An inverse problem with final time additional condition for an abstract evolution equation in an ordered Banach space

1993 ◽  
Vol 27 (1) ◽  
pp. 68-70 ◽  
Author(s):  
A. I. Prilepko ◽  
I. V. Tikhonov
Keyword(s):  
Banach Space ◽  
Inverse Problem ◽  
Evolution Equation ◽  
Additional Condition ◽  
Ordered Banach Space ◽  
Final Time ◽  
Abstract Evolution Equation

scholarly journals On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

2019 ◽  
Vol 17 (1) ◽  
pp. 1082-1112
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
A Priori ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Improvement Effect ◽  
Abstract Evolution Equation ◽  
Necessary And Sufficient

Abstract Given the abstract evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array}$$ with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

scholarly journals On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Spectral Operator ◽  
Scalar Type ◽  
Abstract Evolution Equation ◽  
Necessary And Sufficient ◽  
Space Conditions

For the evolution equation with a scalar type spectral operator in a Banach space, conditions on are found that are necessary and sufficient for all weak solutions of the equation on to be strongly infinite differentiable on or . Certain effects of smoothness improvement of the weak solutions are analyzed.

scholarly journals An inverse problem for an abstract evolution equation

2001 ◽  
Vol 79 (3-4) ◽  
pp. 475-482 ◽  
Author(s):  
A.G. Ramm ◽  
S.V. Koshkin
Keyword(s):  
Inverse Problem ◽  
Evolution Equation ◽  
Abstract Evolution Equation

scholarly journals On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one

2019 ◽  
Vol 17 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Scalar Type ◽  
Abstract Evolution Equation

Abstract It is shown that, if all weak solutions of the evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array} $$ with a scalar type spectral operator A in a complex Banach space are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded.

scholarly journals On an abstract evolution equation with a spectral operator of scalar type

2002 ◽  
Vol 32 (9) ◽  
pp. 555-563 ◽  
Author(s):  
Marat V. Markin
Keyword(s):  
Banach Space ◽  
Evolution Equation ◽  
Weak Solutions ◽  
Operational Calculus ◽  
Complex Banach Space ◽  
Spectral Operator ◽  
Initial Values ◽  
Scalar Type ◽  
Abstract Evolution Equation

It is shown that the weak solutions of the evolution equationy′(t)=Ay(t),t∈[0,T) (0<T≤∞), whereAis a spectral operator of scalar type in a complex Banach spaceX, defined by Ball (1977), are given by the formulay(t)=e tAf,t∈[0,T), with the exponentials understood in the sense of the operational calculus for such operators and the set of the initial values,f's, being∩ 0≤t<TD(e tA), that is, the largest possible such a set inX.

scholarly journals Positive coincidence points for a class of nonlinear operators and their applications to matrix equations

2020 ◽  
Vol 18 (1) ◽  
pp. 858-872
Author(s):  
Imed Kedim ◽  
Maher Berzig ◽  
Ahdi Noomen Ajmi
Keyword(s):  
Banach Space ◽  
Positive Solution ◽  
Positive Definite ◽  
Positive Cone ◽  
Matrix Equations ◽  
Ordered Banach Space ◽  
Coincidence Points ◽  
Nonlinear Operators ◽  
Nonlinear Matrix

Abstract Consider an ordered Banach space and f,g two self-operators defined on the interior of its positive cone. In this article, we prove that the equation f(X)=g(X) has a positive solution, whenever f is strictly \alpha -concave g-monotone or strictly (-\alpha ) -convex g-antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear matrix equations.

Compactness in Lebesgue–Bochner spaces of random variables and the existence of mean-square random attractors

2019 ◽  
Vol 19 (04) ◽  
pp. 1950032
Author(s):  
Yejuan Wang ◽  
Xiangming Zhu ◽  
Peter Kloeden
Keyword(s):  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Separable Banach Space ◽  
Additional Condition ◽  
Probability Space ◽  
Parabolic Partial Differential Equations ◽  
Mean Square ◽  
Probabilistic Argument ◽  
Random Attractors

Let [Formula: see text] be a probability space and let [Formula: see text] be a separable Banach space. It is shown a subset [Formula: see text] of [Formula: see text], where [Formula: see text], is relatively compact in [Formula: see text] if and only if it is uniformly [Formula: see text]-integrable and uniformly tight. The additional condition of scalarly relatively compact required in the literature is shown to hold by a probabilistic argument. The result is then used to establish the existence of a mean-square random attractor for dissipative stochastic differential equations and stochastic parabolic partial differential equations.

Sign in / Sign up

Export Citation Format

Share Document