Cantor's intersection theorem and some generalized fixed point theorems over a locally convex topological vector space

Fixed point theorems for better admissible multimaps on almost convex sets

The present research paper aims to establish some generalized Banach fixed point theorems on closed Hilbert spaces using the sequence of mappings and satisfying the Banach Fixed point Property. The Banach fixed point theory plays an important role in modern areas of mathematics, mathematical Science, and different branches of Science. Koparde P. V; Waghmode B. B [5]; Pandhar, M. D and Waghmode, B. B [6] have established some Banach Fixed point theorems on a sequence of mappings on a closed subset S of a Hilbert space H. After that, Veerapandi, T and Kumar, S. A. [9] have introduced some fruitful results in this line. In this research article, some generalized Banach fixed point theorems of Veerapandi, T and Kumar, S. A. [9] are stated above. The mathematician established a theory on Hilbert spaces in sequences of mappings on closed spaces.

We extend the main results of Aamri and El Moutawakil and Pant to the weakly compatible or R-weakly commuting pair (T, f) of maps, where T is multivalued. As applications, common fixed point theorems are obtained for new class of maps called R-subcommuting maps in the setup of locally convex topological vector spaces. We also study some results on best approximation via common fixed point theorems.2000 MSC 41A65; 46A03; 47H10; 54H25.

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we establish a common fixed point theorem for a family of self set-valued mappings on a compact and convex set in a locally convex topological vector space. As applications, we obtain an existence theorem of solutions for a variational inequality of Stampacchia type and some Ky Fan-type minimax inequalities.

Helly and Klee type intersection theorems for finitary connected paved spaces

In this paper, a new type of contraction for several self-mappings of a metric space, called FM-contraction, is introduced. This extends the one presented for a single map by Wardowski [Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012:94, 2012]. Coincidence and common fixed point of eight self mappings satisfying FM-contraction conditions are established via common limit range property without exploiting the completeness of the space or the continuity of the involved maps. Coincidence and common fixed point of eight self-maps satisfying FM-contraction conditions via the common property (E.A.) are also studied. Our results generalize, extend and improve the analogous recent results in the literature, and some examples are presented to justify the validity of our main results.

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.

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

In this paper, we discuss about different types of α , β -admissible mappings and introduce some new α , β -contraction-type mappings under simulation function. Furthermore, we present the definition of S -metric-like space and its topological properties. Some fixed point theorems in this space are established, proved, and verified with examples.

Some fixed point results are derived for mappings of contractive type in metric and topological spaces.

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.

EXISTENCE OF SOLUTION FOR A VOLTERRA TYPE INTEGRAL EQUATION USING DARBO-TYPE F-CONTRACTION

In this paper, we introduce a new class of operators, for which a fixed point theorem is proven. This class of mappings is very large and unifies several classes of contractive type operators from the literature, including Berinde mappings. Such fact is proven via a comparison with various metrical contractive type mappings.