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

On an Abstract Evolution Equation with a Spectral Operator of Scalar Type

It is shown that the weak solutions of the evolution equation y′(t)=Ay(t), t ∈ [0,T ) (0< T ≤∞), where A is a spectral operator of scalar type in a complex Banach space X, defined by Ball (1977), are given by the formula y(t) = etAf , 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<T D(...

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

Comment on "On the Carleman Classes of Vectors of a Scalar Type Spectral Operator"

Homogenization of Diffusion Equation with Scalar Hysteresis Operator

The paper deals with a scalar diffusion equation c ut = (F [ux])x+f, where F is a Prandtl-Ishlinskii operator and c, f are given functions. In the diffusion or heat conduction equation the linear constitutive relation is replaced by a scalar Prandtl-Ishlinskii hysteresis spatially dependent operator. We prove existence, uniqueness and regularity of solution to the corresponding initial-boundary...

On an Operator Equation Involving Mappings of Monotone Type

Let X be a real reflexive Banach space and A: X — 2A a maximal monotone mapping such that the graph G(A) of A is weaklyclosed in X X X and 0 £/4(0). Also, let T: X — 2A be a quasi-bounded coercive mapping of type (M) such that the effective domain D(T) of T contains a dense linear subspace X of X. Then it is shown that for each * f I co e X there exists a u e X such that co e Au + Tu and the su...