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Leibniz's Rule and Fubini's Theorem Associated with Power International Journal of Mathematical Analysis Vol. 9, 2015, no. 55, 2733 - 2747 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.510243 Leibniz's Rule and Fubini's Theorem Associated with Power Quantum Difference Operators Alaa E. Hamza Department of Mathematics Faculty of Science Cairo University, Giza, Egypt M. H. Al-Ashwal Department of Mathematics Faculty of Science Cairo University, Giza, Egypt Copyright c 2015 Alaa E. Hamza and M. H. Al-Ashwal. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Jackson in 1908 introduced the well{known and the most used quan- tum difference operator Dq f(t) = (f(qt) − f(t))=(qt − t) for a fixed 0 < q < 1. Aldwoah in 2009 introduced the power quantum n; q{ n n difference operator Dn;qf(t) = (f(qt ) − f(t))=(qt − t), where n is an odd natural number and 0 < q < 1 is fixed. Dn;q yields Jackson q{ difference operator, when n = 1. In this paper, we establish Leibniz's rule and Fubini's theorem associ- ated with this power quantum difference operator. Mathematics Subject Classification: 39A13, 39A70 Keywords: n; q-power difference operator, n; q{Integral, n; q{Leibniz,s Rule, n; q{Fubini's Theorem. 2734 Alaa E. Hamza and M. H. Al-Ashwal 1 Introduction The power quantum difference operator is defined by 8f(qtn) − f(t) < n t 2 R\{−θ; 0; θg; Dn;q f(t) = qt − t :f 0 (t) t 2 {−θ; 0; θg; where n is a fixed odd positive integer, 0 < q < 1 is fixed number and θ = 1 1 for n = 1 and θ = q 1−n for n > 1, which was introduced in [3]. See also [1, 2]. If Dn;q f(t) exists, then f(t) is called n; q–differentiable for all t 2 R. Here f is supposed to be defined on a set A ⊆ R for which qtn 2 A whenever t 2 A. This operator unifies and generalizes two difference operators. The first is the well-known and the most used Jackson q–difference operator which was defined by f(qt) − f(t) D f(t) = ; t 6= 0; q t(q − 1) where 0 < q < 1 is fixed. Here f is supposed to be defined on a q-geometric set A, i.e., A is a subset of R (or C) for which qt 2 A whenever t 2 A. The derivative at zero is normally defined to be f 0(0), provided that f 0(0) exists [4, 5, 6, 7]. The second operator is the n{power difference operator 8f(tn) − f(t) < n t 2 R n {−1; 0; 1g; Dn f(t) = t − t :f 0(t) t 2 {−1; 0; 1g; where n is a fixed odd positive integer [3]. In [2], Aldwoah et al. gave a rigorous analysis of the calculus associated with Dn;q. They stated and proved some basic properties of such a calculus. For instance, they defined the inverse of Dn;q which contains the right inverse of Dq and the right inverse of Dn. Then, they proved a fundamental lemma of the power quantum variational calculus. This paper is devoted to establishing Leibniz's rule and Fubini's theorem associated with the power quantum difference operator. We organize this pa- per as follows. Section 2 gives an introduction to power quantum difference calculus. In Section 3, we prove Leibniz's rule which is concerning with differ- entiating under the integral sign. Some related results are obtained. Also, we prove Fubini's theorem in the power quantum difference operator setting, that is, we prove that the iterated integrals are equal. 2 Preliminaries We assume that I is the interval (−θ; θ) and X is a Banach space, endowed 1 with a norm k · k, where θ = 1 for n = 1 and θ = q 1−n for n > 1. An essential n; q{Leibniz's rule and n; q{Fubini's theorem 2735 y O y=qtn y=t θ b h4(b) o a / t −θ p hp5(a) θ −θ Figure 1: The iteration of h(t) = qtn, t 2 I for fixed n > 1 and 0 < q < 1. function which plays an important role in this calculus is h(t) := qtn. The set of fixed points of h(t) is f0g when n = 1 and is {−θ; 0; θg for n > 1. We study this function on the interval I. One can see that the k{th order iteration of h(t) is given by k hk(t) = q[k]n tn ; t 2 I; where, for α 2 C,[k]α is defined by (Pk−1 i i=0 α k 2 N; [k]α = 0 k = 0: k 1 The sequence fh (t)gk=0, see Figure 1, is uniformly convergent to 0 on I. Theorem 2.1 ([2]). Let f; g : I ! R be n; q–differentiable at t 2 I and c1; c2 2 R. Then, (i) Dn;q c1f + c2 g (t) = c1Dn;q f(t) + c2Dn;q g(t). n (ii) Dn;q fg (t) = Dn;q f(t) g(t) + f(qt )Dn;q g(t): f D f(t) g(t) − f(t)D g(t) (iii) D (t) = n;q n;q provided that g(t) g(qtn) 6= n;q g g(t) g(qtn) 0: We notice that (ii) and (iii) are also true for f : I ! X. Also, (i) is true if f; g : I ! X. We need the following prelimiary results in our study 2736 Alaa E. Hamza and M. H. Al-Ashwal Lemma 2.2. Let h(t) be the function defined above. Then, the sequence k of iteration functions fh (t)gk2N0 converges uniformly to 0 on every interval J = [a; b] ⊆ I containing 0. k+1 k Proof. For t 2 [0; b] we have h (t) ≤ h (t) for all k 2 N0. So, the sequence k k fh (t)gk2N0 is decreasing to 0. By Dini's theorem fh (t)gk2N0 is uniformly convergent to 0 on the interval [0; b]. Similarly, we can prove its uniform k convergence on [a; 0]. Consequently, the sequence fh (t)gk2N0 is uniformly convergent on the interval J = [a; b]. Corollary 2.3. The series 1 X k k k [k]n n n n (n−1) q t (q t − 1) k=0 is uniformly convergent to jtj on every interval J = [a; b] ⊆ I containing 0. k k k Pm [k]n n n n (n−1) Proof. We apply Dini's theorem to Sm(t) = k=0 −q t (q t − 1) on both [0; b] and [a; 0], then we get the desired result. The proof of the following lemma is a result of continuity and will be omitted. Lemma 2.4. Let f : I ! X be continuous at 0. Then, the sequence k ff(h (t))gk2N0 converges uniformly to f(0) on every compact interval J ⊆ I containing 0. We need the following theorem to guarantee the convergence of the series in the definition 2.7 of n; q{integral. Theorem 2.5. If the function f : I ! X is continuous at 0, then the series 1 X k k k k [k]n n n n (n−1) [k]n n q t (q t − 1)f(q t ) k=0 is uniformly convergent on every compact interval J ⊆ I containing 0. Proof. Let J ⊆ I be a compact interval containing 0. By Lemma 2.4, there exists k0 2 N such that k [k]n n f(q t ) − f(0) < 1 8t 2 J; k ≥ k0: k [k]n n Then f(q t ) < 1 + kf(0)k for k ≥ k0 and t 2 J which in turn implies that k k k k k k k [k]n n n n (n−1) [k]n n [k]n n n n (n−1) q t (q t − 1) f(q t ) < q t (q t −1) (1 + kf(0)k) n; q{Leibniz's rule and n; q{Fubini's theorem 2737 8t 2 J; k ≥ k0. Consider the sequences m X k k k k [k]n n n n (n−1) [k]n n Gm(t) = q t (q t − 1)f(q t ) k=0 and m X k k k [k]n n n n (n−1) Cm(t) = q t (q t − 1) (1 + kf(0)k): k=0 By Corollary 2.3, Cm(t) is uniformly convergent to jtj(1 + kf(0)k) on J. By Cauchy criterion, given > 0, there exists m0 2 N such that 0 kCm(t) − Cm0 (t)k < 8t 2 J; m ≥ m ≥ m0: This implies that 0 kGm(t) − Gm0 (t)k ≤ kCm(t) − Cm0 (t)k < 8t 2 J; m ≥ m ≥ maxfm0; k0g: 1 X k k k k [k]n n n n (n−1) [k]n n Therefore, q t (q t − 1)f(q t ) is uniformly convergent on k=0 J. Theorem 2.6 (power quantum chain rule). Assume g : I ! R is continuous and n; q–differentiable and f : R ! R is continuously differentiable. Then there exists a constant c between qtn and t with Dn;q(f ◦ g)(t) = f (g(c)) Dn;q g(t): (1) Relation (1) is true at t = 0, by the classical chain rule. Definition 2.7 ([2]).
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