On perturbation of closed operators in a Banach space

Let Hi(i = 1, 2, …, n), be closed operators in a Banach space X. The generalised initialvalue problem of the abstract Cauchy problem is studied. We show that the uniqueness of solution u: [0, T1] × [0, T2] × … × [0, Tn] → X of this n-abstract Cauchy problem is closely related to C0-n-parameter semigroups of bounded linear operators on X. Also as another application of C0-n-parameter semigroups, we prove that many n-parameter initial value problems cannot have a unique solution for some initial values.

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Ranges of Products of Operators

Canadian Journal of Mathematics ◽

10.4153/cjm-1974-137-7 ◽

1974 ◽

Vol 26 (6) ◽

pp. 1430-1441 ◽

Cited By ~ 7

Author(s):

Sandy Grabiner

Keyword(s):

Banach Space ◽

Dual Problem ◽

Bounded Operator ◽

Linear Operators ◽

Bounded Operators ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Closed Operators

Suppose that T and A are bounded linear operators. In this paper we examine the relation between the ranges of A and TA, under various additional hypotheses on T and A. We also consider the dual problem of the relation between the null-spaces of T and AT; and we consider some cases where T or A are only closed operators. Our major results about ranges of bounded operators are summarized in the following theorem.Theorem 1. Suppose that T is a bounded operator on a Banach space E and that A is a non-zero bounded operator from some Banach space to E.

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A functional calculus of closed operators on Banach space. III. certain topics of perturbation theory

Russian Mathematics ◽

10.3103/s1066369x17120039 ◽

2017 ◽

Vol 61 (12) ◽

pp. 19-28 ◽

Cited By ~ 1

Author(s):

A. R. Mirotin

Keyword(s):

Banach Space ◽

Perturbation Theory ◽

Functional Calculus ◽

Closed Operators

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The fundamental solutions for fractional evolution equations of parabolic type

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953304311020 ◽

2004 ◽

Vol 2004 (3) ◽

pp. 197-211 ◽

Cited By ~ 68

Author(s):

Mahmoud M. El-Borai

Keyword(s):

Banach Space ◽

Continuous Dependence ◽

Evolution Equations ◽

Initial Conditions ◽

Parabolic Type ◽

Fundamental Solutions ◽

Parabolic Partial Differential Equations ◽

Fractional Evolution Equations ◽

Closed Operators ◽

Continuous Dependence Of Solutions

The fundamental solutions for linear fractional evolution equations are obtained. The coefficients of these equations are a family of linear closed operators in the Banach space. Also, the continuous dependence of solutions on the initial conditions is studied. A mixed problem of general parabolic partial differential equations with fractional order is given as an application.

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On some functional calculus of closed operators in a Banach space. II

Russian Mathematics ◽

10.3103/s1066369x15050011 ◽

2015 ◽

Vol 59 (5) ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

A. A. Atvinovskii ◽

A. R. Mirotin

Keyword(s):

Banach Space ◽

Functional Calculus ◽

Closed Operators

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Existence of Mild Solutions for a Semilinear Integrodifferential Equation with Nonlocal Initial Conditions

Abstract and Applied Analysis ◽

10.1155/2012/647103 ◽

2012 ◽

Vol 2012 ◽

pp. 1-15 ◽

Cited By ~ 9

Author(s):

Carlos Lizama ◽

Juan C. Pozo

Keyword(s):

Banach Space ◽

Fixed Point ◽

Integrodifferential Equation ◽

Measure Of Noncompactness ◽

Initial Conditions ◽

Mild Solutions ◽

Hausdorff Measure Of Noncompactness ◽

Nonlocal Initial Conditions ◽

Closed Operators ◽

Point Argument

Using Hausdorff measure of noncompactness and a fixed-point argument we prove the existence of mild solutions for the semilinear integrodifferential equation subject to nonlocal initial conditionsu′(t)=Au(t)+∫0tB(t-s)u(s)ds+f(t,u(t)),t∈[0,1],u(0)=g(u), whereA:D(A)⊆X→X, and for everyt∈[0,1]the mapsB(t):D(B(t))⊆X→Xare linear closed operators defined in a Banach spaceX. We assume further thatD(A)⊆D(B(t))for everyt∈[0,1], and the functionsf:[0,1]×X→Xandg:C([0,1];X)→XareX-valued functions which satisfy appropriate conditions.

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