View metadata, citation and similar papers at core.ac.uk brought to you by CORE I S S N 2provided 3 4 7 -by1921 KHALSA PUBLICATIONS Volume 12 Number 06 J o u r n a l of Advances in Mathematics Leibniz’s rule and Fubini’s theorem associated with Hahn difference operators å Alaa E. Hamza † , S. D. Makharesh † † Jeddah University, Department of Mathematics, Saudi, Arabia † Cairo University, Department of Mathematics, Giza, Egypt E-mail: [email protected] å † Cairo University, Department of Mathematics, Giza, Egypt E-mail: [email protected] ABSTRACT In 1945 , Wolfgang Hahn introduced his difference operator Dq, , which is defined by f (qt ) f (t) D f (t) = , t , q, (qt ) t 0 where = with 0 < q <1, > 0. In this paper, we establish Leibniz’s rule and Fubini’s theorem associated with 0 1 q this Hahn difference operator. Keywords. q, -difference operator; q, –Integral; q, –Leibniz Rule; q, –Fubini’s Theorem. 1Introduction The Hahn difference operator is defined by f (qt ) f (t) D f (t) = , t , (1) q, (qt ) t 0 where q(0,1) and > 0 are fixed, see [2]. This operator unifies and generalizes two well known difference operators. The first is the Jackson q -difference operator which is defined by f (qt) f (t) D f (t) = , t 0, (2) q qt t see [3, 4, 5, 6]. The second difference operator which Hahn’s operator generalizes is the forward difference operator f (t ) f (t) f (t) = , t R, (3) (t ) t where is a fixed positive number, see [9, 10, 13, 14]. The associated integral of (3) is the well known Nörlund sum x f (t)t = f (x k), (4) k=0 see [12, 13, 15]. In some literature Nörlund sums are called the indefinite sums, cf. [14]. Then we can define b f (t)t = f (a k) f (b k), (5) a k=0 whenever the convergence of the series is guaranteed. Note that, under appropriate conditions, 6335 | P a g e council for Innovative Research July 2016 www.cirworld.com I S S N 2 3 4 7 - 1921 Volume 12 Number 06 J o u r n a l of Advances in Mathematics lim Dq, f (t) = f (t), q1 lim Dq, f (t) = Dq f (t), (6) 0 lim Dq, f (t) = f (t). q1,0 In [1], A.Hamza et al. gave a rigorous analysis of the calculus associated with Dq, . They stated and proved some basic properties of such a calculus. For instance, they defined the inverse of Dq, which contains the right inverse of Dq and the right inverse of . Then, they proved a fundamental lemma of Hahn calculus. This paper is devoted to establishing Leibniz’s rule and Fubini’s theorem associated with the q, - difference operator. We organize this paper as follows. Section 2 gives an introduction to q, - difference calculus. In Section 3, we prove Leibniz’s rule which is concerning with differentiating under the integral sign. Some related results are obtained. Also, we prove Fubini’s theorem in Hahn difference operator setting, that is, we prove that the iterated integrals are equal. 2 Preliminaries Let N be the set of natural numbers and N0 := N{0}. For k N0 and 0 < q <1, we define the q -numbers 1 qk [k] := . q 1 q Let I is an interval of R containing 0 , where 0 := /(1 q), and h denote the transformation h(t):= qt , t I. One can see that > t, fort < 0 , h(t) = t, fort = 0 , < t, fort > 0. 1 The transformation h has the inverse h (t) = (t )/q,t I . The k th order iteration of h is given by k k h (t) := hhh(t) = q t [k]q , t I, (1) ktimes k 1 k 1 1 1 t [k]q (h (t)) := h (t) := hhh(t) = k , t I. (2) ktimes q k Furthermore, {h (t)}k=1 is a decreasing (an increasing) sequence in k when t > 0 ( t < 0 ) with 6336 | P a g e council for Innovative Research July 2016 www.cirworld.com I S S N 2 3 4 7 - 1921 Volume 12 Number 06 J o u r n a l of Advances in Mathematics k inf h (t), t > 0 , kN 0 = (3) k suph (t), t < 0. kN k The sequence {h (t)}k=1 is increasing (decreasing), t > 0 ( t < 0 ) with k suph (t), t > 0 , kN = (4) k inf h (t)), t < 0. kN Let f be a function defined on I . The Hahn difference operator is defined in [2] by f (qt ) f (t) D f (t) := , if t , (5) q, (qt ) t 0 ' and Dq, f (0 ) = f (0 ) , provided that f is differentiable at0 , where q(0,1) and > 0. In this case, we call Dq, f , the q, -derivative of f . Finally, we say that f is q, -differentiable, i.e. throughout I , if Dq, f (0 ) exists. One can easily check that if f , g are q, -differentiable at t I, then Dq, (f g)(t) = Dq, f (t) Dq, g(t), , C, Dq, ( fg)(t) = Dq, ( f (t))g(t) f (qt )Dq, g(t), f Dq, ( f (t))g(t) f (t)Dq, g(t) Dq, (t) = , g g(t)g(qt ) provided in the last identity g(t)g(qt ) 0, cf. [1]. The right inverse for Dq, is defined in [1] in terms of Jackson-Nörlund sums as follows. Let a,b I the q, -integral of f from a to b is defined to be b b a f (t)dq,t := f (t)dq,t f (t)dq,t, (6) a 0 0 x k k f (t)dq,t := x(1 q) q f (xq [k]q ), x I, (7) 0 k=0 provided that the series converges at x = a and x = b. It is known that if f is continuous at 0 , then the series in (7) is uniformly convergent. In the folloing we present some needed results from [1] concerning the calculus associated with Dq. Corollary 1 The series 6337 | P a g e council for Innovative Research July 2016 www.cirworld.com I S S N 2 3 4 7 - 1921 Volume 12 Number 06 J o u r n a l of Advances in Mathematics qk (t(1 q) ), k=0 is uniformly convergent to | t 0 | on interval I = [a,b] containing 0 . Theorem 2Let X be a Banach space endowed with a norm⃦. .⃦. Assume f : I X is continuous at 0 . Then the following statements are true k • { f ((sq ) [k]q )}kN converges uniformly to f (0 ) on I. k k • q f (sq [k] ) is uniformly convergent on I and consequently f is q, -integrable over I . k=0 q • The function x F(x) = f (t)dq,t, x I, 0 is continuous at0 . Furthermore, Dq, F(x) exists for every x I and Dq, F(x) = f (x). Conversely, b Dq, f (t)dq,t = f (b) f (a) for all a,b I. a Consequently, the q, - integration by parts for continuous function f , g is giving in by b b b f (t)Dq, g(t)dq,t = f (t)g(t) |a Dq, ( f (t))g(qt )dq,t, a,b I. (8) a a Theorem 3 Let f : I R be continuous function at 0 , then for t I t f (s)dq, s = (t h(t)) f (t). h(t) We will apply the time scales calculus tools, see [7, 8], to obtain a q, -analog of the chain rule. Theorem 4 Let g : I R be continuous and q, -differentiable and f :R R be continuously differentiable. Then, there exists c between qt and t such that, ' Dq, ( f g)(t) = f (g(c))Dq, g(t).(9) 3 q, -Differentiation Under The Integral Sign In this section we study the continuity and the q, -differentiation of the integral (t) f (t, s)dq, s. (t) We establish Libnitz’s rule. Finally, we prove that the iterated integrals are equal (this theorem is known by Fubini’s Theorem). Let f : I I R . We begin by the following definitions. 6338 | P a g e council for Innovative Research July 2016 www.cirworld.com I S S N 2 3 4 7 - 1921 Volume 12 Number 06 J o u r n a l of Advances in Mathematics Definition 1 (i) We say that f (t,s) is continuous at t = t0 uniformly with respect to s[a,b] I if f (t,s) = f (t ,s)uniformly with respect to s[a,b] I. limtt0 0 (ii) The q, -partial derivative of f with respect to t is defined by f (t, s) f (h(t), s) t I \ , t h(t) 0 Dq,,t f (t, s) = f (t, s) f (0 , s) lim t = 0 , t 0 t 0 whenever the limit exists. (iii) We say that f (t,s) is uniformly partially differentiable at t = 0 I , with respect to s[a,b] I if f (t, s) f ( , s) lim 0 t 0 t 0 exists uniformly with respect to s[a,b] . Definition 2 Let 0 [a,b] I . We define the q, -interval by k k [a,b]q, ={aq [k]q : k N0}{bq [k]q : k N0}{0}. For any point c I , we denote by k [c]q, ={cq [k]q : k N0}{0}. (t) From now on, we assume some appropriate conditions that imply the integrals of the form g(t, s)dq, s := F(t) exist, (t) where g : I I R and , : I I .