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This paper is devoted to the study of high-order finite difference scheme for the nonlinear Schrodinger equation with wave operator. The difference scheme is three level and a five point stencil is used for spatial variable. Existence of solutions is shown using a variant of Brouwer fixed point theorem. The unconditional stability as well as uniqueness of the difference scheme are also discussed in detail. The convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete maximum norm without any restrictions on the mesh sizes. Finally, some numerical experiments and comparisons with other existing methods in the literature are presented. The numerical results show that the high-order difference scheme of this article improves the accuracy of the space and time direction.

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.

It is shown that the stochastic counterpart of the classical fixed point theorem for continuous maps in a finite dimensional Euclidean space (“Brouwer’s theorem”) is not, in general, true. This result implies, in particular, that a careful choice of invariant sets in the stochastic version of Brouwer’s theorem is necessary in the theory of stochastic nonlinear operators.

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.

Existence of positive solutions for the nonlinear algebraic system$x=\lambda GF ( x ) $x=λGF(x)has been extensively studied when the$n\times n$n×ncoefficient matrixGis positive or nonnegative. However, to the best of our knowledge, few results have been obtained when the coefficient matrix changes sign. In this case, some commonly applied analysis methods such as the cone theory, the Krein–Rutman theorem, the monotone iterative techniques, and so on cannot be directly applied. In this note, we prove the existence of positive solutions for the above nonlinear algebraic system with sign-changing coefficient matrix taking the advantages of the classical Brouwer fixed point theorem combined with a decomposition condition on the coefficient matrix. We provide an example in solving a second-order difference equation with periodic boundary conditions to illustrate the applications of the results.

This chapter reviews several important theorems and formulas which form the bases for the derivation of various equations. Gauss's theorem, Green's theorem, and Stokes' theorem are foundations of electromagnetic theory. The integral theorem of Helmholtz and Kirchhoff and the Fresnel-Kirchhoff diffraction formula are basic theories for solving diffraction problems. The representation of the surface integral on the upper and lower surface is called Gauss's theorem. It states that the summation of the divergence of a vector in a volume space is equal to the sum of the outward normal components on the surface enclosing the space. Using the Helmholtz and Kirchhoff theorem of integration, the amplitude of light at an arbitrary observation point can be obtained by knowing the field distribution of light on the surface enclosing the observation point. The integration theorem of Helmholtz and Kirchhoff is used to find the diffraction pattern of an aperture when illuminated by a point source and projected onto a screen. An important modification of the Fresnel-Kirchhoff diffraction formula implies that if the amplitude distribution of the light across the aperture is known, then the field at the point of observation can be obtained. The chapter also summarizes the formulas in cylindrical and spherical coordinates.

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.

The Brouwer fixed-point theorem in topology states that for any continuous mapping $f$ on a compact convex set into itself admits a fixed point, i.e., a point $x_0$ such that $f(x_0)=x_0$. Under certain conditions, this fixed point corresponds to the throat of a traversable wormhole, i.e., $b(r_0)=r_0$ for the shape function $b=b(r)$. The possible existence of wormholes can therefore be deduced from purely mathematical considerations without going beyond the existing physical requirements., 6 pages, 1 figure

A conservative finite difference scheme is presented for solving the two-dimensional fourth-order nonlinear wave equation. The existence of the numerical solution of the finite difference scheme is proved by Brouwer fixed point theorem. With the aid of the fact that the discrete energy is conserved, the finite difference solution is proved to be bounded in the discrete L ∞ − norm. Then, the difference solution is shown to be second order convergent in the discrete L ∞ − norm. A numerical example shows the efficiency of the proposed scheme.

In this chapter, the first approach is made to establishing the convergence of scaled random sums, considering independent sequences. The classic Lindeberg–Lévy, Khinchine, Lindeberg–Feller, and Liapunov theorems are proved. The main focus is on the treatment of heterogeneous summands, applying the Lindeberg condition, and extensions are given to allow trending (growing or shrinking) variances. The final sections review cases of the central limit theorem under non-standard conditions and α-stable convergence.

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.

In this paper, a Crank–Nicolson difference scheme is first derived for solving the nonlinear time–space fractional Schrodinger equations. The truncation error and stability of the scheme are discussed in detail. The existence of the numerical solution is shown by the Brouwer fixed point theorem. For improving the calculating efficiency, a three-level linearized difference scheme is also proposed and analyzed. Both schemes are subsequently extended to the nonlinear coupled equations, and some similar results are given and proved. Several numerical experiments are included to verify the accuracy and efficiency of the two types of schemes, and comparison with the related work is presented.

In this paper, a mass conserved domain decomposition method is developed for solving the nonlinear parabolic equations with Dirichlet conditions. The domain decomposition is divided into multiple subdomains, two steps are used to solve the numerical solutions on each sub-domain at every time interval [tn, tn+1]. In the first step, the interface fluxes are computed by local multi-points weighted average from the solutions. Secondly, the interior solutions are solved by the solution-flux coupled scheme. By some auxiliary lemmas and the Brouwer fixed point theorem, the existence, uniqueness and convergence of our scheme are analyzed in the discrete L2-norm. Numerical experiments are performed to illustrate the result.

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...

Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies., 14 pages, to appear in Journal of Pure and Applied Algebra

A topological fuzzy fixed point theorem is given, when generalizing and improving the main result by Diamond, Kloeden and Pokrovskii (1997) [1] . Apart from the sole existence, a weak local stability property, called non-ejectivity in the sense of Browder, of fuzzy fixed points is established. This theorem is then applied for obtaining non-ejective fuzzy fractals. An alternative approach via the Knaster–Tarski theorem is also presented.

We consider the energy supercritical wave maps from R d into the d-sphere S d with d ≥ 7 . Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation ∂ t 2 u = ∂ r 2 u + ( d − 1 ) r ∂ r u − ( d − 1 ) 2 r 2 sin ⁡ ( 2 u ) . We construct for this equation a family of C ∞ solutions which blow up in finite time via concentration of the universal profile u ( r , t ) ∼ Q ( r λ ( t ) ) , where Q is the stationary solution of the equation and the speed is given by the quantized rates λ ( t ) ∼ c u ( T − t ) l γ , l ∈ N ⁎ , l > γ = γ ( d ) ∈ ( 1 , 2 ] . The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphael and Rodnianski [49] for the energy supercritical nonlinear Schrodinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.

In his Gottingen dissertation of 1851 Bernard Riemann enunciated (p. 40 of his Werke) and attempted to prove the famous theorem that now bears his name: Theorem 9.1 Every simply-connected region other than ℂ itself is conformal to an open disk.

Many problems arising in science and engineering call for the solving of the Euler–Lagrange equations of functionals. Thus, solving the Euler–Lagrange equations is tantamount to finding critical points of the corresponding functional. An idea that has been very successful is to find appropriate sets that sandwich the functional. This means that the functional is bounded from above on one of the sets and bounded from below on the other. Two sets of the space are said to form a sandwich if they produce a critical sequence whenever they sandwich a functional. If the critical sequence has a convergent subsequence, then that produces a critical point. Finding sets that sandwich a functional is quite easy, but determining whether or not the sets form a sandwich is quite another story. It appears that the only way we can check to see if two sets form a sandwich is to require that one of them be contained in a finite-dimensional subspace. The reason is that in order to verify the definition, we need to invoke the Brouwer fixed point theorem. Our aim is to find a counterpart that holds true when both sets are infinite dimensional. We adjust our definitions to accommodate infinite dimensions. These definitions reduce to the usual when one set is finite dimensional. In order to prove the corresponding theorems, we make adjustments to the topology of the space and introduce infinite dimensional splitting. This allows us to use a form of compactness on infinite dimensional subspaces that does not exist under the usual topology. We lose the Brouwer index, but we are able to replace it with the Leray–Schauder index. We carry out the details in Sections 5, 7, and 8. In Section 6 we solve a system of equations which require infinite dimensional splitting.

International audience; This article is dedicated to insensitization issues of a quadratic functional involving the solution of the linear heat equation with respect to domains variations. This work can be seen as a continuation of [P. Lissy, Y. Privat, and Y. Simpor\'e. Insensitizing control for linear and semi-linear heat equations with partially unknown domain. ESAIM Control Optim. Calc. Var., 25:Art. 50, 21, 2019], insofar as we generalize several of the results it contains and investigate new related properties. In our framework, we consider boundary variations of the spatial domain on which the solution of the PDE is defined at each time, and investigate three main issues: (i) approximate insensitization, (ii) approximate insensitization combined with an exact insensitization for a finite-dimensional subspace, and (iii) exact insensitization. We provide positive answers to questions (i) and (ii) and partial results to question (iii).

For a smooth mapping \(f : \overline D \subset {\mathbb R}^n \to {\mathbb R}^n\) with 0∉f(∂D), D open, bounded with smooth boundary ∂D, the Kronecker index of f on ∂D is defined as the integral over ∂D of some (n − 1)-differential form depending upon f and its partial derivatives. For n = 2, it is the Cauchy index or winding number of f on ∂D, the algebraic number of turns of f around the closed curve ∂D. The Gauss linking number of two curves is a special case of the Kronecker index in \(\mathbb R^3\). Various expressions of the Kronecker index are given, as well as applications to the Ginzburg-Landau theory of liquid crystals. Writing the Kronecker index as an integral over \(\overline D\) leads in a natural way to the more general concept of the Brouwer degree for a continuous mappings \(f : \overline D \to {\mathbb R}^n\) without smoothness assumption upon ∂D. Its fundamental properties are established, including an axiomatic characterization and its localization to a neighborhood of an isolated zero, the Brouwer index. Applications are given to generalized implicit function theorems. An easy extension is made to mappings \(f : \overline D \subset X \to Y\) between oriented normed spaces X and Y of the same finite dimension, and some results on its computation are given, including reductions to situations of smaller dimension. Borsuk’s fruitful topological concept of retract leads to the obtention of the Brouwer fixed point theorem for continuous self-mappings of various classes of sets, and to the definition of an index of fixed points for continuous mappings defined in retracts of \({\mathbb R}^n\) like convex sets, wedges or cones in \({\mathbb R}^n\). The role of dimension is illustrated by the hairy ball theorem and the Brouwer degree is applied to the linking of curves and its use in various minimax theorems of critical point theory.

After extending the homotopy invariance of the Brouwer degree to situations where not only the mapping but also its set of definition may depend upon a parameter, the Leray–Schauder continuation theorem is stated and proved, giving conditions under which a subcontinuum of the set of solutions \({\mathcal S} = F^{-1}(\{0\})\) connects the slice \({\mathcal D}_a\) to the slice \({\mathcal D}_b\) of the open bounded set \({\mathcal D} \subset {\mathbb R}^n \times [a,b]\). Applications are given to the method of a priori bounds for proving the existence of a zero of a mapping, and to the Leray–Schauder alternative. Important special cases are the perturbed linear mappings, the surjectivity of coercive and monotone mappings, boundary conditions on the mapping leading to the existence of a zero in the set, further fixed point theorems and degree conditions implying the uniqueness of the zero or the fixed point. Other applications are given to the bifurcation of the zeros of mappings F(x, μ) depending upon a parameter μ, including the detection of the bifurcation points through linearization of F (theorem of Krasnosel’skii) and the global behavior of the bifurcation branches (theorem of Rabinowitz). Finally, the extension of the Brouwer fixed point theorem to set-valued mappings with convex values (theorem of Kakutani) is considered and applied to the existence of equilibria in game theory and economics following von Neumann and Nash.

The history of the Brouwer fixed point theorem, closely linked to the history of the Brouwer degree, is particularly intricated and is a case story showing the ‘nonlinear’ character of the evolution of mathematics. After describing and commenting the fundamental contribution of Brouwer, it is shown how it has been in one way or another anticipated by Poincare (with his n-dimensional intermediate value theorem), Hadamard (with his first published version of the theorem) and Bohl (with his no-retraction theorem for a n-cube). Subsequent new proofs of the fixed point theorem, by Alexander, by Birkhoff and Kellogg, and by Knaster, Kuratowski, Mazurliewicz, and its infinite-dimensional extensions, by Birkhoff and Kellogg, by Schauder and by Tychonov, are then described. The role of the Brouwer fixed point theorem in the theory of games and economics, in the hands of von Neumann, Kakutani, and Nash, and the numerical approximation of the fixed points by Scarf, and by Kellogg, Li and Yorke are then analyzed. A rediscovery of Poincare’s internmediate value theorem has led, not only to the proof of its equivalence to Brouwer fixed point theorem by Miranda, but also to a long and vivacious quarrel between Cinquini and Scorza-Dragoni for deciding if the Brouwer fixed point theorem is or is not a ‘topological theorem’. The chapter ends with the independent formulation of the Brouwer fixed point theorem in terms of variational inequalities, due to Harmann-Stampacchia and Karamardian, and to the endless quest for an ‘elementary proof’ of a theorem whose applications are exceptionally diverse and numerous.

The results reported in this paper are concerned with the existence and uniqueness of solutions of discrete fractional order two-point boundary value problem. The results are developed by employing the properties of Caputo and Riemann–Liouville fractional difference operators, the contraction mapping principle and the Brouwer fixed point theorem. Furthermore, the conditions for Hyers–Ulam stability and Hyers–Ulam–Rassias stability of the proposed discrete fractional boundary value problem are established. The applicability of the theoretical findings has been demonstrated with relevant practical examples. The analysis of the considered mathematical models is illustrated by figures and presented in tabular forms. The results are compared and the occurrence of overlapping/non-overlapping has been discussed.

It is well-known that a partial order induced from a lower semi-continuous map gives us a clear picture of a proof of the Caristi’s fixed point theorem. The proof utilized Zorn’s lemma to guarantee the existence of a minimal element which turns out to be a desired fixed point. The proof cannot be carried over to prove the Brouwer fixed point theorem. We show that making an idea of ordering, we get a proof of the later one.

Let ℋ be a complex separable Hilbert space, e the collection of orthogonal projections in ℋ, and (X, Ɓ) a Borel space. A function E : Ɓ → e is called a spectral measure if E(X) = I and E(∪ i=1 ∞ A i ) = Σ i=1 ∞ (A i ), for any pairwise disjoint collection A1, A2, A3, …, of sets in Ɓ.

Boundary value problems have wide applications in science and technology. This present work discusses discrete fractional boundary value problem of order 2

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.

In this paper, we study common fixed point properties for a family of mappings. We present the extensions of Caristi's fixed point theorem and Markov–Kakutani fixed point theorem for a family of set-valued mappings.

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.

We show existence theorem of global mild solutions with small initial data and external forces in Lorentz spaces with scaling invariant norms. If the initial data have more regularity in another scaling invariant class, then our mild solution is actually the strong solution. The result on local existence of solutions for large data is also discussed. Our method is based on the maximal regularity theorem on the Stokes equations in Lorentz spaces. Then we apply our theorem to prove existence of self-similar solutions provided both initial data and external forces are homogeneous functions. Since we construct the global solution by means of the implicit function theorem, as a byproduct, its stability with respect to the given data is necessarily obtained.

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.

In this paper, we consider a certain theorem on the global invertibility of a \begin{document} $C^{2n+1}$ \end{document} -mapping between Banach spaces with a singular point in which the derivatives of order up to \begin{document} $2n$ \end{document} vanish. The theorem is illustrated by several applications.

Brezis and Browder gave an important characterization of maximal monotonicity for linear operators, in reflexive Banach spaces. In this paper, we show that this characterization also preserves the maximal p-monotonicity, in finite dimensional spaces.

In 1995, Palfy and Szabo stated the theorem that a projective space satisfies the six-cross theorem if and only if it is Desarguesian. A mistake in their proof is corrected. Moreover, for projective planes, the three-cross theorem of Palfy and Szabo is identified as the Reidemeister condition.

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.

In this paper, a generalized fixed point theorem and its results are established in the concept of multiplicative distance which was introduced by Agamirza et.al [3] to improve the non-Newtonian calculus. Our results include some existing results in the concept of multiplicative metric space.

We present, in a unified framework, conforming and nonconforming virtual element methods for nonlinear Schrodinger equation. The constructed schemes conserve not only the mass but also the energy in the discrete senses. Then, by using the Brouwer fixed point theorem, the Gagliardo–Nirenberg inequality and the classical Ritz projection, we prove the boundedness, unique solvability and optimal convergence of the conforming virtual element scheme. To obtain the boundedness, unique solvability and optimal convergence of the nonconforming virtual element scheme, we utilize a new defined Ritz projection and a new type of Gagliardo–Nirenberg inequality. The optimal rates of convergence in the discrete L 2 -norm are derived without any restrictions on the grid ratio for both types of virtual element methods. Finally, numerical examples on a set of polygonal meshes are given to support the theoretical analysis. As a contrast, we also supply another classical proof method of the optimal convergence which has to limit the time–space grid ratio τ = o ( h 1 ∕ 2 ) .