In the present paper we introduce a new notion of order called b-order. Then we define a bistochasticity quadratic stochastic operator (q.s.o.) with respect to the b-order, and call it a b-bistochastic q.s.o. We include several properties of the b-bistochastic q.s.o. and descriptions of all b-bistochastic q.s.o. defined on a two dimensional simplex. MSC: 46L35; 46L55; 46A37

It was known that orthogonality preserving property and surjectivity of nonlinear Markov operators, acting on finite dimensional simpleces, are equivalent. It turns out that these notions are no longer equivalent when such kind of operators are considered over on infinite dimensional spaces. In the present paper, we find necessary and sufficient condition to be equivalent of these notions, for the second order nonlinear Markov operators. To do this, we fully describe all surjective second order nonlinear Markov operators acting on infinite dimensional simplex. As an application of this result, we provided some sufficient conditions for the existence of positive solutions of nonlinear integral equations whose domain are not compact.

Abstract. In the present paper, we consider nonlinear Markov operators, namely polynomial stochastic operators. We introduce a notion of orthogonal preserving polynomial stochastic operators. The purpose of this study is to show that surjectivity of nonlinear Markov operators is equivalent to their orthogonal preserving property. Mathematics Subject Classification: 47H25, 37A30, 47H60

Abstract. In the present paper, we consider a class of quadratic stochastic operators (q.s.o.) called b−bistochastic q.s.o. We include several properties of b−bistochastic q.s.o. and their dynamical behavior. One of the main findings in this paper is the description on the uniqueness of the fixed points. Besides, we list the conditions on strict contractive b−bistochastic q.s.o. on low dimensional simplices and it turns out that, the uniqueness of the fixed point does not imply strict contraction. Finally, we associated Markov measures with b-bistochastic q.s.o. On a class of b-bistochastic q.s.o. on finite dimensional simplex, the defined measures were proven to satisfy the mixing property. Moreover, we show that Markov measures associated with a class of b−bistochastic q.s.o on one dimensional simplex meets the absolute continuity property.Mathematics Subject Classification: 46L35, 46L55, 46A37.

In the present paper, we are aiming to study limiting behaviour of infinite dimensional Volterra operators. We introduce two classes Ṽ+ and Ṽ− of infinite dimensional Volterra operators. For operators taken from the introduced classes we study their omega limiting sets ω V and ω (w) V with respect to 1 -norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to Ṽ+ , then the sets ω V (x) and ω (w) V (x) coincide for every x ∈ S, and moreover, they are non empty. If Volterra operator belongs to Ṽ− , then ω V (x) could be empty, and it implies the non-ergodicity (w.r.t. 1 -norm) of V, while it is weak ergodic.