Неравенства для некоторых новых квадратурных формул с весом
Автор: Зунг Ф.Т., Чунг Н.Т., Зуй В.Н.
Журнал: Владикавказский математический журнал @vmj-ru
Статья в выпуске: 3 т.16, 2014 года.
Бесплатный доступ
В настоящей работе обобщены неравенства Островского на шкале времени для $n$ точек и $L_p$-норм $m$-й производной, где $m,n \in \mathbb{N}$ и $p \in [1,+\infty]$.
Короткий адрес: https://sciup.org/14318473
IDR: 14318473
Текст научной статьи Неравенства для некоторых новых квадратурных формул с весом
In 1938, Ostrowski proved the following interesting integral inequality which has received considerable attention from many researchers [1, 5, 6, 13-16].
Let f : [a, b] ^ R be eoiitiiinous on [a, b] and differentiable in (a, b) and its derivative f : (a, b) ^ R is bounded in (a, b), that is kf 0k^ := supxG( a,b ) |f (x)| < to. Then for any x G [a, b]. we have the inequality
b j f (t) dt — f (x)(b — a) a
/ (b — a)2
V 4
+
x
-
a + b \ 2
2 )
kf0k∞
(1.1)
In [9]. the following results was obtained: If f : [a,b] ^ R is sudi that f (n 1) is an absolutely continuous function and Yn 6 f (n)(x) 6 Гп for all x G [a, b] for some constants Yn arid Гп. then
b
^--a [ f (a) + 4f (^-a) + f (b) ] — j f (t) dt
6 С п (Г п — Y n )(b — a)n+1,
(1.2)
a where the constants Ci = -2. C2 = 162 and C3 = ^ are sharp in the sense that they cannot be replaced by smaller ones.
Very recently, V. N. Huy et al. [5, 6] have strengthened (1.1) and (1.2) by enlarging the number of knots. More precisely, they proved that b-a n
6 A a,b,m(S — s)
(1.3)
f (x) dx--/ f (a + x i (b — a))
n a i=1
b j f (x) dx —
a
b-a n
n f (a + xi(b - a))
i =1
6 B a,b,m kf m +1k p ,
(1.4)
where s = inf xG[aib] f (m)(x) aiid S = sup xG [ a,b] f (m)(x). and
52 x k = 7++1 (V i = 1’ 2,...,m). (L5)
Note that, (1.5) have the solutions only for n E [1, 9], n E N (see [17-19]).
On time scales,the Ostrowski type inequalities have been generalized in various ways. For example, [2, 3, 10]. It proved in [2] the following result on time scales: Let a,b,x,t E T, a < b and f : [a, b] ^ R be differentiable. M = supa<x|f A(x)|. Then
b j f°(t)A(t) — f (x)(b — a) 6 M(h(x, a) + h(x, b)), a where hk(•, •) is defined in section 2.
In this paper, making use of the above theorem and some simple estimations, we obtain propose a new way of treating a class of quadrature formulas with weight involving n points and the Lp norin of m-th derivative on time scales where m,n E N arid 1 6 p 6 to.
2. Preliminaries on time scales
A time scale is a nonempty closed subset of R and is denoted by T. We define the forward and backward jump operators a, p : T ^ T by
a(t) = inf{s E T : s > t}, p(t) = sup{s E T : s < t} (Vt E T), with inf 0 = supT arid sup0 = inf T. А г>oint t E T is called right-dense, right-scattered, left-dense and left-scattered if a(t) = t. a(t) > t. p(t) = t and p(t) < t. respectively. We now introduce the set Tk which is derived from the time scales T. as follows. If T has a left-scattered maximum m then Tk = T — {m}, otherwise Tk = T. The delta graininess function ^ : T ^ [0, to) is defined by
^(t) : = a(t) — t (V t E T ).
If f : T ^ R is a function then we define the function f° : T ^ R by f°(t) = f (a(t)) (Vt E T).
We say that a function f : T ^ R is delta dijji'rentiable at t E Tk if there exists a number f A (t) such tin it for all e > 0 there is a neighlmrliood U of t (i. e.. U = (t — 5,t + 5) П T for some 5 > 0) such that
If(a(t)) — f (s) — fA(i)(a(i) — s)| 6 e|a(t) — s)| (Vs E U ).
We call f A ( t ) the delta decwatwe- of f at t.
For delta differentiable function f and g. the next formula holds:
(fg)A(t) = f Ag ° (t) + f (t)gA(t) = f Ag(t) + f ° (t)gA (t).
A function f : T ^ R is said to be rd- continuous if it is continuous at right-dense points, and its left-side limits exist at left-dense points.
A function F : T ^ R is cal led a A-antideriinitive of f : T ^ R pros’ided F Л( t ) = f ( t ) holds for all t E Tk. The 11 the A-iiitegral of f is defiired by Rab f ( t )A t = F(b) — F ( a ) .
It is known that every rd-coiitinuous function f has an antiderivative.
The functions hk : T 2 ^ R are defined recursively as follows:
t ho (t, s) = 1,
hk+i(t, s) = j hk ( y,s )A y
( V s,t E T ).
s
Proposition 2.1. If a,b E T, then the assertions hold:
-
1. If a 6 x 6 b then 0 6 hk (x,a) 6 hk (b, a);
-
2. For a 6 b we 1 lave 0 6 hk+1(b, a) 6 (b — a)hk(b, a).
Now, we introduce a. useful result, which is well-known in the literature as Taylor’s formula, with the integral remainder.
Lemma 2.2 [1]. Assume f E Crd(T) aiid xo E T. Then for all x E (a,b) we have f (x) = Tr-1(f, xo, x) + Rr-1(f, xo, x)
where Tr-1 (f, xo, • ) is Taylor's polyilomial of degree r — 1. that is.
r -1
Tr-i(f,xo,x) = ^2 hk(x,xo)fЛ (xo)
k =0
and the remainder can be given by
x
Rr-i(f,xo,x) = j hr-i(x,o(t))fЛГ (t)At.
x 0
We have the Montgomery identity which is stated in the following lemma.
Lemma 2.3 [8]. Let a,b,s,t E T, a < b and f : T ^ R be differentiable. Then bb f (t) = Г— [ f"(s)As + -^ /p(t,s)f Л(s)As, b-a b-a а а where s-a s-b
for a 6 s < t, for t 6 s 6 b.
3. Main results
Let 1 6 m, n and 1 6 p 6 to, 0 6 ai 6 1 satis lies ^i-^i ai = 1. For each i = 1,..., n, let a 6 xi 6 b and we consider the following condition
Hi(xi,x2,... ,xn) = hi+i(b,a) ( V i = 1,2,...,m — 1), (3.1)
where H i( x i , x 2 , ... ,X n ) = ( b — a ) Pn=i ak h i (xk, a). and
b j hm(b,a(t)^t — a
x n i
(b — a)E i =1 a
a i h m—i (x i ,a(t)) At = 0.
(3.2)
We point out the fact that in the continuous case T = R and ai = ... = an = 1, conditions (3.1) and (3.2) become n ■ 1
X Ук = i+1 (v i = 1,2,•••,m), k=i where xi = a + yi(b — a). Before stating our main result, let us introduce the following notations.
b
I (f ) = j f (x)A x, a
n
Q (f,n,m,x i ,...,X n ) = (b — a) X a i f (x i ). i =i
(3.3)
Note that, for the ease a i = ... = a n = n then
1 n
Q (f, n, m,xi,..., xn) = (b — a) X f (a + yi(b — a)) n i=i are also known in [5, 6]. Now, we slightly improve [5, 6] with weights ak on time scales:
Theorem 3.1. Let a,b E T aiid f E Cm(T). Then, under conditions (3.1) and (3.2). we have
| I(f) — Q(f,n,m,x i ,... ,X n ) | 6 2(b — a) 2 (T
-
s)h m-i (b, a),
where s = inf xG[aib] f A m(x). T = (f A m 1 (b) — f A m 1 (a))/(b — a).
C Let ns first define
x
F ( x ) = j f ( x ) A x.
a
Then I(f) = F(b) — F(a). Applying Lemma, 2.2 to the function F(x) with x = b and xo = a. we get mb
F(b) = F(a) + X hk(b,a)FAk(a) + / hm(b,a(t))FAm+1 (t)At k=i which yields that
I (f )= X hk+i(b,a)f Ak (a)+ [ hm{b,r(t))f Am (t)At. k=0a
(3.4)
For each 1 6 i 6 n. applying Lernma 2.2 again to the function f (x) with x = xi aiid xo = a. we get m-i
f (x i ) = X h k (x i ,a)f A k(a) + / h m-i (x i ,a(t))f A m(t)At.
k=0
By applying to i = 1,..., n and then summing up, we deduce that
n n m -1 n x i
Xa i f(X i ) = £ £ a i h k (x i ,a)fA(a) + £ / a i h m i=1 i=1 k=0 i=1 a
1 (x i ,^(t))f Am(t) At
m-1 n n xi
= XX a i h k (X i ,a)f Ak(a) + X / a i h m k =0 i =1 i =1 a
1 (x i ,^(t))f Am(t) At
m-1
= X b k=0
—Hk (X 1 ,X 2 , . .
-a
k n x i
. ,X n )fA (a) + X / a i h m i=1 a
1 (x i ,a(t))f Am(t) At.
Thus, m-1
= Hk(X1 , X2, . . . , Xn k=0
Q(f,n,m,x i , ...,x n )
k n x i
)f Ak(a) + (b - a) X / aihm i=1 a
i (x i ,a(t))f Am(t) At.
Then it follows from condition (3.1) that
Q(f,n,m,x i ,... ,X n )
m -1
= X kk+i(b,a)f Ak (a) + k=0
n x i
(b - a) X / a i h m- i i =1 a
(x i ,a(t))f Am(t) At.
(3.5)
By (3.4), (3.5), we obtain that
I(f) - Q(f,n,m,x i ,... ,X n )
b x ni
(b - a) a i h m -1
i =1 a
j hm (b,a(t))f Am(t) At - a
(x i ,a(t))f Am(t) At
Then, by using condition (3.2), we have
I(f) - Q(f,n,m,x i ,... ,X n )
x n i
(b - a)E i=1 a
a i h m- i (x i , a(t))[f Am(t) - s] At
(3.6)
We estimate the first term of (3.6) as follows
b
| j h m (b,a(t))[f Am (t) - s] At | a
6 h m
b
(b, a) [fAm (t) - s] At
a
= h m (b,a) ( f Am 1 (b) - f Am 1 (a) - s(b - a) )
= (b - a)h m (b, a)(T - s) 6 (b - a)2h m-i (b, a)(T - s).
(3.7)
For the second one, we first have xi
/ ai h m-1
a
(x i , a(t)) f Am (t) - s] At 6 h
,m-1 (x i , a) j | f Am (t) - s ] At a
= h m- 1 (x i , a) ( f Am 1 (b) - f Am 1 (a) - s(b - a) ) 6 (b - a)h m- i (x i ,a)(T - s).
Hence, summing up the above inequalities with i = 1, 2,..., n, using the Proposition 2.6, it implies that
n
x i
n
(b - ")E /
i=1
a i h m— i (x i , a(t)) f Am(t) - s] At
a
n
(3.8)
-
6 (b - a)2 (T - s) X a i h m— i (x i , a) = (b - a)(T - s) h m (b, a) i =i
6 (b - a)2(T - s) h m- i (b,a).
Combining relations (3.6), (3.7) and (3.8), we conclude that
I(f ) - Q(f,n,m,x i , ... ,x n ) | 6 2(b - a)2(T - s) h m- i (b, a)
and the proof of Theorem 3.1 is now completed. B
With the similar arguments as those used in the proof of Theorem 3.1, we also obtain the following theorem.
Theorem 3.2. Let a, b E T aiid f E Cm(T). Then, under conditions (3.1) and (3.2). we
-
1 1 f ) - Q(f, n m, x i ,..., x n )| 6 2(6 - a)2 (S - T )hm -i (6. a),
where S = sup x G[ a,b] f Am(x). T = (f Am-1 (b) - f Am-1 (a))/(b - a).
Since s = inf xG[a,b] f Am (x) 6 T = b-a J ab f Am (x)Ax 6 sup xG[a,b] f Am (x) = S, we have the following corollary.
Corollary 3.3. Let a,b E T aiid f E C m (T). Then, under conditions (3.1) and (3.2). we have
|I(f) - Q(f, n, m, x i ,..., x n )| 6 2(b - a)2(S - s)hm -i (b, a),
Where S = SUP xG[a,b] f Am(x), s = inf xG[a,b] f Am(x)-
Now, we will give a new quadrature formulas with weight involving n points and Lp norm m-th derivative on time scales.
Theorem 3.4. Let 1 6 p 6 to , a,b E T and let f E C m (T). Then, under conditions (3.1).
-
11 (f) - Q(/,n,m,x i ,...,x „ )| 6 2hm -i (6.^)(6 - a) l q+1 > /q kf Am k p , where p + q = 1.
C AVe lurve known that
I(f) - Q(f,n,m,x i , ... ,x n )
b n xi f hm(b,a(t))f Am (t)At - (b - a) X f aihm a i=i a
i (x i ,a(t))f Am(t) At .
(3.9)
The first term of (3.9) can be estimated by using the Holder inequality as follows
b j hm(b,c(t))f Am (x) Ax a
b
b
6 (/
a
b
b
a
1/p
1/p
6 h m (b, a) (b - a)1/q kf A" k p
(3.10)
a
a
6 h m- i (b,a)(b - a)(q+1)/q kf Am k p .
Similarly, we deduce since xi G (0,1) and the Holder inequality that xi
αihm- a
i (x i ,a(t))f Am(t) At 6 a i
x i
( j [h m-1 (x i , ^(t))]q At
a
x i
a
b
b
1/p
6 h m- 1
1 /p
= h m-1 (x i , a) (b — a)1/q kfAk p .
a
a
Now, applying the above inequalities with i = 1, 2,..., n we get
n
x i
n
(b — ")E /
a i h m- 1 (x i i ^ (t))f
(t) At
i=1
a
n
(3.11)
= (b - a)(q+1)/q kf Am k p . X a i h m- 1 (x i , a) i =1
= h m (b, a) (b - a)1/q kf Am k p 6 h m-1 (b, a) (b - a)(q+1)/q kf Am k p .
Relations (3.9), (3.10) and (3.11) imply that
-
1 1 (f) - Q(f,n,m,x 1 , ...,x n ) | 6 2h m-1 (b,a) (b - a)(q+1)/q kf Am k p
and thus Theorem 3.4 is completely proved. B
Next, we define the Chebyshev functional on a time scale by b bb
T A (f,g) = b-a j f (x)g(x) Ax - (b - a) 2 j f (x) Ax j g(x)Ax.
Then
b
TA(f,f) = w^ [ f2(x)Ax -b-a
a
(b - a)2
b j f (x)Ax
a
We also define ctaU) = (b - a) Тд(/,/)• Then, it should be noticed that in [15], N. Ujevic obtained the following result for the ease T = R: Let f : [a, b] ^ R Ire an absolutely continuous function, whose derivative f G L2(a,b). Then it holds that b-a
b
(b-У2 ^(f7).
f(a)+4f(bafa) + f(b) - /f(t)At 6
a
In this article, base on the result of N. Ujevic we will give a new quadrature formulas with weight involving n points and m-th derivative on time scales by using Chebyshev functional.
Theorem 3.5. Let a, b E T and let f E C m (T) be sueh that f Am E L2 (a, b). Then, under conditions (3.1) and (3.2). we have
|I(f) — Q(f,n,m,x i ,... ,x n )| 6 2hm-i(b,a^(b - a)3 o a U Am).
C lie have known that
I(f ) — Q(f,n,m,x i ,... ,xn )
b n x i
[ h m (b,^(t))f Am (t) At - (b - a) X f aihm
1 (x i ,^(t))f Am(t) At
a i =1 a
Then, by using condition (3.2), we have
I (f) — Q(f,n,m,x i ,... ,x n )
b j hm(b,a(t))[f Am(t) — T] a
x n i
At - (b - a) / aihm i=1 a
i(xi,a(t))[f Am(t) - T ] At ,
(3.12)
where T = (f Am 1 (b) - f Am 1 (a))/(b - a). The first term of (3.12) can be estimated by using the Holder inequality as follows
b j hm(b, a(t)) [f Am(t) - T] At a
b


b
1 / 2
6 h m (b, a)Vb - a ( j [ f Am (t) - T ] At
Combining this with the fact that
J [ fAm(t) - T ] 2 At = J [ fAm(t) ] 2 At - 2T j fAm(t)At +(b - a)T2
= j [ fAm (t) ] 2 At - [/fAm (t) At ] 2 = , д (fAm)
aa we obtain that
b j hm(b,a(t))[f Am(t) - T]At a
6 h m (b,a)^(b - a)^ A (f Am)
(3.13)
Similarly, we deduce since xi G (a, b) and the Holder inequality that xi
α i h m-1
a
x i
6 ai^ I [hm-1 (xi, a(t))]2 At a b 1/2 b m
6 h m -1 ( x i , a ) A t f A ( t ) - T
(x^(t)) [f Am(t) - T] At xi
1/2
a
1 / 2
) = h m -i(x i ,a)Vb - a\J^A(f Am).
aa
Now, applying the above inequalities with i = 1, 2,.. ., n, we get
n
x i
n
(b - a)E /
i=1
a i h m -i(x i , a(t)) f Am(t) - T| At
a
= ^(.Ь - a)a^(f Am) X a i h m —i(x i ,a) = h m (b^^(b - а)а д (/Am) ( )
i =1
6 (b - a)h m —i(b,a)^(b - a)aA(f Am).
Relations (3.12), (3.13) and (3.14) imply that
I (f ) - Q(f,n,m,xi, -A ) 6 2(b - a)h m —i (b,a)^ (b - a)ffA(f Am)
and thus Theorem 3.5 is completely proved. B
Base on the inequality in (1.1), by using some simple estimations, we obtain some new quadrature formulas involving n knots on timо scales: For 0 6 xi 6 1. a + xi(b - a) G T. we put
1 n
Q(f,X1,X2,... ,Xn) = - E f(a + Xi(b - a))- n i=1
The next result of this paper can be described as follows.
Theorem 3.6. Let a,b G T a < b f : T ^ R be differentiable. and assume that fA is rd-continuous such that fA G L2(T). Then for 0 6 xi 6 1 with Vi' 1 xi = 2 we have the following estimate bb
Q(f, xi,x2, ...,x n ) - b-Ta j f ° (x) Ax + f ^)_ afya j 2 h(s) As
6 ^/ (b - a) ^ A(f A ) .
aa
C Put tk = a + xk(b - a), then it follows from Lemma 2.3 that f (a + xk(b - a)) -
b
E- / f CT(x)Ax = b - a
a
b
7-^ / p(t k ,s)f A(s) As b - a
a
b
= 77— /p(tk ,s) f A(s) - b - a
f (b) - f (a) " b - a
A s +
b
1 f „ ff (b) - f (a)
7----- P(tk,s) 7-------- b-a b-a
A s.
a
a
Since b2 - a2
-
a b
btb j p(t, s) As — J(s — a)As + J(s — b)As b b tb s + s + ^^(s)^ As— J^^^(s)As— ay As— by
a
I ~ ^(s)As — a(t — a) — b(b — t) — (t
2 \
a
a a + b
-
t
As
b
1 z Z Л
) (b - a) - 2 ^(s) A s,
a
we deduce that f (a + xk (b — a)) —
b
-^ / f ' (x) Ax — b - a
a
b
_1_ [ p(t k ,s) f A (s) — fM b - a b - a
a
As
f (b) — f (a) | 2 / 1 ) b 1
+ (b — a) 2 [( b — a) Vk — 2) "J 2 ^ (s) As
a
Hence, bb f (a + xk(b — a)) — -^ / f CT(x)Ax + f^—f^ / 1 ^(s) As b — a (b — a)2 J 2
aa
b
-
— /p(t k , s) [f A (s) — f( b ) — f(a) 1 As + [f (b) — f (a)] Xх— — .
b — a b — a 2
a
By applying to k — 1,... ,n and then summiiig up. since ^n=1 x— — 2. we obtain that bb
Q(f,x i ,x2 , ... ,xn) — b — a У f ^ (x)Ax + f ((Ь)— af)( 2 a ) У 2 ^ (s)A s aa nb
-
— 1 X t A( f (b) — f (a) A
-
— n(b — a)/L J p(tk ,s) f (s) b — a A s.
k=1 a
We first observe that
Г [,Az x f (b) — f (a) l2 л Г '.a. 2, 1 Г Laz u12 z,Az f (s)--b—a— As — f (s)J As — b—a f (s)As — CTA(f ), which yields bb
Q (f , xi , x2 , ...,x n ) — b — a У fa (x) Ax+ f ((b)—af)( 2 a ) У 2 ^ (s) As aa b 1b 1
b1 n2 njb—^i V-afA) E (j [pftk • s)i2 As) k=1 a nb 1
6 n ( b - a) V-A(f A ) X (/(b—a) 2 As) = v^b—a^ A f)• k =1 a
The proof of Theorem 3.6 is now completed. B
Theorem 3.7. Let a, b E T aiid f E Cd (T). He a Iso let y = inf xGT f A(x) aiid T = f (b)-f (a)
b-a
. Then for 0 6 x i 6 1 with 522=1 x i = 2 we have the following estimate
b
b j 2 ^(s)As 6 (b — a)(T — y). a
Q(f,x i ,x 2 ,...,xn) — --- fа (x)Ax + -^
b-a b-a
a
C Put tk = a + Xk(b — a), then it follows from Lemma 2.3 that f (a + Xk (b — a))
-
b-
b
- [ f' (x) Ax = a
a
1 b-a
b
ЬТ1-- jp(t k , s) [ f A(s) — Y ] As + a
1 b-a
b
J p(t k ,s)f A(s)As a
b j p(tk ,s)Y As.
a
Combining this with the fact that
b
p(t, s) As = t
a
-
a+b 2
b
) (b - a) " j 2 ^ (s) As
a
we get f (a + xk(b — a)) —
b
^ / f CT(x)Ax b-a
a
b-
b a J p(tk,s)[f A(s) — y] As + b
a
γ
-a
b
(b — a ) 2 ^x k — 2^) + У 2 ^ (s) As '
a
Hence,
f (a + X k (b — a))
-
1 b-a
b j f"(x) Ax + b
a
γ
-
b aj 2 , >,
a
b
= v—a f^ ■ s) l fA (s)
a and then by 52n=1 xk = 2 that
- y As + [f (b) - f (a)] x k
-
Q(f, X 1 ,X 2 , -A)
^^^.^^^^ ^^^^^^^^^^^^^^^^^^^^^™ b-
b a If"(x)Ax + b
a
γ
-
b
- j 2 ^(»> A s
a
n(b — a)
n b n
X / p(t k , s) [ fA(s) — Y ] As + [f(b) — f(a)] X (b — a) k —1 a k—1
x k
-
1 n f
= у—) E /pft k ^Т A<s) — y Iv k=1 a
Hence,
Q(f, X 1 ,X 2 , . . . ,X n ) -
bb
A- [ f '(x) Ax + ^ [^ ^(s) As b — a b — a 2
aa nb
-
6 nbTa X /(b - a) [ fA« — Y ] As = (b — a)(T — 7). k=1 a
The proof of Theorem 3.7 is now completed. B
With the similar arguments as thosed used in the proof of Theorem 3.7 we also conclude the following result.
Theorem 3.8. Lei a,b E T aiid f E Cr 1 d(T). He also lei Г = supxeT f A(x) and T = Mil. Theiг for 0 6 x 6 1 wilh Pn x = n и.с htwe b-a " c ^г—у 2
bb
Q(f,X 1 ,X 2 , . . . ,X n )
— Г— [ f " (x)Ax ' /I ^(s) As
6 (b — a)(' — T ).
b—a b—a 2
aa
Acknowledgments. The authors thank the referee for useful remarks and comments that led to improvement of this paper.
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