On infinitedimensional dissipative quadratic stochastic operators
 Farruh Shahidi^{1}Email author
https://doi.org/10.1186/168718472013272
© Shahidi; licensee Springer. 2013
Received: 30 April 2013
Accepted: 13 August 2013
Published: 10 September 2013
Abstract
The purpose of the paper is to extend the notion of dissipativity of maps on infinitedimensional simplex. We study the fixed points of dissipative quadratic stochastic operators on infinitedimensional simplex. Besides, we study the limit behavior of the trajectories of such operators. We also show the difference of dissipative operators defined on finite and infinitedimensional spaces. We obtain the results by using majorization for infinite vectors and ${\ell}_{1}$ convergence.
MSC:15A51, 47H60, 46T05, 92B99.
Keywords
1 Introduction
A quadratic stochastic operators (q.s.o. in short), firstly initiated by Bernstein [1], is a nonlinear difference equation, which has arisen from some problems of population genetics. Further development of this theory belongs to Lyubich [2, 3], Kesten [4, 5], Vallander [6], and Zakharevich [7], where the authors investigate the limit behavior of the trajectories (or dynamics) of q.s.o. It should be noted that the limit behavior of the trajectories of q.s.o. on 1D simplex was fully studied by Lyubich [2, 3], where it was shown that the ωlimit set (see definition below) of any initial point is a finite set. Vallander [6] studied the dynamics of some special q.s.o. on 2D simplex. Vallander’s result was later extended to any finitedimensional space by Ganikhodzhaev in [8, 9]. Later on, this special q.s.o. was called as Volterra q.s.o., which is, in fact the LotkaVolterra predator prey equation in discrete settings. The dynamics of Volterra q.s.o. was somehow studied successfully in [8]. However, not all q.s.o. are of Volterratype, and the dynamics of nonVolterra q.s.o. remains open. Notable results for nonVolterra q.s.o. were obtained by Rozikov and his students [10–13], who introduced different classes of q.s.o., such as ‘strictly nonVolterra,’ ‘Fq.s.o.’ (F stands for a ‘female’ due to its genetic interpretation), ‘separable q.s.o.,’ ‘ℓVolterra’ and studied the limit behavior of the trajectories. A manuscript [14] provides some results and open problem on q.s.o.
A majorization of vectors [15] turned out to be a useful tool for classifying q.s.o. into some of its subclasses. With the help of it, the definition of doubly stochastic and dissipative q.s.o. were introduced in papers [16] and [17], respectively. Further properties of such operators were studied in [18–20]. Of course, Volterra q.s.o. and classes considered in papers [10–13] are different from doubly stochastic and dissipative q.s.o. It is to note that the limit behavior of the trajectories of the dissipative q.s.o. on finitedimensional simplex (the set of vectors with nonnegative components summing up to 1) was fully classified in [20]. Note that a q.s.o. is just a discrete probability distribution of a finite population. However, there are models where the probability distribution is countably infinite, which means that a q.s.o. is defined on infinitedimensional space. In the simplest case, the infinitedimensional space should be the Banach space ${\ell}_{1}$ of absolutely summable sequences. It is worth mentioning that Volterra q.s.o. and doubly stochastic q.s.o. on infinitedimensional space was introduced and studied in papers [21] and [22], respectively.
Therefore, the purpose of the present paper is to introduce a dissipative q.s.o. on infinitedimensional subspace of ${\ell}_{1}$, by using majorization for infinite vectors [23]. We show the difference between finite and infinitedimensional cases. While the existence of fixed point and convergence of Cesaro averages (that is an ergodic theorem) holds for finitedimensional dissipative operators, we show that it fails for infinitedimensional operators. We also provide some regular dissipative q.s.o. in infinitedimensional case.
The paper is organized as follows. The next chapter provides some preliminaries and results from finitedimensional cases. In Section 3, we introduce a dissipative q.s.o. in infinitedimensional simplex and study its properties. Finally, we study the limit behavior of the trajectories of dissipative q.s.o. in Section 4. We use notations and terminology as in [17, 20].
2 Preliminaries
are its vertices. For $\alpha \subset I=\{1,2,\dots ,m\}$, the set ${F}_{\alpha}=\{x\in {S}^{m1}:{x}_{i}=0,i\notin \alpha \}$ is called a face of the simplex.
More information on dissipative operators on ${S}^{m1}$ can be found in [17]. Now, let us recall some terminology. Let ${x}^{0}\in {S}^{m1}$ and $V:{S}^{m1}\to {S}^{m1}$ be an operator. Then the set $\{{x}^{0},V{x}^{0},{V}^{2}{x}^{0},\dots \}$ is called the trajectory of V starting at the point ${x}^{0}$. The point ${x}^{0}$ satisfying $V{x}^{0}={x}^{0}$ is called fixed. The set of all fixed points of the q.s.o. V is denoted by $Fix(V)$. A q.s.o. $V:{S}^{m1}\to {S}^{m1}$ is called regular if the trajectory of any $x\in {S}^{m1}$ converges to a unique fixed point. We may note that regular q.s.o. a priori must have a unique fixed point. Let V be a q.s.o. Then the set $\omega ({x}^{0})={\bigcap}_{k\ge 0}\overline{{\bigcup}_{n\ge k}\{{V}^{n}{x}^{0}\}}$ is called an ωlimit set of trajectory of initial point ${x}^{0}\in {S}^{m1}$. From the compactness of the simplex, one can deduce that $\omega ({x}^{0})\ne \mathrm{\varnothing}$ for all ${x}^{0}\in {S}^{m1}$. V is called ergodic if the following limit exists ${lim}_{n\to \mathrm{\infty}}\frac{x+Vx+\cdots +{V}^{n1}x}{n}$ for any $x\in {S}^{m1}$.
The following facts are known for dissipative q.s.o. on ${S}^{m1}$.

Any dissipative operator is ergodic.

Any dissipative q.s.o. has either unique or infinitely many fixed points.

One of the following statements always holds for a dissipative q.s.o.

The operator is regular. Its unique point is either a vertex of the simplex or the center of its face.

The operator has infinitely many fixed points. ωLimit set of any initial point is contained in the set of fixed points, i.e., $\omega (x)\subset Fix(V)$.
In the next section, we define a dissipative operator on infinitedimensional simplex and study the statements above in infinitedimensional setting.
3 Infinitedimensional dissipative operators
In this section, we define dissipative quadratic stochastic operators on infinitedimensional simplex. We study some properties and examples of dissipative q.s.o.
One can easily see that the sum (2) is convergent. It is also important to note that the operator (2) is well defined, that is, it maps simplex into itself.
This definition of majorization is given in [23]. General definition of majorization differs from the one that is given above in [23]; however, on ${\ell}_{1}$, we can give as above.
Lemma 3.1 Let V be a linear dissipative operator, that is $Vx=Ax$, where $A={({a}_{ij})}_{i,j\in N}$ is an infinite matrix. Then A is $(0,1)$ column stochastic matrix.
Note that here and henceforth N denotes natural numbers.
Proof of Lemma 3.1 Since $Vx\succ x$, then by putting $x={e}_{i}$ we have $A{e}_{i}\succ {e}_{i}$. At the same time, it is easy to see that $A{e}_{i}\prec {e}_{i}$. That is why ${(A{e}_{i})}_{\downarrow}={({e}_{i})}_{\downarrow}$, which means that only one component of the vector $A{e}_{i}$ is 1, and the others are 0. Therefore, the matrix A is $(0,1)$ column stochastic matrix. □
From this lemma, it follows that the class of linear dissipative operators on S is not large. Therefore, we are interested to study nonlinear (that is q.s.o.) dissipative operators. Let us provide some examples of dissipative quadratic operators.
is an evidently infinitedimensional dissipative q.s.o. Because V is dissipative, then the conditions (4) can easily be verified.
is a dissipative q.s.o. Dissipativity can be verified by using the fact that V is dissipative.
One can see that this operator maps infinitedimensional simplex into itself, and since ${(Vx)}_{\downarrow}={x}_{\downarrow}$, then V is dissipative. One can see that V does not have nonzero fixed points. As we consider V acting on S, then V has no fixed points. Therefore, Theorem 2.1 fails in infinitedimensional setting.
does not have a limit. Thus, for dissipative q.s.o., an ergodic theorem fails dramatically. In addition to that, one can see that the trajectory of certain point under V may not converge in general. Indeed, take $x={e}_{1}=(1,0,0,\dots )$, then ${V}^{n}x={e}_{n}$ (${e}_{n}$ is n th vertex of the simplex S), and hence ${\parallel {V}^{n}x{V}^{n+m}x\parallel}_{1}={\parallel {e}_{n}{e}_{n+m}\parallel}_{1}=2$. So, the trajectory is divergent. We see that when we consider operators in infinitedimensional space, all the statements in Theorem 2.1 fail dramatically. This is the difference between finitedimensional and infinitedimensional cases.
We now study some properties of dissipative q.s.o.
Given q.s.o. V, we denote ${a}_{ij}=({p}_{ij,1},{p}_{ij,2},\dots ,{p}_{ij,m},\dots )$ $\mathrm{\forall}i,j\in N$, where ${p}_{ij,k}$ are the coefficients of q.s.o. V. One can see that ${a}_{ij}\in S$, for all $i,j\in N$.
Proof Due to dissipativity of V one has $Vx\succ x$, $\mathrm{\forall}x\in S$. Now by putting $x={e}_{i}$ we get ${e}_{i}\prec V{e}_{i}$. On the other hand, we have ${e}_{i}\succ x$, $\mathrm{\forall}x\in S$. That is why ${(V{e}_{i})}_{\downarrow}={({e}_{i})}_{\downarrow}={e}_{1}$. Then the equality $V{e}_{i}={a}_{ii}$ implies the assertion. □
We call (5) a canonical form of dissipative q.s.o. V.
 (i)
If $j\in {\alpha}_{{k}_{0}}$, then ${p}_{ij,{k}_{0}}={({a}_{ij})}_{[1]}\ge \frac{1}{2}$, $\mathrm{\forall}i\in N$.
 (ii)
For any $k\ge 3$, one has ${({a}_{ij})}_{[k]}=0$, $\mathrm{\forall}i\in N$.
 (ii)Denote ${p}_{ij,{k}^{\ast}}={max}_{t\ne {k}_{0}}{p}_{ij,t}$. This maximum exists as the coefficients of q.s.o. are not greater than 1. One can see that ${(Vx)}_{{k}^{\ast}}={({a}_{ij})}_{[2]}$. Now, from${x}_{[1]}+{x}_{[2]}\le {(Vx)}_{[1]}+{(Vx)}_{[2]},$
From this inequality, we get ${p}_{ij,{k}_{o}}+{p}_{ij,{k}^{\ast}}\ge \frac{2\lambda {\lambda}^{2}}{2\lambda (1\lambda )}=\frac{2\lambda}{2(1\lambda )}=\frac{1}{2}(1+\frac{1}{1\lambda})\ge 1$. This yields ${p}_{ij,{k}_{o}}+{p}_{ij,{k}^{\ast}}=1$ and ${({a}_{ij})}_{[k]}=0$ $\mathrm{\forall}k\ge 3$, $\mathrm{\forall}i\in N$. □
4 The limit behavior of the trajectories
In this section, we study the limit behavior of the trajectories of dissipative q.s.o. We study some criteria for the existence of fixed point. We also provide some examples of regular dissipative q.s.o.
Theorem 4.1 A dissipative q.s.o. V defined on infinitedimensional simplex S has either 0 or 1 or infinitely many fixed points.
The proof is based on expressing V in canonical form, dividing the proof into several cases and using proof method of its finitedimensional counterpart.
Proof of Theorem 4.1 First, we rewrite dissipative q.s.o. in its canonical form and correspond the partition $\{{\alpha}_{k},k\in N\}$ of N to a dissipative q.s.o. V.
 (1)
There is no k such that $k\in {\alpha}_{k}$, that is, $k\notin {\alpha}_{k}$, $\mathrm{\forall}k\in N$.
 (2)
There exists numbers $\{{k}_{i},i\in N\}$ such that ${k}_{i}\in {\alpha}_{{k}_{i}}$, $\mathrm{\forall}i\in N$.
 (1)Since there is no k with $k\in {\alpha}_{k}$ and ${\alpha}_{k}$ is a partition of N, then a particular number ${k}_{1}$ must belong to one of the set other than ${\alpha}_{{k}_{1}}$, say ${\alpha}_{{k}_{2}}$. The number ${k}_{2}$ belongs to some ${\alpha}_{{k}_{3}}$ and so on. Therefore, there exists a sequence $K=\{{k}_{l},l\in N\}$ such that ${k}_{l}\in {\alpha}_{\pi ({k}_{l})}$, where π is a bijection on the set $K=\{{k}_{l},l\in N\}$. In this case, the operators V can be written as follows$\begin{array}{l}{(Vx)}_{{k}_{l}}={x}_{\pi ({k}_{l})}^{2}+{\sum}_{i\in {\alpha}_{{k}_{l}}\setminus \{\pi ({k}_{l})\}}{x}_{i}^{2}+{\sum}_{i<j}2{p}_{ij,{k}_{l}}{x}_{i}{x}_{j},\phantom{\rule{1em}{0ex}}{k}_{l}\in K,\\ {(Vx)}_{k}={\sum}_{i\in {\alpha}_{k}}{x}_{i}^{2}+2{\sum}_{i<j}{p}_{ij,k}{x}_{i}{x}_{j},\phantom{\rule{1em}{0ex}}k\in N\setminus K.\end{array}\}$(7)
The set K can be finite or infinite. We assume that the set K is the largest set, for which ${k}_{l}\in {\alpha}_{\pi ({k}_{l})}$, $\mathrm{\forall}{k}_{l}\in K$. In this case, one can show that ${\alpha}_{k}=\mathrm{\varnothing}$, $\mathrm{\forall}k\in N\setminus K$. Indeed, clearly numbers elements of K do not belong to any of the sets ${\alpha}_{{k}_{l}}$, ${k}_{l}\in N$. This implies that it is possible to find finite or infinite sequence ${K}^{\prime}=\{{k}_{l}^{\prime},l\in N\}$ such that ${k}_{l}^{\prime}\in {\alpha}_{{\pi}^{\prime}({k}_{l}^{\prime})}$, where ${\pi}^{\prime}$ is a bijection on ${K}^{\prime}$. But this implies that we found the set $(K\cup {K}^{\prime})$ larger than K with the property that ${k}_{l}\in {\alpha}_{\pi ({k}_{l})}$ $\mathrm{\forall}{k}_{l}\in K\cup {K}^{\prime}$, which is the contradiction. Therefore, one can consider ${\alpha}_{k}=\mathrm{\varnothing}$, $\mathrm{\forall}k\in N\setminus K$.
 (2)
Let F be the set of numbers ${k}_{l}$ such that ${k}_{l}\in {\alpha}_{{k}_{l}}$ for all ${k}_{l}\in F$.
Both sets F and K can be finite or infinite, and note that $F\ne \mathrm{\varnothing}$, otherwise, the case would coincide with the previous case. We also assume that the sets F and K are largest sets satisfying conditions given in their definitions. Therefore, similar to a previous case one can show that ${\alpha}_{k}=\mathrm{\varnothing}$, $k\in N\setminus F\cup K$.
which implies that ${x}_{k}=0$, $\mathrm{\forall}k\in N\setminus K$. In addition, it follows that all ${x}_{{k}_{l}}$, ${k}_{l}\in K$ are equal.
Using (12), one can find all the fixed points by putting $Vx=x$ and solving the system of equations. We consider the following few cases. First of all, note that $F>0$ and $K>1$ (here $F$ stands for the cardinality of F).
where β appears k times.
If $F=1$ (say, $F=\{1\}$) and K is infinite, then since all ${x}_{{k}_{l}}$, ${k}_{l}\in K$ are equal and ${\sum}_{{k}_{l}\in K}{x}_{{k}_{l}}\le 1$ implies that ${x}_{{k}_{l}}=0$ for all ${k}_{l}\in K$. Therefore, the operator has a unique fixed point $(1,0,0,\dots )$.
If F is finite (say, $F=\{1,2,\dots ,k\}$) and K is infinite, then the unique fixed point is $(\frac{1}{k},\frac{1}{k},\dots ,\frac{1}{k},0,0,\dots )$.
Finally, if both F and K are infinite, then the vertices ${e}_{{k}_{l}}$, ${k}_{l}\in F$ of the simplex are fixed. So, in this case, there are infinitely many fixed points since F is infinite. □
Corollary 4.2 Let dissipative quadratic operator V be given in canonical form (5). Then V has a fixed point if and only if there exists a finite sequence ${k}_{1},{k}_{2},\dots ,{k}_{l}$ ($l\ge 1$) such that ${k}_{l}\in {\alpha}_{\pi ({k}_{l})}$ for some permutation π of the sequence $\{{k}_{l},l\ge 1\}$.
Proof follows from the prof of Theorem 4.1.
Now, we study the limit behavior of the trajectories. We have seen in Section 3 that the trajectory of the point under dissipative operator may not converge in general. Here, we consider a particular case, assuming that a dissipative operator has a unique fixed point.
Theorem 4.3 If dissipative q.s.o. $V:S\to S$ has a unique fixed point, then the operator is regular at this point, i.e., the trajectory of any initial point tends to this unique point.
Proof Let us use the decomposition of V given by (11), and let F and K be the sets defined in (11). We have seen in the previous Theorem 4.1 that an operator has a unique fixed point if and only if one of the conditions is satisfied: (1) $F=1$, $K=\mathrm{\varnothing}$, (2) $F=\mathrm{\varnothing}$, $K<\mathrm{\infty}$.
Note that we set ${\alpha}_{k}=\mathrm{\varnothing}$, $k\ge 2$.
Let us set ${x}^{(n)}={V}^{(n)}x$ and define $\phi (x)={x}_{1}$. Because Lemma 3.3 implies that ${\sum}_{i,j=1}^{\mathrm{\infty}}2{p}_{ij,1}{x}_{i}{x}_{j}{\sum}_{i=2}^{\mathrm{\infty}}{x}_{1}{x}_{i}\ge 0$, one can see that $\phi ({x}^{(1)})\ge \phi (x)$, which means that $\phi ({x}^{(n)})$, $n\in N$ is monotone and bounded sequence, and hence convergent. We put ${lim}_{n\to \mathrm{\infty}}\phi ({x}^{(n)})=C$.
as $n\to \mathrm{\infty}$, we find that the trajectory of any initial point converges to ${e}_{1}$.
Here, ${L}_{i}$ satisfy (9). Since ${L}_{i}\ge 0$, then from the above, it follows that $\phi (Vx)\ge \phi (x)$. Hence, the sequence $\{\phi ({x}^{(n)}):n=1,2,\dots \}$ is nondecreasing and bounded. That is why the limit ${lim}_{n\to \mathrm{\infty}}\phi ({x}^{(n)})$ exists. Let us put $C={lim}_{k\to \mathrm{\infty}}\phi ({x}^{(n)})$.
which implies that ${lim}_{n\to \mathrm{\infty}}{({x}^{(n)})}_{i}=\frac{1}{l}$, $\mathrm{\forall}i=\overline{1,l}$.
as $n\to \mathrm{\infty}$. □
5 Conclusion
In this paper, we study dissipative q.s.o. defined on infinitedimensional simplex. In this case, we have some obstacles. First, an infinitedimensional simplex is not compact in ${\ell}_{1}$ topology, nor it is compact in a weak topology, which makes the study of limit behavior harder. Moreover, the operator may not have fixed points at all. Despite this, we were able to define the dissipative q.s.o. in such space, and we classified the fixed points of dissipative operators. We also studied the limit behavior of the trajectories in some particular cases that can be an impetus to further studies of dissipative q.s.o. on infinitedimensional space. There are some questions left unanswered. First, note that simple examples can show that a dissipative q.s.o. is not nonexpansive operator, so we can not use the general theorems guaranteeing the convergence of Cesaro means. So the questions is: find necessary and sufficient conditions for a dissipative q.s.o. to be mean ergodic (i.e., Cesaro mean of any initial point converges in ${\ell}_{1}$ norm). Second, investigate the limit behavior of the trajectories for arbitrary dissipative q.s.o.
Declarations
Authors’ Affiliations
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