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This paper is devoted to the proof of an equilibrium existence theorem in incomplete markets, concerning the one-period economy with a finite number states of the world. The proof of the main theorem relies on the Brouwer Fixed Point Theorem. The assumption of incomplete markets is crucial for the proof of the main theorem, since the pricing functionals’-space is not one-dimensional. Usually, in General Equilibrium Theory of Incomplete Markets, the utility functions of the consumers -investors are supposed to be smooth-regular. In the present paper, we assume that the utility functions are continuous and the risk-less asset is extremely desirable. This class of utility functions assures that the demand vector is well-defined and unique, as well. The interest of economists about the effect of perturbation of pricing functionals on the existence of equilibrium is satisfied by the reference on the pricing functionals’ dimension. The equilibrium allocation uniqueness is not deduced by the Brouwer Fixed Point Theorem. We also notice that the existence of equilibrium may be deduced in the corresponding multi-period model by using the corresponding event-tree.

This paper studied the Crank-Nicolson(CN) difference scheme for the derivative nonlinear Schrodinger equation with the Riesz space fractional derivative, which generalized the classical Schrodinger equation that was used as a model in quantum mechanics. The existence of this difference solution is proved by the Brouwer fixed point theorem. Since the difference solution of the equation satisfies the mass conservation law, the corresponding convergence is also investigated in the $ L_2 $ norm, which turns out to be the second order accuracy in both temporal and space directions. Especially when the fractional order equals to two, all those results are in accordance with the conclusions for the difference solution developed for the non-fractional derivative Schrodinger equation. Finally, some numerical examples are carried out and further verified the theoretical results.

Stoilow's theorem from 1928 states that a continuous, open, and light map between surfaces is a discrete map with a discrete branch set. This result implies that such maps between orientable surfaces are locally modeled by power maps z -> z(k) and admit a holomorphic factorization. The purpose of this expository article is to give a proof of this classical theorem having readers in mind that are interested in continuous, open and discrete maps. (C) 2019 Elsevier GmbH. All rights reserved.

We study singular second-order boundary value problems with mixed boundary conditions on an infinitely discrete time scale. We prove the existence of a positive solution by means of a lower and upper solutions method and the Brouwer fixed-point theorem, in conjunction with perturbation methods used to approximate regular problems.

The theorems of Hadamard and of Poincare–Miranda give sufficient conditions for the existence of at least one zero for some continuous mappings from a Euclidean space into itself. The note gives el...

We use that the n-sphere for \(n\ge 2\) is simply-connected to prove the Poincare-Birkhoff-Witt Theorem.

In this article we prove a general result on a nef vector bundle $E$ on a projective manifold $X$ of dimension $n$ depending on the vector space $H^{n,n} (X, E). $ It is also shown that $H^{n,n} (X, E)=0$ for an indecomposable nef rank 2 vector bundles $E$ on some specific type of $n$ dimensional projective manifold $X.$ The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.

This paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation f2 + g2 = 1 and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

The aim of this article is to give a proof of the Jordan Curve Theorem via complex analysis.

In this paper, we introduce (weak) Stampacchia and Minty arcwise connected vector variational-like inequalities in the term of right upper-Dini-derivative and establish not only the relations of introduced inequalities with vector optimization problems but also the existence results, by using KKM-Fan theorem and Brouwer fixed point theorem. Examples are provided to illustrate the derived results.

The fact that every combinator has a fixed point is at the heart of the λ -calculus as a model of computation. We consider several aspects of such phenomenon; our specific, perhaps eccentric, point of view focuses on problems and results that we consider worthy of further investigations. We first consider the relation with self application, in comparison with the opposite view, which stresses the role of coding, unifying the first and the second fixed point theorems. Then, we consider the relation with the diagonal argument, a relation which is at the origin of the fixed point theorem itself. We also review the Recursion Theorem, which is considered a recursion theoretic version of the fixed point theorem. We end considering systems of equations which are related to fixed points.

We study the following fractional Yamabe-type equation: { A s ⁢ u = u n + 2 ⁢ s n - 2 ⁢ s , u > 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle A_{s}u&\displaystyle=u^{\frac{n+2s}{n-2s}% },\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right. Here Ω is a regular bounded domain of ℝ n {\mathbb{R}^{n}} , n ≥ 2 {n\geq 2} , and A s {A_{s}} , s ∈ ( 0 , 1 ) {s\in(0,1)} , represents the fractional Laplacian operator ( - Δ ) s {(-\Delta)^{s}} in Ω with zero Dirichlet boundary condition. We investigate the effect of the topology of Ω on the existence of solutions. Our result can be seen as the fractional counterpart of the Bahri–Coron theorem [3].

We prove that, under certain restrictions, every locally compact group equipped with a nonzero, σ-finite, regular left Haar measure can be decomposed into two small sets, one of which is small in the sense of measure and the other is small in the sense of category, and all such decompositions originate from a generalised notion of a Lebesgue point. Incidentally, such class of topological groups for which this happens turns out to be metrisable. We also observe an interesting connection between Luzin sets in such spaces and decompositions of the above type.

The main aim of this article is to establish analogues of Landau’s theorem for solutions to the$\overline{\unicode[STIX]{x2202}}$-equation in Dirichlet-type spaces.

We give a proof for the common fixed point theorem of Jungck in rectangular b-metric spaces. As a corollary, we obtain well known common fixed point theorems in b-metric spaces.

The nonlinear versions of Sturm-Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.

We use suitably Page’s theorem to get effective results for interesting problems, by avoiding the ineffective Siegel’s theorem.

The Valiron–Mohon’ko theorem plays an important role in the theory of complex differential equations. In this paper, a difference analogue of the Valiron–Mohon’ko theorem is established, which can be used to get the characteristic functions on irreducible rational functions of f(z) and its shifts. Using our results and some properties of periodic functions, the proof of [10, Theorem 1.1] can be organized in a short way.

In 1955, M. A. Krasnosel’skii proved a fixed-point theorem for a single-valued map which is a completely continuous contraction (a hybrid theorem). Subsequently, his work was continued in various directions. In particular, it has stimulated the development of the theory of condensing maps (both single-valued and set-valued); the images of such maps are always compact. Various versions of hybrid theorems for set-valued maps with noncompact images have also been proved. The set-valued contraction in these versions was assumed to have closed images and the completely continuous perturbation, to be lower semicontinuous (in a certain sense). In this paper, a new hybrid fixed-point theorem is proved for any set-valued map which is the sum of a set-valued contraction and a compact set-valued map in the case where the compact set-valued perturbation is upper semicontinuous and pseudoacyclic. In conclusion, this hybrid theorem is used to study the solvability of operator inclusions for a new class of operators containing all surjective operators. The obtained result is applied to solve the solvability problem for a certain class of control systems determined by a singular differential equation with feedback.

We prove a theorem on the existence of nonzero periodic solution to a system of differential equations by the method of fixed point of nonlinear operator defined on a topological product of two compact sets.

Assume X is a compact Hausdorff space and C(X) is the space of real-valued continuous functions on X. A version of the Stone–Weierstrass theorem states that a closed subalgebra $$A\subset C(X)$$ , which contains a nonzero constant function, coincides with the whole space C(X) if and only if A separates points of X. In this paper, we generalize this theorem to the case in which two subalgebras of C(X) are involved.

New B-U-cyclic weak contraction C-class function concept has been introduced, B-U-cyclic weak F-φ-Φ-contraction types of mapping are defined and the existence of fixed point for such types has been proved. These results mainly generalize fixed point theorems in some previous research papers.

We present new proof of non-bijective Wigner theorem on symmetries of quantum systems using only basic linear algebra. It is based on showing that any non-zero Jordan ∗-homomorphism between matrix algebras preserving rank-one projections is implemented by either a unitary or an anitiunitary map. As a new application we extend hitherto known results on preservers of quantum relative entropy to infinite quantum systems.

We prove analogues of the Bombieri–Vinogradov Theorem and the Barban–Davenport–Halberstam Theorem on primes in arithmetic progressions for Fourier coefficients of cusp forms.

We consider infinite systems of stochastic differential equations used to describe the motion of interacting particles in random media. It is assumed that the mass of each particle approaches zero and the density of particles infinitely increases in a proper way. It is proved that the sequence of corresponding measure-valued processes converges in distribution. We also prove the existence and uniqueness of a strong solution for the limit equation.

A non-Euclidean analog of the generalized Darboux equation is considered. For the case where its solutions are radial functions of second variable we obtain a uniqueness result (Theorem 1) which deals with zero sets of these solutions. The example of the function in Theorem 2 of the paper shows that Theorem 1 cannot be essentially reinforced.

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano’s ones for real functions on an interval is deduced in a very simple way from Cauchy’s theorem for holomorphic functions. A more complicated proof, using Cauchy’s argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from ℂn to ℂn are obtained using Brouwer degree.

Rouche’s Theorem is a standard topic in undergraduate complex analysis. It is usually covered near the end of the course with applications relating to pure mathematics only (e.g., using it to produce an alternate proof of the Fundamental Theorem of Algebra). The winding number provides a geometric interpretation relating to the conclusion of Rouche’s Theorem, but most undergraduate texts give no geometric insights that lead to an understanding of why Rouche’s Theorem holds. In addition, most texts do not inform students that a stronger version of the theorem exists. In this paper we present a simplified proof of the stronger version, which is a suitable topic for students to pursue as a short project, and provide a geometric argument for the weaker version. Finally, as a project for advanced students, we unpack a standard application of this theorem as used in control systems: the Nyquist stability criterion.

We study the notion of a non-Archimedean fuzzy anti-2-normed space over a non-Archimedean field and prove that Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy anti-2-normed spaces.

In recent years, fixed-point theorems have attracted increasing attention and have been widely investigated by many authors. Moreover, determining a fixed point has become an interesting topic. In this paper, we provide a constructive proof of the general Brouwer fixed-point theorem and then obtain the existence of a smooth path which connects a given point to the fixed point. We also present a non-interior point homotopy algorithm for solving fixed-point problems on a class of nonconvex sets by numerically tricking this homotopy path.

In this paper we study the solvability of the generalized vector equilibrium problem (for short, GVEP) with a variable ordering relation in reflexive Banach spaces. The existence results of strong solutions of GVEPs for monotone multifunctions are established with the use of the KKM-Fan theorem. We also investigate the GVEPs without monotonicity assumptions and obtain the corresponding results of weak solutions by applying the Brouwer fixed point theorem. These results are also the extension and improvement of some recent results in the literature.

In this paper we prove Caristi’s fixed point theorem using only purely metric techniques.

Each one of six general existence principles of compactness (the extreme value theorem), completeness (the Newton method or the modified Newton method), topology (Brouwer’s fixed point theorem), homotopy (on contractions of a sphere to its center), variational analysis (Ekeland’s principle), and monotonicity (the Minty–Browder theorem)) is shown to lead to the inverse function theorem, each one giving some novel insight. There are differences in assumptions and algorithmic properties; some of the propositions have been constructed specially for this paper. Simple proofs of the last two principles are included. The proof by compactness is shorter and simpler than the shortest and simplest known proof, that by completion. This gives a very short self-contained proof of the Lagrange multiplier rule, which depends only on optimization methods. The proofs are of independent interest and are intended as well to be useful in the context of the ongoing efforts to obtain new variants of methods that are based on the inverse function theorem, such as comparative statics methods.

We analyze an elementary theorem of Euclidean geometry, the Droz-Farny Line Theorem, from the point of view of the foundations of geometry. We start with an elementary synthetic proof which is based on simple properties of the group of motions. The proof reveals that the Droz-Farny Line Theorem is a special case of the Theorem of Goormatigh which is, in turn, a special case of the Counterpairing Theorem of Hessenberg. An axiomatic analysis in the sense of Hilbert [14] and Bachmann [2] leads to a study of different versions of the theorems (e.g., of a dual version or of an absolute version, which is valid in absolute geometry) and to a new axiom system for the associated very general plane absolute geometry (the geometry of pencils and lines). In the last section the role of the theorems in the foundations of geometry is discussed.

We prove a comparison theorem for the solutions of Riccati matrix equations in which the diagonal entries of the matrix multiplying the linear term are perturbed by a bounded function. This theorem is used to study optimal trajectories in a pollution control problem stated in the form of a linear regulator over an infinite time horizon with a discount function of the general form.

The purpose of this paper is to show that the Rudin–Carleson interpolation theorem is a direct corollary of Fatou’s much older interpolation theorem (of 1906).

We study the arithmetic three-spheres theorems for subsolutions of subelliptic PDEs of p p -harmonic type in Carnot groups of Heisenberg type for 1 > p > ∞ 1>p>\infty . In the presentation we exhibit the special cases of sub-Laplace equations ( p = 2 p=2 ) and the case p p is equal to the homogeneous dimension of a Carnot group. Corollaries include asymptotic behavior of subsolutions for small and large radii and the Liouville-type theorems.

It was proved earlier that, for multiple Fourier series whose coefficients are monotone in each index, the classicalHardy–Littlewood theorem is not valid for p ≤ 2m/(m+1), where m is the dimension of the space. We establish how the theorem must be modified in this case.