The Sandwich theorem via Pataraia’s fixed point theorem

Positivity - The Sandwich theorem (König in Archiv der Mathematik 23:500–1972, 1972) yields the existence of a linear functional sandwiched in between a given superlinear functional and...     

The Hyers theorem via the Markov–Kakutani fixed point theorem

We present an application of the Markov–Kakutani common fixed point theorem to the theory of stability of functional equation by proving some version of the Hyers theorem concerning approximate homomorphisms.     

The Meir–Keeler fixed point theorem in incomplete modular spaces with application

In this paper, we present a new generalized modular version of the Meir–Keeler fixed point theorem endowed with an orthogonal relation. Our results improve the results of (Eshaghi Gordji et al., On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 2017). Finally, this result is applied to the existence and uniqueness of solutions to perturbed integral equations in modular function spaces.     

A new approach to the Assad–Kirk fixed point theorem

In the present paper, considering the Wardowski’s technique, we give a new approach to the Assad–Kirk fixed point theorem on metrically convex metric spaces.We also provide a nontrivial example showing that our result is a proper extension of the Assad–Kirk fixed point theorem.     

Chasles’ fixed point theorem for Euclidean motions

Chasles’ theorem, a classic and important result of kinematics, states that every orientation-preserving isometry of ${\mathbb{R}^3}$ is a screw motion. We show that this is equivalent to the assertion that each proper Euclidean motion that is not a pure translation, acting on the space of oriented lines, has a unique fixed point (the axis of the screw motion). We use that formulation to derive a simple and novel constructive proof of Chasles’ theorem.     

Around Browder’s fixed point theorem for contractions

We revisit a fixed point theorem for contractions established by Felix Browder in 1968. We show that many definitions of contractive mappings which appeared in the literature after 1968 turn out to be equivalent formulations or even particular cases of Browder’s definition. We also discuss the problem of the existence of approximate fixed points of continuous mappings; in particular, we settle it in the affirmative for Browder contractions. Finally, we recall three problems concerning Browder contractions which remain unsolved. With great respect for Professor Felix E. Browder     

A note on a fixed point theorem and the Hyers–Ulam stability

We present a theorem that is a simultaneous generalization of the classical Banach fixed point theorem and a theorem of G.L. Forti on the Hyers–Ulam stability (of difference and functional equations). We also give an example that shows how to use that result in finding the solutions of some difference equations.     

A soft version of the Knaster–Tarski fixed point theorem with applications

In this paper, first, we define a partial order on a soft set (F, A) and introduce some related concepts. Then, using the concept of a soft mapping introduced by Babitha and Sunil (Comput Math Appl 60(7):1840–1849, 2010), a soft version of Knaster–Tarski fixed point theorem is obtained. Some examples are presented to support the concepts introduced and the results proved herein. As an application of our result, we show that the soft Knaster–Tarski fixed point theorem ensures the existence of a soft common fixed point for a commuting family of order-preserving soft mappings.     

Mizoguchi–Takahashi’s type common fixed point theorem

Recently Kamran extended the result of Mizoguchi and Takahashi for closed multi-valued mappings and proved a fixed point theorem. In this paper we further extend the result concluded by Kamran and prove a common fixed point theorem by using the concept of lower semi-continuity.     

On a generalization of a Greguš fixed point theorem

Let C be a closed convex subset of a complete convex metric space X. In this paper a class of selfmappings on C, which satisfy the nonexpansive type condition (2) below, is introduced and investigated. The main result is that such mappings have a unique fixed point.     

Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces

Aequationes mathematicae - Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are...     

Two applications of Brouwer’s fixed point theorem: in insurance and in biology models

In the first part of the article, a new interesting system of difference equations is introduced. It is developed for re-rating purposes in general insurance. A nonlinear transformation φ of a d-dimensional (d ≥ 2) Euclidean space is introduced that enables us to express the system in the form f^t+1:=φ( f^t), t = 0, 1, 2,. …. Under typical actuarial assumptions, existence of solutions of that system is proven by means of Brouwer’s fixed point theorem in normed spaces. In addition, conditions that guarantee uniqueness of a solution are given. The second, smaller part of the article is about Leslie–Gower’s system of d ≥ 2 difference equations. We focus on the system that satisfies conditions consistent with weak inter-specific competition. We prove existence and uniqueness of the equilibrium of the model under surprisingly simple and very general conditions. Even though the two parts of this article have applications in two different sciences, they are connected with similar mathematics, in particular by our use of Brouwer’s fixed point theorem.     

On Olaleru’s open problem on Gregus fixed point theorem

Let (X, d) be a complete metric space and ${TX \longrightarrow X }$ be a mapping with the property d(Tx, Ty) ≤ ad(x, y) + bd(x, Tx) + cd(y, Ty) + ed(y, Tx) + fd(x, Ty) for all ${x, y \in X}$ , where 0 < a < 1, b, c, e, f ≥ 0, a + b + c + e + f = 1 and b + c > 0. We show that if e + f > 0 then T has a unique fixed point and also if e + f ≥ 0 and X is a closed convex subset of a complete metrizable topological vector space (Y, d), then T has a unique fixed point. These results extend the corresponding results which recently obtained in this field. Finally by using our main results we give an answer to the Olaleru’s open problem.     

Some fixed point results for a generalized ψ-weak contraction mappings in orbitally metric spaces

Samet and Vetro [Samet B, Vetro C. Berinde mappings in orbitally complete metric spaces. Chaos Solitons Fract 2011;44:1075–9.] studied a fixed point theorem for a self-mapping satisfying a general contractive condition of integral type in orbitally complete metric spaces. In this paper, we introduce the notion of a generalized ψ-weak contraction mapping and establish some results in orbitally complete metric spaces. Our results generalize several well-known comparable results in the literature. As an application of our results we deduce the result of Samet and Vetro. Some examples are given to illustrate the useability of our results.     

Jackson’s theorem in Q_p spaces

A Jackson type inequality in Q p spaces is established, i.e., for any f (z) = Σ∞ j=0 ajzj ∈ Qp , 0≤p ∞, a 1, and k-1 ∈ N,where ω(1/k, f, Q p ) is the modulus of continuity in Q p spaces and C(a) is an absolute constant depending only on the parameter a.