Showing papers on "Fixed-point theorem published in 2004"
TL;DR: In this paper, a general logical metatheorem for strong uniformity of non-uniform existence statements is presented. But the proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis are not restricted to this area at all.
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. Here 'uniform' 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, CAT(0)-spaces, normed linear spaces, uniformly convex spaces, as well as inner product spaces.
292 citations
01 Jan 2004
TL;DR: In this article, a Banach fixed point theorem for complete partial metric spaces in the sense of O'Neill was obtained, which is a special case of Matthews' fixed-point theorem.
Abstract: Summary. - In 1994, S.G. Matthews introduced the notion of a partial metric space and obtained, among other results, a Banach contraction mapping for these spaces. Later on, S.J. O’Neill generalized Matthews’ notion of partial metric, in order to establish connections between these structures and the topological aspects of domain theory. Here, we obtain a Banach fixed point theorem for complete partial metric spaces in the sense of O’Neill. Thus, Matthews’ fixed point theorem follows as special case of our result.
250 citations
Posted Content•
TL;DR: In this paper, a unified theory of invariant manifolds for infinite dimensional dynamical systems generated by stochastic partial differential equations is presented, where a random graph transform and a fixed point theorem for non-autonomous systems are introduced.
Abstract: Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite dimensional {\em random} dynamical systems generated by
{\em stochastic} partial differential equations. We first introduce a random graph transform and a fixed point theorem for non-autonomous systems. Then we show the existence of generalized fixed points which give the desired invariant manifolds.
228 citations
TL;DR: The existence of a positive solution to a singular coupled system of nonlinear fractional differential equations based on a nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem in a cone is established.
167 citations
TL;DR: By using the Krasnoselskii fixed point theorem, the existence of one or multiple positive solution of the fourth-order two point boundary value problem y^(^4^)(t)=f(t,y(t),y^'^'(t)), y(0)=y(1)=y+1, y(2)=y+2, y (3) =0, and so on.
125 citations
TL;DR: In this paper, a new fixed point theorem in a cone is applied to obtain the existence of at least one positive solution for the second-order three-point boundary value problem, where f is a nonnegative continuous function, α>0, η∈(0, 1), αη
120 citations
TL;DR: This work studies the existence behavior of positive solutions to a singular two point boundary value problem of second order impulsive equation with fixed moments using a method of upper and lower solutions with fixed point index theorems on a cone.
110 citations
TL;DR: Several existence and multiplicity results are obtained by an application of the Krasnosel'skii fixed-point theorem of cone expansion-compression type for positive solutions of equations that describe the deformations of elastic beams with fixed both endpoints.
106 citations
TL;DR: The notion of T-weak commutativity for a hybrid pair ( f, T ) of single-valued and multivalued mappings was introduced in this paper, where the authors obtained some fixed point theorems for this class of maps and derived an approximation theorem.
102 citations
TL;DR: In this article, it was shown that if is a bounded open set in a complete space, and if is nonexpansive, then always has a fixed point if there exists such that for all.
Abstract: We show that if is a bounded open set in a complete space , and if is nonexpansive, then always has a fixed point if there exists such that for all . It is also shown that if is a geodesically bounded closed convex subset of a complete -tree with , and if is a continuous mapping for which for some and all , then has a fixed point. It is also noted that a geodesically bounded complete -tree has the fixed point property for continuous mappings. These latter results are used to obtain variants of the classical fixed edge theorem in graph theory.
97 citations
TL;DR: In this article, the existence and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks is studied. But, the authors do not apply in solving these problems because there is no maximum principle or conservation laws available to the integral differential equation.
Journal Article•
TL;DR: In this article, a fixed point theorem of Leray-Schauder type for operators defined on Banach algebras is established. But this theorem is only applicable to functional integral equations.
Abstract: This paper establishes an existence theorem for a certain class of functional integral equations via a new fixed point theorem of Leray-Schauder type for operators defined on Banach algebras.
TL;DR: In this article, the non-local and non-variational elliptic problem with fixed point theorems is studied, in two different cases N = 1 and N ⩾ 2.
Abstract: We will study the nonlocal and nonvariational elliptic problem - a ∫ Ω | u | q Δ u = H ( x ) f ( u ) in Ω , u = 0 on ∂ Ω , by considering, in two different results, the cases N = 1 and N ⩾ 2 . In both cases we will use fixed point theorems.
TL;DR: In this paper, the existence of global solutions for the nonlinear and damped extensible plate (or beam) equation is proved by means of the Fixed Point Theorem and continuity arguments, and uniform decay rates of the energy are also obtained by making use of perturbed energy method for domains with finite measure.
Abstract: The nonlinear and damped extensible plate (or beam) equation is considered where Ω is any bounded or unbounded open set of Rn, α>0 and f, g are power like functions. The existence of global solutions is proved by means of the Fixed Point Theorem and continuity arguments. To this end we avoid handling the nonlinearity M(∫Ω|∇u|2dx) in the a priori estimates of energy. Furthermore, uniform decay rates of the energy are also obtained by making use of the perturbed energy method for domains with finite measure.
TL;DR: In this article, a new fixed-point theorem of functional type in a cone is established, and the existence of three positive solutions for the boundary value problemx^''(t)+f(t,x(t),x^'(t)) = 0,0 [0, ~] is continuous.
Abstract: In this paper, a new fixed-point theorem of functional type in a cone is established. With using the new fixed-point theorem and imposing growth conditions on the nonlinearity, the existence of three positive solutions for the boundary value problemx^''(t)+f(t,x(t),x^'(t))=0,0 [0, ~) is continuous. Finally, an example is given to illustrate the importance of results obtained.
TL;DR: By choosing a different fixed-point theorem, it is shown that the measures of noncompactness can be avoided and the existence and stability can be proved under weaker conditions.
Journal Article•
TL;DR: In this article, the existence of positive solu- tions of a boundary value problem for a one dimensional -Laplacian ordinary dierential equation with deviating arguments was proved.
Abstract: We provide sucient conditions for the existence of positive solu- tions of a boundary-value problem for a one dimensional -Laplacian ordinary dierential equation with deviating arguments, where is a sup-multiplicative- like function (in a sense introduced here) and the boundary conditions include nonlinear expressions at the end points. For this end, we use the Krasnoselskii fixed point theorem in a cone. The results obtained improve and generalize known results in (17) and elsewhere.
TL;DR: In this paper, the existence and multiplicity of positive solutions for the m-boundary value problems (p(t)u^')^'-q( t)u+f(t,u)=0,0 are considered.
Book•
05 Aug 2004
TL;DR: In this article, the authors present a functional analysis of convex-valued mapping with fixed points and Aumann Integrals, and show that the fixed points for multivalued contractions are decomposable sets.
Abstract: Introduction.- Part 1. Functional Analysis Background. 1. Preliminaries. 2. Real and Vector Measures.- Part 2. Multifunctions. 3. Preliminary Notions. 4. Upper and Lower Semicontinuous multifunctions. 5. Measurable Multifunctions. 6. Caratheodory type multifunctions. 7. Fixed Points Property for Convex-Valued Mapping.- Part 3. Decomposability. 8. Decomposable Sets. 9. Selections. 10. Fixed Points Property. 11. Aumann Integrals. 12. Selections of Aumann Integrals. 13. Fixed Points for Multivalued Contractions. 14. Operator and Differential Inclusions. 15. Decomposable Analysis.- Bibliography.
TL;DR: In this article, the existence of a positive solution for the three point boundary value problem on time scale T given by y ΔΔ +f(x,y)=0, x∈(0, 1]∩ T, y(0)=0 and y(p)=y σ 2 (1), where p ∈( 0, 1) ∩ T is fixed and f is singular at y=0 and possibly at x=0, y=∞.
TL;DR: In this paper, some multiplicity results for positive solutions of some singular semi-positone three-point boundary value problem can be obtained by using the fixed point index method, and the results are shown in Table 1.
31 Aug 2004
TL;DR: The modal fixed point logic (HFL) as mentioned in this paper extends the modal μ-calculus to allow predicates on states (sets of states) to be specified using recursively defined higher order functions on predicates.
Abstract: We present a higher order modal fixed point logic (HFL) that extends the modal μ-calculus to allow predicates on states (sets of states) to be specified using recursively defined higher order functions on predicates. The logic HFL includes negation as a first-class construct and uses a simple type system to identify the monotonic functions on which the application of fixed point operators is semantically meaningful. The model checking problem for HFL over finite transition systems remains decidable, but its expressiveness is rich. We construct a property of finite transition systems that is not expressible in the Fixed Point Logic with Chop [1] but which can be expressed in HFL. Over infinite transition systems, HFL can express bisimulation and simulation of push down automata, and any recursively enumerable property of a class of transition systems representing the natural numbers.
TL;DR: In this article, the notion of -distance is used to prove fixed point theorems, which are generalizations of fixed point generalizations by Kannan, Meir-Keeler, Edelstein, and Nadler.
Abstract: Using the notion of -distance, we prove several fixed point theorems, which are generalizations of fixed point theorems by Kannan, Meir-Keeler, Edelstein, and Nadler. We also discuss the properties of -distance.
TL;DR: In this paper, the authors apply a cone theoretic fixed point theorem and obtain conditions for the existence of positive solutions to the three-point nonlinear second order boundary value problem with boun...
Abstract: In this paper, we apply a cone theoretic fixed point theorem and obtain conditions for the existence of positive solutions to the three-point nonlinear second order boundary value problem with boun...
Journal Article•
TL;DR: In this article, it is shown that some of the hypotheses of a fixed point theorem of the present author involving three operators in a Banach algebra are redundant, and this claim is also illustrated with the applications to some nonlinear functional integral equations for proving the existence result.
Abstract: In this article, it is shown that some of the hypotheses of a fixed point theorem of the present author involving three operators in a Banach algebra are redundant. Our claim is also illustrated with the applications to some nonlinear functional integral equations for proving the existence result.
TL;DR: In this article, fixed point, domain invariance and coincidence results are presented for single-valued generalized contractive maps of Meir-Keeler type defined on complete metric spaces (or more generally complete gauge spaces).
Abstract: Fixed point, domain invariance and coincidence results are presented for single-valued generalized contractive maps of Meir–Keeler type defined on complete metric spaces (or more generally complete gauge spaces). The maps of Caristi type are also considered. In addition the random analogue of some of these fixed point results will be presented. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this paper, the controllability problem of evolution inclusions with nonlocal conditions is examined, using the set-valued and single-valued Monch fixed-point theorem, which is applicable to a wide class of (impulsive) control systems.
Abstract: In this paper, we examine controllability problems of evolution inclusions with nonlocal conditions. Using the set-valued and single-valued Monch fixed-point theorem, we establish some sufficient conditions for the controllability under convex and nonconvex orientor fields respectively. Our approach is different from all previous approaches; we do not assume that the evolution system generates a compact semigroup; so, our method is applicable to a wide class of (impulsive) control systems and evolution inclusions in Banach spaces.
TL;DR: In this article, the collective fixed points theorem for a family of multivalued maps with or without assuming that the product of these maps is Φ-condensing is established.
TL;DR: In this article, the authors considered the following dynamical system on a measure chain: uΔΔ1(t) + f 1(t, u 1 (σ(t)), u 2 (σ (t)) = 0, t ∈ [a, b], u ΔΔ2(t), f 2 (t), u1 (σ, t), u2 (σ) = 0 for i = 1, 2 for αui(a) - βuΔi(a), γui(σ(b)) + �
TL;DR: By using Sadovskii fixed point theorem, the controllable and the local controllability of abstract neutral functional differential systems with unbounded delay are studied.