Showing papers on "Fixed-point theorem published in 2013"
Book•
21 Mar 2013TL;DR: In this paper, the authors give a systematic grounding in the theory of Hamiltonian differential equations from a dynamical systems point of view and develop a solid foundation for students to read some of the current research on Hamiltonian systems.
Abstract: This book gives a systematic grounding in the theory of Hamiltonian differential equations from a dynamical systems point of view. It develops a solid foundation for students to read some of the current research on Hamiltonian systems. Topics covered include a detailed discussion of linear Hamiltonian systems, an introduction to the theory of integrals and reduction, Poincare's continuation of periodic solution, normal forms and applications of KAM theory. A chapter is devoted to the theory of twist maps and various extensions of the classic Poincare-Birkhoff fixed point theorem.
828 citations
TL;DR: In this article, the authors modify the notions of α-admissible and α-ψ-contractive mappings and establish new fixed point theorems for such mappings in complete metric spaces.
Abstract: The aim of this work is to modify the notions of α-admissible and α-ψ-contractive mappings and establish new fixed point theorems for such mappings in complete metric spaces. Presented theorems provide main results of Karapinar and Samet (Abstr. Appl. Anal. 2012:793486, 2012) and Samet et al. (Nonlinear Anal. 75:2154-2165, 2012) as direct corollaries. Moreover, some examples and applications to integral equations are given here to illustrate the usability of the obtained results.
181 citations
TL;DR: In this paper, the authors investigated the existence of a unique equilibrium for general bidirectional associative memory neural networks with time-varying delays in the leakage terms by the fixed point theorem and established sufficient conditions on the global exponential stability of the equilibrium for such networks.
Abstract: In this paper, we first investigate the existence of a unique equilibrium to general bidirectional associative memory neural networks with time-varying delays in the leakage terms by the fixed point theorem. Then, by constructing a Lyapunov functional, we establish some sufficient conditions on the global exponential stability of the equilibrium for such neural networks, which substantially extend and improve the main results of Gopalsamy [K. Gopalsamy, Leakage delays in BAM, J. Math. Anal. Appl. 325 (2007) 1117–1132].
168 citations
TL;DR: In this article, a particular case of a contractive self-mapping on a complete metric space, namely the F-contraction introduced by Wardowski (fixed point theory Appl. 87, 2012), is considered, and some new properties of it are provided.
Abstract: In this paper we consider a particular case of a contractive self-mapping on a complete metric space, namely the F-contraction introduced by Wardowski (Fixed Point Theory Appl. 87, 2012, doi:10.1186/1687-1812-2012-94), and provide some new properties of it. As an application, we investigate the iterated function systems (IFS) composed of F-contractions extending some fixed point results from the classical Hutchinson-Barnsley theory of IFS consisting of Banach contractions. Some illustrative examples are given. MSC: Primary 28A80; secondary 47H10; 54E50
153 citations
TL;DR: In this article, the fixed point results for closed multi-valued F-contractions were presented for complete metric spaces or complete ordered metric spaces, and two applications for the solution of certain functional and integral equations were given to illustrate the usability of the obtained results.
Abstract: Wardowski (Fixed Point Theory Appl., 2012:94) introduced a new concept of contraction and proved a fixed point theorem which generalizes Banach contraction principle. Following this direction of research, we will present some fixed point results for closed multi-valued F-contractions or multi-valued mappings which satisfy an F-contractive condition of Hardy-Rogers-type, in the setting of complete metric spaces or complete ordered metric spaces. An example and two applications, for the solution of certain functional and integral equations, are given to illustrate the usability of the obtained results.
151 citations
TL;DR: By employing the fixed point theory and the monotone iterative technique, the existence of a unique solution for a class of nonlinear fractional integro-differential equations on semi-infinite domains in a Banach space is investigated.
146 citations
TL;DR: In this article, the Hyers-Ulam stability of the Cauchy equation was improved by using a fixed point theorem, which was proved by T. Aoki, Z. Gajda, and Th. M. Rassias.
Abstract: We show that a very classical result, proved by T. Aoki, Z. Gajda and Th. M. Rassias and concerning the Hyers–Ulam stability of the Cauchy equation f(x+y)=f(x)+f(y), can be significantly improved. We also provide some immediate applications of it (among others for the cocycle equation, which is useful in characterizations of information measures). In particular, we give a solution to a problem that was formulated more than 20 years ago and concerned optimality of some estimations. The proof of that result is based on a fixed point theorem.
125 citations
TL;DR: This formalization was started in 2005 and has been extensively developed since then, partly in direct support of the Flyspeck project, partly out of a general desire to develop a well-rounded and comprehensive theory of basic analytical, geometrical and topological machinery.
Abstract: We describe the library of theorems about N-dimensional Euclidean space that has been formalized in the HOL Light prover. This formalization was started in 2005 and has been extensively developed since then, partly in direct support of the Flyspeck project, partly out of a general desire to develop a well-rounded and comprehensive theory of basic analytical, geometrical and topological machinery. The library includes various `big name' theorems (Brouwer's fixed point theorem, the Stone-Weierstrass theorem, the Tietze extension theorem), numerous non-trivial results that are useful in applications (second mean value theorem for integrals, power series for real and complex transcendental functions) and a host of supporting definitions and lemmas. It also includes some specialized automated proof tools. The library has as planned been applied to the Flyspeck project and has become the basis of a significant development of results in complex analysis, among others.
121 citations
TL;DR: In this article, a few generalizations of the Darbo fixed point theorem are provided, and several interconnections among assumptions imposed in the proved theorems are indicated, showing the applicability of obtained results to the theory of functional integral equations.
Abstract: In the paper we provide a few generalizations of Darbo fixed point theorem. Several interconnections among assumptions imposed in the proved theorems are indicated. We also show the applicability of obtained results to the theory of functional integral equations. A concrete example illustrating the mentioned applicability is also included.
119 citations
Journal Article•
TL;DR: In this article, the authors present some common fixed point theorems for occasionally weakly compatible mappings in fuzzy metric spaces, which is a special case of fuzzy metric space.
Abstract: : This paper presents some common fixed point theorems for occasionally weakly compatible mappings in fuzzy metric spaces. Keywords : Occasionally weakly compatible mappings,fuzzy metric space.
108 citations
TL;DR: By using the fixed point theorem of the mixed monotone operator, the uniqueness of positive solution for a singular fractional differential system involving derivatives is established.
TL;DR: In this article, the existence of solutions of nonlinear fractional pantograph equations was studied in strongly anomalous media and the results were obtained using fractional calculus and fixed point theorems.
TL;DR: In this article, it was shown that Caristi's theorem holds in complete generalized metric spaces without further assumptions, which is noteworthy because Banach's fixed point theorem seems to require more than the quadrilateral inequality.
Abstract: A ‘generalized metric space’ is a semimetric space which does not satisfy the triangle inequality, but which satisfies a weaker assumption called the quadrilateral inequality. After reviewing various related axioms, it is shown that Caristi’s theorem holds in complete generalized metric spaces without further assumptions. This is noteworthy because Banach’s fixed point theorem seems to require more than the quadrilateral inequality, and because standard proofs of Caristi’s theorem require the triangle inequality.
TL;DR: By using the fixed point theory in cone and constructing some available integral operators together with approximating technique, the existence of positive solution for a singular nonlinear semipositone fractional differential system with coupled boundary conditions is established.
TL;DR: In this paper, some basic hybrid fixed point theorems of Banach and Schauder type and some hybrid fixed-point theorem of Krasnoselskii type involving the sum of two operators are proved in a partially ordered normed linear spaces which are further applied to nonlinear Volterra fractional integral equations for proving the existence of solutions under certain mono- tonic conditions blending with the existence either a lower or an upper solution type function.
Abstract: In this paper, some basic hybrid fixed point theorems of Banach and Schauder type and some hybrid fixed point theorems of Krasnoselskii type involving the sum of two operators are proved in a partially ordered normed linear spaces which are further applied to nonlinear Volterra fractional integral equations for proving the existence of solutions under certain mono- tonic conditions blending with the existence of either a lower or an upper solution type function. This research is dedicated in the loving memory of my late father and mother who imbibed in me the honesty, hard-work and services for all.
Book•
11 Jul 2013
TL;DR: Kantorovich Theory for Newton-like Methods Holder Conditions and Newton-type Methods Regular Smoothness Conditions for Iterative Methods Fixed Point Theory and iterative Methods Mathematical Programming fixed point theory for set-valued mapping Special Convergence Conditions Recurrent Functions and Newton Like Methods Recurrent functions and special iterative methods as mentioned in this paper.
Abstract: Kantorovich Theory for Newton-Like Methods Holder Conditions and Newton-Type Methods Regular Smoothness Conditions for Iterative Methods Fixed Point Theory and Iterative Methods Mathematical Programming Fixed Point Theory for Set-Valued Mapping Special Convergence Conditions Recurrent Functions and Newton-Like Methods Recurrent Functions and Special Iterative Methods.
TL;DR: In this paper, the p-Laplacian model involving the Caputo fractional derivative with Dirichlet-Neumann boundary conditions was studied and the existence of at least three solutions of the model was proved.
Abstract: In this paper, we study the p-Laplacian model involving the Caputo fractional derivative with Dirichlet-Neumann boundary conditions. Using a fixed point theorem, we prove the existence of at least three solutions of the model. As an application, an example is included to illustrate the main results.
TL;DR: In this article, the existence of solutions for a sequential integrodifferential equation of fractional order with some boundary conditions is investigated by means of some standard tools of fixed point theory.
Abstract: We investigate the existence of solutions for a sequential integrodifferential equation of fractional order with some boundary conditions. The existence results are established by means of some standard tools of fixed point theory. An illustrative example is also presented.
03 Nov 2013
TL;DR: In this paper, the authors apply Rothe's type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation: in on, where is a bounded domain in,, is an open nonempty subset of, denotes the characteristic function of the set, the distributed control belongs to, and the nonlinear function is smooth enough, and there are, and such that for all under this condition, the system is approximately controllable on.
Abstract: We apply Rothe’s type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation: in on , where is a bounded domain in , , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control belongs to , and the nonlinear function is smooth enough, and there are , and such that for all Under this condition, we prove the following statement: for all open nonempty subset of , the system is approximately controllable on . Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state to an neighborhood of the final state at time .
TL;DR: In this article, the existence and nonexistence of nontrivial traveling wave solutions are determined by the reproduction number in a Kermack-McKendrick epidemic model with nonlocal dispersal.
Abstract: In this paper, we consider a Kermack-McKendrick epidemic model with
nonlocal dispersal. We find that the existence and nonexistence of
traveling wave solutions are determined by the reproduction number.
To prove the existence of nontrivial traveling wave solutions, we
construct an invariant cone in a bounded domain with initial
functions being defined on, and apply Schauder's fixed point theorem
as well as limiting
argument. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Moreover, the
nonexistence of traveling wave solutions is obtained by Laplace transform if the speed is
less than the critical velocity.
TL;DR: In this article, the controllability of a class of abstract impulsive mixed-type functional integro-differential equations with finite delay in a Banach space is established.
Abstract: In this paper, we establish the controllability for a class of abstract impulsive mixed-type functional integro-differential equations with finite delay in a Banach space. Some sufficient conditions for controllability are obtained by using the Monch fixed point theorem via measures of noncompactness and semigroup theory. Particularly, we do not assume the compactness of the evolution system. An example is given to illustrate the effectiveness of our results.
Journal Article•
TL;DR: In this paper, a common fixed point theorem using new continuity condition in fuzzy metric spaces was proved using R-weakly commuting maps in fuzzy matrices, which is based on the reciprocal continuity condition.
Abstract: - The Purpose of this paper, we prove common fixed point theorem using new continuity condition in fuzzy metric spaces Keywords: - Compatible maps, R-weakly commuting maps, reciprocal continuity Mathematics subject classification: - 47H10, 54H25
TL;DR: In this article, the existence and uniqueness of a fixed point for certain α-admissible contraction mappings was proved and extended to the case of fixed points in the form of fixed-points.
Abstract: In this paper, we prove the existence and uniqueness of a fixed point for certain α-admissible contraction mappings Our results generalize and extend some well-known results on the topic in the literature We consider some examples to illustrate the usability of our results MSC: 46N40; 47H10; 54H25; 46T99
TL;DR: In this paper, the fixed point results for single-valued Geraghty and Meir-Keeler-type contractions, as well as multi-valued contractive mappings are presented.
Abstract: Samet et al. (Nonlinear Anal. 75:2154-2165, 2012) introduced α-ψ-contractive mappings and proved some fixed point results for these mappings. More recently Salimi et al. (Fixed Point Theory Appl. 2013:151, 2013) modified the notion of α-ψ-contractive mappings and established certain fixed point theorems. Here, we continue to utilize these modified notions for single-valued Geraghty and Meir-Keeler-type contractions, as well as multi-valued contractive mappings. Presented theorems provide main results of Hussain et al. (J. Inequal. Appl. 2013:114, 2013), Karapinar et al. (Fixed Point Theory Appl. 2013:34, 2013) and Asl et al. (Fixed Point Theory Appl. 2012:212, 2012) as corollaries. Moreover, some examples are given here to illustrate the usability of the obtained results. MSC:46N40, 47H10, 54H25, 46T99.
TL;DR: In this paper, the existence theory for nonlinear fractional differential equations with Riemann-Liouville type boundary conditions involving nonintersecting finite many strips of arbitrary length was developed.
Abstract: We develop the existence theory for nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type boundary conditions involving nonintersecting finite many strips of arbitrary length. Our results are based on some standard tools of fixed point theory. For the illustration of the results, some examples are also discussed.
TL;DR: This article showed that most of the coupled fixed point theorems (on ordered metric spaces) are in fact immediate consequences of well-known fixed point theorem in the literature, and showed that these results can be seen as a direct consequence of the fixed point conjecture.
Abstract: In this paper, we show that, unexpectedly, most of the coupled fixed point theorems (on ordered metric spaces) are in fact immediate consequences of well-known fixed point theorems in the literature. MSC: 47H10, 54H25.
TL;DR: The existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a Hilbert space is studied by using semigroup theory, fractional calculus and Schauder's fixed point theorem.
Abstract: We study the existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a Hilbert space. The results are obtained by using semigroup theory, fractional calculus and Schauder’s fixed point theorem. Multi-delay controls and a fractional nonlocal condition are introduced. Furthermore, we present an appropriate set of sufficient conditions for the considered fractional nonlocal multi-delay control system to be approximately controllable. An example to illustrate the abstract results is given.
DOI•
01 Jan 2013TL;DR: A survey of the most important basic properties of measures of noncompactness on bounded sets of complete metric spaces and Banach spaces can be found in this paper, along with a characterization of classes of compact operators between these spaces.
Abstract: This paper contains a survey of the most important basic properties of certain measures of noncompactness on bounded sets of complete metric spaces and Banach spaces, and of the Hausdorff measure of noncompactness of operators between Banach spaces. We also demonstrate how the theory of measures of noncompactness can be applied in fixed point theory, the theory of differential and integral equations, and the characterizations of classes of compact operators between certain Banach spaces.
TL;DR: Caballero et al. as discussed by the authors improved the best proximity point theorem for Geraghty non-self contraction by presenting a short and simple proof, which satisfies weak P-property but not P-Property.
Abstract: In Caballero et al. (Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-231), the authors prove a best proximity point theorem for Geraghty nonself contraction. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.
TL;DR: In this paper, the existence of positive solutions of a nonlinear fractional heat equation with nonlocal boundary conditions depending on a positive parameter was studied and the authors extended the second-order thermostat model to the non-integer case.
Abstract: We study the existence of positive solutions of a nonlinear fractional heat equation with nonlocal boundary conditions depending on a positive parameter. Our results extend the second-order thermostat model to the non-integer case. We base our analysis on the known Guo-Krasnosel’skii fixed point theorem on cones.