Unbounded Linear Operators and Evolution Equations

2018 ◽  
pp. 1-8
Author(s):  
Alain Haraux
Keyword(s):  
Evolution Equations ◽  
Linear Operators ◽  
Unbounded Linear Operators

scholarly journals Local attractivity in nonautonomous semilinear evolution equations

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Joël Blot ◽  
Constantin Buşe ◽  
Philippe Cieutat
Keyword(s):  
Banach Space ◽  
Exponential Stability ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Linear Part ◽  
Linear Operators ◽  
Nonlinear Mapping ◽  
Solutions Of Equations ◽  
Unbounded Linear Operators ◽  
Semilinear Evolution Equations

AbstractWe study the local attractivity of mild solutions of equations in the form u’(t) = A(t)u(t) + f (t, u(t)), where A(t) are (possible) unbounded linear operators in a Banach space and where f is a (possible) nonlinear mapping. Under conditions of exponential stability of the linear part, we establish the local attractivity of various kinds of mild solutions. To obtain these results we provide several results on the Nemytskii operators on the space of the functions which converge to zero at infinity

Periodic solutions of time-dependent, semilinear evolution equations of compact type

1983 ◽  
Vol 95 (1-2) ◽  
pp. 7-24 ◽  
Author(s):  
Herbert C. Sager
Keyword(s):  
Evolution Equations ◽  
Homogeneous Problem ◽  
Zero Solution ◽  
Weak Sense ◽  
Linear Operators ◽  
Bounded Linear ◽  
Compact Semigroups ◽  
Image Position ◽  
Unbounded Linear Operators ◽  
Semilinear Evolution Equations

SynopsisWe establish the existence of solutions in a weak sense ofwhere t Є J = [0, T] and′ = d/dt. It is supposed that the unbounded, linear operators A(t) generate analytic and compact semigroups on a Hilbert space H and that B(t, x) are bounded linear operators. The function f(t, x) with values in H may have asymptotically sublinear growth.We prove the existence of a periodic solution with the help of Schauder’s fixed point theorem.Accordingly, we first verify that the corresponding linearized version of (0.1),has a unique solution for each square integrable ψ(t), provided that the homogeneous problem has only the zero solution.

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

scholarly journals The numerical ranges of unbounded linear operators

1975 ◽  
Vol 12 (1) ◽  
pp. 23-25 ◽  
Author(s):  
Béla Bollobás ◽  
Stephan E. Eldridge
Keyword(s):  
Banach Space ◽  
Scalar Field ◽  
Banach Spaces ◽  
Unbounded Operator ◽  
Numerical Range ◽  
Linear Operators ◽  
Density Property ◽  
Unbounded Linear Operators

Giles and Joseph (Bull. Austral. Math. Soc. 11 (1974), 31–36), proved that the numerical range of an unbounded operator on a Banach space has a certain density property. They showed, in particular, that the numerical range of an unbounded operator on certain Banach spaces is dense in the scalar field. We prove that the numerical range of an unbounded operator on a Banach space is always dense in the scalar field.

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