Showing papers on "Fixed-point theorem published in 2003"
18 Sep 2003
TL;DR: In this paper, an analogue of Banach's fixed point theorem in partially ordered sets is proved, and several applications to linear and nonlinear matrix equations are discussed, including the application of the Banach theorem to the Partially ordered Set (POPS) problem.
Abstract: An analogue of Banach's fixed point theorem in partially ordered sets is proved in this paper, and several applications to linear and nonlinear matrix equations are discussed.
1,323 citations
Journal Article•
379 citations
TL;DR: A matroid-generalization of the stable marriage theorem is formulated and results of Vande Vate and Rothblum are extended on the bipartite stable matching polytope and the lattice structure of generalized stable matchings are studied.
Abstract: We describe a fixed-point based approach to the theory of bipartite stable matchings. By this, we provide a common framework that links together seemingly distant results, like the stable marriage theorem of Gale and Shapley, the Mendelsohn-Dulmage theorem, the Kundu-Lawler theorem, Tarski's fixed-point theorem, the Cantor-Bernstein theorem, Pym's linking theorem, or the monochromatic path theorem of Sands et al. In this framework, we formulate a matroid-generalization of the stable marriage theorem and study the lattice structure of generalized stable matchings. Based on the theory of lattice polyhedra and blocking polyhedra, we extend results of Vande Vate and Rothblum on the bipartite stable matching polytope.
293 citations
TL;DR: In this article, the existence of periodic solutions of the second-order Caratheodory problem is studied, by combining some new properties of Green's function together with Krasnoselskii fixed point theorem on compression and expansion of cones.
290 citations
TL;DR: In this article, an iterative algorithm is proposed to generate a sequence (xn) from an arbitrary initial point x0∈H, which converges in norm to the unique solution u* of the variational inequality.
Abstract: Assume that F is a nonlinear operator on a real Hilbert space H which is η-strongly monotone and κ-Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed point sets of a finite number of nonexpansive mappings on H. We devise an iterative algorithm which generates a sequence (xn) from an arbitrary initial point x0∈H. The sequence (xn) is shown to converge in norm to the unique solution u* of the variational inequality
$$\left\langle {F(u*),\user1{v} - u*} \right\rangle \geqslant 0$$
Applications to constrained pseudoinverse are included.
289 citations
TL;DR: The approximate controllability of the abstract semilinear deterministic and stochastic control systems under the natural assumption that the associated linear control system is approximately controllable is shown.
Abstract: Various sufficient conditions for approximate controllability of linear evolution systems in abstract spaces have been obtained, but approximate controllability of semilinear control systems usually requires some complicated and limited assumptions. In this paper, we show the approximate controllability of the abstract semilinear deterministic and stochastic control systems under the natural assumption that the associated linear control system is approximately controllable. The results are obtained using new properties of symmetric operators (which are proved in this paper), compact semigroups, the Schauder fixed point theorem, and/or the contraction mapping principle.
263 citations
TL;DR: In this article, the authors established two fixed-point theorems for mappings satisfying a general contractive inequality of integral type, which substantially extend the theorem of Branciari (2002).
Abstract: We establish two fixed-point theorems for mappings satisfying a
general contractive inequality of integral type. These results substantially extend the theorem of Branciari (2002).
204 citations
01 Jan 2003
TL;DR: In this paper, the authors present a scheme for the relationship of single sections in locally convex spaces, and a topological dimension of fixed point sets, and an approximation method for the fixed point theory of multivalued mappings.
Abstract: Preface. Scheme for the relationship of single sections. I: Theoretical Background. I.1. Structure of locally convex spaces. I.2. ANR-spaces and AR-spaces. I.3. Multivalued mappings and their selections. I.4. Admissible mappings. I.5. Special classes of admissible mappings. I.6. Lefschetz fixed point theorem for admissible mappings. I.7. Lefschetz fixed point theorem for condensing mappings. I.8. Fixed point index and topological degree for admissible maps in locally convex spaces. I.9. Noncompact case. I.10. Nielsen number. I.11. Nielsen number: Noncompact case. I.12. Remarks and comments. II: General Principles. II.1 Topological structure of fixed point sets: Aronszajn Browder Gupta-type results. II.2. Topological structure of fixed point sets: inverse limit method. II.3. Topological dimension of fixed point sets. II.4. Topological essentiality. II.5. Relative theories of Lefschetz and Nielsen. II.6. Periodic point principles. II.7. Fixed point index for condensing maps. II.8. Approximation method for the fixed point theory of multivalued mappings. II.9. Topological degree defined by means of approximation methods. II.10. Continuation principles based on a fixed point index. II.11. Continuation principles based on a coincidence index. II.12. Remarks and comments. III: Application to Differential Equations and Inclusions. III.1. Topological approach to differential equations and inclusions. III.2. Topological structure of solution sets: initial value problems. III.3. Topological structure of solution sets: boundary value problems. III.4. Poincare operators. III.5. Existence results. III.6. Multiplicity results. III.7. Wazewski-type results. III.8. Bounding and guiding functions approach. III.9. Infinitely many subharmonics. III.10. Almost-periodic problems. III.11. Some further applications. III.13.Remarks and comments. Appendices. A.1. Almost-periodic single-valued and multivalued functions. A.2. Derivo-periodic single-valued and multivalued functions. A.3. Fractals and multivalued fractals. References. Index.
178 citations
TL;DR: In this paper, the authors investigated the multiple and infinitely solvability of positive solutions for nonlinear fractional differential equation Du(t)=tνf(u), 0 0, γ⩾0, 0 −β(γ+1).
154 citations
TL;DR: In this paper, it was proved that if T :M→M satisfies d T n (x),T n (y) ⩽φ n d(x,y), x,y∈M, where each φn is continuous and φ n→φ uniformly on the range of d, then T has a unique fixed point, and moreover all of the Picard iterates of T converge to this fixed point.
153 citations
TL;DR: By using the Banach fixed point theorem and constructing suitable Lyapunov function, some sufficient conditions are obtained ensuring existence, uniqueness and global stability of almost periodic solution of the BAM neural networks with variable coefficients and delays.
TL;DR: The aim in this paper is to discuss the difference between two different type of Cauchy sequences used in the literature to prove fixed point theorems in fuzzy metric spaces.
TL;DR: In this article, the weak approximate and complete controllability properties of semilinear stochastic systems were studied by using the Banach fixed point theorem, and applications to the Stochastic heat equation were given.
TL;DR: Using the technique of fixed-point theorem of Darbo type associated with measures of noncompactness, an existence result for some functional-integral equation is obtained and a generalization of the classical Banach fixed- point principle is created.
TL;DR: In this paper, the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions is studied. And the results are a generalization and continuation of the recent results on this issue.
Abstract: In this paper, by using fractional power of operators and Sadovskii's fixed point theorem, we study the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions. The results we obtained are a generalization and continuation of the recent results on this issue. In the end, an example is given to show the application of our results.
TL;DR: In this paper, the authors established the existence of traveling warships for delayed reaction diffusion systems without quasimonotonicity in the reaction term, by using Schauder's fixed point theorem.
Abstract: In this paper, we establish the existence of traveling
wavefronts for delayed reaction diffusion systems without quasimonotonicity
in the reaction term, by using Schauder's fixed point theorem. We show the
merit of our result by applying it to
the Belousov-Zhabotinskii reaction model with two delays.
TL;DR: In this article, the data dependence of the fixed point set for a special class of weakly Picard operators is studied and an application to a Fredholm integral inclusion is given. And the existence and data dependence for some Reich-type multivalued operators are also proved.
Abstract: In this paper we study data dependence of the fixed point set for a special class of multivalued weakly Picard operators. Existence and data dependence of the common fixed points for some Reich-type multivalued operators are also proved. Finally, an application to a Fredholm integral inclusion is given.
TL;DR: A generalization of the fixed point theorems of Khan, Swaleh and Sessa, Pathak and Rekha Sharma, and Sastry and Babu for a self-map on a metric space was obtained in this article.
Abstract: A generalization is obtained for some of the fixed point theorems of Khan, Swaleh and Sessa, Pathak and Rekha Sharma, and Sastry and Babu for a self-map on a metric space, which involve the idea of alteration of distances between points.
TL;DR: In this article, the authors studied the problem of the existence and uniqueness of solutions to the Bellman equation in the presence of unbounded returns and provided sufficient conditions for the existence of solutions that can be applied to fairly general models.
Abstract: We study the problem of the existence and uniqueness of solutions to the Bellman equation in the presence of unbounded returns. We introduce a new approach based both on consideration of a metric on the space of all continuous functions over the state space, and on the application of some metric fixed point theorems. With appropriate conditions we prove uniqueness of solutions with respect to the whole space of continuous functions. Furthermore, the paper provides new sufficient conditions for the existence of solutions that can be applied to fairly general models. It is also proven that the fixed point coincides with the value function and that it can be approached by successive iterations of the Bellman operator.
TL;DR: In this paper, the authors have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis, where ''uniform'' here means independence from parameters in compact spaces.
Abstract: In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. `Uniform' here means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, normed linear spaces, uniformly convex spaces as well as inner product spaces.
TL;DR: In this article, the authors apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the three point boundary value problem.
TL;DR: In this article, a new twin fixed point theorem is applied to obtain the existence of at least two positive solutions for the boundary value problem, and two corollaries and an example are given to illustrate the main results.
TL;DR: In this article, the authors considered the nonlinear fourth order ordinary differential equation (E) with boundary conditions (B) and obtained some results on the existence and nonexistence of positive solutions.
TL;DR: In this article, the existence of positive solutions of p-Laplacian difference equations is considered by means of fixed point theorem in a cone, and the authors show that positive solutions are possible.
19 Aug 2003
TL;DR: In this article, a nonempty closed convex subset of a real Banach space is constructed for which ∥x n - Tx n ∥ → 0 as n → oc.
Abstract: Let K be a nonempty closed convex subset of a real Banach space E and T be a Lipschitz pseudocontractive self-map of K with F(T) := {x E K : Tx = x} ¬= O. An iterative sequence {x n } is constructed for which ∥x n - Tx n ∥ → 0 as n → oc. If, in addition, K is assumed to be bounded, this conclusion still holds without the requirement that F(T) ¬= O. Moreover, if, in addition, E has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of K has the fixed point property for nonexpansive self-trappings, then the sequence {x n } converges strongly to a fixed point of T. Our iteration method is of independent interest.
TL;DR: The existence of Nash and Walras equilibrium is proved via Brouwer's fixed point theorem without recourse to Kakutani's Fixed Point Theorem for correspondences as discussed by the authors. But the authors of this paper assume that the domain of the Walras fixed point map is confined to the price simplex, even when there is production and weakly quasiconvex preferences.
Abstract: The existence of Nash and Walras equilibrium is proved via Brouwer’s Fixed Point Theorem, without recourse to Kakutani’s Fixed Point Theorem for correspondences. The domain of the Walras fixed point map is confined to the price simplex, even when there is production and weakly quasi-convex preferences. The key idea is to replace optimization with “satisficing improvement,” i.e., to replace the Maximum Principle with the “Satisficing Principle.”
TL;DR: In this paper, the existence theorem for monotone positive solutions of nonlinear second-order ODEs was obtained by using the Schauder-Tikhonov fixed point theorem.
Abstract: We obtain an existence theorem for monotone positive solutions of nonlinear second-order ordinary differential equations by using the Schauder-Tikhonov fixed point theorem. The result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in R-n (n greater than or equal to 3).
01 Jun 2003
TL;DR: In this article, the existence of positive solutions to boundary value problems was proved based on the fixed-point theorem in cones, and the results extended some of the existing literature on superlinear semipositone problems.
Abstract: Abstract In this paper we consider the existence of positive solutions to the boundary-value problems \begin{align*} (p(t)u')'-q(t)u+\lambda f(t,u)\amp=0,\quad r\ltt\ltR, \\[2pt] au(r)-bp(r)u'(r)\amp=\sum^{m-2}_{i=1}\alpha_iu(\xi_i), \\ cu(R)+dp(R)u'(R)\amp=\sum^{m-2}_{i=1}\beta_iu(\xi_i), \end{align*} where $\lambda$ is a positive parameter, $a,b,c,d\in[0,\infty)$, $\xi_i\in(r,R)$, $\alpha_i,\beta_i\in[0,\infty)$ (for $i\in\{1,\dots m-2\}$) are given constants satisfying some suitable conditions. Our results extend some of the existing literature on superlinear semipositone problems. The proofs are based on the fixed-point theorem in cones. AMS 2000 Mathematics subject classification: Primary 34B10, 34B18, 34B15
TL;DR: By constructing a special cone and using cone compression and expansion fixed point theorem, some existence results of positive solutions of singular boundary value problem (BVP) on half-line for a class of second order differential equations are presented.
TL;DR: In this paper, the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed and a nonlinear alternative of Leray-Schauder type and on a fixed point theorem in cones are presented.