We give a definition of the spectral flow for continuous paths in the space of bounded and essentially hyperbolic operators. We provide a homotopical characterization of the spectral flow in terms of a group homomorphism of the fundamental group of the projectors of the Calkin algebra with the infinite cyclic group Z. This characterization helps us to exhibit examples of infinite-dimensional Banach spaces where the spectral flow is not injective nor surjective. We prove that a path with spectral flow equal to an integer m exists if and only if there exists a projector P connected by an arc to a projector Q such that Range(Q) has co-dimension m in Range(P). We prove that if A is an asymptotically hyperbolic and essentially splitting path the differential operator F(u) = du/dt - Au is Fredholm. Moreover if A is also essentially hyperbolic the Fredholm index coincides with minus the spectral flow of A.
Motivated by the study of dynamics of interacting spins for infinite particle systems, we consider an infinite family of first order differential equations in a Euclidean space, parameterized by elements $x$ of a fixed countable set. We suppose that the system is row-finite, that is, the right-hand side of the $x$-equation depends on a finite but in general unbounded number $n_x$ of variables. Under certain dissipativity-type conditions on the right-hand side and a bound on the growth of $n_x$, we show the existence of the solutions with infinite life-time, and prove that they live in an incre…
In this paper, we investigate the existence and uniqueness of $(ω,ρ)$-periodic solutions for a class of the abstract impulsive integro-differential equations on Banach space.
In this paper we are concerned with existence results for a coupled system of quadratic functional differential equations. This system is reduced to a fixed point problem for a block operator matrix with nonlinear inputs. To prove the existence we are established some fixed point theorem of Dhage's type for the block matrix operator acting in partially ordered Banach algebras.
Consider a smooth vector field $f\colon \mathbb{R}^n\to\mathbb{R}^n$ and a maximal solution $γ\colon \,]a,b[\,\to \mathbb{R}^n$ to the ordinary differential equation $x'=f(x)$. It is a well-known fact that, if $γ$ is bounded, then $γ$ is a global solution, i.e., $\,]a,b[\,=\mathbb{R}$. We show by example that this conclusion becomes invalid if $\mathbb{R}^n$ is replaced with an infinite-dimensional Banach space.
We prove results of existence of a solution (resp. existence and uniqness of a solution) for nonlinear differential equations of type $x'(t) +G(x,t) x(t) = F(x,t),$ in an abstract Banach subspace $X$ of the space of bounded real-valued continuous functions, satisfying some general and natural property. In our work, the functions $F$ and $G$ jointly depend on the variables $(x,t)\in X\times \R$. Several examples will be given, in various function spaces, to illustrate our results. The vector-valued framework is also considered.
We deduce trace properties for modulation spaces of Gelfand-Shilov distributions. We use these properties to show that pseudo-differential operators with amplitudes in suitable modulation spaces, agree with pseudo-differential operators of normal type whose symbols belong to (other) modulation spaces.
We introduce global wave-front sets $\operatorname{WF}_{\mathcal B} (f)$, $f\in {\mathscr S}^\prime(\textbf{R}^d)$, with respect to suitable Banach or Fréchet spaces ${\mathcal B}$. An important special case is given by the modulation spaces ${\mathcal B}=M(ω,\mathscr B)$, where $ω$ is an appropriate weight function and $\mathscr B$ is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to $\operatorname{WF}_{\mathcal B} (f)$. In particular, we prove that micro locality and microellipticity hold for a class of globally …
The aim of this work is to study the existence of a periodic solutions of integro-differential equations d dt [x(t)-- L(x t)] = A[x(t)-- L(x t)]+ G(x t)+ t --$\infty$ a(t-- s)x(s)ds+ f (t), (0 $\le$ t $\le$ 2$π$) with the periodic condition x(0) = x(2$π$), where a $\in$ L 1 (R +). Our approach is based on the M-boundedness of linear operators, Fourier type, B s p,q-multipliers and Besov spaces.
In this work, we define a new class of fractional analytic functions containing functional parameters in the open unit disk. By employing this class, we introduce two types of fractional operators, differential and integral. The fractional differential operator is considered to be in the sense of Ruscheweyh differential operator, while the fractional integral operator is in the sense of Noor integral. The boundedness and compactness in a complex Banach space are discussed. Other studies are illustrated in the sequel.
We present two new forms in which the Frechet differential of a power series in a Banach algebra can be expressed in terms of absolutely convergent series involving the commutant $C(T):A\mapsto [A,T]$.Then we apply the results to the analytic functional calculus in a complex Banach space.
We show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing the harmonic extension we prove existence and uniqueness of a bounded solution (i.e. of the harmonic extension).
We establish continuity and Schatten-von Neumann properties for matrix operators with matrices satisfying mixed quasi-norm estimates. These considerations also include the case when the Lebesgue and Schatten parameters are allowed to stay between $0$ and $1$. We use the results to deduce continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in a broad class of modulation spaces.
We consider the effect of perturbations to a quasi-linear parabolic stochastic differential equation set in a UMD Banach space $X$. To be precise, we consider perturbations of the linear part, i.e. the term concerning a linear operator $A$ generating an analytic semigroup. We provide estimates for the difference between the solution to the original equation $U$ and the solution to the perturbed equation $U_0$ in the $L^p(Ω;C([0,T];X))$-norm. In particular, this difference can be estimated $|| R(λ:A)-R(λ:A_0) ||$ for sufficiently smooth non-linear terms. The work is inspired by the desire to pr…
Let $ A$ be a complex unital Banach algebra. An element $a \in A$ is said to be Hermitian, if $ \| \exp (ita) \| =1$ for all $t\in R$. In the case of the algebra of bounded linear operators in a Hilbert space this Hermitian property agrees with the ordinary selfadjointness. If $a \in A$ is Hermitian, then $|a|=||a||$, where $|a|$ denotes the spectral radius of $a$. A function $F: R\to \C$ is called the universal symbol if $ \|| F(a)||= |F(a)|$\ for each $ A$ and all Hermitian $a\in A$. We characterize universal symbols in terms of positive definite functions.
Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces $M_{φ,w}$ and $m_{φ,w}$ respectively, introduced earlier in \cite{HKM} for describing the Köthe dual of ordinary Orlicz-Lorentz spaces in a large variety of cases ($φ$ is an Orlicz function and $w$ a {\it decreasing} weight). We study these $M_{φ,w}$ classes in the most general setting…
In this paper, we investigate the existence and uniqueness of fixed point for partially ordered contraction type operators in Banach Space. We also present applications to integral and differential equations.
Module-valued ordinary differential equations and structure of solution spaces
The Key Laboratory of Cognition and Decision Intelligence for Complex Systems, Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；School of Mathematics and Statistics, Guizhou University（贵州大学数学与统计学院）；School of Mathematics, Nanjing University（南京大学数学系）
In this paper we introduce the notion of weak differential subordination for martingales and show that a Banach space $X$ is a UMD Banach space if and only if for all $p\in (1,\infty)$ and all purely discontinuous $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb E \|N_{\infty}\|^p \leq C_p\mathbb E \|M_{\infty}\|^p$. As a corollary we derive the sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.
System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of viscoelasticity and in the system identfica- tion theory. Also, the existence and uniqueness of a solution to a general linear fractional differential equation in the space of tempered distributions is given.
We deduce continuity properties for pseudo-differential operators with symbols in quasi-Banach Orlicz modulation spaces when rely on other quasi-Banach Orlicz modulation spaces. In particular we extend certain results in \cite{GH1,GH2,Toft2,Toft10,Toft16}.
We present a unified approach to the processes of inversion and duality for quasilinear and $1$-quasilinear maps; in particular, for centralizers and differentials generated by interpolation methods.
We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano-Poincaré-Miranda theorem to infinite dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.
It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [3] and [10]). In this framework, the novelty of the present paper represents the development of some fixed point results regarding sequences of contractions in the setting of cone metric spaces over Banach algebras. Furthermore, some examples are given in order to strengthen our new concepts. Also, based on the powerful notion of a cone metric space over a Banach algebra, we present important applications to systems …
We construct dense Banach subalgebras $A$ of the null sequence algebra $c_0$ which are spectral invariant, but do not satisfy the $D_1$-condition $\|ab \|_A \leq K(\|a\|_{\infty} \|b\|_A + \|a \|_A \|b \|_{\infty})$, for all $a, b \in A$. The sequences in $A$ vanish in a skewed manner with respect to an unbounded function $σ\colon {\mathbb N} \rightarrow [1, \infty)$.
This work is devoted to the study of a class of linear time-inhomogeneous evolution equations in a scale of Banach spaces. Existence, uniquenss and stability for classical solutions is provided. We study also the associated dual Cauchy problem for which we prove uniqueness in the dual scale of Banach spaces. The results are applied to an infinite system of ordinary differential equations but also to the Fokker-Planck equation associated with the spatial logistic model in the continuum.
In this paper we consider Meyer-Yoeurp decompositions for UMD Banach space-valued martingales. Namely, we prove that $X$ is a UMD Banach space if and only if for any fixed $p\in (1,\infty)$, any $X$-valued $L^p$-martingale $M$ has a unique decomposition $M = M^d + M^c$ such that $M^d$ is a purely discontinuous martingale, $M^c$ is a continuous martingale, $M^c_0=0$ and \[ \mathbb E \|M^d_{\infty}\|^p + \mathbb E \|M^c_{\infty}\|^p\leq c_{p,X} \mathbb E \|M_{\infty}\|^p. \] An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingales into a sum of a quasi-l…
Geodesic contraction in vector-valued differential equations is readily verified by linearized operators which are uniformly negative-definite in the Riemannian metric. In the infinite-dimensional setting, however, such analysis is generally restricted to norm-contracting systems. We develop a generalization of geodesic contraction rates to Banach spaces using a smoothly-weighted semi-inner product structure on tangent spaces. We show that negative contraction rates in bijectively weighted spaces imply asymptotic norm-contraction, and apply recent results on asymptotic contractions in Banach s…
We investigate mapping properties for the Bargmann transform on modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are well-known when the weights are moderated. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators in the context of these modulation spaces.
Known investigations of nonlinear evolution equations $${dx\over dt} + A(t)x(t) = f(t)\ ,\quad x(t_{0}) = x^{0},\ \quad t_{0} \le t < \infty\ , \eqno(0.1)$$ with monotone operators $A(t)$ acting from reflexive Banach space $B$ to dual space $B^*$, usually assume that along with $B$ and $B^*$ there is a Hilbert space $H$ and continuous imbedding $B \hookrightarrow H$ in the triplet $$B \hookrightarrow H \hookrightarrow B^*\ ; \eqno(0.2)$$ and that $B$ is dense in $H$. The stabilization of solutions of evolution equations has been proven either in the sense of weak convergence in $B$ or in the n…
We show that if $X$ is an infinite-dimensional separable Banach space (or more generally a Banach space with an infinite-dimensional separable quotient) then there is a continuous mapping $f\colon X\to X$ such that the autonomous differential equation $x'=f(x)$ has no solution at any point.
We study a class of nonlinear implicit fractional differential equations subject to nonlocal boundary conditions expressed in terms of nonlinear integro-differential equations. Using the Krasnosel'skii fixed point theorem we prove, via the Kolmogorov--Riesz criteria, existence of solutions. The existence results are established in a specific fractional derivative Banach space and they are illustrated by two numerical examples.
In this note we provide a self-contained proof of an existence and uniqueness result for a class of Banach space valued evolution equations with an additive forcing term. The framework of our abstract result includes, for example, finite dimensional ordinary differential equations (ODEs), semilinear deterministic partial differential equations (PDEs), as well as certain additive noise driven stochastic partial differential equations (SPDEs) as special cases. The framework of our general result assumes somehow mild regularity conditions on the involved semigroup and also allows the involved sem…
In this paper, we are concerned with the mild solutions of Riemann-Liouville fractional differential equations with nonlocal conditions in Banach space. We use Banach contraction principle to prove the existence and uniqueness. Moreover, we derive the existence by using Krasnoselkii's theorem. An illustrative example is presented.
If a differential equation in a Banach manifold is invariant or quasi-invariant under the action of one or more Lie groups, then its stationary points cannot be isolated, so that classical linearized stability theorem does not apply to it. The first main purpose of this paper is to establish a linearized stability theorem for parabolic differential equations in Banach manifolds which are either invariant or quasi-invariant under actions of a number of Lie groups. The second purpose of this paper is to apply this theorem to analyze stability of stationary solutions of some free boundary problem…
