By M. Giaguinta, J. Shatah, S. R. S. Varadhan

This quantity comprises the complaints of a convention held in get together of the 70th birthdays of Peter Lax and Louis Nirenberg at Villa los angeles Pietra in Florence (Italy). audio system from worldwide gave talks on matters regarding the mathematical parts within which Lax and Nirenberg labored: research, partial differential equations, utilized arithmetic and clinical computing. the 2 males performed seminal roles in those parts and had major impression at the improvement of many different mathematicians. This quantity provides testomony to the most important position performed by way of Lax and Nirenberg within the improvement of mathematical research

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**Additional info for Differential Equations: LA Pietra 1996 : Conference on Differential Equations Marking the 70th Birthdays of Peter Lax and Louis Nirenberg, July 3-7,**

**Sample text**

8 to study existence of periodic solution in some classes of systems with random impulsive effects, more exactly, systems with small random perturbations. Let us find a relation between an asymptotically stable, compact invariant set of a deterministic autonomous system and periodic solutions of a perturbed system obtained from the deterministic system by introducing small continuous and impulsive random perturbations. 50) dt defined on a domain D ⊂ Rn . 50) has an asymptotically stable, compact invariant set S.

G. [51, p. 115]. g. [51, p. 127], that solutions of the equation satisfy ln ξ(t) = 1 = 1. P lim t→∞ t Hence, there exists ∆(ω) > 0 such that ξ(t, ω) > t2 for arbitrary t > ∆(ω). Then i(t, ω) = [η(t)] + 1 > ξ(t) > t2 . Consider the scalar equation dx = f (t, x, ω) = dt −x, x, t < ∆, t ≥ ∆. It is clear that its solutions are unstable. Consider now the impulsive system dx = f (t, x, ω), dt ∆x|t=τi = 0, 1 − 1 x, e ∆x|t=τi = t = τi , τi < ∆(ω), τi ≥ ∆(ω) . It is easy to see that all solutions of this system have the form x(t) = x0 exp{−t}, t < ∆, x0 exp{−2∆(ω) + t − i(t) + i(∆)}, t ≥ ∆, and, since i(t) > t2 , we have that x(t) → 0 for t → ∞ with probability 1 and, hence, the impulsive system is totally asymptotically stable with probability 1.

Lq +p ∈ Bq } for arbitrary t > 0, A0 , A1 , . . Am , B1 , B2 , . . Bq , s1 , . . sm , l1 , . . lq . 1) satisfying x(0) = ζ(ω). 42) hold for k > 0. Denote by η(t) a random process that coincides with ηi on the intervals [ti , ti+1 ). Clearly, it is stochastically right continuous and periodically connected with ξ(t). Introduce a random variable τk , which is independent of ξ(t), ηi , and y(0) such that 1 (n = 0, 1, . . , k) P{τk = nT } = k+1 and xk0 = y(τk ), xk (t) = y(t + τk ), ξk (t) = ξ(t + τk ) , ηk (t) = η(t + τk ) .