Analytic solutions of functional equations cheng li

[...]...December 18, 2007 15:40 World Scientific Book - 9.75in x 6.5in ws-book975x65 Analytic Solutions of Functional Equations x 4 Functional Equations without Differentiation 4.1 4.2 4.3 4.4 4.5 4.6 83 Introduction Analytic Implicit Function Theorem Polynomial and Rational Functional Equations Linear Equations 4.4.1 Equation I 4.4.2 Equation II ... ws-book975x65 Analytic Solutions of Functional Equations 20 Ω Ω relative to any Ψ if, and only if, it belongs to l1 Furthermore, if f ∈ l1 , then f = Ω Ω u+ − u− + i Ω Ω k→∞ i=0 v− Ω u+ − lim Ψ(i) k→∞ k k k k = lim v+ − i u− + i lim Ψ(i) k→∞ i=0 i=0 + vΨ(i) − i lim k→∞ − vΨ(i) i=0 k fΨ(i) , = lim k→∞ i=0 where f = u + iv, that is, if f is absolutely summable, then its sum is independent of the ordering... It is well known that the set of all open balls can be used to generate the Euclidean topology for Fκ In particular, a subset Ω of December 18, 2007 15:40 World Scientific Book - 9.75in x 6.5in ws-book975x65 Analytic Solutions of Functional Equations 8 Fκ is said to be open if every point in Ω is the center of an open ball lying inside Ω Besides the open balls, polycylinders are also natural in future... 6.5in ws-book975x65 Analytic Solutions of Functional Equations 14 Theorem 2.1 (Lebesgue Monotone Convergence Theorem) Let g ∈ l Ω and let {f (j) }j∈N be a sequence of nonnegative sequences f (j) ∈ lΩ such that (0) 0 ≤ fk (1) ≤ fk ≤ · · · < ∞, k ∈ Ω, and (j) lim f j→∞ k = gk , k ∈ Ω, then f (j) = lim j→∞ Ω g Ω Indeed, since 0≤ f (j) ≤ Ω f (j+1) ≤ Ω g, Ω thus lim j→∞ Ω f (j) ∈ [0, ∞] and lim j→∞ Ω f (j)... ≤ (m) lim hk m→∞ (m) · · · ≤ hk (n) = lim inf fk n→∞ f (n) Ω (m) hk (n) = inf n≥m fk (m) ≤ fk < ∞, for each k ∈ Ω for each k ∈ Ω and m ≥ 0, k ∈ Ω, so that lim inf f (n) = Ω n→∞ lim h(m) = lim Ω m→∞ = lim inf m→∞ m→∞ h (m) h(m) Ω ≤ lim inf m→∞ Ω f (m) Ω κ We have mentioned that any discrete set Ω in Z can be linearly ordered Note however, that for each linear ordering, the corresponding sum of a sequence... solution exists and is given by (1.2) Such solutions often reveal important quantitative as well as qualitative information which can help us understand the complex behavior of the physical systems represented by these equations In this book, we intend to provide some elementary properties of power series functions and its applications to finding solutions of equations involving unknown functions and/or... There are a large of number of properties of uniformly convergent sequence of functions and uniformly convergent functional series By means of the generalized partial sums introduced in the last section, we can carry some of these properties to functional series with multiple indices Let Λ be a nonempty set in Fκ and let f (λ) ∈ lΩ for each λ ∈ Λ We now have a family (λ) f (λ) λ∈Λ of sequences in l... n  n  ∞  (j) (j) = lim g (j) = = lim gk gk   n→∞   n→∞ k∈Ω j=0 j=0 Ω j=0 k∈Ω ∞ g (j) , (2.5) j=0 Ω where we have used the linearity of the Lebesgue sum in the second equality As another corollary, we have Fatou’s lemma: If {f (n) }n∈N is a sequence of nonnegative sequences in l Ω such that (n) lim inf fk n→∞ < ∞, k ∈ Ω, then Ω To see this, let h (m) = lim inf f (n) ≤ lim inf n→∞ (m) hk (0)... that is, j j→∞ ∞ fk = lim k=0 fk = k=0 f N and j lim m,j→∞ fk = k=−m ∞ k=−∞ fk = f Z respectively Limits of partial sum sequences will be discussed in details in the next section We remark also that our definition of a sum of infinite sequence is a special case of the Lebesgue integral for measurable functions Thus standard results from the theory of Lebesgue integrals can be applied In particular, Lebesgue’s... iterate of f Note that f [n] may not be defined if the range of f [n−1] does not lie inside the domain of f The n-th derivative of a function is defined by f (z + w) − f (z) f (z) = f (1) (z) = lim w→0 w December 18, 2007 15:40 World Scientific Book - 9.75in x 6.5in ws-book975x65 Analytic Solutions of Functional Equations 4 and f (k) (z) = (f (k−1) ) (z) for k ≥ 2 As is customary, we will also define f (0) (z) . existence of power solutions. Some of the material in this book is based on classical theory of analytic func- tions, and some on theory of functional equations. However, a large number of material. differential equations is treated as an application. Then several selected functional differential equations are discussed and their analytic solutions found. • In Chapter 6 analytic solutions for functional. ws-book975x65 x Analytic Solutions of Functional Equations 4. Functional Equations without Differentiation 83 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Analytic