Fredholm Theory Research Articles

Why bootstrapping for J-holomorphic curves fails in C^k

We present a simple example for the failure of the Calderón–Zygmund estimate for the overline{partial }-operator when the Sobolev (k, p)-norms are replaced by the C^k-norms. This example is discussed in the context of elliptic bootstrapping, Fredholm theory, and the regularity of J-holomorphic curves.

Asymptotic behaviors and existence of traveling wave solutions to the Keller-Segel model with logarithmic sensitivity

<p style='text-indent:20px;'>In this paper, we are concerned with the existence of traveling wave solutions for the Keller-Segel model with logarithmic sensitivity. By the Hopf-Cole transformation and traveling wave transformation, the degenerate Keller-Segel system is transformed into a singularly perturbed system. By constructing an invariant region to prove the existence of the traveling wave solutions for the degenerate system, we obtain the traveling wave solutions for Keller-Segel system with small parameter by using geometric singular perturbation theory and Fredholm theory. Finally, we discuss the asymptotic behaviors of the traveling wave solutions.</p>

Existence of traveling waves in a delayed convecting shallow water fluid model

<abstract><p>This paper investigates a delayed shallow water fluid model that has not been studied in previous literature. Applying geometric singular perturbation theory, we prove the existence of traveling wave solutions for the model with a nonlocal weak delay kernel and local strong delay convolution kernel, respectively. When the convection term contains a nonlocal weak generic delay kernel, the desired heteroclinic orbit is obtained by using Fredholm theory and linear chain trick to prove the existence of two kink wave solutions under certain parametric conditions. When the model contains local strong delay convolution kernel and weak backward diffusion, under the same parametric conditions to the previous case, the corresponding traveling wave system can be reduced to a near-Hamiltonian system. The existence of a unique periodic wave solution is established by proving the uniqueness of zero of the Melnikov function. Uniqueness is proved by utilizing the monotonicity of the ratio of two Abelian integrals.</p></abstract>

Fredholm theory of families of discrete dynamical systems and its applications to bifurcation theory

In a previous work, we proved an index theorem for families of asymptotically hyperbolic discrete dynamical systems and obtained applications to bifurcation theory. A weaker and far more common assumption than asymptotic hyperbolicity is the existence of an exponential dichotomy. In this paper we generalize all our previous results to the latter setting, which requires substantial modifications of our arguments. In addition, we generalize previous results on continuity and differentiability of Nemitski operators for discrete dynamical systems to further weaken the assumptions of our bifurcation theorems.

Analysis of contact Cauchy–Riemann maps III: Energy, bubbling and Fredholm theory

In [Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps I: A priori [Formula: see text] estimates and asymptotic convergence, Osaka J. Math. 55(4) (2018) 647–679; Y.-G. Oh and R. Wang, Analysis of contact Cauchy-Riemann maps II: Canonical neighborhoods and exponential convergence for the Morse-Bott case, Nagoya Math. J. 231 (2018) 128–223], the authors studied the nonlinear elliptic system [Formula: see text] without involving symplectization for each given contact triad [Formula: see text], and established the a priori [Formula: see text] elliptic estimates and proved the asymptotic (subsequence) convergence of the map [Formula: see text] for any solution, called a contact instanton, on [Formula: see text] under the hypothesis [Formula: see text] and [Formula: see text]. The asymptotic limit of a contact instanton is a ‘spiraling’ instanton along a ‘rotating’ Reeb orbit near each puncture on a punctured Riemann surface [Formula: see text]. Each limiting Reeb orbit carries a ‘charge’ arising from the integral of [Formula: see text]. In this paper, we further develop analysis of contact instantons, especially the [Formula: see text] estimate for [Formula: see text] (or the [Formula: see text]-estimate), which is essential for the study of compactification of the moduli space and the relevant Fredholm theory for contact instantons. In particular, we define a Hofer-type off-shell energy [Formula: see text] for any pair [Formula: see text] with a smooth map [Formula: see text] satisfying [Formula: see text], and develop the bubbling-off analysis and prove an [Formula: see text]-regularity result. We also develop the relevant Fredholm theory and carry out index calculations (for the case of vanishing charge).

Fredholm theory for demicompact linear relations

We first attempt to determine conditions on a linear relation T such that μT becomes a demicompact linear relation for each μ ∈ [0,1)(see Theorems 2.4 and 2.5). Second, we display some results on Fredholm and upper semi-Fredholm linear relations involving a demicompact one(see Theorems 3.1 and 3.2). Finally, we provide some results in which a block matrix of linear relations becomes a demicompact block matrix of linear relations (see Theorems 4.2 and 4.3).

Khoja Akhmet Yassawi International Kazakh-Turkish University

When considering non-local boundary value problems for functional-differential equations, when the derivative of the desired function is contained in the right side, one could use the resolvent of the integral equation. But, as is known, the resolvent of an integral equation of the second kind of the Fredholm type cannot always be uniquely determined. In some cases, you can use the properties of the kernel of the integro-differential equation. In this paper, we consider a nonlocal boundary value problem for systems of integro-differential equations with involution, when the kernel of the integral term containing the derivative has a partial derivative. Using the properties of an involutive transformation, the problem is reduced to the study of a multipoint boundary value problem for systems of integro-differential equations. The parameterization method proposed by Professor D. Dzhumabaev was applied to this problem. New parameters are introduced, and based on these parameters, we pass to new variables. When passing to new variables, we obtain the initial conditions for the initial equation. With the help of this condition, it is possible to determine the solution of the resulting Cauchy problem, as well as the system of linear equations. Applying the Fredholm theory to solve the obtained systems of integral equations, i.e. the unique solvability of the problem under study, we reduce to the reversibility of the matrix, which depends on the initial data. An example was shown as an illustration of the proposed method.

Existence of Traveling Wave Fronts for a Generalized Nonlinear Schrodinger Equation

In the presented paper, a generalized nonlinear Schr o dinger equation without delay convolution kernel and with special delay convolution kernel is investigated. By using the geometric singular perturbation theory, the existence of traveling wave fronts is proved. Firstly, we show that such traveling wave fronts exist without delay by non-Hamiltonian qualitative analysis. Then, for the generalized nonlinear Schr o dinger equation with a special local strong delay convolution kernel, the desired heteroclinic orbit is obtained by using the Fredholm theory.

Spectral Properties of Compact Operators

The spectral properties of a compact operator \(T : X \longrightarrow Y\) on a normed linear space resemble those of square matrices. For a compact operator, the spectral properties can be treated fairly completely in the sense that Fredholm's famous theory of integral equations may be extended to linear functional equations \(T x -\lambda\) \(= y\) with a complex parameter \(\lambda\) . This paper has studied and investigated the spectral properties of compact operators in Hilbert spaces. The spectral properties of compact linear operators are relatively simple generalization of the eigenvalues of finite matrices. As a result, the paper has given a number of corresponding propositions and interesting facts which are used to prove basic properties of compact operators. The Fredholm theory has been introduced to investigate the solvability of linear integral equations involving compact operators.

Perturbation Ideals and Fredholm Theory in Banach Algebras

In this paper we characterize perturbation ideals of sets that generate the familiar spectra in Fredholm theory.

Travelling Waves for Adaptive Grid Discretizations of Reaction Diffusion Systems III: Nonlinear Theory

In this paper we consider a spatial discretization scheme with an adaptive grid for the Nagumo PDE and establish the existence of travelling waves. In particular, we consider the time dependent spatial mesh adaptation method that aims to equidistribute the arclength of the solution under consideration. We assume that this equidistribution is strictly enforced, which leads to the non-local problem with infinite range interactions that we derived in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021). Using the Fredholm theory developed in Hupkes and Van Vleck (J Dyn Differ Eqn, 2021) we setup a fixed point procedure that enables the travelling PDE waves to be lifted to our spatially discrete setting.

Simultaneous recovery of a locally rough interface and the embedded obstacle with its surrounding medium

Consider the problem of scattering of time-harmonic point sources by an infinite locally rough interface with bounded obstacles embedded in the lower half-space. The model problem is first reduced to an equivalent integral equation formulation defined in a bounded domain, where the well-posedness is obtained in L p by the classical Fredholm theory. Then a global uniqueness theorem is proved for the inverse problem of recovering the locally rough interface, the embedded obstacles and the wave number in the lower-half space by means of near-field measurements above the interface.

Spectral description of Fredholm operators via polynomially Riesz operators perturbation

Abstract In this paper, we establish some spectral analysis in Fredholm theory involving the concept of a polynomially Riesz operators perturbation. Moreover, we apply the obtained results to analyze the incidence of some essential spectra of a perturbed operator via this kind of perturbation. Our approach allows us to investigate a new description in the theory of unbounded operator matrices defined with maximal domain via this kind of perturbation. Finally, we conclude by presenting some progress in the theory of polynomially operators and the theory of semi-Fredholm and semi-Browder operators.

A Fredholm theory on Krein spaces and its application to Weyl-type theorems and homogeneous equations

In this paper, we review the approach presented by An and Heo on the study of Weyl-type theorems for self-adjoint operators on Krein spaces and show that this approach is not appropriate due to a fallacy. Motivated by this fact, we define a new modification of the kernel of a bounded linear operator on a Krein space, namely $J$-kernel, which allows us to successfully introduce a Fredholm theory in this context and study some variations of Weyl-type theorems for bounded linear operators defined on these spaces. In addition, we will describe the $J$-index in terms of solution sets of homogeneous equations.

Об условиях нётеровости и индексе некоторых двумерных сингулярных интегральных операторов

The main problems in the theory of singular integral operators are the problems of boundedness, invertibility, Noethericity, and calculation of the index. The general theory of multidimensional singular integral operators over the entire space E_n was constructed by S.G. Mikhlin. It is known that in the two-dimensional case, if the symbol of an operator does not vanish, then the Fredholm theory holds. For operators over a bounded domain, the boundary of this domain significantly affects the solvability of the corresponding operator equations. In this paper, we consider two-dimensional singular integral operators with continuous coefficients over a bounded domain. Such operators are used in many problems in the theory of partial differential equations. In this regard, it is of interest to establish criteria for the considered operators to be Noetherian in the form of explicit conditions on their coefficients. The paper establishes effective necessary and sufficient conditions for two-dimensional singular integral operators to be Noetherian in Lebesgue spaces L_p (D) (considered over the field of real numbers), 1<p<∞, and formulas for calculating indices are given. The method developed by R.V. Duduchava [Duduchava R. On multidimensional singular integral operators. I: The half-space case; II: The case of compact manifolds // J. Operator Theory, 1984, v. 11, 41–76 (I); 199–214 (II)]. In this case, the study of the Noetherian properties of operators is reduced to the factorization of the corresponding matrix-functions and finding their partial indices.

Demicompactness, selection of linear relation and application to multivalued matrix

In this paper, we investigate the concept of demicompactness and we establish some new results in Fredholm theory connected with the existence of selections of a given linear relation and we explore the possibility of finding a selection demicompact for some linear relation demicompact. Moreover, we give the relationship of the resolvent set between the linear relation with its selection. Furthermore, we give an application to matrix linear relation.

Long-Time Estimates for Heat Flows on Asymptotically Locally Euclidean Manifolds

Abstract We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel $L^p - L^q$ decay estimates are established, allowing the Schrödinger operator to have a non-trivial $L^2$-kernel. We also prove new decay estimates for spatial derivatives of arbitrary order, in a general geometric setting. Our main motivation is the application to stability of non-linear geometric equations, primarily Ricci flow, which will be presented in a companion paper. The arguments in this paper use that many geometric Schrödinger operators can be written as the square of Dirac-type operators. By a remarkable result of Wang, this is even true for the Lichnerowicz Laplacian, under the assumption of a parallel spinor. Our analysis is based on a novel combination of the Fredholm theory for Dirac-type operators on ALE manifolds and recent advances in the study of the heat kernel on non-compact manifolds.

Fredholm theory for pseudoholomorphic curves with brake symmetry

We study the pseudoholomorphic curves with brake symmetry in symplectization of a closed contact manifold. We introduce the pseudoholomorphic curves with brake symmetry and the corresponding moduli space. Then we get the virtual dimension of the moduli space.

Exponential dichotomies for nonlocal differential operators with infinite range interactions

We show that functional differential equations of mixed type (MFDEs) with infinite range discrete and/or continuous interactions admit exponential dichotomies, building on the Fredholm theory developed by Faye and Scheel for such systems. For the half line, we refine the earlier approach by Hupkes and Verduyn Lunel. For the full line, we construct these splittings by generalizing the finite-range results obtained by Mallet-Paret and Verduyn Lunel. The finite dimensional space that is ‘missed’ by these splittings can be characterized using the Hale inner product, but the resulting degeneracy issues raise subtle questions that are much harder to resolve than in the finite-range case. Indeed, there is no direct analogue for the standard ‘atomicity’ condition that is typically used to rule out degeneracies, since it explicitly references the smallest and largest shifts.We construct alternative criteria that exploit finer information on the structure of the MFDE. Our results are optimal when the coefficients are cyclic with respect to appropriate shift semigroups or when the standard positivity conditions typically associated to comparison principles are satisfied. We illustrate these results with explicit examples and counter-examples that involve the Nagumo equation.

Continuous Fredholm Theory, Regularities and Semiregularities

Continuous Fredholm Theory, Regularities and Semiregularities