Issue 1, Volume 28, April 2003

2003 · Open access

1. MILD SOLUTIONS OF NONLINEAR EVOLUTION FUNCTIONAL DIFFERENTIAL INCLUSIONS IN BANACH SPACE A. SGHIR PDF

   ▸ Abstract

   The existence of a mild solution of the abstract functional differential inclusion
   $$P(\varphi)\begin{cases}u^\prime(t)+Au(t)-F(t,u_t)\ni 0, & t\in I=[0,T]\\ u(t)=\varphi(t),& t\in J=[-r,0]\end{cases}$$
   is obtained by a Filippov technique. Here $A$ is an operator such that $A +\omega\mathcal I$ is m-accretive for some $\omega\in\mathbb R^+$, and the multimapping $F(t, .)$ is $h(t)$-Lipschitzian. The result is applied to a nonlinear functional control problem.

   1-9
2. HYPER-GROUPS OF ORTHOGONAL POLYNOMIALS NGUYEN VAN THU PDF

   ▸ Abstract

   In the present paper we give a new condition for existence of dual weak hypergroups of hypergroups generated by orthogonal polynomials. In the case of Jacobi polynomials we prove a representation theorem for Lévy type processes in terms of their infinitesimal operators.

   11-15
3. DIRECTIONAL KUHN-TUCKER CONDITION AND DUALITY FOR QUASIDIFFERENTIABLE PROGRAMS NGUYEN DINH, LE ANH TUAN PDF

   ▸ Abstract

   In this paper a notion called “directional Kuhn-Tucker condition” for quasidifferentiable programs with inequality constraints is introduced. This is a version of the Lagrange multiplier rule where the Lagrange multipliers depend on the directions. It is proved that this condition is a necessary condition for optimality. Under the assumption that the problem is directionally $\eta$-invex, it is also a sufficent condition for optimality. Some results on duality of the class of problems are obtained.

   17-38
4. ON THE WEAK TAUTNESS AND THE LOCALLY WEAK TAUTNESS OF A DOMAIN IN A BANACH SPACE LE MAU HAI, PHAM KHAC BAN PDF

   ▸ Abstract

   It is shown that every Banach analytic manifold $X$ which is weakly taut is hyperbolic in the sense of Kobayashi. A relation between the locally weak tautness and the weak  tautness of a domain in a Banach space is also established.

   39-50
5. ON THE ASYMPTOTIC STABILITY OF TIME-VARYING DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS AND APPLICATIONS NGUYEN SINH BAY, NGUYEN THE HOAN, VU NGOC PHAT PDF

   ▸ Abstract

   This paper studies the asymptotic stability of some class of retarded functional differential equations. Based on the stability of the underlying linear system, new sufficient conditions for asymptotic stability of linear retarded systems as well as systems with nonlinear perturbations are derived. The class of systems is allowed to be time-varying and time-delay. The results are applied to the input feedback stabilization problem of some class of linear control systems.

   51-64
6. ANOTHER CLASSIFICATION OF QUASI-MARTINGALES IN THE LIMIT TRAN QUANG VINH PDF

   ▸ Abstract

   Given a stochastic basic $(\mathcal A_n)$, a sequence $(X_n)$ of integrable random variables, adapted to $(\mathcal A_n)$ is said to be a quasi-martingale in the limit if for every $\varepsilon > 0$, there exists $p\in N$ such that for every $m\geq p$ there exists $p_m\geq m$ such that for all $n\geq p_m$ we have
   $$P\left(\sup\limits_{p\leq q\leq m}|X_q(n)-X_q| > \varepsilon\right) < \varepsilon.$$
   The main aim of this note is to prove that the class of all quasi-martingales in the limit would be classified into a nondecreasing directed family of subclasses whose smallest element is just the class of mils introduced by M. Talagrand (1985).

   65-71
7. LYAPUNOV’S INEQUALITY FOR LINEAR DIFFERENTIAL ALGEBRAIC EQUATION NGUYEN DINH CONG, HOANG NAM PDF

   ▸ Abstract

   We introduce a concept of Lyapunov exponents and Lyapunov spectrum of a linear differential algebraic equation (DAE) and derive Lyapunov’s inequality for a DAE of index 1. We derive some estimates for sum of Lyapunov exponents from a fundamental solution matrix of a DAE by means of its coefficients.

   73-88
8. FIXED POINTS OF SEMIGROUPS OF LIPSCHITZIAN MAPPINGS LE ANH DUNG, DO HONG TAN PDF

   ▸ Abstract

   In this paper we establish three results on fixed points of semigroups of Lipschitzian mappings. The first one generalizes Lifschitz’s fixed point theorem for uniformly Lipschitzian mappings in metric spaces. The second one generalizes Kirk’s fixed point theorem for mappings of asymptotically nonexpansive type. The last one generalizes Lim-Xu’s fixed point theorem for uniformly Lipschitzian mappings satisfying the Casini-Maluta condition.

   89-100
9. EXISTENCE OF ABSTRACT SOLUTIONS OF INTEGRO-DIFFERENTIAL OPERATOR EQUATIONS A. E. HAMZA, G. MURAZ PDF

   ▸ Abstract

   We consider the integro-differential operator equations having the form
   $$\sum\limits_{i=1}^n\alpha_iu^{(i)}(t)-\alpha Au(t)+\beta \mu \ast u(t)=f(t), \quad t\in\mathbb R,$$
   where the free term $f$ belongs to a closed subspace $M$ of $L^{\infty}(\mathbb R,X)$, $A$ is the generator of a $C_0$-semigroup of operators defined on a Banach space $X$, $\mu$ is a bounded Borel measure on $\mathbb R$ and $\alpha,\beta,\alpha_i\in\mathbb C$, $i=1,2,\dots,n$. Certain conditions will be imposed to guarantee the existence of solutions in the class $M$.

   101-109
10. TWO COINCIDENCE THEOREMS OF VIETORIS MAPS LIANG-JU CHU, CHING-YAN LIN PDF

    ▸ Abstract

    In 1959, Nikaidô established a remarkable coincidence theorem in a compact Hausdorff topological space, which generalizes and gives a unified treatment to the results of Gale regarding the existence of economic equilibrium and theorems in game problems. The main purpose of the present paper is to deduce several generalized key results based on this very powerful result together with some KKM property. Indeed, we shall simplify and reformulate a few coincidence theorems on acyclic multifunctions as well as some Gòrniewicz-type fixed point theorems. Beyond the realm of monotonicity nor metrizability, the results derived here generalize and unify various earlier ones from classic optimization theory. In the sequel, we shall deduce two versions of Nikaidô’s coincidence theorem about Vietoris maps from different approaches.

    111-122