Browder's Theorem Research Articles

On exact observability of a nonlinear evolutionary equation with a bounded right-hand side operator on a small interval

For the Cauchy problem associated with a nonlinear ordinary differential equation in a Hilbert space $X$, we obtain sufficient conditions for exact observability on a small interval. By means of the condition of boundedness from below by a positive constant on the unit sphere with respect to a linear observer (observation operator) and with the help of Minty–Browder's theorem, the observability problem is reformulated as an operator (integral) equation with the right-hand side involving (in addition to the Volterra-type term, which is “local” in time) a nonlocal term as well. The unique solvability of the obtained operator equation (the equation of state reconstruction from observation) is proved with the help of the contraction map principle and the hypothesis about smallness of the observation interval. Moreover, we prove two theorems on global reconstruction of a state: 1) from observation on a small interval and under the condition of global solvability of some majorant integral equation in the space $\mathbb{R}$; 2) from a series of observations on small intervals in the presence of a priori information on the belonging of the state values to a bounded ball in $X$. As an example (of an independent interest), a semilinear equation of the global electric circuit in the Earth's atmosphere is considered.

Homogenization of a semilinear elliptic problem in a thin composite domain with an imperfect interface

In this paper, we consider the asymptotic behavior of a semilinear elliptic problem in a thin two‐composite domain with an imperfect interface, where the flux is discontinuous. For this thin domain, both the height and the period are of order . We first use Minty–Browder theorem to prove the well‐posedness of the problem and then apply the periodic unfolding method to obtain the limit problems and some corrector results for three cases of a real parameter , and , respectively. To deal with the semilinear terms, the extension operator and the averaged function are used.

The quasilinear Schrödinger–Poisson system

This paper deals with the (p, q)-Schrödinger–Poisson system, which is new and has never been considered in the literature. The uniqueness of solutions of the quasilinear Poisson equation is obtained via the Minty–Browder theorem. The variational framework of the quasilinear system is built, and nontrivial solutions of the system are obtained via the mountain pass theorem.

Properties (RB) and (gRB) for Bounded Linear Operators

In this article, we consider pseudo invertible operators for study of the relationship between the space of relatively regular operators and some generalizations of Weyl and Browder theorems. By using the analysis and representation of pseudo invertible operators, some new properties in connection with Browder’s type theorems, were presented for bounded linear operators T ∈ B(X). These properties, which we refer to as property (RB), imply that All poles of the resolvent of T of finite rank in the typical spectrum are precisely those places of the spectrum for which a reasonably regular operator with its pseudo inverse operator is surjective. (γI – T)† ∈ SU(X), In the usual spectrum, the set of all poles of the resolvent is exactly those points of the spectrum for which we call property (gRB). γI – T is a B-relatively regular operator with its pseudo inverse operator is surjective (γI – T)† ∈ SU(X). In addition, several sufficient and necessary conditions for which properties (RB) and (gRB) hold are given.

Spectral theory of B-Weyl elements and the generalized Weyl's theorem in primitive C*-algebra

Let $\mathcal{A}$ be a unital primitive C*-algebra. This paper studies the spectral theories of B-Weyl elements and B-Browder elements in $\mathcal{A}$, including the spectral mapping theorem and a characterization of B-Weyl spectrum. In addition, we characterize the generalized Weyl's theorem and the generalized Browder's theorem for an element $a\in\mathcal{A}$ and $f(a)$, where $f$ is a complex-valued function analytic on a neighborhood of $\sigma(a)$. What's more, the perturbations of the generalized Weyl's theorem under the socle of $\mathcal{A}$ and quasinilpotent element are illustrated.

A Sub‐Supersolution Method for p‐Laplacian Equation with Non‐Local Term

This paper is concerned with the existence of solutions for p‐Laplace problems with non‐local term. We prove the sub‐supersolution theorem using the pseudomonotone operator theorem and Minty–Browder theorem with appropriate assumptions on M, gi(i = 1,2). Then, using the sub‐supersolution approach, we determine the existence of positive solutions to specific problems.

On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces

<p style='text-indent:20px;'>A class of differential quasi-variational-hemivariational inequalities (DQVHI, for short) is studied in this paper. First, based on the Browder's result, KKM theorem and monotonicity arguments, we prove the superpositionally measurability, convexity and strongly-weakly upper semicontinuity for the solution set of a general quasi-variational-hemivariational inequality. Further, by using optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of (DQVHI) is nonempty and compact. This kind of evolutionary problems incorporates various classes of problems and models.</p>

A second order convergent difference scheme for the initial‐boundary value problem of Korteweg–de Vires equation

AbstractIn this article, we present a two‐level implicit difference scheme for Korteweg–de Vires equation with the initial and boundary conditions by the method of order reduction. The truncation error of the difference scheme is analyzed in detail. In the practical computation, the introduced intermediate variable is decoupled in order to reduce the computational cost. It is proved that the difference scheme is solvable by the Browder theorem. The conservation, boundedness, and the unconditional convergence of the numerical solution are also analyzed at length. The convergence order is two both in space and in time in L2‐norm. The numerical solution is proved to be unique under the optimal step ratio condition. Numerical examples demonstrate that the theoretical analysis is correct.

On Properties of Cobordism Groups of Skew-Framed Immersions

In this paper, we introduce a geometric technique of working with skew-framed manifolds. It allows us to study stable homotopy groups of some Thom spaces by geometric means. We schematically describe how our results (which are also of independent interest) can be applied to obtain a proof of the Baum–Browder theorem stating nonimmersibility of ℝP10 to ℝ15.

Ekeland’s variational principle with weighted set order relations

The main results of the paper are a minimal element theorem and an Ekeland-type variational principle for set-valued maps whose values are compared by means of a weighted set order relation. This relation is a mixture of a lower and an upper set relation which form the building block for modern approaches to set-valued optimization. The proofs rely on nonlinear scalarization functions which admit to apply the extended Brezis–Browder theorem. Moreover, Caristi’s fixed point theorem and Takahashi’s minimization theorem for set-valued maps based on the weighted set order relation are obtained and the equivalences among all these results is verified. An application to generalized intervals is given which leads to a clear interpretation of the weighted set order relation and versions of Ekeland’s principle which might be useful in (computational) interval mathematics.

Anisotropic system with singular and regular nonlinearities

ABSTRACTWe consider in this paper the following system: where Ω is a bounded regular domain in , we will assume that , , and . Using approximation techniques, a result of Rabinowitz, and Minty–Browder Theorem, we obtain the existence of positive solutions to the considered problem.

Existence and uniqueness of solutions of an A-harmonic elliptic equation

Abstract This paper deals with the existence and uniqueness of weak solution of a problem which involves a class of A-harmonic elliptic equations of nonhomogeneous type. Under appropriate assumptions on the function f, our main results are obtained by using Browder Theorem.

Existence of solutions for a higher order fractional boundary value problem posed on the half-line

The aim of this paper is to study the existence and uniqueness of solutions for a boundary value problem associated with a fractional nonlinear differential equation with higher order posed on the half-line. An appropriate continuous embedding for suitable Banach spaces are proved and the Minty–Browder theorem for monotone operators is used in the proof of existence of solutions for a boundary value problem of fractional order posed on the half-line.

Spherical classes in $$H_*(\Omega ^lS^{l+n};{\mathbb {Z}}/2)$$ H ∗ ( Ω l S l + n ; Z / 2 ) for $$4\pmb {\leqslant } l \pmb {\leqslant } 8$$ 4 ⩽ l ⩽ 8

We show that for $$l\in \{4,5,6,7,8\}$$ , the only spherical classes in $$H_*\Omega ^lS^{n+l}$$ arise from the inclusion of the bottom cell, or the Hopf invariant one elements. Together with our previous work in Zare (Topol Appl 224:1–18, 2017), this verifies Eccles conjecture when we restrict to finite loop spaces $$\Omega ^lS^{n+l}$$ with $$l<9$$ and $$n>0$$ . Moreover, we prove a generalised Browder theorem for spherical classes in $$H_*\Omega ^lS^{n+l}$$ showing that for $$1\leqslant l<+\infty $$ , if $$\xi ^2\in H_*\Omega ^lS^{n+l}$$ is given with $$\dim \xi +1\ne 2^t$$ and $$\dim \xi +1\equiv 2\text { mod }4$$ and $$n>l$$ , then $$\xi ^2$$ is not spherical. We call this as a generalised Browder theorem. We also record an observation, probably well known, that for $$n>0$$ and $$1\leqslant l\leqslant +\infty $$ , if $$\xi \in H_*\Omega ^lS^{n+l}$$ is spherical with $$\sigma _*^l\xi =0$$ where $$\sigma _*^i$$ is the iterated homology suspension for $$i\leqslant l$$ , then for some $$j<+\infty $$ we have $$\sigma _*^j\xi =\zeta ^2$$ for some A-annihilated primitive class where $$\zeta $$ is odd dimensional. This reduces study of spherical classes to those which are square. We call this the reduction theorem as it reduces the problem to the study of spherical classes that are square. As an application, we show that if $$f\in {_2\pi _{2d}}QS^n$$ is given with $$h(f)\ne 0$$ , $$\sigma _*h(f)=0$$ , and $$d+1\equiv 2{\mathrm {mod}}4$$ then the dimension of the sphere of origin of f is bounded below in the sense that if f pulls back to an element of $${_2\pi _{2d}}\Omega ^lS^{n+l}$$ then $$l>n$$ . This is the first type of this result that we know of in the existing literature providing a lower bounded on the dimension of sphere of origin of an element of $${_2\pi _*^s}$$ .

The relationship of generalized manifolds to Poincaré duality complexes and topological manifolds

The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincaré duality complexes (PD complexes). The problem is that an arbitrary generalized manifold X is always an ENR space, but it is not necessarily a complex. Moreover, finite PD complexes require the Poincaré duality with coefficients in the group ring Λ (Λ-complexes). Standard homology theory implies that X is a Z-PD complex. Therefore by Browder's theorem, X has a Spivak normal fibration which in turn, determines a Thom class of the pair (N,∂N) of a mapping cylinder neighborhood of X in some Euclidean space. Then X satisfies the Λ-Poincaré duality if this class induces an isomorphism with Λ-coefficients. Unfortunately, the proof of Browder's theorem gives only isomorphisms with Z-coefficients. It is also not very helpful that X is homotopy equivalent to a finite complex K, because K is not automatically a Λ-PD complex. Therefore it is convenient to introduce Λ-PD structures. To prove their existence on X, we use the construction of 2-patch spaces and some fundamental results of Bryant, Ferry, Mio, and Weinberger. Since the class of all Λ-PD complexes does not contain all generalized manifolds, we appropriately enlarge this class and then describe (i.e. recognize) generalized manifolds within this enlarged class in terms of the Gromov–Hausdorff metric.

A remark on the Brézis–Browder theorem in finite dimensional spaces

Brézis 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.

Differential variational inequalities in infinite banach spaces

In this paper, we consider a new differential variational inequality (DVI, for short) which is composed of an evolution equation and a variational inequality in infinite Banach spaces. This kind of problems may be regarded as a special feedback control problem. Based on the Browder's theorem and the optimal control theory, we show the existence of solutions to the mentioned problem.

Inverse Functions and Existence Principles

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.

Green functions for weighted subelliptic p-Laplace operator constructed by Hörmander's vector fields

This paper deals with the weighted subelliptic p-Laplace operator constructed by Hörmander's vector fields:Lpu=divX(〈A(x)Xu(x),Xu(x)〉p−22A(x)Xu(x)), where u∈W1,p(U,w),1<p<Q,andA(x) is a bounded measurable and m×m symmetric matrix satisfyingλ−1w(x)2/p|ξ|2⩽〈A(x)ξ,ξ〉⩽λw(x)2/p|ξ|2,ξ∈Rm,w(x)∈Ap. We first prove existence of the modified Green function for Lp by virtue of Minty–Browder theorem and then existence of the Green function for Lp by checking the convergence of sequence of modified Green functions. Next, we derive upper bounds of the modified Green function for Lp by establishing the interpolation inequality in the weighted weak Lp spaces. Finally, the bounds of the Green function for Lp are also obtained by virtue of results for the modified Green function and the compact weighted embedding theorem.

Maximum Principle and Existence of Weak Solutions for Nonlinear System Involving Singular (p,q)- Laplacian Operators

We investigate in this work necessary and sufficient conditions for having the maximum principle for nonlinear system involving singular (p,q)-Laplacian operators on bounded domain Ω of R^{N}. Moreover, we prove the existence of positive weak solutions by the Browder theorem method for the considered system