# Banach's fixed point theorem

## Top PDF Banach's fixed point theorem: ### A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces

Recently, many authors studied the problems of finding a common element of the set of fixed points for a mapping and the set of solutions of equilibrium problem in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively see, e.g., 4–21 and the references therein. In a Hilbert space H, S. Takahashi and W. Takahashi 17 introduced the iteration as follows: sequence {x n } generated by u, x 1 ∈ C, ### Fixed Points for Multivalued Mappings in Uniformly Convex Metric Spaces

In 1974, Lim 1 developed a result concerning the existence of fixed points for multivalued nonexpansive self-mappings in uniformly convex Banach spaces. This result was extended to nonself-mappings satisfying the inwardness condition independently by Downing and Kirk 2 and Reich 3. This result was extended to weak inward mappings independently by Lim 4 and Xu 5. Recently, Dhompongsa et al. 6 presented an analog of Lim-Xu’s result in CAT0 spaces. In this note, we extend the result to uniformly convex metric spaces which improve results of both Lim-Xu and Dhompongsa et al. In addition, we also give a new proof of a result of Lim 7 by using Caristi’s theorem 8. Finally, we give some basic properties of fixed point sets for quasi-nonexpansive mappings for these spaces. ### Solvability of Chandrasekhar’s Quadratic Integral Equations in Banach Algebra

Tx t = Cx t + Ax t ⋅ Bx t (2) Hence the existence of solutions of the FIE (1) is equivalent to finding a fixed point to the operator Equation (7) in  ( J ,  ) . We shall prove that A , B and C satisfy all the conditions of Theorem 1. ### Common Fixed Point Theorem in Cone Metric Space for Rational Contractions

 Mehdi Asadi, S. Mansour Vaezpour, Vladimir Rakocevic, Billy E. Rhoades, Fixed point theorems for contractive mapping in cone metric spaces, Math. Commun. 16 (2011) 147-155.  Mahpeyker OzturkOn, Metin Basarr, Some common fixed point theorems with rational ex- pressions on cone metric spaces over a Banach algebra, Hacettepe Journal of Mathematics and Statistics, 41 (2) (2012) 211-222. ### Vol 8, No 3 (2017)

In 1922, S. Banach The first important and significant result was proved a fixed point theorem for contraction mappings in complete metric space and also called it Banach fixed point theorem / Banach contraction principle which is considered as the mile stone in fixed point theory. This theorem states that, A mapping 𝑇𝑇: 𝑋𝑋 → 𝑋𝑋 where (𝑋𝑋, 𝑑𝑑) is a metric space, is said to be a contraction if there exists 𝑘𝑘 ∈ [0,1) such that ### On generalized weakly directional contractions and approximate fixed point property with applications

In , the author established some new fixed point theorems for nonlinear multiva- lued contractive maps by using τ 0 -function, τ 0 -metrics and R -functions. Applying those results, the author gave the generalizations of Berinde-Berinde ’ s fixed point theo- rem, Mizoguchi-Takahashi ’ s fixed point theorem, Nadler ’ s fixed point theorem, Banach contraction principle, Kannan ’ s fixed point theorems and Chatterjea ’ s fixed point theo- rems for nonlinear multivalued contractive maps in complete metric spaces; for more details, we refer the reader to . ### Some new fixed point theorems in rectangular metric spaces

Fixed point theory is a vital and genuine theme of nonlinear analysis. Furthermore, it’s well established that the contraction mapping principle substantiated doctoral thesis of Banach  is one of the most prominent theorems in functional analysis. Since 2010, this theorem has exposed to multifarious generalization either by easing circumstance on contractivity or ### A Generalized Metric Space And Related Fixed Point Theorems

In this work, we present a new generalized metric spaces and extend some well-known related fixed point theorems that recovers a large class of topological spaces including standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces. In such spaces, we establish new versions of some known fixed point theorems in standard metric spaces including Banach contraction principle, Ćirić’s fixed point theorem, a fixed point result due to Ran and Reurings, and a fixed point result due to Nieto and Rodíguez-López. ### Application of the Banach Fixed-Point Theorem to the Scattering Problem at a Nonlinear Three-Layer Structure with Absorption

was not solved till now. In the following we propose a solution based on the Banach fixed- point theorem contraction principle of the functional analysis 12. To demonstrate the broad range of applicability of this theorem we consider a nonlinear lossy dielectric film with spatially varying saturating permittivity. In Section 2 we reduce Maxwell’s equations to a Volterra integral equation 2.12 for the intensity of the electric field Ey and give a solution in form of a uniform convergent sequence of iterate functions. Using these solutions, we determine the phase function ϑy of the electric field, and, evaluating the boundary conditions in Section 3, we derive analytical expressions for reflectance, transmittance, absorptance, and phase shifts on reflection and transmission. ### Quasicontraction Mappings in Modular Spaces without Δ2-Condition

for any x, y ∈ M. In 1974, ´ Ciri´c 1 introduced these mappings and proved an existence fixed point result very similar to the original Banach contraction fixed point theorem. Recently, the authors 2 tried to extend their ideas to modular spaces. Though their conclusions are very similar to ´ Ciri´c’s results proved in metric spaces, they were unable to escape the Δ 2 -condition. ### Common fixed point results for generalized contraction mappings in b - metric space

Fixed point theory is rapidly moving into the mainstream of Mathematics mainly because of its applications in diverse fields which include numerical methods like Newton-Raphson method, establishing Picard’s existence theorem, existence of solution of integral equations and a system of linear equations. In 1922, S. Banach , The first important and significant result was proved a fixed point theorem for contraction mappings in complete metric space and also called it Banach fixed point theorem / Banach contraction principle which is considered as the mile stone in fixed point theory. This theorem states that, A mapping : → where( , ) is a metric space, is said to be a contraction if there exists ∈ [0,1) such that ### Ray's Theorem for Firmly Nonexpansive-Like Mappings and Equilibrium Problems in Banach Spaces

Let C be a subset of a Banach space E. A mapping T : C → E is nonexpansive if Tx − Ty ≤ x − y for all x, y ∈ C. In 1965, it was proved independently by Browder 1, G ¨ohde 2, and Kirk 3 that if C is a bounded closed convex subset of a Hilbert space and T : C → C is nonexpansive, then T has a fixed point. Combining the results above, Ray 4 obtained the following interesting result see 5 for a simpler proof. ### A Generalized Metric Space and Related Fixed Point Theorems

In this work, we present a new generalization of metric spaces that recovers a large class of topological spaces including standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces. In such spaces, we establish new versions of some known fixed point theorems in standard metric spaces including Banach contraction principle, Ćirić’s fixed point theorem, a fixed point result due to Ran and Reurings, and a fixed point result due to Nieto and Rodíguez -López. 2. A generalized metric space: ### Contractive gauge functions in strongly orthogonal metric spaces

Existence of fixed point in orthogonal metric spaces has been initiated recently by Eshaghi and et al. [On orthogonal sets and Banach fixed Point theorem, Fixed Point Theory, in press]. In this paper, we introduce the notion of the strongly orthogonal sets and prove a genuine generalization of Banachfixed point theorem and Walter’s theorem. Also, we give an example showing that our main theorem is a real generalization of these fixed point theorems. ### Existence results of solutions for some fractional neutral functional integro-differential equations with inﬁnite delay

Abstract: In this paper, by means of the Banach fixed point theorem and the Krasnoselskii’s fixed point theorem, we investigate the existence of solutions for some fractional neutral functional integro-differential equations involving infinite delay. This paper deals with the fractional equations in the sense of Caputo fractional derivative and in the Banach spaces. Our results generalize the previous works on this issue. Also, an analytical example is presented to illustrate our results Keywords: Fractional neutral integro-differential equations; Initial value problem; Caputo fractional derivative; Krasnoselskii’s fixed point theorem ### The existence result of a fuzzy implicit integro-differential equation in semilinear Banach space

Theorem 2.11. (, Schauder Fixed point theorem for semilinear spaces) Let B be a nonempty, closed, bounded and convex subset of a semilinear Banach space S having the cancelation property, and suppose that P : B → B is a compact operator. Then P has at least one ﬁxed point in B. ### Cyclic Contraction on S- Metric Space

Abstract. In this paper we introduced the concepts of cyclic contraction on S- metric space and proved some fixed point theorems on S- metric space. Our presented results are proper generalization of Sedghi et al. . We also give an example in support of our theorem. ### Approximation Method for Hybrid Functional Differential Equations

is very rare in the literature. Very recently, the study along this line has been initiated via fixed point theorem. See Dhage and Regan (4) and Dhage (3) and the references there in. The HFDE (1.1) is new to the literature and the study of this problem will definitely contribute immensely to the area of hybrid functional differential equation. The specialty of the results of the present paper lies in our HFDE (1.1) on𝐽. ### Vol 2, No 12 (2011)

Lemma (S. Kasahara): Let be an L- space which is d- complete for a non negative real valued function d on + if is separated then, / % / % - 67 ( % % During the past few years many great mathematicians Yeh, Singh, Pathak, and Dubey, Sharma, and Agrawa, Patel,Sahu, and Sao, Patel and Patel, worked for L- Space. In this chapter, we similar investigation for the study of Fixed Point ### Nonexpansive mappings on complex C*-algebras and their fixed points

Let T : E −→ E be a self-map on the nonempty set E. We denote {x ∈ E : T (x) = x} by Fix(T ) and call the fixed points set of T . The symbol K denote a field that can be either C or R . Let (X, k·k) be a normed linear space over K . A mapping T : E ⊆ X −→ X is nonexpansive if kT (x)− T (y)k 6 kx − yk for all x, y ∈ E. We say that the normed linear space (X, k · k) over K has the fixed point property (or weak fixed point property) if for every nonempty bounded closed convex (or weakly compact convex, respectively) subset E of X and every nonexpansive mapping T : E −→ E we have Fix(T ) 6= ∅.