Some Common Fixed Point Theorems on Complete Hilbert Space

In this present paper, we establish Cantor's intersection like theorem in a locally convex topological vector spaces. Some fixed point and common fixed point theorems are proved for Reich and Caccioppoli type contractive mappings in such a locally convex topological vector space. Also in this setting we prove a fixed point theorem for some mapping which is the uniform limit of a sequence of Reich type contractive mappings therein. <br><br> Встановлена теорема, подібна теоремі Кантора про перетин, у випадку локально опуклих векторних просторів. Для стискуючих відображень типу Райха і Каччіополі відповідних просторів доведені теореми про нерухому точку та спільну нерухому точку. Також у цій постановці доведена теорема про нерухому точку для відображення, яке є рівномірною границею послідовності стискуючих відображень типу Райха.

In this paper, the solution of classical and generalized linear and nonlinear elliptic partial differential equations of the second order with non-local boundary conditions is studied. The basic concepts and theorems related to the study are included, which includes the main result of this research. Many basic definitions, theorems and observations about Sobolev spaces H^k (Ω) and H_0^k (Ω) are discus-sed. The existence and uniqueness of the solution of the Poisson equation with non-local boundary conditions are considered. The existence and uniqueness of the solution of a second-order quasi-linear elliptic differential equation with non-linear integral boun-dary condition are also discussed. The argument for proving the previous problem is based on Banach's fixed point theorem in the complete metric space, the maxima and minima principle, and the comparison principle. Key word: Sobolev Spaces, Poisson Equation, Quesilinear Ellipitic Differential Equation, Boundary, and Banach's fixed Point Theorem.

We study the viscosity approximation method due to Moudafi for a fixed point problem of quasinonexpansive mappings in a Hilbert space. First, we establish a strong convergence theorem for a sequence of quasinonexpansive mappings. Then we employ our result to approximate a solution of the variational inequality problem over the common fixed point set of the sequence of quasinonexpansive mappings. MSC:47H09, 47H10, 41A65.

Analytical properties and the box-counting dimension of nonlinear hidden variable recurrent fractal interpolation functions constructed by using Rakotch's fixed point theorem

It has been a century since the Banach fixed point theorem was established, and because of this, the result is the progenitor in some ways.This seems essential to revisit fixed point theorems in specific and in light of most of those.Those are numerous and prevalent in mathematics, as we will demonstrate.Fixed point theorems can be noticed in advanced mathematics, economics, micro-structures, geometry, dynamics, computational mathematics, and differential equations.G -metric space is the broaden and extrapolate the paradigm of the concept of metric space.The characteristic of a G -metric space, in essence, is to comprehend the topological features of three points ratherthan two points via the perimeter of a triangle, where the metric indicates the distance between two points.The domain of G b -metric space is significantly larger than that of the class of G -metric space.Hence we utilised this generalized space inorder to obtain common tripled fixed point for three mappings using rational type contractions in the setting of G b -metric spaces.Recently, Khomadram et al have developed coupled fixed point theorems in G b -metric spaces via rational type contractions.The main aim of our paper is to broaden and extrapolate the paradigm of Khomadram's results into tripled fixed point theorems.Therefore, examples are offered to support our findings.

In this research article we generalize and improve the results of K. Jha and V. Pant [K. Jha and V. Pant, Some Common fixed point theorem in fuzzy metric space with property (E.A.), Thai Journal of Mathematics, Vol 15, No.1 (2017), 217-226] using weak compatible maps along with property (CLRg).We also demonstrate an example in support of our main result. K.Jha and V.Pant proved their main result that is Theorem-A by applying the concept of property (E-A),to prove this result they take closeness of range of subspaces that is they proved their main result by using stronger contractive conditions, while in this paper we prove the same result as a Theorem 3.1 by removing the condition of ‘closeness of range of subspaces’ and instead of applying the concept of property (E-A) we do apply the concept of property (CLRg).The importance of property ‘CLR’(Common Limit in Range) ensures that one does not require the closeness of range subspaces. That is we prove the same result for a weaker contractive conditions. By proving the result as Theorem-A, Jha and Pant improved and generalized many similar results on fixed points. The purpose of this paper is to further improve and generalize the result of Jha and Pant and some earlier similar results on fixed point.

In this article, we prove some coincidence and common fixed point theorems under the relation-theoretic Meir-Keeler contractions in a metric space endowed with a locally finitely T -transitive binary relation. Our newly proved results generalize, extend, and sharpen some existing coincidence point as well as fixed point theorems existing in the literature. Moreover, we give some examples to affirm the efficacy of our results.

Banach contraction principle is a fundamental result in Metric Fixed Point Theory. It is a very popular and powerful tool in solving the existence problems in pure and applied sciences. In this chapter, Banach contraction principle and its converse are presented. Moreover, various applications of this famous principle, including mixed Volterra–Fredholm-type integral equations and systems of nonlinear matrix equations, are provided. Some results of this chapter appeared in [3, 5, 13, 19].

The flourishing field of fixed point theory started in the early days of topology with seminal contributions by Poincare, Lefschetz-Hopf, and Leray-Schauder at the turn of the 19th and early 20th centuries. The theory vigorously developed into a dense and multifaceted body of principles, results, and methods from topology and analysis to algebra and geometry as well as discrete and computational mathematics. This interdisciplinary theory par excellence provides insight and powerful tools for the solvability aspects of central problems in many areas of current interest in mathematics where topological considerations play a crucial role. Indeed, existence for linear and nonlinear problems is commonly translated into fixed point problems; for example, the existence of solutions to elliptic partial differential equations, the existence of closed periodic orbits in dynamical systems, and more recently the existence of answer sets in logic programming. The classical fixed point theorems of Banach and Brouwer marked the development of the two most prominent and complementary facets of the theory, namely, the metric fixed point theory and the topological fixed point theory.Themetric theory encompasses results and methods that involve properties of an essentially isometric nature. It originates with the concept of Picard successive approximations for establishing existence and uniqueness of solutions to nonlinear initial value problems of the 1st order and goes back as far as Cauchy, Liouville, Lipschitz, Peano, Fredholm, and most particularly, Emile Picard. However, the Polish mathematician Stefan Banach is credited with placing the underlying ideas into an abstract framework suitable for broad applications well beyond the scope of elementary differential and integral equations. Metric fixed point theory for important classes of mapping gained respectability and prominence to become a vast field of specialization partly and not only because many results have constructive proofs, but also because it sheds a revealing light on the geometry of normed spaces, not to mention its many applications in industrial fields such as image processing engineering, physics, computer science, economics, and telecommunications. A particular interest in fixed points for set-valued operators developed towards the mid-20th century with the celebrated extensions of the Brouwer and Lefschetz theorems by Kakutani and Eilenberg-Montgomery, respectively. The Banach contraction principle was later on extended to multivalued contractions by Nadler. The fixed point theory for multivalued maps found numerous applications in control theory, convex and nonsmooth optimization, differential inclusions, and economics. The theory is also used prominently in denotational semantics (e.g., to give meaning to recursive programs). In fact, it is still too early to truly estimate the importance and impact of set-valued fixed point theorems in mathematics in general as the theory is still growing and finding renewed outlets. This special issue adds to the development of fixed point theory by focusing on most recent contributions. It includes works on nonexpansive mappings in Banach and metric spaces, multivalued mappings in Banach and metric spaces, monotone mappings in ordered spaces, multivalued mappings in ordered spaces, and applications to such nonmetric spaces as modular spaces, as well as applications to logic programming and directed graphs.

In this paper, we propose a method to prove the existence and the local uniqueness of solutions to infinite-dimensional fixed-point equations using computers. Choosing a set which possibly includes a solution, we transform it by an approximate linearization of the operator appearing in the equation. Then we calculate the radii of the transformed set in order to check sufficient conditions for Banach's fixed-point theorem. This method is applied to elliptic problems and numerical examples are given.

Recently Suzuki (2008) and then Kikkawa and Suzuki (2008) gave a new generalization of the Banach contraction principle. Then, Mot and Petrusel (2009), Dhompapangasa and Yingtaweesttikulue (2009), Bose and Roychowd- hury (2011), Singh and Mishra (2010), and Doric and Lazovic (2011) further extended their work. Recently Bose (2012) obtained some Suzuki-type common fixed point theorems for generalized contractive multivalued mappings using a result of Bose and Mukherjee(1977) which extend the previously obtained re- sults. Recently Damjanovic and Doric(2011) obtained a multivalued generaliza- tion of theorems Kikkawa and Suzuki (2008) concerning Kannan mappings. Also Singh and Mishra (2010) considered coincidence and fixed point theorems for a class of hybrid pair of single-valued and multi-valued maps in metric space setting and Singh et al (2012) prsented a common fixed point theorem for a pair of multi-valued maps in a complete metric space extending a recent theorem of Doric and Lazovic. First we generalize the theorem of Singh et al which also extend the theorem of Singh and Mishra(2010). Then we extend the theorem of Damjanovic and Doric to a common fixed point theorem of a pair of mul- tivalued mappings. As an application, we consider the existence of a common solution for a class of functional equations arising in dynamic programming.

Let $ H$ be a Hilbert space and $C$ be a closed, convex and nonempty subset of $H$. Let $T:C rightarrow H$ be a non-self and non-expansive mapping. V. Colao and G. Marino with particular choice of the sequence ${alpha_{n}}$ in Krasonselskii-Mann algorithm, ${x}_{n+1}={alpha}_{n}{x}_{n}+(1-{alpha}_{n})T({x}_{n}),$ proved both weak and strong converging results. In this paper, we generalize their algorithm and result, imposing some conditions upon the set $C$ and finite many mappings from $C$ in to $H$, to obtain a converging sequence to a common fixed point for these non-self and non-expansive mappings.

On the existence of common fixed points for a pair of lipschitzian mappings in banach spaces

In this paper we prove some fixed point theorems for weakly compatible mappings with the notation of A-distance and E-distance in uniform space.

The concept of a W function was introduced very recently by Chen (1) and was used to prove several common fixed point theorems in complete generalized metric spaces. Using a a subclass of these W functions introduced in (1), we prove some fixed point theorems with one, and a pair of self maps on a G-metric space