The uniqueness of a solution to the inverse Cauchy problem for a fractional differential equation in a Banach space

We consider linear fractional differential operator equations involving the Caputo derivative. The goal of this paper is to establish conditions for the unique solvability of the inverse Cauchy problem for these equations. We use properties of the Mittag-Leffler function and the calculus of sectorial operators in a Banach space. For equations with operators in a general form we obtain sufficient conditions for the unique solvability, and for equations with densely defined sectorial operators we obtain necessary and sufficient unique solvability conditions.     

On the solvability of an abstract Cauchy problem

We consider the Cauchy problem for a first-order linear inhomogeneous differential equation for functions ranging in a Banach space in the case of a sectorial operator coefficient. We find conditions for the solvability of the Cauchy problem for the case in which the right-hand side is not necessarily a Hölder function.     

Volterra equations in Banach spaces with completely monotone kernels

We consider a class of infinite delay equations in Banach spaces that models arising in the theory of viscoelasticity, for instance. The equation involves a completely monotone convolution kernel with a singularity at t = 0 and a sectorial linear spatial operator. Our main goal here is the construction of a semigroup formulation for the integral equation; in the last part of the paper, we illustrate the potentiality of the approach by considering a stochastic perturbation of the problem. Existence and uniqueness of a weak solution is established. The corresponding evolutionary solution process is Markovian, and the tools of linear analytic semigroup theory can be utilized.     

Difference schemes for solving the Cauchy problem for a second-order operator differential equation

A class of finite-difference schemes for solving an ill-posed Cauchy problem for a second-order linear differential equation with a sectorial operator in a Banach space is studied. Time-uniform estimates of the convergence rate and the error of such schemes are obtained. Previously known estimates are improved due to an optimal choice of initial data for a difference scheme.     

Abstract parabolic Cauchy problem in complex interpolation scales

We show that if the values of the inhomogeneous part of an autonomous parabolic linear equation belong to the complex interpolation scale associated with the sectorial operator of the equation, then there exists a unique classical solution of the Cauchy problem. This fact generalizes the well-known result obtained by Da Prato and Grisvard for real interpolation scales to the case of complex scales.     

Comments on Some Recent Results on Controllability of Abstract Differential Problems

In this note we comment on some recent papers treating exact controllability of abstract control problems with a linear part dominated by a sectorial operator.     

Extremal maximal sectorial extensions of sectorial relations

The extremal maximal sectorial extensions of a not necessarily densely defined sectorial relation (multivalued linear operator) in a Hilbert space are characterized in terms of a construction which goes back to Sebestyén and Stochel. In particular the two extreme maximal sectorial extensions, namely the Friedrichs extension and the Kreĭn extension, are characterized. For this purpose a survey is given of the connection between closed sectorial forms and maximal sectorial relations.     

On a complete discretization scheme for an ill-posed Cauchy problem in a Banach space

A complete discretization scheme for an ill-posed Cauchy problem for abstract firstorder linear differential equations with sectorial operators in a Banach space is validated. The scheme combines a time semidiscretization of the equations and a finite-dimensional approximation of the spaces and operators. Regularization properties of the scheme are established. Error estimates are obtained in the case of approximate initial data under various a priori assumptions concerning the solution.     

The $H^{\infty}-$calculus and sums of closed operators

We develop a very general operator-valued functional calculus for operators with an calculus. We then apply this to the joint functional calculus of two sectorial operators when one has an calculus. Using this we prove theorem of Dore-Venni type on sums of sectorial operators and apply our results to the problem of maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of spaces and spaces, to obtain some more detailed results in this setting. Received: 3 July 2000 / Revised version: 31 January 2001 / Published online: 23 July 2001     

The regular part of second-order differential sectorial forms with lower-order terms

We present a formula for the regular part of a sectorial form that represents a general linear second-order differential expression that may include lower-order terms. The formula is given in terms of the original coefficients. It shows that the regular part is again a differential sectorial form and allows to characterise when also the singular part is sectorial. While this generalises earlier results on pure second-order differential expressions, it also shows that lower-order terms truly introduce new behaviour.     

The regular part of sectorial forms

We study the regular part of a densely defined sectorial form, first in the abstract setting and then under mild conditions for a differential sectorial form. The regular part of the latter turns out to be again a differential sectorial form. Moreover, we characterize when taking the real part of a differential sectorial form commutes with taking the regular part. An example shows that these two operations do not commute in general.     

On series of sectorial forms

We prove a convergence theorem for partial sums of sectorial forms with vertex zero and a common semi-angle. As an example, we prove an absorption theorem for sectorial forms.     

Continuation of multiple power series in a sectorial domain

For multiple power series centered at the origin we consider the problem of its analytic continuability into a sectorial domain. The condition for continuability is formulated in terms of a holomorphic function that interpolates the series coefficients. For series in one variable this problem has been studied in the works of E. Lindelöf, N. Arakelian, and others.     

弹性平面扇形域问题及哈密顿体系*

通过变量代换及变分原理,将平面弹性扇形域的方程导向哈密顿体系,从而可用分离变量法、本征函数展开等方法求解扇形域的分析单元,这样便可以与有限元的程序系统相结合。显示了哈密顿体系、辛数学的应用潜力。     

On a linearized problem arising in the Navier–Stokes flow of a free liquid jet

In this work, we analyze a Stokes problem arising in the study of the Navier–Stokes flow of a liquid jet. The analysis is accomplished by showing that the relevant Stokes operator accounting for a free surface gives rise to a sectorial operator which generates an analytic semigroup of contractions. Estimates on solutions are established using Fourier methods. The result presented is the key ingredient in a local existence and uniqueness proof for solutions of the full nonlinear problem.     

Asymptotic type for sectorial operators and an integral of fractional powers

There is a standard notion of type for a sectorial linear operator acting in a Banach space. We introduce a notion of asymptotic type for a linear operator V, involving estimates on the resolvent ⁻¹(λI+V) as λ→0. We show, for example, that if V is sectorial and of asymptotic type ω then the fractional power Vα is of asymptotic type αω for a suitable range of positive α. Moreover, we establish various properties of the operator ; in particular, this operator is of asymptotic type 0, for a sectorial operator V. This result has an application to the construction of operators satisfying the well-known Ritt resolvent condition.     

On a multistep method approximating a linear sectorial evolution equation

This paper concerns the linear multistep approximation of anabstract dissipative linear sectorial evolution equation ona Banach space X. We study how well the semigroup generatedby a sectorial operator A is approximated by the numerical semigroupgenerated by a q-step, strictly A ()-stable multistep method.An optimal order error bound is obtained.     

A note on immersions of domains of fractional powers of certain sectorial operators in Sobolev spaces

In this note we give a proof of a result on immersions of domains of fractional powers of certain sectorial operators associated to strongly elliptic operators in Sobolev spaces; such immersions preserve information on fractional derivatives. We also briefly comment on the application of this result to a problem of optimal control of mosquito populations.     

Solvability of Elliptic Differential Equations,Set in Three Habitats with Skewness Boundary Conditions at the Interfaces

In this work, we study an elliptic differential equation set in three habitats with skewness boundary conditions at the interfaces. It represents the linear stationary case of dispersal problems of population dynamics which incorporate responses at interfaces between the habitats. Existence, uniqueness and regularity of the solution of these problems are obtained in Hölder spaces under necessary and sufficient conditions on the data. Our techniques are based on the semigroup theory, the fractional powers of linear operators, the \(H^{\infty }\) functional calculus for sectorial operators in Banach spaces and some properties of real interpolation spaces.