We prove Banach, Newton-Raphson and Brouwer fixed point theorems in the framework of generalized smooth functions, a minimal extension of Colombeau's theory (and hence of classical distribution theory) which makes it possible to model nonlinear singular problems, while at the same time sharing a number of fundamental properties with ordinary smooth functions, such as the closure with respect to composition and several non trivial classical theorems of the calculus. The proved results allows one to deal with equations of the form F(x)=0, where F is a generalized smooth function, in particular, a Sobolev-Schwartz distribution. We consider examples with singularities that are not included in the classical version of these theorems.
We obtain sufficient conditions for existence of unique fixed point of Kannan type mappings on complete metric spaces and on generalized complete metric spaces depended an another function.
In this paper, we study a new iterative method for finding the fixed point of a weak Bregman relatively nonexpansive mapping and the set of solutions of generalized mixed equilibrium problems in Banach spaces.
The aim of this paper is to present some fixed point theorems for generalized contractions by altering distance functions in a complete cone metric spaces endowed with a partial order. We also generalize fixed point theorems of J. Harjani, K. Sadarangani [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Analysis 72 (2010) 1188-1197] from metric spaces to cone metric spaces.
Let (X, d) be a complete metric space, and let S, T : X to CB(X) be a duality of multi-valued generalized weak contraction mappings or a duality of generalized weak contraction mappings. We discuss the common fixed points and endpoints of the two kinds of multi-valued weak mappings. Our results extend and improve some results given by Daffer and Kaneko (1995), Rouhani and Moradi (2010), and Moradi and Khojasteh (2011).
In this paper, we introduce a commutative mappings satisfying the class of generalized non-expansive mappings which is wider than the class of mappings satisfying the condition (C), so called Condition B gamma, mu. The results obtained in this paper extend and generalized non-expansive mappings and other results in this direction. Different properties and some fixed point results for the mappings are obtained here. Keywords: Non expansive mappings, Condition (C), Condition B gamma mu, Commuting mappings, Fixed point.
In this paper we introduce generalized symmetric Meir-Keeler contractions and prove some coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ in partially ordered metric spaces. The obtained results extend, complement and unify some recent coupled fixed point theorems due to Samet [B. Samet, \textit{Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces}, Nonlinear Anal. \textbf{72} (2010), 4508-4517], Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordere…
In this work, we prove an existence theorem of the Hyers-Ulam stability for the nonlinear Volterra integral equations which improves and generalizes Castro-Ramos theorem by using some weak conditions.
In this paper, we consider a wider class of simulation functions and present some coincidence and common fixed point results in metric spaces. Results obtained in this paper extend, generalize and unify some well-known fixed and common fixed point results. Some illustrative examples are presented. We also discuss some applications to system of integral equations and functional equations arising in dynamic programming.
In this paper, we introduce a new class of Bregman generalized $α$-nonexpansive mappings in terms of Bregman distances, and investigate the Ishikawa and Noor iterations for these mappings. We establish weak and strong convergence theorems of Ishikawa and Noor iterative schemes for Bregman generalized $α$-nonexpansive mappings in Banach spaces. Furthermore, we propose an example of our generated mapping and some numerical examples which support our main theorem. Our results are new and improve the recent ones in the literature.
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.
Conjugation, or Legendre transformation, is a basic tool in convex analysis, rational mechanics, economics and optimization. It maps a function on a linear topological space into another one, defined in the dual of the linear space by coupling these space by meas of the duality product. Generalized conjugation extends classical conjugation to any pair of domains, using an arbitrary coupling function between these spaces. This generalization of conjugation is now being widely used in optima transportation problems, variational analysis and also optimization. If the coupled spaces are equal, gen…
This paper is devoted to prove the S. L. Singh's common fixed point Theorem for commuting mappings in cone metric spaces. In this framework, we introduce the notions of Generalized Kannan Con- traction, Generalized Zamfirescu Contraction and Generalized Weak Contraction for a pair of mappings, proving afterward their respective fixed point results.
We consider a relatively new hybrid generalized F-contraction involving a pair of mappings and utilize the same to prove a common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying generalized $(F,φ)$-contraction condition under common limit range property in complete metric spaces. A similar result involving a hybrid pair of mappings satisfying a Rational type Hardy-Rogers $(F,φ)$-contractive condition is also proved. Our results generalize and improve several results of the existing literature. As applications of our results, we prove two theo…
In this paper, a metric on $S_b$-metric space analogous to the Hausdorff metric has been introduced and some basic properties are obtained on multi-valued $S_b$-metric space. Further, the fundamental multi-valued contraction of Nadler(1962) has been extended to the $S_b$-metric space setting, and two results have been established. The entire study is supported by suitable examples.
In this paper we propose a new iteration process, called the K iteration process, for approximation of fixed points. We show that our iteration process is faster than the existing leading iteration processes like Picard-S iteration process, Thakur New iteration process and Vatan Twostep iteration process for contraction mappings. We support our analytic proof by a numerical example. Stability of K iteration process and data dependence result for contraction mappings by employing K iteration process is also discussed. Finally we prove some weak and strong convergence theorems for the Suzuki gen…
In this paper, we introduce a new iterative method to find a common solution of a generalized mixed equilibrium problem, a variational inequality problem and a hierarchical fixed point problem for a demicontinuous nearly nonexpansive mapping. We prove that the proposed method converges strongly to a common solution of above problems under the suitable conditions. It is also noted that the main theorem is proved without usual demiclosedness condition. Also, under the appropriate assumptions on the control sequences and operators, our iterative method can be reduced to recent methods. So, the re…
In this paper, we present some common fixed point theorems for a commuting pair of mappings, including a generalized nonexpansive single valued mapping and a generalized nonexpansive multivalued mapping in strictly convex Banach spaces. The results obtained in this paper extend and improve some recent known results.
In this paper some generalized mappings due to L.B. Ciric over a generalized metric space have been studied with illustrative examples. Some theorems with necessary and sufficient conditions have been proved for such mappings on that space.
A new, simple and unified approach in the theory of contractive mappings was recently given by Samet \emph{et al.} (Nonlinear Anal. 75, 2012, 2154-2165) by using the concepts of $α$-$ψ$-contractive type mappings and $α$-admissible mappings in metric spaces. The purpose of this paper is to present a new class of contractive pair of mappings called generalized $α$-$ψ$ contractive pair of mappings and study various fixed point theorems for such mappings in complete metric spaces. For this, we introduce a new notion of $α$-admissible w.r.t $g$ mapping which in turn generalizes the concept of $g$-m…
Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^*$. We present a novel hybrid method for finding a common solution of a family of equilibrium problems, a common solution of a family of variational inequality problems and a common element of fixed points of a family of a general class of nonlinear nonexpansive maps. The sequence of this new method is proved to converge strongly to a common element of the families. Our theorem and its applications complement, generalize, and extend various results in literature.
We introduce a large class of contractive mappings, called Suzuki Berinde type contraction. We show that any Suzuki Berinde type contraction has a fixed point and characterizes the completeness of the underlying normed space. A fixed point theorem for multivalued mapping is also obtained. These results unify, generalize and complement various known comparable results in the literature.
This article is the natural continuation of the paper: Mukhammadiev A.~et al Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring $\tilde{R}$ of Robinson-Colombeau is non-Archimedean, a classical series $\sum_{n=0}^{+\infty}a_{n}$ of generalized numbers $a_{n}\in\tilde{R}$ is convergent if and only if $a_{n}\to0$ in the sharp topology. Therefore, this property does not permit us to generalize several classical results, mainly in the study of analytic generalized functions (as well as, e.g., in the study of sigma-additivity in integration of general…
We use a method similar to ultraproducts to study the common fixed point of a left reversible semitopological semigroup acting on a Banach space. As an application, we prove a Bruck-type theorem for nearly uniformly convex Banach spaces to the effect that such spaces have weak fixed point property for left reversible semigroups.
In this paper, we prove the existence of fixed points of mappings satisfying the condition (Da), a kind of generalized nonexpansive mappings, on a weakly compact convex subset in a Banach space satisfying Opial's condition. And we use Sahu([6]) and Thakur([10])'s iterative scheme to establish several convergence theorems in uniformly convex Banach spaces and give an example to show that this scheme converges faster than the scheme in [1]
In this paper, we show that the generalized fixed-point algebra of a proper groupoid dynamical system, under certain assumptions, may be fibered over any locally compact Hausdorff space to which a continuous map exists from the unit space of the underlying groupoid. We will also provide some important examples.
Gähler ([4],[5]) introduced and investigated the notion of 2-metric spaces and 2-normed spaces in sixties. These concepts are inspired by the notion of area in two dimensional Euclidean space. In this paper, we choose a fundamentally different approach and introduce a possible generalization of usual norm retaining the distance analogue properties. This generalized norm will be called as $G$-norm. We show that every $G$-normed space is a $G$-metric space and therefore, a topological space and develop the theory for $G$-normed spaces. We also introduce $G$-Banach spaces and obtain some fixed po…
The famous Banach Contraction Principle holds in complete metric spaces, but completeness is not a necessary condition -- there are incomplete metric spaces on which every contraction has a fixed point. The aim of this paper is to present various circumstances in which fixed point results imply completeness. For metric spaces this is the case of Ekeland variational principle and of its equivalent - Caristi fixed point theorem. Other fixed point results having this property will be also presented in metric spaces, in quasi-metric spaces and in partial metric spaces. A discussion on topology and…
We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler-Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi's theorem on conjugate points and Noether's theorem. We close with an application to low regularity Riemannian geometry.
We present a derivation of classical Hermite, Laguerre, and Jacobi orthogonal polynomials directly through the Gram-Schmidt orthogonization process. The derivation uses certain generalized Vandermonde determinants with entries defined by Gamma and Beta functions. We also provide a geometric formulation of Gram-Schmidt orthogonalization using the Hodge star operator.
