Regularized cosine existence and uniqueness families for second order abstract Cauchy problems

Jizhou Zhang

Displaying similar documents to “Regularized cosine existence and uniqueness families for second order abstract Cauchy problems”

Existence of the fundamental solution of a second order evolution equation

Jan Bochenek (1997)

Annales Polonici Mathematici

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We give sufficient conditions for the existence of the fundamental solution of a second order evolution equation. The proof is based on stable approximations of an operator A(t) by a sequence $A_{n} (t)$ of bounded operators.

Existence of solutions of the dynamic Cauchy problem on infinite time scale intervals

Ireneusz Kubiaczyk, Aneta Sikorska-Nowak (2009)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In the paper, we prove the existence of solutions and Carathéodory’s type solutions of the dynamic Cauchy problem $x^{Δ} (t) = f (t, x (t))$ , t ∈ T, x(0) = x₀, where T denotes an unbounded time scale (a nonempty closed subset of R and such that there exists a sequence (xₙ) in T and xₙ → ∞) and f is continuous or satisfies Carathéodory’s conditions and some conditions expressed in terms of measures of noncompactness. The Sadovskii fixed point theorem and Ambrosetti’s lemma are used to prove the main result. The...

Forms, functional calculus, cosine functions and perturbation

Wolfgang Arendt, Charles J. K. Batty (2007)

Banach Center Publications

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In this article we describe properties of unbounded operators related to evolutionary problems. It is a survey article which also contains several new results. For instance we give a characterization of cosine functions in terms of mild well-posedness of the Cauchy problem of order 2, and we show that the property of having a bounded $H^{\infty}$ -calculus is stable under rank-1 perturbations whereas the property of being associated with a closed form and the property of generating a cosine function...

On the Cauchy problem for hyperbolic functional-differential equations

Adrian Karpowicz, Henryk Leszczyński (2015)

Annales Polonici Mathematici

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We consider the Cauchy problem for a nonlocal wave equation in one dimension. We study the existence of solutions by means of bicharacteristics. The existence and uniqueness is obtained in $W_{l o c}^{1, \infty}$ topology. The existence theorem is proved in a subset generated by certain continuity conditions for the derivatives.

The distance between fixed points of some pairs of maps in Banach spaces and applications to differential systems

Cristinel Mortici (2006)

Czechoslovak Mathematical Journal

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Let $T$ be a $γ$ -contraction on a Banach space $Y$ and let $S$ be an almost $γ$ -contraction, i.e. sum of an $(ε, γ)$ -contraction with a continuous, bounded function which is less than $ε$ in norm. According to the contraction principle, there is a unique element $u$ in $Y$ for which $u = T u .$ If moreover there exists $v$ in $Y$ with $v = S v$ , then we will give estimates for $∥ u - v ∥ .$ Finally, we establish some inequalities related to the Cauchy problem.

Comparison of solutions and successive approximations in the theory of the equation $\partial^{2} z / \partial x \partial y = f (x, y, z, \partial z / \partial x, \partial z / \partial y)$

J. Kisyński, A. Pelczar

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CONTENTSIntroduction........................................................................................................................................................................................................... 5I. THE CAUCHY-DARBOUX PROBLEM IN FUNCTION CLASSES $C^{1}' * (Δ_{a, b}; E)$ AND $L_{1}^{1, *} (Δ_{a, b}; E)$ ......................... 71. Basic function classes ......................................................................................................................................................................................

An inequality for spherical Cauchy dual tuples

Sameer Chavan (2013)

Colloquium Mathematicae

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Let T be a spherical 2-expansive m-tuple and let $T^{}$ denote its spherical Cauchy dual. If $T^{}$ is commuting then the inequality $\binom{\sum_{| β | = k} {(β!)}^{- 1} {(T^{})}^{β} (T^{}) *^{β} \leq (k + m - 1}{k) \sum_{| β | = k} {(β!)}^{- 1} (T^{}) *^{β} {(T^{})}^{β}}$ holds for every positive integer k. In case m = 1, this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a 2-expansive (or concave) operator are hyponormal.

Local and global solutions of well-posed integrated Cauchy problems

Pedro J. Miana (2008)

Studia Mathematica

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We study the local well-posed integrated Cauchy problem $v^{'} (t) = A v (t) + (t^{α}) / Γ (α + 1) x$ , v(0) = 0, t ∈ [0,κ), with κ > 0, α ≥ 0, and x ∈ X, where X is a Banach space and A a closed operator on X. We extend solutions increasing the regularity in α. The global case (κ = ∞) is also treated in detail. Growth of solutions is given in both cases.

On the semilinear integro-differential nonlocal Cauchy problem

Piotr Majcher, Magdalena Roszak (2005)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we prove an existence theorem for the pseudo-non-local Cauchy problem $x^{'} (t) + A x (t) = f (t, x (t), \int_{t ₀}^{t} k (t, s, x (s)) d s)$ , x₀(t₀) = x₀ - g(x), where A is the infinitesimal generator of a C₀ semigroup of operator ${T (t)}_{t > 0}$ on a Banach space. The functions f,g are weakly-weakly sequentially continuous and the integral is taken in the sense of Pettis.

Local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation in Besov spaces

Gang Wu, Jia Yuan (2007)

Applicationes Mathematicae

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We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation $\partial_{t} u - \partial ³_{t x x} u + 2 κ \partial_{x} u + \partial_{x} [g (u) / 2] = γ (2 \partial_{x} u \partial ²_{x x} u + u \partial ³_{x x x} u)$ for the initial data u₀(x) in the Besov space $B_{p, r}^{s} (ℝ)$ with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given $C^{m}$ -function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.

On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

Sameer Chavan (2011)

Studia Mathematica

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The Cauchy dual operator T’, given by $T {(T * T)}^{- 1}$ , provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T’ admits the Wold-type decomposition (see Definitions 1.1...

Bounded Solutions for the Degasperis-Procesi Equation

Giuseppe Maria Coclite, Kenneth H. Karlsen (2008)

Bollettino dell'Unione Matematica Italiana

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This paper deals with the well-posedness in $L^{1}\cap L^{\infty}$ of the Cauchy problem for the Degasperis-Procesi equation. This is a third order nonlinear dispersive equation in one spatial variable and describes the dynamics of shallow water waves.

The Cauchy kernel for cones

Sławomir Michalik (2004)

Studia Mathematica

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A new representation of the Cauchy kernel $_{Γ}$ for an arbitrary acute convex cone Γ in ℝⁿ is found. The domain of holomorphy of $_{Γ}$ is described. An estimation of the growth of $_{Γ}$ near the singularities is given.

On the initial-boundary value problems for a degenerate parabolic equation

Jan Goncerzewicz (2008)

Banach Center Publications

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For an equation of the type of porous media equation the Cauchy-Dirichlet and Cauchy-Neumann problems are considered. The existence and uniqueness results in the case of $L^{\infty}$ initial and boundary data are given.

On $L^{p}$ -estimates for solutions of the Cauchy problem for parabolic differential equations

P. Besala (1971)

Annales Polonici Mathematici

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A new characteristic property of Mittag-Leffler functions and fractional cosine functions

Zhan-Dong Mei, Ji-Gen Peng, Jun-Xiong Jia (2014)

Studia Mathematica

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A new characteristic property of the Mittag-Leffler function $E_{α} (a t^{α})$ with 1 < α < 2 is deduced. Motivated by this property, a new notion, named α-order cosine function, is developed. It is proved that an α-order cosine function is associated with a solution operator of an α-order abstract Cauchy problem. Consequently, an α-order abstract Cauchy problem is well-posed if and only if its coefficient operator generates a unique α-order cosine function.

Convergence results for unbounded solutions of first order non-linear differential-functional equations

Henryk Leszczyński (1996)

Annales Polonici Mathematici

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We consider the Cauchy problem in an unbounded region for equations of the type either $D_{t} z (t, x) = f (t, x, z (t, x), z_{(t, x)}, D_{x} z (t, x))$ or $D_{t} z (t, x) = f (t, x, z (t, x), z, D_{x} z (t, x))$ . We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.

О голоморфном продолжении $C R$ -гиперфункций в фиксированную область.

А.М. Кытманов, И.А. Цих (1997)

Sibirskij matematiceskij zurnal

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Maximal regularity for second order non-autonomous Cauchy problems

Charles J. K. Batty, Ralph Chill, Sachi Srivastava (2008)

Studia Mathematica

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We consider some non-autonomous second order Cauchy problems of the form ü + B(t)u̇ + A(t)u = f(t ∈ [0,T]), u(0) = u̇(0) = 0. We assume that the first order problem u̇ + B(t)u = f(t ∈ [0,T]), u(0) = 0, has $L^{p}$ -maximal regularity. Then we establish $L^{p}$ -maximal regularity of the second order problem in situations when the domains of B(t₁) and A(t₂) always coincide, or when A(t) = κB(t).

Less than one implies zero

Felix L. Schwenninger, Hans Zwart (2015)

Studia Mathematica

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In this paper we show that from an estimate of the form $s u p_{t \geq 0} | | C (t) - c o s (a t) I | | < 1$ , we can conclude that C(t) equals cos(at)I. Here ${(C (t))}_{t \geq 0}$ is a strongly continuous cosine family on a Banach space.

Non-autonomous stochastic Cauchy problems in Banach spaces

Mark Veraar, Jan Zimmerschied (2008)

Studia Mathematica

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We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $d U (t) = A (t) U (t) d t + B (t) d W_{H} (t)$ , t ∈ [0,T], U(0) = u₀. Here, $W_{H}$ is a cylindrical Brownian motion on a real separable Hilbert space H, ${(B (t))}_{t \in [0, T]}$ are closed and densely defined operators from a constant domain (B) ⊂ H into E, ${(A (t))}_{t \in [0, T]}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting...

Semilinear perturbations of Hille-Yosida operators

Horst R. Thieme, Hauke Vosseler (2003)

Banach Center Publications

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The semilinear Cauchy problem (1) u’(t) = Au(t) + G(u(t)), $u (0) = x \in \bar{D (A)}$ , with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that ||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A) with some linear B: D(A) → X, $B {(λ - A)}^{- 1} x = λ \int_{0}^{\infty} e^{- λ t} V (s) x d s$ , where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on $[0, \infty) \times \bar{D (A)}$ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)| ≤...

Divergent solutions to the 5D Hartree equations

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We consider the Cauchy problem for the focusing Hartree equation $i u_{t} + Δ u + {(| \cdot |}^{- 3} * | u | ²) u = 0$ in ℝ⁵ with initial data in H¹, and study the divergence property of infinite-variance and nonradial solutions. For the ground state solution of $- Q + Δ Q + (| \cdot |^{- 3} * | Q | ²) Q = 0$ in ℝ⁵, we prove that if u₀ ∈ H¹ satisfies M(u₀)E(u₀) < M(Q)E(Q) and ||∇u₀||₂||u₀||₂ > ||∇Q||₂||Q||₂, then the corresponding solution u(t) either blows up in finite forward time, or exists globally for positive time and there exists a time sequence tₙ → ∞ such that ||∇u(tₙ)||₂...

The analytic $α$ -capacity and the Cauchy integral

K. Adžievski (1986)

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On a functional equation with derivative and symmetrization

Adam Bobrowski, Małgorzata Kubalińska (2006)

Annales Polonici Mathematici

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We study existence, uniqueness and form of solutions to the equation $α g - β g^{'} + γ g_{e} = f$ where α, β, γ and f are given, and $g_{e}$ stands for the even part of a searched-for differentiable function g. This equation emerged naturally as a result of the analysis of the distribution of a certain random process modelling a population genetics phenomenon.

Cauchy-Poisson transform and polynomial inequalities

Mirosław Baran (2009)

Annales Polonici Mathematici

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We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in $ℝ^{N}$ is Hölder continuous then E admits a Szegö type inequality with weight function $d i s t {(x, \partial E)}^{- (1 - κ)}$ with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

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We study the Cauchy problem for the 3D MHD system with damping terms ${ε | u |}^{α - 1} u$ and ${δ | b |}^{β - 1} b$ (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

On the Cauchy problem in linear viscoelasticity

Pasquale Renno (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Con riferimento all’operatore integrodifferenziale della viscoelasticità lineare nella formulazione creep, si determina la soluzione fondamentale $E$ in corrispondenza di un’arbitraria funzione di memoria. Di conseguenza viene risolto esplicitamente il problema di Cauchy relativo al moto unidimensionale di un sistema viscoelastico $\mathcal{B}$ , omogeneo ed isotropo, determinato da dati iniziali e storia di stress comunque prefissati. Successivamente, nell’ambito di opportune ipotesi di memoria labile,...