to Nonlinear Analysis, Grad. Texts in Math., Springer-Verlag, Berlin, 1998, p. 17; S.S. Zhang, Q. Luo, Set-valued Caristi fixed point theorem and Ekeland's variational principle, Appl. Math. Mech. 10 (2) (1989) 111-113 (in Chinese), English translation: Appl. Math. Mech. (English Ed.) 10 (2) (1989) 119-121], etc.     

An iterative algorithm for finding a common solution of fixed points and a general system of variational inequalities for two inverse strongly accretive operators

In this paper, we introduce an iterative scheme based on a viscosity approximation method with a modified extragradient method for finding a common solutions of a general system of variational inequalities for two inverse-strongly accretive operator and solutions of fixed point problems involving the nonexpansive mapping in Banach spaces. Consequently, we obtain new strong convergence theorems in the frame work of Banach spaces. Our results extend and improve the recent results of Qin et al. (J Comput Appl Math 233:231–240, 2009) and many others.     

Convergence theorems for mixed equilibrium problems, variational inequality problem and uniformly quasi-?-asymptotically nonexpansive mappings

We introduce a modified block hybrid projection algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi-?-asymptotically nonexpansive mappings, the set of the variational inequality for an α-inverse-strongly monotone operator, the set of solutions of the mixed equilibrium problems. Then, we obtain strong convergence theorems for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Our results extend and improve ones from several earlier works.     

Suzuki?s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces

Recently, Suzuki [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008) 1861-1869] proved a fixed point theorem that is a generalization of the Banach contraction principle and characterizes the metric completeness. In this paper we prove an analogous fixed point result for a self-mapping on a partial metric space or on a partially ordered metric space. Our results on partially ordered metric spaces generalize and extend some recent results of Ran and Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443], Nieto and Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239]. We deduce, also, common fixed point results for two self-mappings. Moreover, using our results, we obtain a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends Suzuki?s characterization of metric completeness.     

Projection algorithms for the system of generalized mixed variational inequalities in Banach spaces

Some projection algorithms are suggested for solving the system of generalized mixed variational inequalities, and the convergence of the proposed iterative methods are proved without any monotonicity assumption for the mappings in Banach spaces. Our theorems generalize some known results.