On Lyapunov Functions for Infinite Dimensional Volterra Quadratic Stochastic Operators

In the present paper, we study the existence of Lyapunov functions for Volterra qudratic stochastic operators(QSO) on infinite dimensional simplex. We provide a construction of Lyapunov functions for such kind of operators, which allows us to estimate their limiting points.


Introduction
An original work on quadratic stochastic operators (in short QSOs) was done by Bernstein [1] where such kind of operators appeared from the problems of population genetics (see also [2]). These operator appear to have tremendous applications especially in modelings in many different fields such as biology [3] (population and disease dynamics), physics [4,5](nonequilibrium statistical mechanics) , economics and mathematics [2,5] (replicator dynamics and games). A quadratic stochastic operator is usually used to present the time evolution of species in biology, which arises as follows. Consider evolution of species in biology as given in the following situation. Let I = {1, 2, . . . , n} be the n type of species (or traits) in a population and we denote x (0) = (x (0) 1 , . . . , x (0) n ) to be the probability distribution of the species in an early state of that population. By P ij,k we mean the probability of an individual in the i th species and j th species to cross-fertilize and produce an individual from k th species (trait). Given x (0) = (x n ) by using a total probability, i.e., This relation defines an operator which is denoted by V and it is called quadratic stochastic operator (QSO). Each QSO can be reinterprate as evolutionary operator that describes the sequence of generations in terms of probabilistic distribution if the values of P ij,k and the distribution of the current generation are given.
Currently theory of QSO were done on finite set of all probabilistic distributions. However, there are models where the probability distribution is given on countable set, which means that the corresponding QSO is defined on infinite-dimensional space. The simplest case, the infinite-dimensional space should be the Banach space 1 of absolutely summable sequences. It is worth mentioning that Volterra QSOs, orthogonal preserving QSO on infinite-dimensional space were studied in [6], [7]. For more information on recent achievements and open problems in the theory of QSO, one may refer to [8,9]. One of the main problems in mathematical biology is to study the asymptotic behavior of the trajectories. In [10][11][12][13] this problem was solved for the finite dimensional Volterra QSO's by using the theories of the Lyapunov function and tournaments. The functions constructed were based on finite dimensional case. Therefore, it is natural to continue this study by constructing Lyapunov function for infinite dimensional Volterra QSO which allows us to estimate their limit points for future study. Note that, up to now there is no paper yet is devoted to construct Lyapunov function for infinite dimensional Volterra QSOs.

Preliminaries
In this paper we are going to consider the set of absolutely summable sequences i.e., Next, let V be a mapping defined by Here, P ij,k are hereditary coefficients and satisfy It is important to see that the mapping V is well-defined i.e., V (x) k < ∞ for any k ∈ N.
Moreover one can check that V maps S into itself. Such kind of mapping V is called Quadratic Stochastic Operator (QSO).
Recall that a QSO V : S → S is called Volterra if one has where a ki = 1 − 2P ik,k . From (2.6), one can deduce that all faces of the simplex S are invariant with respect to V . In particular, all the vertices of S are the fixed points. Several properties of infinite dimensional Volterra operators have been investigated in [6].
Next we recall Young's inequality: Here and henceforth, we use V (n) (x) to denote the iterations of the given operator V at x ∈ S i.e., By ω V (x) we denote the set of limit points with respect to 1 -norm of the trajectory. Namely, for x * ∈ ω V (x) means, there exist a subsequence {n k } such that Obviously, if ω V (x) consists of a single point, then the trajectory converges, to a fixed point of (2.5). However, looking ahead, we remark that convergence of the trajectories is not the typical case for the dynamical systems (2.5). Therefore, it is of particular interest to obtain an upper bound for x 0 ∈ S, i.e., to determine a sufficiently "small" set containing x 0 ∈ S.
Obviously, if lim Consequently, to estimate of ω V (x 0 ) we should construct the set of Lyapunov functions that is as large as possible.

Finite Dimensional Volterra QSO
Before, considering infinite dimensional Volterra QSO, let us observe some facts on finite dimensional Volterra QSO which turns to be very useful in the investigation one of the classes of infinite dimensional Volterra QSO. Denote for any r ∈ (0, 1]. Note that, a huge number of papers on the theory of QSO were done on S n−1 1 (or simply we write S n−1 ). Therefore, it is self-interest to examine Volterra QSO defined on S n−1 r for any r ∈ (0, 1). We need an auxiliary result to further in this topic. Denote a ki x i ≥ 0 for every k ∈ {1, . . . , n} .
a ki x i ≥ 0 for every k ∈ N .
We stress that the set P m−1 is non-empty (see [10]), in contrast to the set P could be empty [6]. Now, we recall some of known facts on Volterra QSO over a finite dimensional simplex.
Using the constructed Lyapunov function, it was proved the following result  (for any r ∈ (0, 1]) be a Volterra QSO associated with a skew-symmetric matrix A = (a ki ) n k,i=1 , then its dynamics is the same as V : S n−1 → S n−1 where the associated skew-symmetric matrix for V is given by A = rA i.e., Proof. Let us prove the statement by showing V and V are conjugate to each other i.e., there exist a homeomorphism function T : S n−1 → S n−1 r such that Indeed, V is well-defined on S n−1 r i.e., it maps from S n−1 r to itself. Next, we define a linear operator by T r (y) := r (y) = (ry 1 , ry 2 , . . . , ry n ) for any y ∈ S n−1 . One can check that T r (S n−1 ) = S n−1 r and T −1 (S n−1 r ) = S n−1 , where T −1 (x) is the pre-image of x under mapping T . Moreover, for any y ∈ S n−1 we have The last statement means V : S n−1 r → S n−1 r is conjugate to V : S n−1 → S n−1 . Hence it proves the statement.
In [6], it has been proved that every infinite dimensional Volterra QSO can be constructed by means of compatible sequence of finite dimensional Volterra QSO. Therefore, it is interesting to know what is the relationship between the sequence of compatible Volterra QSO and the corresponding Lyapunov functions. Let us consider a sequence V n] : S n−1 → S n−1 finite dimensional Volterra operators, i.e.
Remark 3.4. Let V n] (x) be a sequence of compatible Volterra QSO and let p = (p 1 , . . . , p m ) ∈ P m−1 for m < ∞. If there exists l 0 < m such that
We emphasize that |a ij | ≤ 1 for any i, j ∈ {1, . . . , n}. Then, for any x ∈ S, the operator V can be written as follows Thus, the investigation of dynamics on V (x) reduces to V (x) k for 1 ≤ k ≤ n which defines a finite Volterra QSO V on S n−1 r for any r ∈ (0, 1]. Here V is associated to skew-syammetric matrix A = (a ij ) n i,j=1 . Since all faces of the simplex S n−1 r are invariant with respect to V , therefore it is enough to study V on riS n−1 r . Thanks to Proposition 3.3 and Theorem 3.2 we conclude that for any x 0 = x 0 1 , x 0 2 , . . . , x 0 n ∈ riS n−1 r and x 0 / ∈ F ix( V ), Define y 0 = x 0 1 , x 0 2 , . . . , x 0 n , y 0 n+1 , . . . ∈ riS such that ∞ k=n+1 y 0 k = 1 − r. Due to (4.1) and (4.2) one has ω V (y 0 ) ∈ ∂S Note that, the limiting points with respect to 1 and point-wise are the same because of (4.1). Now, we want to establish Lyapunov function for infinite dimensional Volterra QSO which has the same form as given by ( then the functional Proof. Using the same argument as before, we can choose x ∈ riS without the loss of generality. Next, one sees that Since p satisfies (4.3), then Due to the facts x ∈ riS and a kk = 0, we have a ki x i > 0 for any k ∈ N.
a ki x i converges absolutely.
From simple algebra and the fact a ki = −a ik , we have Since p ∈ P, then we infer that ∞ k=1 a ik p k ≥ 0 for any i ∈ N. Hence, we show that Hence, from (4.4) one concludes that for any x ∈ riS. Therefore, lim n→∞ ϕ p V (n) (x) exist. This completes the proof.
From the last theorem, it seems hard to choose p ∈ P that satisfied condition (4.3) due to infiniteness. This problem is solved by the next theorem. Let x ∈ riS, then for any p ∈ P m−1 , the following functional Proof. For any x ∈ riS, ϕ p (V (x)) can be written as follows (see (4.4)) Keeping in mind (4.5), hence using Young's Inequality and fact a ki = −a ik , we have x i m k=1 a ik p k (4.9) Due to p ∈ P m−1 one infers m i=1 x i m k=1 a ik p k ≥ 0 (4.10) and assumption (4.8) yields ∞ i=m+1 x i m k=1 a ik p k ≥ 0 (4.11) From (4.9), (4.10) and (4.11) we conclude that ϕ p (V (x)) ≤ ϕ p (x) Therefore, the limit lim n→∞ ϕ p V (n) (x) exists. This completes the proof.  where 0 < a, b, c ≤ 1 and a ij ≥ 0 for all i ≥ 4, j = 1, 3. One sees that the set is a Lyapunov function for V .

Acknowledgments
The present work is supported by the UAE "Start-Up" Grant, No. 31S259.