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17COMPOSITION OF MIXED FRACTIONAL INTEGRAL AND MIXED
FRACTIONAL DERIVATIVE
Dilshod Barakaev
Bukhara Technological Institute of Engineering
ABSTRACT
We study the question of the composition of the mixed fractional integral and the mixed fractional derivative in sufficiently broad class of functions. The treatment formula for mixed fractional derivative is obtained.
Keywords: mixed fractional integral, mixed fractional derivative, function of two variables, Riemann-Liouville integrals.
Introduction
Various forms of fractional integrals and derivatives are known. Fractional integrals and Riemann-Liouville derivatives are the most common in the scientific literature [1]. Operators of generalized fractional integro-differentiation with Gauss hypergeometric function.
Direct extension of the Riemann-Liouville fractional integro-differentiation operations to the case of many variables, when these operators are applied for each variable or some of them, gives the so-called partial and mixed fractional integrals and derivatives. They are known [1], as well as [4], [5], [6], [7], [8], [9], [10]. Thus, in [2], using the two-dimensional Laplace transform, a solution of the two-dimensional Abel integral equation was obtained.
In this paper, we study the question of the composition of the mixed fractional integral and the mixed fractional derivative in sufficiently broad class of functions. The treatment formula for the mixed fractional derivative is obtained. The results obtained can be applied in the theory of differential equations containing the mixed fractional derivatives.
Lemma 3 on the representability of f (x, y)e ACn,m (q) function in the form of (6) and Lemma 4 generalized is the previously known Lemmas 1 and 2 for the two-dimensional case. Lemmas 3, 4 permit to prove the theorem (a necessary and sufficient condition for the representability of f (x, y) function as the mixed fractional integral of a summable function) and Theorems 2 and 3 about the composition of a mixed fractional integral and a mixed fractional derivative. Note that Theorems 2 and 3 generalize the results of Theorem 2.4 [1, p. 44] for the two-dimensional case.
Preliminaries
The important role in the theory of fractional integro differentiation is played by absolutely continuous functions. Let Q = {(x, y):a < x < b, c < y < d},
— ro < a < b < — ro < c < d <
Definition 1 [1, p. 2]. f(x) functionis called absolutely non-discontinuous into segments [a, b], if for any s> 0 there exists 5> 0 such that for any finite set of pairwise
__m
non-intersecting intervals [ak, b ] e [a, b], k = 1, m, such that £ (b — %) < 5, the inequality
k=1
m
£1 f (b)—f (ak) < s holds. The space of these functions is denote by Ac([a, b]).
k=1
Definition 2 [1, p. 2]. Let us denote by ACn([a,b]), where n = 1,2,..., the spaces of functions f (x) which have continuous derivatives up to order n — 1 on [a,b] with
f in—1\x )e AC ([a, b]).
Definition 3. A function f (x, y) is called absolutely continuous in q , if for any s > 0 there exists 5> 0 such that for any finite set of pairwise non-intersecting intervals Ak ={(x, y): x1k < x < x2k, y1k < y < y2k}, the sum of the areas of which is less 5, the inequality holds
n
£ |f (x2k , y2k ) — f (x2k , y1k ) — f (x1k , y2k ) + f (x1k , y1k } < s ,
k=1
(1)
and if, moreover, f (a, y)e AC ([c, d ]) and f (x, c)e AC ([a, b]). The class of all such functions is indicated AC (q).
Definition 4. ByACn,m(q), where n = 1,2,..., let us denote the class of functions continuously differentiable on q up to order (n—1, m—1), and its mixed partial derivative
^n+m—2 r
-:—J— is absolutely continuous in Q.
dxn—1dym—1 J
It is known that the class ACn ([a, b]) belongs to those and only those functions f (x) that are representable as antiderivatives of Lebesgue summable functions:
x
f (x) = J v(x )dx + C, y(x) e L (a, b ).
a
(2)
Lemma 1 [1, p. 39]. The space ACn ([a, b]) consists of those and only those functions f (x), which are represented in the form
i x n—1
f (x) = J(x — t)n—1 v(t)dt + £Ck(x — a)k ,
V /'a k=0
(3)
where ^(x)e L([a,b]), Ck being arbitrary constants. Uzbekistan www.scientificprogress.uz Page 350
In the formula (3)
<p(t ) = /■>(,) Ct = ^.
(4)
The last equality uses the notation f(n'(x) = d f(x). A similar property of the functions
f (x, y )e AC (q) is as follows.
Lemma 2 [3, p. 238]. The class AC(q) consists of those and only those functions f (x, y) which are represented in the form
x y x y
f (x, y) = J J ^(t, ^)dtds+J y (t)dt + J ^(s)ds + C,
a c a
(5)
where ^(x, y)e L (Q), v(x)e L ([a, b]), ^(y)e L ([c, d]), and c is an arbitrary constant. In order to generalize the last lemma to the case of a class ACn (q), we need the following lemma. Lemma 3. Let f (x, y )e AC (q) , then
/ 1 x y f(n,m)(t,s)dtds
f (xy)=(n- 1)!(m -1)! Ja J (x-t)1n(y-s)1m
+
n-1 f('>0)(„ m-1 f{0, k )/ \ n-1 m-1 Ai, k)( \
+ Z LJ&d. (x - a) + EZ-rfcl (y - c)fsp (x - a) (y - tf. (6)
=0 i ! k=0 k! i=0 k=0 i!k!
>_d'+kf (x, y )
In formula (6) the notation used f ^k J(x, y )=
dx'dy
^n+m-2 /*
Proof. Let be-:—^ e AC (q). By virtue of Lemma 2, we have
dx dy
ßn+m-2 r x y x y
—f = J J^(t, s)dtds + Jy(t)dt + J^(s)ds + C0 (7)
cxn-1dy"
Integrating sequentially (7) times n -1 by x and times m -1 by y, we get
^ x y / ^\m-1 x
f (x, y) = 7-^-r- f f (x -1)n-1 (y - s)m-1 rn(t, s)dtds + , (y - ,)-r- f (x -1)n-1 w(t)dt +
X ' (n- 1)!(m-1)!J J (n- 1)!(m-1)!JV '
(y _ /7 V-1 y n-1 m-1
+ / (x - a ^ J(y - s)m-1 ^(s)ds + X T; (y)(x - a) + (x)(y - cj , (8)
(n l)!(m 1)! c i=0 k=0
where xi (y) (i = 0, n — 1), ~ (x)(k = 0,m -1) is arbitrary function. When integrating, the well-known for n - multiple integral formula is used [1]
x x x ^ x
J dx J dx... J F (x )dx = -J (x -1 )n-1 F (t )dt,
a a a (n )! a
(9)
c
proof, which is easy to implement by mathematical induction. It will be clear from the proof that an arbitrary constant in formula (7) is associated with arbitrary functions of formula (8) by the relation (n — 1)\^](c)+{m — 1)!i;()":11)(a) = C0.
d '+kf dx' dyk
Since f (x, y)e ACn,m(q), then derivatives —(0 <' < n, 0 < k < m)exist and are
continuous in Q. Calculating the derivatives with x respect to the order 0,n—1 of the function f (x, y) given by formula (8), and assuming in them x = a, we obtain the equalities
= /n (y)+Z tPM - cy, ' = , dx t0
(10)
dfyl = t\ ir^fm + (n — 1)! W + Z ~in—1 ^ — c)k. (11)
dx (m — 1)! Jc (y — s) k=0
Similarly, differentiating (8) by y and assuming y = c, we obtain the equality
= k!~k(x) + Z T!k)(c)(x — a), k = , (12)
dy
'=0
f1 = J(^P + (m - D'V.i^O+Z.-Mx - a)'. (13)
Expressing from formulas (10) - (13) xl (y) and ~k (x) respectively, we get
l' ( :
(x - a)' ( d'f (a, y) m
(y )(x - a) + Z ~ (xKy - c)k -Z )(x )(y - c)
k=0 '=0 '!
_ z )(x K y - c ) 1 +
V dx k=0
+ Z (y-CL ^d^fC^.C)-Z xjk )(y )(x - a) V, (x - f' J-^
k! t dyk Z ' ' ("-!)!(«-1)! Jc (y-S)
x-a' -1)!(m-1)! J (y-s)1-m
(y - c)m-1 f = (x - a) d 'f (a, y) | ^ (y - c)k dkf (x, c)
(n - 1)!(m -1)!J (x - if" Z i! dx' ¿0 k! dy
n-1 y i \ / / \m-1 x
(n-i)!(m-1j!-_ (x-ir-" 1=0 i! dx' k=0 ^ ^
(x - a)" 1 y ^(s)ds (y - c)m 1 x y(i)di -("-l)!(m-1)!J (y-s)1-m -("- 1)!(m-1)!J (x-i)1-" -
)(c) )(a
=0 k=0
Z Z(x - a)(y - c)k^ + . (14)
■\i+k
d f
Calculating the mixed derivatives —-f of the function (8) at point (a, c), we get
dx' dy
1 d'+fa, c)_ x!k \c) + )(a)
' !k! dx' dyk k! '!
(15)
Substituting (14), (15) into (8), we get
f (x v)= 1 } } 9(t, s)dtds + v1 d'f & y) (x +
f(xy)=(n — 1)!(m — 1)!J J (x — t)1—n(y — s)1—m +Z(x — a) +
=0
"-1 m-1
+ £1 -£ (x-a) {y-cf . (16)
k=0k! ay t0 k=0z!k! dxdy
Equality (6) follows from (16) and from the fact that ^(x, y ) = -—f (a'c). The lemma is proved.
The following lemma gives a description of the class AC"'m (q) . It generalizes Lemma 1 to the case of two variables and Lemma 2 to the case n + m > 2.
Lemma 4. The space AC"'m(q) consists of those and only those functions f (x,y), which are represented in the form
i x y m-i / \k x
f(x,y)=(w_xy{m_x),i J(x-t)"-1 (y"*)m-1 ^s)dtds + (x-1)"-1 v№+
1 (x - a )
n-1 _^ ) X n-1 m-1
+ -fe J (v - 5 )™-1 Vs + X X Q (x - « )! (y - c Y
î=0 i !(m -1)! c 1=0 k=0
(17) ___
where 9(x, y) e L1 (Q), vk (a)e L1 ([a, b]) (k = 0,m -1), ^(y) e L(c, d]) ( = 0 " - l), Ck being arbitrary constants. Proof. Necessity. Let f (a, y)e AC"'m (q). According to the lemma 3
Si \ 1 H f ^ )(t, s) ^ ^ f(i '0)(a, y)( v
f (x, y ) = 7-w-rl It— M / M dtds + X"-^^(x - a ) +
fV'y (n - 1)!(m -1)! J J (x - t)1-n (y - s )1-m X i !
a c
m-1 /(0,k V V "-1 m-1 Ai,k)( \
(y - c)-z z^-rlp (a - a) o- - c)k.
k=0 k! i=0 k=0 i k!
(18) _ Because f ("-1,m-1^(x, y) e AC (q), then f ("-1m-1)(a, y) e AC ([c, d ]), consequently, f ("-1,0)(a,y)e ACm([c,d]), from here f M)(a,y)e ACm([c,d])(i = 6T"-i). Use lemma [1, c.39]
y n M m-1 Ai,k)
r n-(s) ds+yf-
(m -1)! J (y - s)1-m g k!
where ^(y)e L ([c, d ]). Then
(19)
g f^^ - a) = gJ^Ljl&L ds + Z g f ,(t )(a,c) (a - a>(y - c)k. (20)
g i! ¿0 i! (m - 1)!J (y - s )1-m z z i! k! V ' ( )
Similarly, it is proved that
gf^^k =g-fc£U7ЦL dt + g (a - a) (y-c)k. (21)
Z k! U 7 gik! (n-1)!J (x-t)1-m g i! k! V 7 ( )
where wk(x)e L([a,b]). Substituting (20), (21) into (18), we obtain the formula (17), in which
Clk =— f(ik )(a, c). i !k! v '
di+k f
Sufficiency. When calculating directly—f (0 < i < n,0 < k < m), it is easy to make sure
dx1 dy
that they are all continuous in Q, and
^n+m-2 r x y x y
^n—¿T = j j^t,s)dtds + jy(0dt + + (n-1)! (m- Ok,^. (23)
'ac a c
^n+m-2 r
Obviously —-—J— e AC(q), from where it follows f (x, y) g ACn
dxn dym
The theorem is proven completely. Notice, that 9(x, y ) = f{n,m )(x, y);
(24)
V(x) = f(nk)(x,c), k = 0,m-1; (25)
^(y) = f(hm)(a,y), i = 0,n-1;
(26)
Jik
= f^ c).
i !k!'
(27)
Definition 5 [5]. Let f (x, y )g L (q) . The integral
(la p f)x y) =_1_1 f f (t,s)dtds (28)
Va+>cJ^y) r(a)r(p) j j (x -t)1-a(y - s)1-p , ( )
where a > 0, p > 0, is called a left-hand sided mixed Riemann-Liouville fractional integral of order (a, p).
The fractional integral (28) is obviously defined on functions f (x, y)e L (Q), existing almost everywhere. Using the Fubini theorem, the semi group property is proved.
Let f(x, y)e L (Q), a, p, y, 5 be positive numbers, then equality holds almost everywhere in Q
ra,p Tl,5 r _ ra+y,p+S r
1 a+,c+1 a+,c+f 1 a+,c+ f .
(29)
It can be shown that if a > 0 function f (x, y) is defined in Q and f (x, y )e L (Q), then
(C,)(x,y)e L([c,d]) Vx e (a,b); fe, y )(x, y)e L ([a, b]) Vy e (c, d). In the last equations f, f are partial Riemann - Liouville fractional integrals with respect to the variables x and y , respectively. Taking these equalities into account, it is directly verified that (iavcu)(x, y)=(iija+jXx, y)=(c+pc+f )(x, y). (30)
Definition 6 [6]. For function f (x,y), given on Q, formula
(d-P /-Y*V)= 1 an+m x y f(t,s)dtds (D+,C+J ^ y) r(n _ a)r(w _ p) ax-ay^ J J (x _ ty--+1 (y _ s)P--1
(31)
where a > 0, ß > 0, is called a mixed Riemann-Liouville fractional derivative of order (a, ß), n = [a] +1, m = [ß] + 1.
If the function f (x, y ) has a property /";"f- p f e AC"'m (q) , then the order of taking the derivatives in (31) does not matter, and (d";pc+ f )(x, y )e L1 (q) . Definition 7 is a two-dimensional analogue of Definition 2.3 [1, p. 43].
Compositions of mixed fractional integral and mixed fractional derivative of the same order
Following [1, p. 44], we define the following classes of functions.
Definition 7. Let /^(A ) denote the space of function f (x, y), represented by the left-
sided mixed fractional integral of order (a, p) of a summable function: f = /0+pc+9, 9 e l^q) .
Definition 8. Leto <a< 1,0 <p< 1. The function f (x, y)e L (q) is said to have a summable fractional derivative D^ f, if /;;ac+m-p f e AC"m (q) .
The following theorem defines the necessary and sufficient condition for the unique solvability of the two-dimensional Abel integral equation.
Theorem 1. In order that f (x, y )e /0+^(4 ), a> 0, p> 0, it is necessary and sufficient that
f"-am-p e AC",m(q),
(32)
where n = [a] +1, m = [p] +1, and that
Kz,0)
ÄLp(a, y)- 0, i = 0, n -1;
(33)
Äm>, c) = 0, k = 0, m -1 ;
(34)
fn-a^m-p(a,c)-0, i = 0,n-1, k = 0,m-1. (35)
Proof. Necessity. Let f = /0+pc+9, 9 e L (q) . In view of the semi group property
f (x v) = /"-a,m-p f = /a,p 9
fn-a, m-pVA, y/ = /a+,c+ f = /a+,c+9 ,
(36)
where 9 e L(q). From here follow feasibility conditions (33) - (35). Feasibility condition (32) follow from Lemma 4.
This implies the fulfillment of conditions (33) - (35). The fulfillment of condition (32) follows from Lemma 4. Sufficiency. Under condition (32), we can present fn_a^m_p according to Lemma 3, in
the form
X ( \ 1 X y fn-'Zm-ß I, i , fn-a,m-ß <a y h v
R<x, y )=7-W--7-M-/ -r;—dtds +> ---<x - a ) +
Jn-a,m-ßV,y <n-l)!(m-1)!J J(x-t)1-n(y-s)1-m Z i! V '
m-1 f<Q,k) (Y n-1 m-1 f<i,k) (n
Jn-c, m-ßVx> c)( U V^ ST^Jn-a, m-ß Va,c/
f ( \ _ _1_ C f Jn-a,m-ß(t, s) , ,
y ^iW^iïlJ J V-m dtdS
+ !' 0' -c)k-S S (x-a)(y-c)k, (37)
k=0 k ! i=0 k=0 k !
where /¿¿.p e L(fi)• Taking into account conditions (33) - (35), the last equality is written in the form
x y An m) (t \ _ f f Jn-a,m-pV^ s)
(n -l)!(m -1)! J J (x - t)1-n(y - s)1
(38)
Using the semigroup property (29), we can write
rn-a,m-p r _ rn,m /*(n,m) _ rn-a,m-^a,p /*(n,m)
a+,c+ J a+,c+Jn-a,m-p a+,c+ a+,c+J n-a,m-p *
(39)
From here /n-af-p(/ - C+/i-nam2-p )=0. Applying the integral to this equality ia;pc+, we get
iz+{/ - cpc/xLpyx«y=o. (40)
From here / = C/ami-p, /¿¿Lp e L (fi) • The theorem is proved.
Note that Theorem 1 is a generalization of Theorem 2.3 [1, p. 43] in the case of two variables. From it, in particular, it follows that the class of functions having a summable fractional derivative Da;pc+/ in the sense of Definition 8 is wider than the class of
functions iai+(L ). Namely, the class ia;pc+(4 ) owns only those functions that have a
summable fractional derivative D^ /, for which equalities (33) - (35) hold.
Theorem 2. Let a > o, p > o. Then equality
Dat+Cpc+f = / (x, y )
(41)
performed for any summable function / (x, y ). Proof. We have
dxndy"
-pva,ß 7"a,ß r _ u rn-a,m-ß ra,ß r _
a+,c+ a+,c+J ^ n^ m a+,c+ a+,c+J
_1__dn+m x y dtds r s f (u, v)dudv
r(a)r(ß)r(n-a)r(m-ß)dxndym J J (x-t)a(y-s)ß J J (t-u)n-a(s-v)m-ß
Changing the order of integration, we get
p a,p = (r(g)r(p)) 1 jrm_} \f{uvdudJ \_dtds__
Da+,c+/a+,c+f =r(" - a)r(m-$)dx"dym J J f (U' ^ J J (x - t)a(y - s)p(t — ^
m-p)axn-ym J r JB Jv (x -1 )a(y - s )p(t - « )n-a(s - v)
-n+m r r ,( ^ 1 x dt__1 y ds
dxndym J Jf (U V)d« V r(a)r(n-a)J (x - t)a(t - u )n-a r(p)r(m-p)J (y - s)p(s - v)
i ^n+m x y f(% v)
H7-f / \i— dudv = f (x, y ),
r(")r(m) 5x"&ym J J (x - u)1-" (y - v)1-m
(43) Q.E.D.
Theorem 3.For any function f (x, y )e /^(L) the equality
/at*Dat+f=f (x, y),
(44)
and for any function that has summable derivative Da;pc+f (in the sense of definition 8), the equality
"-1 (v _ „Y-i-1 m-1 (, ._ „f-k-1
f= f(x, v)_y (x ,a) f(":г-1'0)(a, v^Z (y , c) , f(°:Tk-1)(x,c)
i=0 k=0
/aa+pc+ D^f = f (x, yyVx^^ft-rKa, y)-!^^^^^ c)+
1 1 7=0 r(a- i ) 1 k=0 r(p- k) 1 p
\a-i-1 / \p-k
- a) (y - c)p
r(a - i )r(p- k )
n 1 m-1 (y nY^ÎM /^p_k_1 ,
+££ (x ;a) .(y:c)- ftmr^a c), (45)
where f^x, y) = /J^f.
Proof. Let f (x, y) e /aa;pc+ (l1 ), then f (x, y) = /^9, 9(x, y) e L(Q). Based on Theorem 2, we have
R t-iv R T-^iv R t-IV R ,a„
^9 = 1 a+,c+ 9
(46) _
jl-^1^ f ^ Jf^iC) 1 A rmrHma tr\ T pmmQ ^ tVip intporcil -f .
-am-p '
can be represented as
CPc+D^+f = /^pc+ D^ /-Pc+9 = CPc+9 = f (x, y).
(46) _
Let now /^ac^ f e AC(q). According to Lemma 3, the integral f"-a m-p(x, y) = /";a+m-p f
n-1 fM) (a y) m-1 f (0-k) (x c)
fU-p(x, y ) = +£ ^""-r y ) (x - a) + £ c ) (y - c)k -
i=0 Z ! k=0 k !
n-1 m-1 f(ik) (n
£ £ c ) (x - a) (y - c)k .
i=0 k=0
(47)
By the semigroup property, the equality
T",m r(",m) _ г"_a'm_^a,p r(",m)
a+,c+f "-a,m-p a+,c+ a+,c+f "-a,m-p *
(48) Further,
(x - a) (i
fiÎiLp(a, y ) = /
^n-a,m-p L a+,c+
Dn
(x - a ) i !
Dm-p f(!,0) (a y)
Dc+,yf n-a,m-p\U, J )
+
+
(x - a)i (y - c)m ß 1 f<i,0)/ )_ jn-a,m-ß Jn-a,1\a, C) = 1 a
i !l(m -ß)
f i V-n+c ^
(X - a)_f (i 0) (a y)
• , \Jn-a,0\U,y)
1(1 + i - n + a)
+
m-ß-1
+
-fÜ2 (a, c ).
(x - a) (y - c) i !T(m -p)
From the last equality it follows that
Ki,0 )
n 1 fn-a,m-ß (a
+ Z
Z^
i=0
1 (x - a )i (y - c)m-ß-1 Ai,0)
(a, y)
n-1
i !r(m -ß) (50)
i !
Äi (a, c ),
(x - a)i = Cac+m-ß Z
(x - a )i
Z r<1 + i - n + a)
fi-a0)0 (a, y )
+
from where, redesignting the summation index, we get
+Z
1 (x - a)i (y - c)m-ß-1 <i-0)
fn!0)1 (a, c ).
i=0 i!
^m-p
i !r(m -p)
(51)
Equality is obtained similarly
n-1 f (<U) (x c)
1 f^0) (n v) f n-1 (v ^y-'-1
I Jn-a,m-ßVa, ■yJi„ „V _ /-n-c,m-ß y (x a ) f<"-i-1,0) (a y)
(x - a )i = I
i=0 r(c- i )
+
i=0
a,m-ß '
7!
(y - c) = I
k _ rn-a,m-ß
f m-
V1 (y - c) f (0,m-k-1)(x )
Z r(ß- k) f0 m-ß (x, c)
k=0
+
1 (x - a)n-c-1 (y - c)k
flmtKa, c).
HO k!r(n -a)
(52)
It is not difficult to see that
-1 n-1 f ) (a c)
Z Z "'T, , (x - a) (y - c) = I^ Z Z CXß
i=0 i=0
n-1 m-
i !k !
n-1 m-1
(49)
i=0 k=0
f{x-a){y-cl_\ (,k) < ^
.| Jn-a,m-ßVW, c/
_ y-n-a,m-ß
ZZ
1 (x - „r^iy - c f
V
i=0 k=0
r(a - i )r(ß- k )
ftZ17k-1)(a, c )
(53)
Taking into account equalities (48), (51) - (53), equality (47) is written in the form
n-a,m-ß
r«-a,m-^ _ rn-a,m-p 7"a,p y-xa,p x , t
1 a+,c+ J = a+,c+ a+,c+D a+,c+J a+,c+
f n-1 (v ^a-i-1 ^
ix-aLr fi-::--1,",(a, y )
Z
r(a - i )
+
+Z
i=0
m-1
+Z
k=0
1 (x - a)(y - c)m-ß-1
i !r(m -ß)
i \n-a-1i \k
(x - a) (y - c) k !r(n -a)
fn-a,1 (a, c) + Om-ß Z
- (y - c)ß-k-1
k=0
r(ß- k )
fa<mmßk-1)(x, c)
+
f», c )-
n
i=0
m
i=0
n
m
-I
a+,c+
(n-\ ^ (x - a)a-i-1 (y-c)ß-k-1 _ . „......^ c ^
n-1 m-
n- a,m-ß
V V (X a) C) f(n-,-1,m-k-l)( ) (54)
h h r(a-i)r(ß-k) (a'C)J ■ (54)
By grouping the terms, we get
n-a,m-ß
1 a+,c+
C n-l(v_ „Y*-i-1 m-1 / Jvß-k-1
r ra,ß na,ß r ^ (X a) An-i-1,0)/ ) ^ C) f(0,m-k-1)( ) J 1 a+,c+ Da+,c+J h W A Jn-a,0 Va,h Wr> i \ J 0, m-ß Vx, ^
1:0 r(a-,) t0 r(ß-k) ß
n-1 m-1/v _\a-i-_\ß-k-1 \ n-1 /v „V^, ^Y"-ß-1
1 X1 (x-a) (y-c) f(n-i-1,m-k-1)( ) _ (x-a)(y-c) (,,0
0 r(a - i)r(ß- k) Jn-a m-ß (a'c)l h i! r(m-k) Jn-a
i:0 k:
+ g(y-cK-aT'1' fm{a,c)
k=0 k! r(n - a)
(55)
In the right-hand side of equality (55), under the integral is a summable function. Applying the operator /'+pc+ to both parts of equality (55), we obtain
jn,m
1 a+,c+
f n-1fv_ „V-i-1 m-1^, k-1
J-T a,ß Da,ß j-y (X a) f (n-i -1,0)(a y) V (> c) J(0,m-k-1)(x
J 1 a+,c+ Da+,c+J h tV A Jn-a,0 V"'/ . Wrt , \ J 0, m-ß VX' V ^
V
r(a-i) ¿0 T(p-k)
n-1 m-1 ,,\a-i, k-1 A n-1 ^Y+a A, -1
+ ^ ¿WWo A fn-a, m-P Va, , , tXt^/ \ fn-a,na,c) +
£0 t0 r(a-i)r(p-k) P ) to r(i + a + 1)r(m)
+m1 (y-c)k+p(x-a)n-1 (o,k )(a c) +¿0 r(k+1+p)r(n) fl,m-p(a,c).
(56)
Under the integral on the left side of the equality is the summable function, and the
^n+m
right side of the equality is absolutely continuous. Finding the mixed derivative -
dx"dym
of both parts of the equality, we get
n-1 (v _ n\a-i-1 m-1 „\ß-k-1
f ja,ß na,ß r y(x a) f(n-i-1,0)( .Ay (y c) f(0,m-k-1)( )
J 1 a+,c+ Da+,c+J h W \ Jn-a,0 K", S / h T-^n ; \ J0, m-ß Vx,^
, , 1:0 r(a-i) , ■k:0 r(ß-k) , ß
+ h J^aim1,m-k-1)(a,c):0. (57)
i:0 k:0
n_1 m_1 (x - a)a-i-1 (y-c )P-k-1 k r(a - i )r(p- k) fn-a,m-P
The theorem is proved.
References
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2. S.Vasilache, Asuprauneiecuati i integrale de tip Abel cu doua variable.//Comun. Acad.R.P.Romane, 1953. Vol. 3. P. 109-113 (journal style)
3. V.I. Smirnov, Higher mathematics course. T. 5. M.: OGIZ, 1947. 584 p. (book style)
4. T. Mamatov, Mixed Fractional Integration Operators in Mixed Weighted Holder Spaces. LAPLAMBERT Academic Publishing, 2018, 73 p. (monograph style)
5. T. Mamatov, Composition of mixed Riemann-Liouville fractional integral and mixed fractional derivative. "Journal of Global Research in Mathematical Archives", vol. 6, No.11, Noveber 2019. Available online at http://www.j grma.info. Issn 2320-5822. India, p. 23-32
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7. T. Mamatov, F. Homidov, and D. Rayimov, On Isomorphism Implemented by Mixed Fractional Integrals In Holder Spaces, International Journal of Development Research, Vol. 09, Issue, 05 (2019) pp. 27720-27730 (journal style)
8. T. Mamatov. Mapping Properties Of Mixed Fractional Integro-Differentiation in Holder Spaces. Journal of Concrete and Applicable Mathematics (JCAAM), vol. 12 (34). 2014. P. 272-290
9. Mamatov T. Mixed Fractional Integro-Differentiation Operators in Holder Spaces. «The latest research in modern science: experience, traditions and innovations» Proceedings of the VII International Scientific Conference North Charleston, SC, USA, 20-21 June, 2018. P. 6-9
10. Mamatov T. Fractional integration operators in mixed weighted generalized Holder spaces of function of two variables defined by mixed modulus of continuity. "Journal of Mathematical Methods in Engineering" Auctores Publishing - vol.1(1)-004 www.auctoresonline.org. ( D0I:10.31579/jmme.2019/004) 2019, p. 1-16
