A rate of convergence for asymptotic contractions

In [P. Gerhardy, A quantitative version of Kirk's fixed point theorem for asymptotic contractions, J. Math. Anal. Appl. 316 (2006) 339-345], P. Gerhardy gives a quantitative version of Kirk's fixed point theorem for asymptotic contractions. This involves modifying the definition of an asymptotic contraction, subsuming the old definition under the new one, and giving a bound, expressed in the relevant moduli and a bound on the Picard iteration sequence, on how far one must go in the iteration sequence to at least once get close to the fixed point. However, since the convergence to the fixed point needs not be monotone, this theorem does not provide a full rate of convergence. We here give an explicit rate of convergence for the iteration sequence, expressed in the relevant moduli and a bound on the sequence. We furthermore give a characterization of asymptotic contractions on bounded, complete metric spaces, showing that they are exactly the mappings for which every Picard iteration sequence converges to the same point with a rate of convergence which is uniform in the starting point.     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

A quantitative version of Kirk's fixed point theorem for asymptotic contractions

In [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645-650], W.A. Kirk introduced the notion of asymptotic contractions and proved a fixed point theorem for such mappings. Using techniques from proof mining, we develop a variant of the notion of asymptotic contractions and prove a quantitative version of the corresponding fixed point theorem.     

A definitive result on asymptotic contractions

We generalize a fixed point theorem for asymptotic contractions due to Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645-650]. Our result is the final generalization in some sense.     

A note on a paper of I.D. Arand?elovi? on asymptotic contractions

W.A. Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645-650] defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the Boyd-Wong fixed point theorem. Subsequently, I.D. Arand?elovi? [I.D. Arand?elovi?, On a fixed point theorem of Kirk, J. Math. Anal. Appl. 301 (2005) 384-385] established somewhat more general version of Kirk's result and he gave an elementary proof of it. However, our purpose is to show that there is an error in this proof and, moreover, Arand?elovi?'s theorem is false. We also explain how to correct this result.     

On the asymptotic integration of nonlinear differential equations

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.     

On existence and asymptotic behaviour of solutions of a functional integral equation

The paper contains a result on the existence and asymptotic behaviour of solutions of a functional integral equation. That result is proved under rather general hypotheses. The main tools used in our considerations are the concept of a measure of noncompactness and the classical Schauder fixed point principle. The investigations of the paper are placed in the space of continuous and tempered functions on the real half-line. We prove an existence result which generalizes several ones concerning functional integral equations and obtained earlier by other authors. The applicability of our result is illustrated by some examples.     

On asymptotic pointwise nonexpansive mappings in modular function spaces

We prove the existence of fixed points of asymptotic pointwise nonexpansive mappings in modular function spaces.     

Leray-Schauder不动点定理的一个新证明

给出Leray-Schauder不动点定理的一个新证明.我们首先给出集值映射的焊接引理,利用集值映射的焊接引理和Kakutani不动点定理证明Leray-Schauder不动点定理,并证明Leray-Schauder不动点定理与Brouwer不动点定理等价.     

The global attractivity and asymptotic stability of solution of a nonlinear integral equation

In this paper, we employ the fixed point theorem to study the existence of an integral equation and obtain the global attractivity and asymptotic stability of solutions of the equation. Some new results are given.     

Approximate controllability of semilinear functional equations in Hilbert spaces

In this paper approximate and complete controllability for semilinear functional differential systems is studied in Hilbert spaces. Sufficient conditions are established for each of these types of controllability. The results address the limitation that linear systems in infinite-dimensional spaces with compact semigroup cannot be completely controllable. The conditions are obtained by using the Schauder fixed point theorem when the semigroup is compact and the Banach fixed point theorem when the semigroup is not compact.     

Fixed points of contractive multivalued maps

For a class of contractive multivalued maps defined on a complete absolute retract and with closed bounded values, the set of fixed points is proved to be an absolute retract. This result unifies and extends to arbitrary absolute retracts both Theorem 1 by Ricceri [Atti Accad. Naz. Lincei Rend. Cl.Sci. Fis. Mat. Natur. (8) 81 (1987), 283--286] and Theorem 1 by Bressan, Cellina, and Fryszkowski [Proc. Amer. Math. Soc. 112 (1991), 413--418].

     

Fixed point theorems for set-valued contractions in complete metric spaces

The fixed point theory of set-valued contractions which was initiated by Nadler [S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969) 475-488] was developed in different directions by many authors, in particular, by [S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5 (1972) 26-42; N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188; Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103-112]. In the present paper, the concept of contraction for set-valued maps in metric spaces is introduced and the conditions guaranteeing the existence of a fixed point for such a contraction are established. One of our results essentially generalizes the Nadler and Feng-Liu theorems and is different from the Mizoguchi-Takahashi result. The second result is different from the Reich and Mizoguchi-Takahashi results. The method used in the proofs of our results is inspired by Mizoguchi-Takahashi and Feng-Liu's ideas. Comparisons and examples are given.     

Uniform asymptotic regularity for Mann iterates

In Numer. Funct. Anal. Optim. 22 (2001) 641-656, we obtained an effective quantitative analysis of a theorem due to Borwein, Reich, and Shafrir on the asymptotic behavior of general Krasnoselski-Mann iterations for nonexpansive self-mappings of convex sets C in arbitrary normed spaces. We used this result to obtain a new strong uniform version of Ishikawa's theorem for bounded C. In this paper we give a qualitative improvement of our result in the unbounded case and prove the uniformity result for the bounded case under the weaker assumption that C contains a point x whose Krasnoselski-Mann iteration (x_k) is bounded. We also consider more general iterations for which asymptotic regularity is known only for uniformly convex spaces (Groetsch). We give uniform effective bounds for (an extension of) Groetsch's theorem which generalize previous results by Kirk, Martinez-Yanez, and the author.     

Some results in asymptotic field point theory

In 1944, Levinson ([22]) introduced the concept of dissipativeness for a map T in a finite-dimensional space which leads to the existence of a fixed point of some iterate T ⁿ for n large, rather than a fixed point of T. Browder ([3]) gave an asymptotic field point theorem which proved that T itself had a field point. Although Browder’s result was a big step, it was not suitable for hyperbolic PDEs and neutral functional differential equations because, in those cases, the map T is not compact. For α-contraction maps the result was extended by Nussbaum ([25]) and Hale and Lopes ([13]) using different methods. In this paper, we review these ideas and some more recent applications. Dedicated to Felix Browder on the occasion of his 80th birthday