On growth and covering theorems of quasi-convex mappings in the unit ball of a complex banach space

A class of biholomorphic mappings named “quasi-convex mapping” is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to ℂⁿ, the “quasi-convex mapping” is exactly the “quasi-convex mapping of type A” introduced by K. A. Roper and T. J. Suffridge.     

Coupled fixed point results for nonlinear integral equations

In this paper, we prove some coupled coincidence point theorems for such nonlinear contraction mappings having a mixed monotone property in partially ordered metric spaces by dropping the condition of commutative. We also prove coupled common fixed point theorem for w-compatible mappings. An example of a nonlinear contraction mapping which is not applied by Lakshmikantham and ?iri?’s theorem [1] but applied by our result is given. Further, we apply our results to the existence theorem for solution of nonlinear integral equations.     

Coupled coincidence point theorems in ordered metric spaces

In this paper we extend the coupled contraction mapping theorem proved in partially ordered metric spaces by Gnana Bhaskar and Lakshmikantham (Nonlinear Anal. TMA 65:1379–1393, 2006) to a coupled coincidence point result for a pair of compatible mappings. A control function has been used in our theorem. The mappings are assumed to satisfy a weak contractive inequality. Our theorem improves the results of Harjani et al. (Nonlinear Anal. TMA 74:1749–1760, 2011). The result we have established is illustrated with an example which also shows that the improvement is actual.     

Stability of coincidence points and properties of covering mappings

Properties of closed set-valued covering mappings acting from one metric space into another are studied. Under quite general assumptions, it is proved that, if a given α-covering mapping and a mapping satisfying the Lipschitz condition with constant β < α have a coincidence point, then this point is stable under small perturbations (with respect to the Hausdorff metric) of these mappings. This assertion is meaningful for single-valued mappings as well. The structure of the set of coincidence points of an α-covering and a Lipschitzian mapping is studied. Conditions are obtained under which the limit of a sequence of α-covering set-valued mappings is an (α–?)-covering for an arbitrary ? > 0.     

Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations

We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing mappings as well as non monotone mappings. We also present some applications to first–order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution. Research partially supported by Ministerio de Educación y Ciencia and FEDER, Project MTM2004-06652-C03-01, and by Xunta de Galicia and FEDER, Projects PGIDIT02PXIC20703PN and PGIDIT05PXIC20702PN     

Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces

We introduce the concept of a mixed g-monotone mapping and prove coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces. Presented theorems are generalizations of the recent fixed point theorems due to Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393] and include several recent developments.     

On coincidence points for vector mappings

For mappings acting in the product of metric spaces we propose a concept of vector covering. This concept is a natural extension of the notion of covering formappings inmetric spaces. The statements on the solvability of systems of operator equations are proved for the case when the left-hand side of an equation is a value of a vector covering mapping and the right-hand side is Lipschitzian vector mapping. In the scalar case the obtained statements are equivalent to the coincidence point theorems by A. V. Arutyunov. As an application, we prove a statement on the existence of n-fold coincidence points and obtain estimates of the points. The sufficient conditions for n-fold fixed points existence, including the well-known theorems on double fixed point, follow from the obtained results.     

A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings

In this paper, we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mapping in Hilbert spaces. We prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main result improve and extend Plubtieng and Punpaeng’s corresponding result [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Applied Mathematics and Computation 197 (2008), 548–558]. Using this theorem, we obtain three corollaries.     

An iterative method of solution for equilibrium and optimization problems

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of solutions of an equilibrium problem in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets. The results of this paper extended and improved the results of H. Iiduka and W. Takahashi [Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515]. Therefore, by using the above result, an iterative algorithm for the solution of a optimization problem was obtained.     

Strong convergence of an iterative method for pseudo-contractive and monotone mappings

In this paper, we introduce an iterative process which converges strongly to a common element of fixed points of pseudo-contractive mapping and solutions of variational inequality problem for monotone mapping. As a consequence, we provide an iteration scheme which converges strongly to a common element of set of fixed points of finite family continuous pseudo-contractive mappings and solutions set of finite family of variational inequality problems for continuous monotone mappings. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.     

Monotone generalized contractions in partially ordered probabilistic metric spaces

In this paper, a concept of monotone generalized contraction in partially ordered probabilistic metric spaces is introduced and some fixed and common fixed point theorems are proved. Presented theorems extend the results in partially ordered metric spaces of Nieto and Rodriguez-Lopez [Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239; Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205-2212], Ran and Reurings [A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443] to a more general class of contractive type mappings in partially ordered probabilistic metric spaces and include several recent developments.     

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions.     

Strong convergence theorems by hybrid methods for a finite family of nonexpansive mappings and inverse-strongly monotone mappings

In this paper, we introduce and study an iterative scheme by a hybrid method for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping in a real Hilbert space. Then, we prove that the iterative sequence converges strongly to a common element of the three sets. Using this result, we consider the problem of finding a common fixed point of a finite family of nonexpansive mappings and a strictly pseudocontractive mapping and the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of zeros of an inverse-strongly monotone mapping. The results obtained in this paper extend and improve the several recent results in this area.     

Norm order and geometric properties of holomorphic mappings in\mathbb{C}^n

We introduce a new notion of the order of a linear invariant family of locally biholomorphic mappings on then-ball. This order, which we call the norm order, is defined in terms of the norm rather than the trace of the “second Taylor coefficient operator” of mappings in a family. Sharp bounds on ‖Df(z)‖ and ‖f(z)‖, a general covering theorem for arbitrary LIFs and results about convexity, starlikeness, injectivity and other geometric properties of mappings given in terms of the norm order illustrate the useful nature of this notion. The norm order has a much broader range of influence on the geometric properties of mappings than does the “trace” order that the present authors and many others have used in recent years.     

Variation of diametrically symmetric or elliptically schlicht conformal mappings

A domain is called diametrically symmetric if it contains with each point its antipodal point on the Riemann sphere. We derive a variational formula for schlicht conformal mappings of such domains onto domains of the same type. This gives an analogue of a classical variational formula of Duren and Schiffer, which is in some sense an “elliptic analogue” of the “hyperbolic case” of Duren and Schiffer.     

Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.     

A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping

The purpose of this paper is to present an iterative scheme by a hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings in the framework of a Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets under appropriate conditions. Additionally, the idea of our results are applied to find a zero of a maximal monotone operator and a strictly pseudocontractive mapping in a real Hilbert space.     

A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings

The purpose of this paper is to consider a new hybrid relaxed extragradient method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a finite family of nonexpansive mappings and the set of solutions of variational inequalities for an inverse-strongly monotone mapping in Hilbert spaces. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions. Our results extend and improve the recent results of Cai and Hu [G. Cai, C.S. Hu, A hybrid approximation method for equilibrium and fixed point problems for a family of infinitely nonexpansive mappings and a monotone mapping, Nonlinear Anal. Hybrid Syst., 3(2009) 395–407], Kangtunyakarn and Suantai [A. Kangtunyakarn, S. Suantai, A new mapping for finding common solution of equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Anal., 71(2009) 4448–4460] and Thianwan [S. Thianwan, Strong convergence theorems by hybrid methods for a finite family of nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. Hybrid Syst., 3(2009) 605–614] and many others.     

Towards variational analysis in metric spaces: metric regularity and fixed points

The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin’s theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping; (b) a “double fixed point” theorem for a couple of mappings, one from X into Y and another from Y to X which implies a fairly general version of the set-valued contraction mapping principle and also a certain (different) version of the first theorem.     

On coincidences of families of mappings on ordered sets

New results on fixed points and coincidences of families of set-valued mappings of partially ordered sets obtained without commutativity assumptions are presented. These results develop theorems on fixed points of an isotone self-mapping of an ordered set (for families of set-valued mappings) and theorems about coincidences of two set-valued mappings one of which is isotone and the other is covering (for finite families of set-valued mappings).