Showing papers on "Fixed-point theorem published in 2001"
TL;DR: The hybrid steepest descent (HST) method as mentioned in this paper is a simple algorithmic solution to the variational inequality problem defined over the nonempty intersection of multiple fixed point sets of nonexpansive mappings in a real Hilbert space.
Abstract: This paper presents a simple algorithmic solution to the variational inequality problem defined over the nonempty intersection of multiple fixed point sets of nonexpansive mappings in a real Hilbert space. The algorithmic solution is named the hybrid steepest descent method, because it is constructed by blending important ideas in the steepest descent method and in the fixed point theory, and generates a sequence converging strongly to the solution of the problem. The remarkable applicability of this method to the convexly constrained generalized pseudoinverse problem as well as to the convex feasibility problem is demonstrated by constructing nonexpansive mappings whose fixed point sets are the feasible sets of the problems.
706 citations
6 Mar 2001
686 citations
1 Jan 2001
TL;DR: In this article, the authors introduce the concept of fixed point free mapping and define a set of non-linear non-expansive Mappings for metric fixed point theory, which they call fixed points of holomorphic mapping.
Abstract: Preface. 1. Contraction Mappings and Extensions W.A. Kirk. 2. Examples of Fixed Point Free Mappings B. Sims. 3. Classical Theory of Nonexpansive Mappings K. Goebel, W.A. Kirk. 4. Geometrical Background of Metric Fixed Point Theory S. Prus. 5. Some Moduli and Constants Related to Metric Fixed Point Theory E.L. Fuster. 6. Ultra-Methods in Metric Fixed Point Theory M.A. Khamsi, B. Sims. 7. Stability of the Fixed Point Property for Nonexpansive Mappings J. Garcia-Falset, A. Jimenez-Melado, E. Llorens-Fuster. 8. Metric Fixed Point Results Concerning Measures of Noncompactness T. Dominguez, M.A. Japon, G. Lopez. 9. Renormings of l1 and c0 and Fixed Point Properties P.N. Dowling, C.J. Lennard, B. Turett. 10. Nonexpansive Mappings: Boundary/Inwardness Conditions and Local Theory W.A. Kirk, C.H. Morales. 11. Rotative Mappings and Mappings with Constant Displacement W. Kaczor, M. Koter-Morgowska. 12. Geometric Properties Related to Fixed Point Theory in Some Banach Function Lattices S. Chen, Y. Cui, H. Hudzik, B. Sims. 13. Introduction to Hyperconvex Spaces R. Espinola, M.A. Khamsi. 14. Fixed Points of Holomorphic Mappings: A Metric Approach T. Kuczumow, S. Reich, D. Shoikhet. 15. Fixed Point and Non-Linear Ergodic Theorems for Semigroups of Non-Linear Mappings A. To-Ming Lau, W. Takahashi. 16. Generic Aspects of Metric Fixed Point Theory S. Reich, A.J. Zaslavski. 17. Metric Environment of the TopologicalFixed Point Theorms K. Goebel. 18. Order-Theoretic Aspects of Metric Fixed Point Theory J. Jachymski. 19. Fixed Point and Related Theorems for Set-Valued Mappings G. X.-Z. Yuan. Index.
561 citations
TL;DR: In this paper, the existence of traveling wavefronts of reaction-diffusion systems with quasimonotonicity reactions was shown to be equivalent to an admissible pair of supersolution and subsolution which are easy to construct in practice.
313 citations
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TL;DR: In this article, the (n − 1)-norm can be derived from the n-norm in such a way that the convergence and completeness in the derived norm is equivalent to those in the n − 1 norm.
Abstract: Given an n-normed space with n ≥ 2, we offer a simple way to derive an (n−1)- norm from the n-norm and realize that any n-normed space is an (n − 1)-normed space. We also show that, in certain cases, the (n − 1)-norm can be derived from the n-norm in such a way that the convergence and completeness in the n-norm is equivalent to those in the derived (n − 1)-norm. Using this fact, we prove a fixed point theorem for some n-Banach spaces.
260 citations
Book•
9 Apr 2001TL;DR: In this article, the theorems of Brouwer, Svhauder and Monch for non-linear alternatives of Leray-Schauder type for condensing maps are discussed.
Abstract: Preface 1 Contradictions 2 Non-expansive maps 3 Continuation methods for contractive and non-expansive mapping 4 The theorems of Brouwer, Svhauder and Monch 5 Non-linear alternatives of Leray-Schauder type 6 Continuation principles for condensing maps 7 Fixed point theorems in conical shells 8 Fixed point theory in Hausdorff locally convex linear topological spaces 9 Contractive and non-expansive multivalued maps 10 Multivalued maps with continuous selections 11 Multivalued maps with closed graph 12 Degree theory Bibliography Index
210 citations
TL;DR: In this paper, the authors studied the space lp, 1 ≤ p ≤ ∞, and its natural n-norm, which can be viewed as a generalisation of its usual norm.
Abstract: We study the space lp, 1 ≤ p ≤ ∞, and its natural n-norm, which can viewed as a generalisation of its usual norm. Using a derived norm equivalent to its usual norm, we show that lp is complete with respect to its natural n-norm. In addition, we also prove a fixed point theorem for lp as an n-normed space.
126 citations
1 Jan 2001
TL;DR: The notion of hyperconvexity is due to Aronszajn and Panitchpakdi [1] as mentioned in this paper who proved that any metric space is a non-expansive retract of any space in which it is isometrically embedded.
Abstract: The notion of hyperconvexity is due to Aronszajn and Panitchpakdi [1] (1956) who proved that a hyperconvex space is a nonexpansive absolute retract, i.e. it is a non-expansive retract of any metric space in which it is isometrically embedded. The corresponding linear theory is well developed and associated with the names of Gleason, Goodner, Kelley and Nachbin (see for instance [19, 29, 42, 46]). The nonlinear theory is still developing. The recent interest into these spaces goes back to the results of Sine [54] and Soardi [57] who proved independently that fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results have been shown to hold in hyperconvex spaces.
103 citations
TL;DR: In this paper, the existence of positive solutions of boundary value problems on the half-line for differential equations of second order was studied and the Krasnoselskii fixed point theorem on cone compression and expansion was used.
83 citations
TL;DR: In this article, the Fan-Browder fixed point theorem was used to establish the continuity property for multimaps and generalized Berge's theorem for multi-paths, and then they applied these results, and the Fan and Browder fixed-point theorem to the existence theorems of quasi-equilibrium problems and generalized quasi-qualities for multimap problems.
60 citations
TL;DR: In this article, the authors considered the discrete focal boundary value problem and proved the existence of three positive solutions under various assumptions on f and the integers a, t 2, and b. To prove their results, they applied a generalization of the Leggett-Williams fixed-point theorem.
Abstract: We are concerned with the discrete focal boundary value problem Δ 3 x ( t − k ) = f ( x ( t )), x ( a ) = Δ x ( t 2 ) = Δ 2 x ( b + 1 = 0. Under various assumptions on f and the integers a , t 2 , and b we prove the existence of three positive solutions of this boundary value problem. To prove our results, we will apply a generalization of the Leggett-Williams fixed-point theorem.
28 Aug 2001
TL;DR: In this article, the generalized Banach contraction conjecture (GBCC) was proved for arbitrary J in the case when T is assumed to be continuous, and also derived a result which enables us to prove the GBCC when J = 3 without the assumption of continuity.
Abstract: One of the most fundamental fixed-point theorems is Banach's Contraction Principle, of which the following conjecture is a generalization. Generalized Banach Contraction Conjecture (GBCC). Let T be a self-map of a complete metric space (X, d), and let 0 < M < 1. Let J be a positive integer. Assume that for each pair x, y ∈ X, min{d(T k x,T k y): 1 ≤ k ≤ J} ≤ M d(x, y). Then T has a fixed point. Unlike Banach's original theorem (the case J = 1), the above hypothesis does not compel T to be continuous. In this paper we use Ramsey's Theorem from combinatorics to establish the GBCC for arbitrary J in the case when T is assumed to be continuous, and also derive a result which enables us to prove the GBCC when J = 3 without the assumption of continuity; it is known that the case J = 3 includes instances where T is not continuous.
TL;DR: In this article, Ahmed and Imdad and Ahmed and Khan have studied the fixed points of non-self hybrid contractions on metrically convex spaces and showed that most of their main theorems contain errors and admit counterexamples.
TL;DR: In this paper, the existence of 8xed points for a more general class of mappings: uniformly Lipschitzian mappings was studied and the main tool in their approach is the coeAcient of normal structure Ñ(L ).
Abstract: The theory of modular spaces was initiated by Nakano [14] in 1950 in connection with the theory of order spaces and rede8ned and generalized by Musielak and Orlicz [13] in 1959. De8ning a norm, particular Banach spaces of functions can be considered. Metric 8xed theory for these Banach spaces of functions has been widely studied (see, for instance, [15]). Another direction is based on considering an abstractly given functional which controls the growth of the functions. Even though a metric is not de8ned, many problems in 8xed point theory for nonexpansive mappings can be reformulated in modular spaces (see, for instance, [8] and references therein). In this paper, we study the existence of 8xed points for a more general class of mappings: uniformly Lipschitzian mappings. Fixed point theorems for this class of mappings in Banach spaces have been studied in [2,3] and in metric spaces in [11,12] (for further information about this subject, see [1, Chapter VIII] and references therein). The main tool in our approach is the coeAcient of normal structure Ñ(L ). We prove that under suitable conditions a k-uniformly Lipschitzian mapping has a 8xed point if k ¡ ( Ñ(L ))−1=2. In the last section we show a class of modular spaces where Ñ(L )¡ 1 and so, the above theorem can be successfully applied.
TL;DR: In this article, sufficient conditions for controllability of functional semilinear integrodifferential systems in a Banach space were established by using the Schaefer fixed-point theorem.
TL;DR: In this paper, the authors present a criteria for the existence and uniqueness of solutions to two point boundary value problems associated with second order non-linear fuzzy differential equations, using estimates on Green's function, Ascoli's Lemma and a fixed point theorem of Banach.
Abstract: This paper presents a criteria for the existence and uniqueness of solutions to two point boundary value problems associated with a second order non-linear fuzzy differential equations. The main tools employed are estimates on Green's function, Ascoli's Lemma and a fixed point theorem of Banach.
TL;DR: In this paper, a class of nonlinear two-dimensional nonlinear difference systems are classified in terms of their asymptotic magnitudes, and necessary and sufficient conditions for the existence of these solutions are also provided.
Abstract: Classification schemes for nonoscillatory solutions of a class of nonlinear two-dimensional nonlinear difference systems are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also provided.
4 Jun 2001
TL;DR: In this article, the existence of solutions to functional differential inclusions on compact intervals was investigated, using the xed point theorem introduced by Covitz and Nadler for contraction multi-valued maps.
Abstract: In this note we investigate the existence of solutions to functional differential inclusions on compact intervals. We use the xed point theorem introduced by Covitz and Nadler for contraction multi-valued maps.
TL;DR: In this paper, the necessary and sufficient conditions for the existence of stable periodic response for a type of catenary anchor leg mooring system (CALM) were derived using the Schauder's fixed-point theorem.
Abstract: It has been shown that there exists the necessary and sufficient condition for the existence of stable periodic response for a type of catenary anchor leg mooring system, (CALM). The mathematical model shows that the governing equation of motion for the system is a non-linear parametric second-order ordinary differential equation. The above-mentioned conditions have been obtained using the Schauder's fixed-point theorem. The validity of the assumptions has been fully demonstrated by analyzing a few examples.
Journal Article•
TL;DR: A general fixed point theorem for weakly compatible mappings satisfying an implicit relation in compact metric spaces is proved in this article, which generalizes the results by [1] and others.
Abstract: A general fixed point theorem for weakly compatible mappings satisfying an implicit relation in compact metric spaces is proved generalizing the results by [1],[3],[13],[14] and others.
TL;DR: In this article, the authors apply a cone theoretic fixed-point theorem and obtain sufficient conditions for the existence of positive solutions to some boundary value problems for a class of functional difference equations.
Abstract: In this paper, we apply a cone theoretic fixed-point theorem and obtain sufficient conditions for the existence of positive solutions to some boundary value problems for a class of functional difference equations. We consider analogues of sublinear or superlinear growth in the nonlinear terms.
TL;DR: Some random fixed point theorems for set-valued random operators under very mild conditions are established in this paper, where the discussion in this paper underlines, in addition to generality, the unifying aspects of our result.
Abstract: Some random fixed point theorems for set-valued random operators under very mild conditions are established Some recent results of O'Regan ([Proc Amer Math Soc 126 (1998), 3045-3053] and [Computers Math Applic 35 (1998), 27-34]) are improved significantly The discussion in this paper underlines, in addition to generality, the unifying aspects of our result
Posted Content•
TL;DR: In this article, it was shown that a continuous map or a continuous flow with a certain recurrence relation must have a fixed point in a compact set W with the property that the forward orbit of every point in W intersects W, and if the omega limit set of W is nonempty and uniformly bounded, then there is a fixed node in W.
Abstract: We show that a continuous map or a continuous flow on $\R^{n}$ with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set W with the property that the forward orbit of every point in $\R^{n}$ intersects W then there is a fixed point in W. Consequently, if the omega limit set of every point is nonempty and uniformly bounded then there is a fixed point.
1 Feb 2001
TL;DR: In this article, sufficient conditions for boundary controllability of integrodifferential systems in Banach spaces are established by using the strongly continuous semigroup theory and the Banach contraction principle.
Abstract: Sufficient conditions for boundary controllability of integrodifferential systems in Banach spaces are established. The results are obtained by using the strongly continuous semigroup theory and the Banach contraction principle. Examples are provided to illustrate the theory.
TL;DR: Existence theorems about the positive solution for the singular equation ( ϕ p ( y ′))′ + f ( t , y ) = 0, y (0) = y (1) = 0 are established by using a fixed-point theorem in cones.
TL;DR: An intersection theory for definable C p -manifolds in an o-minimal expansion of a real closed field is developed and the invariance of the intersection numbers under definableC p -homotopies is proved.