The Leray-Schauder condition for 1-set weakly-contractive and (ws)-compact operators

The main purpose of this article is to prove a collection of new fixed point theorems for (ws)-compact and so-called 1-set weakly contractive operators under Leray-Schauder boundary condition. We also introduce the concept of semi-closed operator at the origin and obtain a series of new fixed point theorems for such class of operators. As consequences, we get new fixed point existence for (ws)-compact (in particular nonexpansive) self mappings unbounded closed convex subset of Banach spaces. The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness. Later on, we give an application to generalized Hammerstein type integral equations.     

Banach空间中半闭1-集压缩算子的不动点定理及其应用(英文)

本文通过定义在Menger-PN的算子建立不同的边界条件,考虑半闭1-集压缩算子的不动点的存在性问题,得到了半闭1-集压缩算子的一些不动点定理.本文中所采用的方法与相关文献中的方法完全不同.     

THE LERAY-SCHAUDER CONDITION FOR 1-SET WEAKLY-CONTRACTIVE AND （ws）-COMPACT OPERATORS

Abstract The main purpose of this article is to prove a collection of new nxea point theorems for （ws）-compact and so-called 1-set weakly contractive operators under Leray- Schauder boundary condition. We also introduce the concept of semi-closed operator at the origin and obtain a series of new fixed point theorems for such class of operators. As consequences, we get new fixed point existence for （ws）-compact （in particular nonexpansive） self mappings unbounded closed convex subset of Banach spaces. The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness. Later on, we give an application to generalized Hammerstein type integral equations.     

一类单调算子的不动点定理

研究了范围广泛的半闭1-集压缩算子及凝聚算子,获得了一些新的不动点定理,所获结果推广了已知的结论.     

Krasnoselskii定理几种新的推广

利用随机不动点指数,研究了随机半闭1-集压缩算子方程的随机解,同时,从不同角度推广了著名的Krasnoselskii定理,得出若干新的结果.     

Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces

Let X be a non-empty set and F:X×X→X be a given mapping. An element (x,y)∈X×X is said to be a coupled fixed point of the mapping F if F(x,y)=x and F(y,x)=y. In this paper, we consider the case when X is a complete metric space endowed with a partial order. We define generalized Meir-Keeler type functions and we prove some coupled fixed point theorems under a generalized Meir-Keeler contractive condition. Some applications of our obtained results are given. The presented theorems extend and complement the recent fixed point theorems due to Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379-1393].     

Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces

In this paper, we extend the coupled fixed point theorems for mixed monotone operators F:X×X→X obtained in [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379–1393] by significantly weakening the contractive condition involved. Our technique of proof is essentially different and more natural. An example as well as an application to periodic BVP is also given in order to illustrate the effectiveness of our generalizations.     

Contractive multifunctions, fixed point inclusions and iterated multifunction systems

We study the properties of multifunction operators that are contractive in the Covitz-Nadler sense. In this situation, such operators T possess fixed points satisfying the relation x∈Tx. We introduce an iterative method involving projections that guarantees convergence from any starting point x₀∈X to a point x∈XT, the set of all fixed points of a multifunction operator T. We also prove a continuity result for fixed point sets XT as well as a “generalized collage theorem” for contractive multifunctions. These results can then be used to solve inverse problems involving contractive multifunctions. Two applications of contractive multifunctions are introduced: (i) integral inclusions and (ii) iterated multifunction systems.     

Z-C-X空间中一类随机方程解的存在性定理

在Z-C-X空间中利用优锥的概念,研究了随机半闭1-集压缩算子的若干问题.首先,证明了一个重要的不等式,其次,利用随机拓扑度理论证明了几个新的定理,从而得到一些新的结果.     

Rate of approximation for the Bézier variant of Balazs Kantorovich operators

In the present paper we study the Bézier variant of the well known Balazs-Kantorovich operators L _n,α (f,x), α ≥ 1. We establish the rate of convergence for functions of bounded variation. For particular value α = 1, our main theorem completes a result due to Agratini [Math. Notes (Miskolc) 2 (2001), 3–10]. Communicated by Michal Zajac     

Faber-hypercyclic operators

Let Ω be a bounded domain of the complex plane whose boundary is a closed Jordan curve and (F _n ) _n≥0 the sequence of Faber polynomials of Ω. We say that a bounded linear operator T on a separable Banach space X is Ω-hypercyclic if there exists a vector x of X such that {F _n (T)x: n ≥ 0} is dense in X. We show that many of the results in the spectral theory of hypercyclic operators involving the unit disk or its boundary have Ω-hypercyclic counterparts which involve the domain Ω or its boundary. The influence of the geometry of Ω or the smoothness of its boundary on Faber-hypercyclicity is also discussed.     

二元φ-序压缩算子的不动点定理(英文)

本文引进了一类新的压缩算子,即二元φ-序压缩算子,并且在完备的半序度量空间(其中的半序由φ所导出)上证明了几个二元φ-序压缩算子的不动点定理.本文所得的部分结论推广了最近一些文献中相应的结论.     

Contractible Lie groups over local fields

Let G be a Lie group over a local field of characteristic p > 0 which admits a contractive automorphism α : G → G (i.e., α ⁿ (x) → 1 as n → ∞, for each x ∈ G). We show that G is a torsion group of finite exponent and nilpotent. We also obtain results concerning the interplay between contractive automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon. Some of the results extend to Lie groups over arbitrary complete ultrametric fields. Supported by the German Research Foundation (DFG), grants GL 357/2-1 and GL 357/6-1.     

On weighted approximation by Bernstein-Durrmeyer operators

In this paper, we consider weighted approximation by Bernstein-Durrmeyer operators in L_p[0, 1] (1≤p≤∞), where the weight function w(x)=x^α(1−x)^β,−1/p<α, β<1-1/p. We obtain the direct and converse theorems. As an important tool we use appropriate K-functionals. Supported by Zhejiang Provincial Science Foundation.     

Some common fixed point theorems for selfmappings satisfying two contractive conditions of integral type in a uniform space

In this paper, we establish some common fixed point theorems for selfmappings of a uniform space by employing both the concepts of an A—distance and an E—distance introduced by Aamri and El Moutawakil [1] and two contractive conditions of integral type. Our results are generalizations and extensions of the classical Banach’s fixed point theorem of [2, 3, 19], some results of Aamri and El Moutawakil [1], Theorem 2.1 of Branciari [5] as well as a result of Jungck [7].      

Generalized projection and approximation of fixed points of nonself maps

Let K be a nonempty closed convex proper subset of a real uniformly convex and uniformly smooth Banach space E; T:K→E be an asymptotically weakly suppressive, asymptotically weakly contractive or asymptotically nonextensive map with F(T){xK: Tx=x}≠. Using the notion of generalized projection, iterative methods for approximating fixed points of T are studied. Convergence theorems with estimates of convergence rates are proved. Furthermore, the stability of the methods with respect to perturbations of the operators and with respect to perturbations of the constraint sets are also established.     

Nonlinear operator equations with a functional perturbation of the argument of neutral type

We construct small solutions x(t) → 0 as t → 0 of nonlinear operator equations F(x(t), x(α(t)),t) = 0 with a functional perturbation α(t) of the argument. By the Newton diagram method, we reduce the problem to quasilinear operator equations with a functional perturbation of the argument. We show that the solutions of such equations can have not only algebraic but also logarithmic branching points and contain free parameters. The number of free parameters and the form of the solution depend on the properties of the Jordan structure of the operator coefficients of the equation.     

Averaging sequences and modulated ergodic theorems for weakly almost periodic group representations

LetT be a weakly almost periodic (WAP) representation of a locally compact Σ-compact groupG by linear operators in a Banach spaceX, and letM = M(T) be its ergodic projection onto the space of fixed points (i.e.,Mx is the unique fixed point in the closed convex hull of the orbit ofx). A sequence of probabilities Μ_n is said toaverage T [weakly] if ∫T(t)x dΜ _n converges [weakly] toM(T)x for eachx ∃X. We callΜ _n [weakly]unitarily averaging if it averages [weakly] every unitary representation in a Hilbert space, and [weakly]WAPRaveraging if it averages [weakly] every WAP representation. We investigate some of the relationships of these notions, and connect them with properties of the regular representation (by translations) in the spaceWAP(G). Research partially supported by the Israel Science Foundation.     

The density of rational points on non-singular hypersurfaces

LetF(x) =F[x₁,…,x_n]∈ℤ[x₁,…,x_n] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ ⁿ ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X ^n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan     

On nonlinear Volterra equations of convolution type

Weakly nonlinear and strongly nonlinear convolution-type Volterra equations u(x) = (K * ϕ(u))(x) are studied in new classes of weakly synchronous and quasiconcave functions f under assumptions less restrictive than the classical ones. Existence and uniqueness theorems, as well as theorems on the absence of solutions, are proved. Smoothness issues for solutions of both weakly nonlinear and strongly nonlinear equations are considered. An integral inequality is obtained for the weight function of a metric ensuring that a nonlinear operator is a contraction, and a number of other results are obtained.