Markovian operators, non-negative linear operators and its subgroups play a significant role for the description of phenomena observed in the nature. Research on asymptotic stability is one of the main issues in this respect. A. Lasota and J. A. Yorke proved in 1982 that the necessary and sufficient condition of the asymptotic stability of a Markovian operator is the existence of a non-trivial lower-bound function. In the present paper it is shown how the method of lower-bound function can be applied to the investigation of genetic algorithms. Genetic algorithms considered used for solving of non-smooth optimization problems are compositions of two random operators: selection and mutation. The compositions are Markovian matrices.

The simple genetic algorithm (SGA) and its convergence analysis are main subjects of the article. The SGA is defined on a finite multi-set of potential problem solutions (individuals) together with mutation and selection operators, and appearing with some prescribed probabilities. The selection operation acts on the basis of the fitness function defined on individuals, and is fundamental for the problem considered. Generation of new population is realized by iterative actions of those operators written in the form of a transition operator acting on probability vectors. The transition operator is a Markov one. Conditions for convergence and asymptotic stability of the transition operator are formulated.

The simple genetic algorithm (SGA) and its convergence analysis are main subjects of the article. A particular SGA is defined on a finite multi-set of individuals (chromosomes) together with mutation and proportional selection operators, each of which with some prescribed probability. The selection operation acts on the basis of the fitness function defined on individuals. Generation of a new population from a given one is made by iterative actions of those operators. Each iteration is written in the form of a transition operator acting on probability vectors which describe probability distributions of all populations. The transition operator is power of Markovian matrix. Based on the theory of Markov operators [1-3] new conditions for asymptotic stability of the transition operator are formulated.

The simple genetic algorithm (SGA) and its convergence analysis are main subjects of the article. The SGA is defined on a finite multi-set of potential problem solutions (individuals) together with random mutation and selection operators. The selection operation acts on the basis of the fitness function defined on potential solutions (individuals), and is fundamental for the problem considered. Generation of a new population from the given one, is realized by the iterative actions of those operators. Each iteration is written in the form of a transition operator acting on probability vectors which describe probability distributions of each population. The transition operator is a Markov one. Thanks to the well-developed theory of Markov operators [5,8,9] new conditions for stability of the transition operator are formulated. The obtained results are related to the class of genetic operators and are not restricted to binary operators.

Asymptotic convergence theorems for semigroups of nonnegative operators on a Banach lattice, on $C(X)$ and on $L^p(X)$ $(1\le p \le \infty )$ are proved. The general results are applied to a class of semigroups generated by some differential equations.