Showing papers on "Fixed-point theorem published in 2011"
Book•
26 Apr 2011TL;DR: This book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space, and a concise exposition of related constructive fixed point theory that allows for a wide range of algorithms to construct solutions to problems in optimization, equilibrium theory, monotone inclusions, variational inequalities, and convex feasibility.
Abstract: This book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space. A concise exposition of related constructive fixed point theory is presented, that allows for a wide range of algorithms to construct solutions to problems in optimization, equilibrium theory, monotone inclusions, variational inequalities, best approximation theory, and convex feasibility. The book is accessible to a broad audience, and reaches out in particular to applied scientists and engineers, to whom these tools have become indispensable.
4,365 citations
1 Jun 2011
TL;DR: The material presented provides a survey of the state-of-the-art theory and practice in fixed-point algorithms, identifying emerging problems driven by applications, and discussing new approaches for solving these problems.
Abstract: "Fixed-Point Algorithms for Inverse Problems in Science and Engineering" presents some ofthe most recent work from top-notch researchers studying projection and other first-order fixed-point algorithms in several areas of mathematics and the applied sciences. The material presented provides a survey of the state-of-the-art theory and practice in fixed-point algorithms, identifying emerging problems driven by applications, and discussing new approaches for solving these problems. This book incorporates diverse perspectives from broad-ranging areas of research including, variational analysis, numerical linear algebra, biotechnology, materials science, computational solid-state physics, and chemistry. Topics presented include: Theory of Fixed-point algorithms: convex analysis, convex optimization, subdifferential calculus, nonsmooth analysis, proximal point methods, projection methods, resolvent and related fixed-point theoretic methods, and monotone operator theory. Numerical analysis of fixed-point algorithms: choice of step lengths, of weights, of blocks for block-iterative and parallel methods, and of relaxation parameters; regularization of ill-posed problems; numerical comparison of various methods. Areas of Applications: engineering (image and signal reconstruction and decompression problems), computer tomography and radiation treatment planning (convex feasibility problems), astronomy (adaptive optics), crystallography (molecular structure reconstruction), computational chemistry (molecular structure simulation) and other areas. Because of the variety of applications presented, this book can easily serve as a basis for new and innovated research and collaboration.
633 citations
TL;DR: In this article, the existence and uniqueness of mild solutions for semilinear fractional evolution equations and optimal controls in the α -norm were proved by means of fractional calculus, singular version Gronwall inequality and Leray-Schauder fixed point theorem.
Abstract: This paper concerns the existence of mild solutions for semilinear fractional evolution equations and optimal controls in the α -norm. A suitable α -mild solution of the semilinear fractional evolution equations is introduced. The existence and uniqueness of α -mild solutions are proved by means of fractional calculus, singular version Gronwall inequality and Leray–Schauder fixed point theorem. The existence of optimal pairs of system governed by fractional evolution equations is also presented. Finally, an example is given for demonstration.
411 citations
TL;DR: In this article, complex valued metric spaces are introduced and sufficient conditions for the existence of common fixed points of a pair of mappings satisfying contractive type conditions are obtained. But these conditions are not sufficient for all mappings.
Abstract: We introduce complex valued metric spaces and obtain sufficient conditions for the existence of common fixed points of a pair of mappings satisfying contractive type conditions.
311 citations
TL;DR: In this paper, Bhaskar and Lakshmikantham proved coupled fixed point theorems for mappings having a mixed monotone property in partially ordered metric spaces.
Abstract: In this paper, we prove some coupled fixed point theorems for mappings having a mixed monotone property in partially ordered metric spaces. The main results of this paper are generalizations of the main results of Bhaskar and Lakshmikantham [T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA 65 (2006) 1379–1393]. As an application, we discuss the existence and uniqueness for a solution of a nonlinear integral equation.
292 citations
TL;DR: Some common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces are proved by using the new property of Kramosil and Michalek.
Abstract: We prove some common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces both in the sense of Kramosil and Michalek and in the sense of George and Veeramani by using the new property and give some examples Our results improve and generalize the main results of Mihet in (Mihet, 2010) and many fixed point theorems in fuzzy metric spaces
271 citations
TL;DR: In this article, the existence of P C-mild solutions for Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving Caputo fractional derivative is discussed.
Abstract: In this paper we discuss the existence of P C-mild solutions for Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving Caputo fractional derivative. By utilizing the theory of operators semigroup, probability density functions via impulsive conditions, a new concept on a P C-mild solution for our problem is introduced. Our main techniques based on fractional calculus and fixed point theorems. Some concrete applications to partial differential equations are considered.
242 citations
TL;DR: In this article, the authors give some fixed point results on these interesting spaces, which are based on the concept of partial metric space, a distance on a nonempty set which is called partial metric.
Abstract: Matthews (1994) introduced a new distance on a nonempty set , which is called partial metric. If is a partial metric space, then may not be zero for . In the present paper, we give some fixed point results on these interesting spaces.
222 citations
TL;DR: In this article, the existence of fixed points and their asymptotic behavior was studied in the context of generalized nonexpansive mapping and fixed points were introduced. But they were not considered in this paper.
220 citations
TL;DR: The controllability result of a class of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems in a Banach space has been established by using the theory of fractionalist calculus, fixed point technique and a new concept called (@a,u)-resolvent family is introduced.
Abstract: In this work, the controllability result of a class of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems in a Banach space has been established by using the theory of fractional calculus, fixed point technique and also we introduced a new concept called (@a,u)-resolvent family. As an application that illustrates the abstract results, an example is given.
215 citations
TL;DR: It is shown that a self-mapping T on a complete metric space X has a fixed point if it satisfied cyclic weak ϕ -contraction.
TL;DR: In this paper, it was shown that some generalizations in fixed point theory are not real generalizations, i.e., they are not generalizations of fixed point generalizations.
Abstract: In this paper, we shall prove that some generalizations in fixed point theory are not real generalizations.
TL;DR: In this article, Bhaskar and Lakshmikantham extended the coupled fixed point theorems for mixed monotone operators F : X × X → X by weakening the contractive condition involved.
Abstract: In this paper, we extend the coupled fixed point theorems for mixed monotone operators F : X × X → X obtained in [T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (7) (2006) 1379–1393] by significantly weakening the contractive condition involved. Our technique of proof is essentially different and more natural. An example as well as an application to periodic BVP is also given in order to illustrate the effectiveness of our generalizations.
TL;DR: Coupled fixed point theorems in a partially ordered G-metric space are established and the results proved are illustrated with an example.
TL;DR: The existence theorems of common fixed points for two weakly increasing mappings satisfying an almost generalized contractive condition in ordered metric spaces are proved.
Posted Content•
TL;DR: Bhaskar and Lakshmikantham as discussed by the authors extended the coupled fixed point theorems for mixed monotone operators by weakening the involved contractive condition, which is essentially different and more natural.
Abstract: In this paper we extend the coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ obtained in [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordered metric spaces and applications}, Nonlinear Anal. TMA \textbf{65} (2006) 1379-1393] by significantly weakening the involved contractive condition. Our technique of proof is essentially different and more natural. An example as well an application to periodic BVP are also given in order to illustrate the effectiveness of our generalizations.
TL;DR: In this article, the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem is studied. And the existence criteria for singular and nonsingular fractional DDE boundary value problems are established by the properties of the Green function, lower and upper solution method and fixed point theorem.
TL;DR: Coupled coincidence and coupled common fixed point theorems for a mixed g -monotone mapping satisfying nonlinear contractions in partially ordered G -metric spaces are proved.
TL;DR: Some coupled fixed point theorems involving a (@j,@f)-weakly contractive condition for mapping having the mixed monotone property in ordered partial metric spaces are proved.
Abstract: In this paper, we prove some coupled fixed point theorems involving a (@j,@f)-weakly contractive condition for mapping having the mixed monotone property in ordered partial metric spaces. These results are analogous to theorems of Van Luong and Xuan Thuan (2011) [10] on the class of ordered partial metric spaces. Also, an application is given to support our results.
TL;DR: In this article, the authors present coupled fixed point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions.
Abstract: The purpose of this paper is to present some coupled fixed point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions. We also present an application to integral equations.
TL;DR: This work studies fixed point results for new extensions of Banach’s contraction principle to partial metric space, and gives some generalized versions of the fixed point theorem of Matthews.
TL;DR: In this paper, the existence of positive solutions for the nonlinear fractional differential equation boundary value problem was studied and sufficient conditions for the nonexistence and existence of at least one or two positive solutions were established.
TL;DR: In this paper, the existence and uniqueness of these fixed point results follows from the Hardy-Rogers theorem in the induced usual metric space (X, dG), where (D, G) need not be symmetric.
Abstract: In this paper several fixed point theorems for a class of mappings defined on a complete G-metric space are proved. In the same time we show that if the G-metric space (X, G) is symmetric then the existence and uniqueness of these fixed point results follows from the Hardy-Rogers theorem in the induced usual metric space (X, dG). We also prove fixed point results for mapping on a G-metric space (X, G) by using the Hardy-Rogers theorem where (X, G) need not be symmetric.
TL;DR: A Szemeredi-Trotter type theorem is studied and used to obtain a different proof of Garaev's sum-product estimate in finite fields.
Abstract: We study a Szemeredi-Trotter type theorem in finite fields. We then use this theorem to obtain a different proof of Garaev's sum-product estimate in finite fields.
TL;DR: The aim of this paper is to extend and generalize the main results of Cho et al.
Abstract: Recently, Cho et al. [Y.J. Cho, R. Saadati, S.H. Wang, Common fixed point theorems on generalized distance in ordered cone metric spaces, Comput. Math. Appl. 61 (2011) 1254–1260] introduced the concept of the c -distance in a cone metric space and established some fixed point theorems on c -distance. The aim of this paper is to extend and generalize the main results of Cho et al. [20] and also show some examples to validate our main results.
TL;DR: An appropriate compactness criterion is established, such that Schauder's fixed point theorem on an unbounded domain to obtain the existence result for solutions, and a suitable choice of a Banach space allows the solutions to be unbounded.
Abstract: This paper deals with a boundary value problem of a fractional differential equation with the nonlinear term dependent on a fractional derivative of lower order on the semi-infinite interval. An appropriate compactness criterion is established, such that we can use Schauder's fixed point theorem on an unbounded domain to obtain the existence result for solutions. Moreover, a suitable choice of a Banach space allows the solutions to be unbounded. An example illustrating our main result is also given.
TL;DR: In this paper, the authors investigated the global stability and the global asymptotic stability independent of the sizes of the delays of linear time-varying Caputo fractional dynamic systems of real fractional order possessing internal point delays.
Abstract: This paper investigates the global stability and the global asymptotic stability independent of the sizes of the delays of linear time-varying Caputo fractional dynamic systems of real fractional order possessing internal point delays. The investigation is performed via fixed point theory in a complete metric space by defining appropriate nonexpansive or contractive self-mappings from initial conditions to points of the state-trajectory solution. The existence of a unique fixed point leading to a globally asymptotically stable equilibrium point is investigated, in particular, under easily testable sufficiency-type stability conditions. The study is performed for both the uncontrolled case and the controlled case under a wide class of state feedback laws.
TL;DR: It is proved that the proposed iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and theSet of fixed points of an infinite family of nonexpansive mappings and theset of a variational inclusion in a real Hilbert space has strong convergence under some mild conditions imposed on algorithm parameters.
TL;DR: The existence of mild solutions for a class of abstract fractional neutral integro-differential equations with state-dependent delay is studied by using the Leray-Schauder alternative fixed point theorem.
Abstract: In this paper we study the existence of mild solutions for a class of abstract fractional neutral integro-differential equations with state-dependent delay. The results are obtained by using the Leray-Schauder alternative fixed point theorem. An example is provided to illustrate the main results.
TL;DR: Moudafi’s viscosity approximations with continuous strong pseudocontraction semigroup are considered and a strong convergence theorem of fixed points is established in the framework of Banach spaces.