Showing papers on "Fixed-point theorem published in 1997"
Book•
01 Jan 1997
TL;DR: In this paper, the authors present a real analysis of the Hamilton-Jacobi Equation and the classical N-body Hamiltonians, as well as pseudo-differential and Fourier Integral operators.
Abstract: 0. Introduction.- 1. Classical Time-Decaying Forces.- 2. Classical 2-Body Hamiltonians.- 3. Quantum Time-Decaying Hamiltonians.- 4. Quantum 2-Body Hamiltonians.- 5. Classical N-Body Hamiltonians.- 6. Quantum N-Body Hamiltonians.- A. Miscellaneous Results in Real Analysis.- A.1 Some Inequalities.- A.2 The Fixed Point Theorem.- A.3 The Hamilton-Jacobi Equation.- A.4 Construction of Some Cutoff Functions.- A.5 Propagation Estimates.- A.6 Comparison of Two Dynamics.- A.7 Schwartz's Global Inversion Theorem.- B. Operators on Hilbert Spaces.- B.1 Self-adjoint Operators.- B.2 Convergence of Self-adjoint Operators.- B.3 Time-Dependent Hamiltonians.- B.4 Propagation Estimates.- B.5 Limits of Unitary Operators.- B.6 Schur's Lemma.- C. Estimates on Functions of Operators.- C.1 Basic Estimates of Commutators.- C.2 Almost-Analytic Extensions.- C.3 Commutator Expansions I.- C.4 Commutator Expansions II.- D. Pseudo-differential and Fourier Integral Operators.- D.0 Introduction.- D.1 Symbols of Operators.- D.2 Phase-Space Correlation Functions.- D.3 Symbols Associated with a Uniform Metric.- D.4 Pseudo-differential Operators Associated with a Uniform Metric.- D.5 Symbols and Operators Depending on a Parameter.- D.6 Weighted Spaces.- D.7 Symbols Associated with Some Non-uniform Metrics.- D.8 Pseudo-differential Operators Associated with the Metric 91.- D.9 Essential Support of Pseudo-differential Operators.- D.10 Ellipticity.- D.12 Non-stationary Phase Method.- D.13 FIO's Associated with a Uniform Metric.- D.14 FIO's Depending on a Parameter.- References.
408 citations
Book•
26 Nov 1997
TL;DR: The fixed point theorems of Brouwer and Schauder measures of noncompactness minimal sets for a measure of non-convexity and smoothness near uniform convexity, and near uniform smoothness fixed point for nonexpansive mappings for normal structure uniformly Lipschitzian mappings were studied in this paper.
Abstract: The fixed point theorems of Brouwer and Schauder measures of noncompactness minimal sets for a measure of noncompactness convexity and smoothness near uniform convexity and near uniform smoothness fixed point for nonexpansive mappings and normal structure fixed point theorems in the absence of normal structure uniformly Lipschitzian mappings asymptotically regular mappings packing rates and 0-contractiveness constants.
352 citations
Journal Article•
TL;DR: In this paper, the authors give a short survey on fixed point theorems which are generalizations of the classical Banach-Caccioppoli principle of contractive mappings.
Abstract: We give a short survey on some fixed point theorems which are generalizations of the classical Banach-Caccioppoli principle of contractive mappings. All these results are gathered in three theorems about existence and uniqueness of fixed points for operators which act in $K$-metric or $K$-normed linear spaces and, in particular, in local convex spaces and scales of Banach spaces. Three fixed point theorems presented in this article cover numerous applications in numerical methods, theory of integral equations, some results on iterative methods for construction of periodic solution to ordinary differential equations, existence and uniqueness results on solvability for Cauchy and Goursat problems of Ovsjannikov - Treves - Nirenberg type and so on.
170 citations
01 Jan 1997
TL;DR: In this paper, the existence of positive solutions of the (g(u′))′ + a(t)f(u) = 0, where g(v) = |v|p−2v, p > 1, subject to nonlinear boundary conditions was studied.
Abstract: In this paper we study the existence of positive solutions of the equation (g(u′))′ + a(t)f(u) = 0, where g(v) = |v|p−2v, p > 1, subject to nonlinear boundary conditions. We show the existence of at least one positive solution by a simple application of a Fixed Point Theorem in cones and the Arzela-Ascoli Theorem.
151 citations
TL;DR: A survey of the latest and new results on some topics in the study of nonlinear analysis, which were obtained by the author and some Chinese mathematicians, is presented in this article.
Abstract: This paper is presented as a survey of the latest and new results on some topics in the study of nonlinear analysis, which were obtained by the author and some Chinese mathematicians.
150 citations
Journal Article•
TL;DR: The notion of weak commutativity was introduced by Sessa as mentioned in this paper as a sharper tool to obtain common fixed points of mappings and a flood of common fixed point theorems was produced by various researchers.
Abstract: It was the turning point in the "fixed point arena" when the notion of weak commutativity was introduced by Sessa [9] as a sharper tool to obtain common fixed points of mappings. As a result, all the results on fixed point theorems for commuting mappings were easily transformed in the setting of the new notion of weak commutativity of mappings. It gives a new impetus to the studying of common fixed points of mappings satisfying some contractive type conditions and a number of interesting results have been found by various authors. A bulk of results were produced and it was the centre of vigorous research activity in "Fixed Point Theory and its Application in various other Branches of Mathematical Sciences" in last two decades. A major break through was done by Jungck [3] when he proclaimed the new notion what he called "compatibility" of mapping and its usefulness for obtaining common fixed points of mappings was shown by him. There-after a flood of common fixed point theorems was produced by various researchers by using the improved notion of compatibility of mappings. of compatibility of mappings.
96 citations
Posted Content•
TL;DR: In this paper, the authors extend Himmelberg's fixed point theorem by replacing the usual convexity in topological vector spaces by an abstract topological notion, which generalizes classical convexness as well as several metric convexities found in the literature, and prove the existence of a fixed point for a compact approachable map.
Abstract: The purpose of this paper is to extend Himmelberg's fixed point theorem replacing the usual convexity in topological vector spaces by an abstract topological notion of convexity which generalizes classical convexity as well as several metric convexity structures found in the literature. We prove the existence, under weak hypothese, of a fixed point for a compact approachable map and we provide sufficient conditions under which this result applies to maps whose values are convex in the abstract sense mentionned above.
95 citations
TL;DR: Fixed point theorems for fuzzy mappings satisfying contractive-type conditions and a rational inequality in complete metric spaces are proved.
94 citations
Book•
30 Sep 1997
TL;DR: In this article, the KKM-map principle is used for fixed point theory and the best approximation theorem of Ky Fan's best approximate theorem is derived from the principle of fixed points.
Abstract: 1: Fixed Point Theory and Best Approximation: The KKM-Map Principle. 1. Introductory Concepts and Fixed Point Theorems. 2. Ky Fan's Best Approximation Theorem. 3. Principle and Applications of KKM-Maps. 4. Partitions of Unity and Applications. 5. Application of Fixed Points to Approximation Theory. Bibliography.
92 citations
TL;DR: In this article, the equivalence between variational inclusions and a generalized type of the Weiner-Hopf equation is established, which is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators.
Abstract: In this paper, the equivalence between variational inclusions and a generalized type of Weiner–Hopf equation is established. This equivalence is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators. Special attention is given to the case where one of the operators is Lipschitz continuous and either is strongly monotone or satisfies the Dunn property. Moreover, when the problem has a nonempty solution set, a fixed-point procedure is proposed and its convergence is established provided that the Brezis–Crandall–Pazy condition holds true. More precisely, it is shown that this allows reaching the element of minimal norm of the solution set.
80 citations
TL;DR: In this paper, a fixed point theorem for operators that are decreasing with respect to a cone is presented for the boundary value problem, where f (x, y ) is singular at y = 0.
TL;DR: In this paper, sufficient conditions for the existence, uniqueness and global asymptotic stability of periodic solutions are established by combining the theory of monotone flow generated by FDEs, Horn's asymPTotic fixed point theorem and linearized stability analysis.
Abstract: We consider here a general Lotka-Volterra type ^-dimensional periodic functional differential system. Sufficient conditions for the existence, uniqueness and global asymptotic stability of periodic solutions are established by combining the theory of monotone flow generated by FDEs, Horn's asymptotic fixed point theorem and linearized stability analysis. These conditions improve and generalize the recent ones obtained by Freedman and Wu (1992) for scalar equations. We also present a nontrivial application of our results to a delayed nonautonomous predator-prey system.
TL;DR: The role of quasi-metrics in the fixed point semantics of logic programs is considered, examining in detail a quite general process by which fixed points of immediate consequence operators can be found.
Abstract: Quasi-metrics have been used in several places in the literature on domain theory and the formal semantics of programming languages In this paper, we consider the role of quasi-metrics in the fixed point semantics of logic programs, examining in detail a quite general process by which fixed points of immediate consequence operators can be found This work takes as its starting point: (i) Fitting's recent application of the Banach contraction mapping theorem in logic programming; (ii) a theorem of Rutten which generalizes both the contraction mapping theorem and the Knaster-Tarski theorem; (iii) Smyth's work on totally bounded spaces and compact ordered spaces as domains of computation Our results therefore are theoretical and to be viewed as a contribution to the mathematical foundations of computer science
01 Jan 1997
TL;DR: In this paper, the authors give a new fixed theorem of lower semicontinuous multivalued mappings, and then obtain some new equilibrium theorems for abstract economies and qualitative games.
Abstract: In this paper, we first give a new fixed theorem of lower semicontinuous multivalued mappings, and then, as its applications we obtain some new equilibrium theorems for abstract economies and qualitative games.
TL;DR: In this article, some random fixed point theorems for random 1-set-contraction and random continuous condensing mappings defined on closed balls of a separable Banach space, or on separable closed convex subsets of a Hilbert space or on spheres of infinite dimensional separable spaces are established.
Abstract: In this paper, we first prove some random fixed point theorems for random nonexpansive operators in Banach spaces. As applications, some random approximation theorems for random 1-set-contraction or random continuous condensing mappings defined on closed balls of a separable Banach space, or on separable closed convex subsets of a Hilbert space or on spheres of infinite dimensional separable Banach spaces are established. Our results are generalizations, improvements or stochastic versions of the corresponding results of Bharucha-Reid (1976), Lin (1988, 1989), Lin and Yen (1988), Massatt (1983), Sehgal and Waters (1984) and Xu (1990).
TL;DR: In this article, the authors extend the classical inverse and implicit function theorems of Lyusternik and Graves and the results of Clarke and Pourciau to the situation when the given function is not smooth, but it has a convex strict prederivative whose measure of noncompactness is smaller than its measure of surjectivity.
TL;DR: In this paper, it was shown that there are necessary and sufficient conditions for the existence of at least one periodic solution for a type of parametric second-order ordinary differential equations, known as the Mathieu-Duffing equation.
Abstract: It is found that there exists necessary and sufficient conditions for the existence of at least one periodic solution for a type of parametric second-order ordinary differential equations, known as the Mathieu-Duffing equation. The correctness of the conditions have been pointed out by the Schauder's fixed point theorem, and the validity of the assumptions has been shown by the analysis of an illustrative example in non-linear vibration.
01 Jan 1997
TL;DR: In this article, the authors prove several fixed point theorems, which are generalizations of the Banach contraction principle and Kannan's fixed point theorem, and discuss a characterization of metric completeness.
Abstract: In this paper, we prove several fixed point theorems, which are generalizations of the Banach contraction principle and Kannan’s fixed point theorem. Further we discuss a characterization of metric completeness.
TL;DR: In this article, the stability of the procedure of successive approximations for Banach contractive maps has been studied and generalized by using a more general contractive definition introduced by F. Browder.
Abstract: A. M. Ostrowski established the stability of the procedure of successive approximations for Banach contractive maps. In this paper we generalize the above result by using a more general contractive definition introduced by F. Browder. Further, we study the case of maps on metrically convex metric spaces and compact metric spaces, obtaining results relative to fixed point theorems of D. W. Boyd and J. S. W. Wong, and M. Edelstein. Finally, as a by-product of our basic lemma, we extend a recent result of T. Vidalis concerning the convergence of an iteration procedure involving an infinite sequence of maps.
01 Jan 1997
TL;DR: In this paper, it was shown that an area-preserving diffeomorphism of an annulus which shifts the boundary circles at opposite directions has at least two fixed points, provided the shift of the boundaries is large enough.
Abstract: Poincare’s geometric theorem claims that an area-preserving diffeomorphism of an annulus which shifts the boundary circles at opposite directions has at least two fixed points. The present paper consists of two parts. In the first one, we show that such a diffeomorphism has more than just two fixed points provided the shift of the boundaries is large enough. In the second part, we prove symplectic fixed point theorems which can be viewed as generalizations of Poincare’s geometric theorem to higher dimensions.
TL;DR: The seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject.
Abstract: Set theory, it has been contended, developed from its beginnings through a progression of mathematical moves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions of the power set of a set into the set in the fundamental work of Zermelo on set theory. His first proof in 1904 of his Well-Ordering Theoremis a central articulation containing much of what would become familiar in the subsequent development of set theory. Afterwards, the motif is cast by Kuratowski as a fixed point theorem, one subsequently abstracted to partial orders by Bourbaki in connection with Zorn's Lemma. Migrating beyond set theory, that generalization becomes cited as the strongest of fixed point theorems useful in computer science. Section 1 describes the emergence of our guiding motif as a line of development from Cantor's diagonal proof to Russell's Paradox, fueled by the clarification of the inclusion vs. membership distinction. Section 2 engages the motif as fully participating in Zermelo's work on the Well-Ordering Theorem and as newly informing on Cantor's basic result that there is no bijection . Then Section 3 describes in connection with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science.
TL;DR: A constructive version of the Stone-Weierstrass theorem is proved in this paper, allowing a globalisation of the Gelfand duality theorem to any Grothendieck topos.
26 Mar 1997
TL;DR: This paper employs Banach's fixed point theory in metric spaces as mathematical foundation to show that the theory used for discrete functional specifications smoothly carries over to real-time and hybrid systems.
Abstract: Functional specifications have been used to specify and verify designs of a number of reactive, discrete systems. In this paper we extend this specification style to deal with real-time and hybrid systems. As mathematical foundation we employ Banach's fixed point theory in metric spaces. The goal is to show that the theory used for discrete functional specifications smoothly carries over to real-time and hybrid systems. An example of a thermostat specification illustrates the method.
TL;DR: In this article, the boundary value problem is solved for operators that are decreasing with respect to a cone, where f(x,y) is a singular operator at y = 0.
Abstract: Solutions are obtained for the boundary value problem, y(n) + f(x,y) = 0, y(i)(0) = y(1) = 0, 0 ≤ i ≤ n − 2, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.
TL;DR: By virtue of this concept, some theorems about common fixed degree of a sequence of fuzzy mappings in probabilistic metric spaces are obtained and these new results are a unified approach to generalize several fixed point theorem for fuzzymappings.
TL;DR: In this article, a generalized form of the Poincare-birkhoff theorem and a fixed point theorem for the equation x + g (x ) = p ( t ), p t )≡ p t + 2 π was presented.
TL;DR: The existence of positive solutions to a three-point boundary value problem for the one-dimensional p-Laplacian is proved by a simple application of a Fixed Point Theorem in cones due to Krasnoselskii and the Arzela-Ascoli Theorem as discussed by the authors.
Abstract: The existence of positive solutions to a three-point boundary value problem for the one-dimensional p-Laplacian is proved by a simple application of a Fixed Point Theorem in cones due to Krasnoselskii and the Arzela-Ascoli Theorem.
TL;DR: In this paper, the authors prove three approximate selection theorems and give an improved version of the Michael selection theorem for generalized games, as well as a fixed point theorem and equilibrium theorem.