The paper deals with semigroups of operators associated with delay diierential equation: _ x(t) = Ax(t) + L 1 x(t ? h) + L 2 xt; where A is the innnitesimal generator of an analytic semigroup on a Hilbert space X and L 1 , L 2 are densely deened closed operators in X and L 2 (?h; 0;X) respectively. The adjoint semigroupof the solution semigroupof the delay diierentialequation is characterized. ...

Semigroups whose congruences form a chain are often termed ∆-semigroups. The commutative ∆-semigroups were determined by Schein and by Tamura. A natural generalization of commutativity is permutativity: a semigroup is permutative if it satisfies a non-identity permutational identity. We completely determine the permutative ∆-semigroups. It turns out that there are only six noncommutative exampl...

Recently, Y.-Y. Huang and C.-C. Hong [15, 16], T.-J. Huang and Y.-Y. Huang [14], and Y.-Y. Huang et al. [17] obtained a few fixed point theorems for left reversible and nearcommutative semigroups of contractive self-mappings in compact and complete metric spaces, respectively. These results subsume some theorems in Boyd and Wong [1], Edelstein [3], and Liu [20]. In this paper, motivated by the ...

One of the most natural approaches to the study of regular semigroups is to impose restrictions on the partial ordering of their idempotents (erg/^e/—jfe==e). The principal object of this note is to describe the structure of the classes of regular semigroups whose idempotents form a tree or a unitary tree, respectively (see Definition 1). We determine, among other things, a complete set of inva...

in this paper we study the existence of commuting regular elements, verifying the notion left (right) commuting regular elements and its properties in the groupoid g(n) . also we show that g(n) contains commuting regular subsemigroup and give a necessary and sucient condition for the groupoid g(n) to be commuting regular.

The paper is concerned with the Cauchy problem for a nonlinear, strictly hyperbolic system with small viscosity: u t + A(u)u x = " u xx ; u(0; x) = u(x): () We assume that the integral curves of the eigenvectors r i of the matrix A are straight lines. On the other hand, do not require the system () to be in conservation form, nor we make any assumption on genuine linearity or linear degeneracy ...

We study the existence of almost periodic mild solutions of a class of partial functional differential equations via semilinear almost periodic abstract functional differential equations of the form (∗) x′ = f(t, x, xt) . To this end, we first associate with every almost periodic semilinear equation (∗∗) x′ = F (t, x) a nonlinear semigroup in the space of almost periodic functions. We then give...

Following Bertoin who considered the ergodicity and exponential decay of Lévy processes in a finite domain [4], we consider general Lévy processes and their ergodicity and exponential decay in a finite interval. More precisely, given Ta = inf{t > 0 : Xt / ∈ (0, a)}, a > 0 and X a Lévy process then we study from spectral-theoretical point of view the killed process P (Xt ∈ ., Ta > t). Under gene...

Throughout this paper, we denote by N, Z, Q and R the sets of all positive integers, all integers, all rational numbers and all real numbers, respectively. We put R+ = [0,∞)n and ej = (0, 0, · · · , 0, 0, (j) 1 , 0, 0, · · · , 0) ∈ R for j ∈ N with 1 ≤ j ≤ n. Let C be a subset of a Banach space E, and let T be a nonexpansive mapping on C, i.e., ‖Tx−Ty‖ ≤ ‖x− y‖ for all x, y ∈ C. We know that T ...

Let T{t) be a Co-semigroup of linear operators on a Banach space X, and let X® , resp. X® , denote the closed subspaces of X" consisting of all functionals x* such that the map t •-» T*{t)x* is strongly continuous for t > 0, resp. t > 0. Theorem. Every nonzero orbit of the quotient semigroup on X*/X® is nonseparably valued. In particular, orbits in X*/X® are either zero for t > 0 or nonseparabl...