Point Theorem Research Articles

A generalization of Lappan’s five point theorem

In this paper, we prove the following result: Let mathcal {F} be a family of meromorphic functions on a domain D and let S=left{ varphi _i:1le i le 5right} be a set of five distinct meromorphic functions on D. If for each f in mathcal {F} and z_0 in D, there is a constant M>0 such that f^{#}(z_0) le M whenever f(z_0)= varphi (z_0) for some varphi in S and if f(z_0) ne varphi (z_0) for all varphi in S whenever varphi _i(z_0) = varphi _j(z_0) for some i,j in left{ 1,2,3,4,5right} with i ne j, then mathcal {F} is normal on D. Further we extend this result to the case where the set S contains fewer functions. In particular, our result generalizes the most significant theorem of Lappan (i.e. Lappan’s five point theorem).

Sadovskii Type Best Proximity Point (Pair) Theorems with an Application to Fractional Differential Equations

Sadovskii Type Best Proximity Point (Pair) Theorems with an Application to Fractional Differential Equations

Existence of a Unique Solution and the Hyers–Ulam‐H‐Fox Stability of the Conformable Fractional Differential Equation by Matrix‐Valued Fuzzy Controllers

In this paper, we consider a conformable fractional differential equation with a constant coefficient and obtain an approximation for this equation using the Radu–Mihet method, which is derived from the alternative fixed‐ point theorem. Considering the matrix‐valued fuzzy k‐normed spaces and matrix‐valued fuzzy H‐Fox function as a control function, we investigate the existence of a unique solution and Hyers–Ulam‐H‐Fox stability for this equation. Finally, by providing numerical examples, we show the application of the obtained results.

Contractive Mappings on Metric Spaces with Graphs

We establish fixed point, stability and genericity theorems for strict contractions on complete metric spaces with graphs.

Existence and uniqueness solutions for nonlinear fractional differential equations with fractional integral boundary conditions

The theory of Schuder fixed point and Banach contraction theorems are used to explain the behavior of solution for differential equation of fractional order β ∈ (2, 3] with fractional integral limit conditions given by .We prove the existence and uniqueness solution of the above equations.

Upper and lower solutions for second-order with m-point impulsive boundary value problems

In this paper, we consider a second-order m-point impulsive boundary value problem. By applying the upper and lower solutions method and the Schauder?s fixed point theorem, we obtain the existence of at least one positive solution. We also give an example to illustrate our main result.

A unified common fixed point theorem for a family of mappings

The main objective of the paper is to prove some unified common fixed point theorems for a family of mappings under a minimal set of sufficient conditions. Our results subsume and improve a host of common fixed point theorems for contractive type mappings available in the literature of the metric fixed point theory. Simultaneously, we provide some new answers in a general framework to the problem posed by Rhoades (Contemp Math 72, 233-245, 1988) regarding the existence of a contractive definition which is strong enough to generate a fixed point, but which does not force the mapping to be continuous at the fixed point. Concrete examples are also given to illustrate the applicability of our proved results.

Smith Normal Form of a distance matrix inspired by the four-point condition

The four point theorem is a condition for distances to arise from trees. Based on this condition, for any tree T on n vertices, we associate an (n2)×(n2) matrix MT.We find the rank and the Smith Normal Form (SNF) of the matrix MT and show that it only depends on n and is independent of the structure of the tree T. Curiously, the non-zero part of the SNF of MT coincides with the SNF of the distance matrix of T. Many such “tree independent” results are known and this result is yet another such result.

Common Fixed Point and Endpoint Theorems for a Countable Family of Multi-Valued Mappings

We prove some common fixed point and endpoint theorems for a countable infinite family of multi-valued mappings, as well as Allahyari et al. (2015) did for self-mappings. An example and an application to a system of integral equations are given to show the usability of the results.

Existence of Nontrivial Solutions for Fractional Differential Equations with p-Laplacian

Combining the properties of the Green function with some point theorems, we consider the existence of nontrivial solutions for fractional equations with p-Laplacian operator D0+βϕp[D0+α(ptu′(t))]+f(t,u(t))=0, 0<t<1, au(0)-bp0u′(0)=0, and cu(1)+dp1u′(1)=0, D0+αptu′tt=0=0, where a,b,c,d are constants and p(·):[0,1]→(0,+∞) is continuous.

一类奇异三阶三点边值问题正解的存在性

讨论非线性三阶三点边值问题 其中 ，非线性项允许在t=0, t=1及u=0 处奇异，利用锥上的不动点定理在较弱的条件下得到了上述边值问题至少存在一个、两个、n个正解的存在性结果。 This paper is concerned with the following nonlinear third-order three-point boundary value problem where , the nonlinear term may be singular at t=0, t=1 and u=0 . By using fixed- point theorem in cone, the existence of one or two or n positive solutions is obtained with the weaker conditions.

Графическое доказательство основной теоремы неевклидовой геометрии

The tragedy for the pioneers of non-Euclidean geometry (N. Lobachevsky and J. Boyai) was their quarrel with the scientific tradition. Figuratively speaking, in the judgment of the scientific world they could not provide proof of their views, and substantive law of science was not on their side despite the efforts of such an influential advocate as Karl Friedrich Gauss. They lost the civil process to the scientific layman, who sincerely believed that the earth is flat. Traditionally mathematical logic considers a new idea proven, if it is derived by inference from already proven ones, or recognized as obvious, or recognized without proof (postulates). Yet the founders of non-Euclidean geometry could not imagine such traditional evidence at all desire, because it had not yet been developed, and most importantly respective starting points (axioms, postulates, and theorems) had not been recognized by mathematicians. The paper outlines the original concept of non-Euclidean geometries. Hyperbolic geometry of Lobachevsky is considered based on viewing the sphere as a surface of zero curvature. In this case, the plane will have a real curvature properties of hyperboloid or a pseudosphere depending on the absolute and space anisotropy index, which replaces the concept of curvature of space; i.e. the notion of the curvature of the surface is converted to purely analytical attributes. Parabolic geometry of Euclid with degenerate absolute becomes a special case of geometries with non-degenerate absolute. The geometry of Riemann having the absolute of imaginary surface with negative Gaussian curvature at all points is declared not real but imaginary, since, according to the authors, it is impossible for plotting. References to textbooks of mechanics and mathematics departments of universities.

A generalization of Ciric fixed point theorems

In this paper, we state and prove a generalization of Ciric fixed point theorems in metric space by using a new generalized quasi-contractive map. These theorems extend other well known fundamental metrical fixed point theorems in the literature (Banach [1], Kannan [11], Nadler [13], Reich [15], etc.) Moreover, a multi-valued version for generalized quasi-contraction is also established.

The smallest sets of points not determined by their X-rays

Let $F$ be an $n$-point set in $\mathbb{K}^d$ with $\mathbb{K}\in\{\mathbb{R},\mathbb{Z}\}$ and $d\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define $\psi_{\mathbb{K}^d}(m)$ as the minimum number $n$ for which there exist $m$ directions $s_1,...,s_m$ (pairwise linearly independent and spanning $\mathbb{R}^d$) such that two $n$-point sets in $\mathbb{K}^d$ exist that have the same X-rays in these directions. The bound $\psi_{\mathbb{Z}^d}(m)\leq 2^{m-1}$ has been observed many times in the literature. In this note we show $\psi_{\mathbb{K}^d}(m)=O(m^{d+1+\varepsilon})$ for $\varepsilon>0$. For the cases $\mathbb{K}^d=\mathbb{Z}^d$ and $\mathbb{K}^d=\mathbb{R}^d$, $d>2$, this represents the first upper bound on $\psi_{\mathbb{K}^d}(m)$ that is polynomial in $m$. As a corollary we derive bounds on the sizes of solutions to both the classical and two-dimensional Prouhet-Tarry-Escott problem. Additionally, we establish lower bounds on $\psi_{\mathbb{K}^d}$ that enable us to prove a strengthened version of R\'enyi's theorem for points in $\mathbb{Z}^2$.

Some fixed/coincidence point theorems under ( ψ , φ ) -contractivity conditions without an underlying metric structure

In this paper we prove a coincidence point result in a space which does not have to satisfy any of the classical axioms that define a metric space. Furthermore, the ambient space need not be ordered and does not have to be complete. Then, this result may be applied in a wide range of different settings (metric spaces, quasi-metric spaces, pseudo-metric spaces, semi-metric spaces, pseudo-quasi-metric spaces, partial metric spaces, G-spaces, etc.). Finally, we illustrate how this result clarifies and improves some well-known, recent results on this topic.

Minimal simplices inscribed in a convex body

Measuring how far a convex body \(\mathcal{K }\) (of dimension \(n\)) with a base point \({O}\in \,\text{ int }\,\mathcal{K }\) is from an inscribed simplex \(\Delta \ni {O}\) in “minimal” position, the interior point \({O}\) can display regular or singular behavior. If \({O}\) is a regular point then the \(n+1\) chords emanating from the vertices of \(\Delta \) and meeting at \({O}\) are affine diameters, chords ending in pairs of parallel hyperplanes supporting \(\mathcal{K }\). At a singular point \({O}\) the minimal simplex \(\Delta \) degenerates. In general, singular points tend to cluster near the boundary of \(\mathcal{K }\). As connection to a number of difficult and unsolved problems about affine diameters shows, regular points are elusive, often non-existent. The first result of this paper uses Klee’s fundamental inequality for the critical ratio and the dimension of the critical set to obtain a general existence for regular points in a convex body with large distortion (Theorem A). This, in various specific settings, gives information about the structure of the set of regular and singular points (Theorem B). At the other extreme when regular points are in abundance, a detailed study of examples leads to the conjecture that the simplices are the only convex bodies with no singular points. The second and main result of this paper is to prove this conjecture in two different settings, when (1) \(\mathcal{K }\) has a flat point on its boundary, or (2) \(\mathcal{K }\) has \(n\) isolated extremal points (Theorem C).

Fixed Point and Equilibrium Theorems in a Generalized Convexity Framework

We study the fixed point property of set-valued maps and the existence of equilibria in the framework of \(\mathbb{B}\)-convexity, recently defined by W. Briec and Ch. Horvath. We introduce some classes of the set-valued maps with generalized convexity and prove continuous selection and fixed point properties for them. Finally, we obtain results concerning the existence of quasi-equilibria for W.K. Kim’s new model.

Three nonzero periodic solutions for a differential inclusion

We prove the existence of three non-zero periodic solutions for an ordinary differential inclusion. Our approach is variational and based on a multiplicity theorem for the critical points of a nonsmooth functional, which extends a recent result of Ricceri.

Linear Subspace Spanned by Principal Points of a Mixture of Spherically Symmetric Distributions

For each positive integer k, a set of k-principal points of a distribution is the set of k points that optimally represent the distribution in terms of mean squared distance. However, explicit form of k-principal points is often difficult to obtain. Hence a theorem established by Tarpey et al. (1995) has been influential in the literature, which states that when the distribution is elliptically symmetric, any set of k-principal points is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem is called a “principal subspace theorem”. Recently, Yamamoto and Shinozaki (2000b) derived a principal subspace theorem for 2-principal points of a location mixture of spherically symmetric distributions. In their article, the ratio of mixture was set to be equal. This article derives a further result by considering a location mixture with unequal mixture ratio.

A folk theorem for endogenous reference points

We establish a folk theorem for infinitely repeated game protocols with players whose preferences exhibit backward-looking reference dependence.