A limit point is another term for "clauster point". The Collins dictionary of mathematics defines it as: a point where neighbourhoods intersect with the set other than at the point itself.

Dear RENE

Thanks for your reply.

Probably you are talking about limit point of subset and not a limit point of a function. Your definition seems to be same as my first definition in the original question ”a point A is a limit point of subset S if and only if the any ε-neighborhood of A contains some points of S different from A”

What I would like to know is this type of definition could be applied to the definition of a limit point of a function and whether there is another formal definition for a function (not subset).

Let me explain why I would like to know another definition.

In the Arrow and Hahn’s textbook ‘General competitive analysis’, a limit point appears relating to the discussion on stability of a general equilibrium in order to analyze the convergence of f(t) when t→∞ where f() means a price vector and t means time.

They use the property of a limit point because the limit point could be analyzed without knowing whether a price path f(t) converges or not.

Since the limit point coincides with the limit value of price path under some conditions, they could prove the stability of a general equilibrium under these conditions.

So what matters is a limit point as t approaches infinity which is not necessarily a limit value in general.

If my first definition or your definition of a limit point is used for {f(t)|t>=0} as the subset S, any point belonging to {f(t){t>=0} becomes a limit point as far as f(t) is a continuous function(continuity is always assumed since f(t) is a solution of some differential equation) and we will lose the connection of a limit point to a limit value to result in failure to prove the stability.

 Best regards,

Dear Mesahiro,

The Limit Point in Arrow-Debreu model is used to find the equilibrium prices in the economy. As demonstrated by Kevin Roberts paper, “The Limit Points of Monopolistic Competition,” Journal of Economic Theory, 22,256--278 (1980), the limit price under Perfect Competition is attained pathologically. However, with the presence of Monopolistic Competition in a large economy with replications over different endowment and preferences, output may be adjusted to reach a limit point equilibrium price, on the one hand, or not reach a limits point equilibrium price because price will be too low, on the other hand.

Another great general equilibrium theorists, Lionel Mackenzie, has also dealt with the condition necessary to reach a limit point solution, p. 835. See his “The Classical Theorem on Existence of Competitive Equilibrium,” Econometrica, Vol. 49, No. 4 (Jul., 1981), pp. 819-841

Dear Masahiro,

You should not put any effort in seeking limit points. This sort of "sciennce" has nothing to do with economics, it is just a nonsense exercise in mathematics.

Dear Masahiro,

First off all, you are right. In the continuous case, it appears that all point would seem to satisfy the definition of limit points. So, your time sequence is right.

In particular in the context of stability, I would refer to LaSalle’s works, not the 1950-1960 works that “everybody knows,” but rather the 1976-1980 works which deal with nonautonomous systems.

See if you can get any of

[1] J. P. LaSalle, Stability Theory for ordinary differential equations, J. Differ. Equations 4, (1968) 57–65.

[2] Joseph P. LaSalle, Stability of nonautonomous systems, Nonlinear Anal., Theory, Methods and Appl., 1 (1976) 83–90.

[3] Joseph P. LaSalle, The Stability of Dynamical Systems, 2nd ed., SIAM, New York, 1976.

(in particular [2]).

On page 84 of [3] I see

A point p is a positive limit point of x(t) if there is a sequence tn such that tn→∞ as n→∞ and x(t) →p.

However, I prefer the definition that I saw elsewhere:

A point p is a positive limit point of x(t) if any neighborhood of p, arbitrarily small, contains an infinite number of points of x(t). This of course, implies defining the discrete time sequence tn  and the corresponding x(tn).

Now, note that the lim may imply two different things:

a)     that x(tn) comes close to p and keeps coming closer and closer to p as n→∞.

b)     that x(tn) comes close to p, then leaves the neighborhood, yet keeps coming and leaving and coming back to p as n→∞.

Hope it helps.

Best regards,

Itzhak

Dear Itzhak,

Thanks for your reply.

Your answer is very helpful and I will read references.

Furthemore two implications of lim are very suggestive since Arrow and Hahn used this property to exclude the second situation under some condition and to prove convergence of price path(i.e. the first meaning).

Best regards,

Hi Masahiro,

I myself am, first of all, an Engineer and, based on my long experience, I should also say (and even used to say) "forget these abstract theories." 

However, although basic knowledge, intuition and experience remain very important (most probably, in any real-world field), when the system is not simple, in particular when its representation contains nonstationary and/or nonlinear terms, you "discover" that previous knowledge is just not enough and you may surprised by things that may look "alright" yet may just blow up.

So, you have to learn some methods (and even to develop some yourself), to make sure that things that seem to go to a desired situation, go there indeed and also to stay there, not to just pass them.

Although I am mainly dealing with stability of Control systems, while your field is different, in case you have specific questions related to stability of systems, please feel free to ask any question, anytime.

Best regards

As I said, I am looking at limit points in the context of stability of nonlinear systems. 

In other words, having a differential equation, you want to know how it will evolve in time, without having to actually solve it. IOW, you want to know the where its solutions (for me, the system trajectories) are going.  

I am not sure if this is your interest, yet if it is,  you may want to follow the Research Gate discussion at 

"Is anyone here familiar with the Barbalat lemma and it's application in control engineering?"

Research Gate. system tells me that this is available from the link:

https://www.researchgate.net/post/Is_anyone_here_familiar_with_the_Barbalat_lemma_and_its_application_in_control_engineering#view=56e81ba83d7f4bc87d4a9801.

Dear Sir

This presentation is very useful

www.math.harvard.edu/~nate/.../lecture_12.pdf