Integral equations involving generalized Mittag-Leffler function
UDC 517.5 The paper deals with solving the integral equation with a generalized Mittag-Leffler function $E_{\alpha ,\beta }^{\gamma ,q}(z)$ that defines a kernel using a fractional integral operator. The existence of the solution is justified and necessary conditions on the integral equation admitin...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512216154898432 |
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| author | Desai, Rachana Salehbhai, I. A. Shukla, A. K. Desai, Rachana Salehbhai, I. A. Shukla, A. K. Desai, Rachana Salehbhai, I. A. Shukla, A. K. |
| author_facet | Desai, Rachana Salehbhai, I. A. Shukla, A. K. Desai, Rachana Salehbhai, I. A. Shukla, A. K. Desai, Rachana Salehbhai, I. A. Shukla, A. K. |
| author_sort | Desai, Rachana |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:44Z |
| description | UDC 517.5
The paper deals with solving the integral equation with a generalized Mittag-Leffler function $E_{\alpha ,\beta }^{\gamma ,q}(z)$ that defines a kernel using a fractional integral operator. The existence of the solution is justified and necessary conditions on the integral equation admiting a solution are discussed. Also, the solution of the integral equation is derived. |
| doi_str_mv | 10.37863/umzh.v72i5.6014 |
| first_indexed | 2026-03-24T03:25:15Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i5.6014
UDC 517.5
R. Desai (K. J. Somaiya College Engineering, Univ. Mumbai, India; S. V. Nat. Inst. Technology, Surat, India),
I. A. Salehbhai (Government Engineering College, Bharuch, India),
A. K. Shukla (S. V. Nat. Inst. Technology, Surat, India)
INTEGRAL EQUATIONS INVOLVING
GENERALIZED MITTAG-LEFFLER FUNCTION
IНТЕГРАЛЬНI РIВНЯННЯ, ЩО МIСТЯТЬ УЗАГАЛЬНЕНУ ФУНКЦIЮ
МIТТАГ-ЛЕФФЛЕРА
The paper deals with solving the integral equation with a generalized Mittag-Leffler function E\gamma ,q
\alpha ,\beta (z) that defines a kernel
using a fractional integral operator. The existence of the solution is justified and necessary conditions on the integral
equation admiting a solution are discussed. Also, the solution of the integral equation is derived.
Розглядається розв’язнiсть iнтегрального рiвняння з узагальненою функцiєю Мiттаг-Леффлера E\gamma ,q
\alpha ,\beta (z), що визна-
чає ядро з використанням дробового iнтегрального оператора. Iснування розв’язку обгрунтовано, обговорюються
необхiднi умови для iнтегрального рiвняння, що допускають розв’язок. Також наведено розв’язок такого iнтеграль-
ного рiвняння.
1. Introduction and preliminaries. The Mittag-Leffler function was defined by the Swedish mathe-
matician, G. Mittag-Leffler [5], by
E\alpha (z) =
\infty \sum
n=0
zn
\Gamma (\alpha n+ 1)
, (1)
where z is a complex variable and \alpha \geq 0 that occurs as the solution of fractional order differential
equations. The Mittag-Leffler function is a direct generalization of exponential function, hyperbolic
functions, and trigonometric functions as E1(z) = ez, E2(z
2) = \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h} z and E2( - z2) = \mathrm{c}\mathrm{o}\mathrm{s} z. For
0 < \alpha < 1 and | z| < 1, it interpolates between the exponential function ez and a geometric function
1
1 - z
=
\sum \infty
k=0
zk.
Due to its vast involvement in the field of physics, engineering and applied sciences, many authors
defined and studied different generalizations of Mittag-Leffler function, namely, E\alpha ,\beta (z) introduced
by Wiman [16], E\gamma
\alpha ,\beta (z) suggested by Prabhakar [6], E\gamma ,q
\alpha ,\beta (z) defined and studied by Shukla and
Prajapati [9], etc.
Shukla and Prajapati [9] defined the generalization of (1) as E\gamma ,q
\alpha ,\beta (z), for \alpha , \beta , \gamma \in C, \mathrm{R}\mathrm{e} (\alpha ),
\mathrm{R}\mathrm{e} (\beta ), \mathrm{R}\mathrm{e} (\gamma ) > 0, and q \in (0, 1) \cup N, in the form
E\gamma ,q
\alpha ,\beta (z) =
\infty \sum
n=0
(\gamma )qn
\Gamma (\alpha n+ \beta )
zn
n!
, (2)
where (\gamma )qn =
\Gamma (\gamma + qn)
\Gamma (\gamma )
denote the generalized Pochhammer symbol, which in particular, reduces
to qqn
\prod q
r=1
\biggl(
\gamma + r - 1
q
\biggr)
n
if q \in N, and, moreover, it is the generalization of the exponential
function as E1,1
1,1(z) = \mathrm{e}\mathrm{x}\mathrm{p}(z). Also, E1,1
\alpha ,1(z) = E\alpha (z), E
1,1
\alpha ,\beta (z) = E\alpha ,\beta (z) and E\gamma ,1
\alpha ,\beta (z) = E\gamma
\alpha ,\beta (z).
Further, Shukla and Prajapati [10] studied some properties of the generalized Mittag-Leffler-type
function and generated integral operator
c\bigcirc R. DESAI, I. A. SALEHBHAI, A. K. SHUKLA, 2020
620 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
INTEGRAL EQUATIONS INVOLVING GENERALIZED MITTAG-LEFFLER FUNCTION 621
E\gamma ,q
\alpha ,\beta ,w;a+f(x) =
x\int
a
(x - t)\beta - 1E\gamma ,q
\alpha ,\beta
\bigl\{
w(x - t)\alpha
\bigr\}
f(t)dt (3)
for \alpha , \beta , \gamma , w \in C, \mathrm{R}\mathrm{e} (\alpha ), \mathrm{R}\mathrm{e} (\beta ), \mathrm{R}\mathrm{e} (\gamma ) > 0, and q \in (0, 1) \cup N.
Fractional integral operators play an important role in the solution of several problems of science
and engineering. Many fractional integral operators like Riemann – Liouville, Weyl, Kober, Erde-
lyi – Kober and Saigo operators are studied by various authors due to their application in physical,
engineering and technological sciences such as reaction, diffusion, viscoelasticity, etc. A detailed
account of these operators can be found in the survey paper by Srivastava and Saxena [13]. Various
properties of family of Mittag-Leffler functions using fractional integral operators have been obtained
by many researchers (see, for example, [1 – 3, 8, 10, 14, 15]).
In this paper, we apply the results obtained by Shukla and Prajapati [10] to prove the existence
and uniqueness of the solution of the following integral equation, involving generalized Mittag-
Leffler function (2). For \alpha , \beta , \gamma , \lambda \in C, \mathrm{R}\mathrm{e} (\alpha ),\mathrm{R}\mathrm{e} (\beta ),\mathrm{R}\mathrm{e} (\gamma ),\mathrm{R}\mathrm{e} (\lambda ) > 0, q \in (0, 1) \cup N and for
any real number a \geq 0,
x\int
a
(x - t)\beta - 1E\gamma ,q
\alpha ,\beta
\bigl\{
\lambda (x - t)\alpha
\bigr\}
f(t)dt = g(x). (4)
In Section 2, integral on left-hand side of (4) is considered as an operator and its existence is justified.
In Section 3, properties of the integral operator are derived. In Section 4, necessary condition for
the solution is obtained and the integral equation (4) is solved. In Section 5, we discuss the integral
equation
b\int
x
(t - x)\beta - 1E\gamma ,q
\alpha ,\beta
\bigl\{
\lambda (x - t)\alpha
\bigr\}
f(t)dt = g(x) (5)
for \alpha , \beta , \gamma , \lambda \in C, \mathrm{R}\mathrm{e} (\alpha ),\mathrm{R}\mathrm{e} (\beta ),\mathrm{R}\mathrm{e} (\gamma ),\mathrm{R}\mathrm{e} (\gamma ) > 0, q \in (0, 1) \cup N and for any real number
0 \leq a \leq x \leq b.
In the sequel that follow, some additional properties of E\gamma ,q
\alpha ,\beta are derived, and we enumerate some
terminologies and definitions, required in the investigation.
Definition 1.1. Lebesgue measurable functions [7]: L(a, b) denote the linear space of real (or
complex) valued functions f(x) on [a, b] for b > a, i.e., f is L-integrable if
L(a, b) =
\left\{ f : \| f\| 1 \equiv
b\int
a
| f(t)| dt < \infty
\right\} .
Definition 1.2. Riemann – Liouville fractional integrals of order \mu [4]: Let f(x) \in L(a, b),
b > a, \mu \in C, \mathrm{R}\mathrm{e}(\mu ) > 0. Then
I\mu f(x) = aI
\mu
x f(x) = I\mu a+f(x) =
1
\Gamma (\mu )
x\int
a
f(t)
(x - t)1 - \mu dt, x > a, (6)
is called fractional integral of order \mu . I\mu is bounded. Also I\mu f = 0 \Rightarrow f = 0. Hence, inverse
operator exists on subspace L\mu of L. Similarly,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
622 R. DESAI, I. A. SALEHBHAI, A. K. SHUKLA
J\mu f(x) = xI
\mu
b f(x) = I\mu b - f(x) =
1
\Gamma (\mu )
b\int
x
f(t)
(t - x)1 - \mu dt, x < b.
We intend to maintain following properties of E\gamma ,q
\alpha ,\beta [10], which are employed to prove theorems
in this paper.
If \mathrm{R}\mathrm{e}(\alpha ), \mathrm{R}\mathrm{e}(\beta ), \mathrm{R}\mathrm{e} (\gamma ), \mathrm{R}\mathrm{e}(\lambda ) > 0, \mathrm{R}\mathrm{e}(\mu ) > 0, and q \in (0, 1) \cup N, then
1
\Gamma (\mu )
x\int
t
(x - s)\mu - 1(s - t)\beta - 1E\gamma ,q
\alpha ,\beta \{ \lambda (s - t)\alpha \} ds = (x - t)\beta - \mu - 1E\gamma ,q
\alpha ,\beta +\mu \{ \lambda (x - t)\alpha \}
and
1
\Gamma (\mu )
x\int
t
(s - t)\mu - 1(x - s)\beta - 1E\gamma ,q
\alpha ,\beta \{ \lambda (x - s)\alpha \} ds = (x - t)\beta - \mu - 1E\gamma ,q
\alpha ,\beta +\mu \{ \lambda (x - t)\alpha \} .
2. The integral operator. We employ the operator E\gamma ,q
\alpha ,\beta ,\lambda ;a+f(x), defined by (3) to repre-
sent left-hand side of the integral equation (4) and, hence, forth denote it as I(\alpha , \beta , \gamma , q;\lambda )f(x).
Therefore,
I(\alpha , \beta , \gamma , q;\lambda )f(x) = E\gamma ,q
\alpha ,\beta ,\lambda ;a+f(x) =
x\int
a
(x - t)\beta - 1E\gamma ,q
\alpha ,\beta \{ \lambda (x - t)\alpha \} f(t)dt. (7)
We may often use I(\beta ) instead of I(\alpha , \beta , \gamma , q;\lambda ) owing to reason that, in the operator I(\alpha , \beta , \gamma , q;\lambda ),
all other parameters remain unaltered except \beta .
In the later part of the paper we discuss the integral equation (5) for which we define the operator
as follows:
I\ast (\alpha , \beta , \gamma , q;\lambda )f(x) =
b\int
x
(t - x)\beta - 1E\gamma ,q
\alpha ,\beta
\bigl\{
\lambda (x - t)\alpha
\bigr\}
f(t)dt. (8)
To justify the existence of the integral defined by the left-hand side of the equation (4), we
employ the following theorem from [10].
Theorem 2.1 (existence of the operator). Let f \in L(a, b), b > a; \alpha , \beta , \gamma , \lambda \in C, \mathrm{R}\mathrm{e} (\alpha ),
\mathrm{R}\mathrm{e} (\beta ), \mathrm{R}\mathrm{e} (\gamma ), \mathrm{R}\mathrm{e} (\lambda ) > 0, and q \in (0, 1) \cup N. Then
x\int
a
(x - t)\beta - 1E\gamma ,q
\alpha ,\beta \{ \lambda (x - t)\alpha \} f(t)dt
defines a function in L. Moreover, it is bounded on L and\bigm\| \bigm\| I(\alpha , \beta , \gamma , q;\lambda )f\bigm\| \bigm\|
1
\leq B\| f\| 1,
where
B = (b - a)Re (\beta )
\infty \sum
n=0
| (\gamma )qn|
| \Gamma (\alpha n+ \beta )| [\mathrm{R}\mathrm{e} (\alpha )n+\mathrm{R}\mathrm{e} (\beta )]
\bigm| \bigm| \lambda (b - a)Re(\alpha )
\bigm| \bigm| n
n!
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
INTEGRAL EQUATIONS INVOLVING GENERALIZED MITTAG-LEFFLER FUNCTION 623
3. Properties of the operator.
Theorem 3.1 (composition property). If \alpha , \beta , \gamma , \beta \prime , \lambda \in C, q \in (0, 1) \cup N, \mathrm{R}\mathrm{e} (\alpha ), \mathrm{R}\mathrm{e} (\beta ),
\mathrm{R}\mathrm{e} (\gamma ), \mathrm{R}\mathrm{e} (\lambda ), \mathrm{R}\mathrm{e} (\beta \prime ) > 0, then the relation
I(\alpha , \beta , \gamma , q;\lambda )I
\bigl(
\alpha , \beta \prime , \gamma \prime , q;\lambda
\bigr)
= I
\bigl(
\alpha , \beta + \beta \prime , \gamma + \gamma \prime , q;\lambda
\bigr)
(9)
is valid for any summable function f \in L(a, b). In particular,
I(\alpha , \beta , \gamma , q;\lambda )I - 1
\bigl(
\alpha , \beta \prime , \gamma \prime , q;\lambda
\bigr)
f(x) = I\beta +\beta \prime
f(x). (10)
Proof. Let f \in L and x \in (a, b). Then
I(\alpha , \beta , \gamma , q;\lambda )I
\bigl(
\alpha , \beta \prime , \gamma \prime , q;\lambda
\bigr)
f(x) =
=
x\int
a
(x - u)\beta - 1E\gamma ,q
\alpha ,\beta \lambda (x - u)\alpha du
u\int
a
(u - t)\beta
\prime - 1E\gamma \prime ,q
\alpha ,\beta \prime \lambda (u - t)\alpha f(t)dt.
Changing the order of integration, which is justified by Fubini’s theorem, we get
I(\alpha , \beta , \gamma , q;\lambda )I
\bigl(
\alpha , \beta \prime , \gamma \prime , q;\lambda
\bigr)
f(x) =
=
x\int
a
f(t)dt
u\int
a
(u - t)\beta
\prime - 1(x - u)\beta - 1E\gamma ,q
\alpha ,\beta \{ \lambda (x - u)\alpha \} E\gamma \prime ,q
\alpha ,\beta \prime \{ \lambda (u - t)\alpha \} du.
Further simplification yields
= (x - t)\beta +\beta \prime - 1
\infty \sum
m=0
\infty \sum
n=0
(\gamma )qm(\gamma \prime )qn
\Gamma (\alpha m+ \beta )\Gamma (\alpha n+ \beta \prime )
\lambda m+n(x - t)\alpha m+\alpha n\Gamma (\alpha m+ \beta )\Gamma (\alpha n+ \beta \prime )
m!n!\Gamma (\alpha (m+ n) + \beta + \beta \prime )
=
= (x - t)\beta +\beta \prime - 1E\gamma +\gamma \prime ,q
\alpha ,\beta +\beta \prime \{ \lambda (x - t)\alpha \} ,
which leads to (9). On substituting \gamma \prime = - \gamma in (9), this reduces to (10), in accordance with the
following remark.
Remark 3.1. It should be noted that for \gamma = 0, the operator defined by (7), coincides with the
Riemann – Liouville fractional integral of order \beta ,
I(\alpha , \beta , 0, q;\lambda )f(x) = I\beta f(x). (11)
Similarly, under same techniques, used in Theorem 3.1, we prove commutative property.
Theorem 3.2 (commutative property). If \alpha , \beta , \gamma , \mu , \lambda \in C, q \in (0, 1) \cup N and \mathrm{R}\mathrm{e} (\alpha ), \mathrm{R}\mathrm{e} (\beta ),
\mathrm{R}\mathrm{e} (\gamma ), \mathrm{R}\mathrm{e} (\lambda ) > 0, then, for almost all x belongs to (a, b) and f \in L,
I\mu I(\beta )f(x) = I(\beta )I\mu f(x), (12)
that justifies that the operator I(\beta ) commutes with I\mu .
One can easily prove this theorem by using the results (6) and (7). Hence, details are avoided.
Further, we will need the following property given by Shukla and Prajapati [10] to obtain the solution
of the integral equation (4).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
624 R. DESAI, I. A. SALEHBHAI, A. K. SHUKLA
Theorem 3.3 (shifting property). Let \alpha , \beta , \gamma , \mu , \lambda \in C, q \in (0, 1) \cup N, \mathrm{R}\mathrm{e} (\mu ) > - \mathrm{R}\mathrm{e} (\beta ) and
\mathrm{R}\mathrm{e} (\alpha ) > 0, \mathrm{R}\mathrm{e} (\beta ) > 0, \mathrm{R}\mathrm{e} (\gamma ) > 0, \mathrm{R}\mathrm{e} (\lambda ) > 0, \mathrm{R}\mathrm{e} (\mu ) > 0. Then
I\mu I(\alpha , \beta , \gamma , q;\lambda ) = I(\alpha , \beta + \mu , \gamma , q;\lambda ). (13)
It can be observed that the composition of fraction integral operator I\mu and the operator I(\beta )
result in to shifting of the second parameter \beta of (7) by the order of fraction integral operator \mu ,
where as all other parameters remain unaltered. Thus, we call this property as a shifting property.
4. Solution of the integral equation. To obtain the solution of integral equation (4), we need
to prove the following lemma.
Lemma 4.1. For \alpha , \beta , \gamma , \lambda \in C, \mathrm{R}\mathrm{e} (\alpha ),\mathrm{R}\mathrm{e} (\beta ),\mathrm{R}\mathrm{e} (\gamma ),\mathrm{R}\mathrm{e} (\lambda ) > 0, and f \in L,
I - \beta I(\alpha , \beta , \gamma , q;\lambda )f(x) = f(x) + \alpha \lambda (\gamma )q
x\int
a
(x - t)\alpha - 1E\gamma +q,q
\alpha ,\alpha +1 \{ \lambda (x - t)\alpha \} f(t)dt.
Proof. By shifting property (13), we get
I1 - \beta I(\alpha , \beta , \gamma , q;\lambda )f(x) = I(\alpha , \beta + 1 - \beta , \gamma , q;\lambda )f(x) =
x\int
a
E\gamma ,q
\alpha ,1 \{ \lambda (x - t)\alpha \} f(t)dt.
Therefore,
I(\alpha , \beta , \gamma , q;\lambda )f(x) = I\beta - 1
x\int
a
E\gamma ,q
\alpha ,1 \{ \lambda (x - t)\alpha \} f(t)dt =
= I\beta
d
dx
x\int
a
E\gamma ,q
\alpha ,1 \{ \lambda (x - t)\alpha \} f(t)dt =
= I\beta
x\int
a
(\gamma )qE
\gamma +q,q
\alpha ,1+\alpha \{ \lambda (x - t)\alpha \} \lambda \alpha (x - t)\alpha - 1f(t)dt =
= I\beta
\left[ f(x) + \alpha \lambda (\gamma )q
x\int
a
(x - t)\alpha - 1E\gamma +q,q
\alpha ,\alpha +1 \{ \lambda (x - t)\alpha \} f(t)dt
\right] . (14)
Lemma 4.1 is proved.
Theorem 4.1 (the necessary condition for existence of solution). The existence of I - \beta g in L is
a necessary condition for the integral
x\int
a
(x - t)\beta - 1E\gamma ,q
\alpha ,\beta \{ \lambda (x - t)\alpha \} f(t)dt = g(t) (15)
to admit a solution f in L .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
INTEGRAL EQUATIONS INVOLVING GENERALIZED MITTAG-LEFFLER FUNCTION 625
Proof. Suppose the integral equation (15) has a solution f \in L. Then the equation (14) can be
written as
I(\alpha , \beta , \gamma , q;\lambda )f(x) = I\beta
\left[ f(x) + \alpha \lambda (\gamma )q
x\int
a
(x - t)\alpha - 1E\gamma +q,q
\alpha ,\alpha +1 \{ \lambda (x - t)\alpha \} f(t)dt
\right] = g(x).
(16)
For f \in L and by Theorem 2.1, it can be observed that the integral in (16) exists in L. Consequently,
I - \beta g exists in L. This completes the justification.
Once the necessary conditions are justified, inwhat follow is the solution of the integral equa-
tion (4).
Theorem 4.2 (the solution). If \mathrm{R}\mathrm{e} (\mu ) > \mathrm{R}\mathrm{e} (\beta ) > 0 and I - \mu g exists in L, then the integral
equation
x\int
a
(x - t)\beta - 1E\gamma ,q
\alpha ,\beta \{ \lambda (x - t)\alpha \} f(t)dt = g(t), (17)
for a < x < b, possesses a solution or a class of equivalent solutions f \in L given by
f(x) =
x\int
a
(x - t)\mu - \beta - 1E - \gamma ,q
\alpha ,\mu - \beta \{ \lambda (x - t)\alpha \} I - \mu g(t)dt. (18)
Proof. The equation (17) can be written as
I(\alpha , \beta , \gamma , q;\lambda )f(x) = g(x). (19)
On writing equation (18) as
I(\alpha , \mu - \beta , - \gamma , q;\lambda )I - \mu g(x) = f(x). (20)
Substituting for f(x) from result (20) to the left-hand side of result (19), we get
I(\alpha , \beta , \gamma , q;\lambda )f(x) = I(\alpha , \beta , \gamma , q;\lambda )I(\alpha , \mu - \beta , - \gamma , q;\lambda )I - \mu g(x). (21)
By composition property (9), we write
I(\alpha , \beta , \gamma , q;\lambda )f(x) = I(\alpha , \mu , 0, q;\lambda )I - \mu g(x).
By the Remark 3.1 and invoking (11), we arrive at
I(\alpha , \beta , \gamma , q;\lambda )f(x) = I\mu I - \mu g(x) = g(x).
Thus, f(x) as defined in (18) is proved to be a solution of the integral equation (4).
Corollary 4.1. Under the conditions of the above theorem, (17) and (18) imply each other.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
626 R. DESAI, I. A. SALEHBHAI, A. K. SHUKLA
It can be easily verified by Theorem 3.2 that the operator I (\beta ) commutes with I\mu . Now, to prove
(17) imply (18), we substitute the value of g(x) from (19) to the left-hand side of (20), we have
I(\alpha , \mu - \beta , - \gamma , q;\lambda )I - \mu I(\alpha , \beta , \gamma , q;\lambda )f(x).
By commutative property (12), above result becomes
I(\alpha , \mu - \beta , - \gamma , q;\lambda )I(\alpha , \beta , \gamma , q;\lambda )I - \mu f(x).
On applying result (21), we obtain f(x).
The converse can also be proved in a similar manner.
Remark 4.1. If q = 1, Theorem 4.2 reduces to the solution of integral equation due to Prab-
hakar [6].
Remark 4.2. For \alpha = 1, a = 0 and \gamma is a positive integer, Theorem 4.2 yield the transform pair
due to Wimp [17].
5. The integral equation (5). The integral operator defined in (8), assists in discussing (5). Actu-
ally, it can be verified that all results analogous to Theorems 2.1, 3.1 and 3.2 hold for I\ast (\alpha , \beta , \gamma , q;\lambda ),
because J\mu plays the same role as I\mu does for I(\alpha , \beta , \gamma , q;\lambda ).
Theorem 5.1. The existence of J - \beta g in L is a necessary condition for the integral
I\ast (\alpha , \beta , \gamma , q;\lambda ) =
b\int
x
(t - x)\beta - 1E\gamma ,q
\alpha ,\beta
\bigl\{
\lambda (x - t)\alpha
\bigr\}
f(t)dt = g(x)
to admit a solution f in L. Whereas, the existence of J - rg for \mathrm{R}\mathrm{e} (r) > \mathrm{R}\mathrm{e} (\beta ) > 0 is a sufficient
condition for equation (5) to admit a unique solution.
Theorem 5.2. If \mathrm{R}\mathrm{e} (r) > \mathrm{R}\mathrm{e} (\beta ) > 0 and J - rg exists in L, then the integral equations
b\int
x
(t - x)\beta - 1E\gamma ,q
\alpha ,\beta
\bigl\{
\lambda (x - t)\alpha
\bigr\}
f(t)dt = g(x) (22)
and
f(x) =
b\int
x
(t - x)r - \beta - 1E - \gamma ,q
\alpha ,r - \beta \{ \lambda (t - x)\alpha \} I - rg(t)dt
imply each other.
Owing to the similarity of the proof to Theorem 4.1 and Corollary 4.1, details are avoided.
Remark 5.1. For q = 1, result of integral equations becomes an apparent special case due to
Prabhakar [6].
Remark 5.2. When \alpha = 1, and parameters are specialized, (22) reduces to the integral equations
due to Srivastava [11, 12].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
INTEGRAL EQUATIONS INVOLVING GENERALIZED MITTAG-LEFFLER FUNCTION 627
6. Some additional result for \bfitE \bfitgamma ,\bfitq
\bfitalpha ,\bfitbeta .
Theorem 6.1. If \alpha , \beta , \gamma \in C, \mathrm{R}\mathrm{e} (\alpha ),\mathrm{R}\mathrm{e} (\beta ),\mathrm{R}\mathrm{e} (\gamma ) > 0, and q \in N, then\biggl(
qz
d
dz
+ \gamma
\biggr)
E\gamma ,q
\alpha ,\beta (z) = \gamma E\gamma +1,q
\alpha ,\beta (z). (23)
Proof. We have\biggl(
qz
d
dz
+ \gamma
\biggr)
E\gamma ,q
\alpha ,\beta (z) =
\infty \sum
n=0
(\gamma )qn
\Gamma (\alpha n+ \beta )
zn
n!
(nq + \gamma ) =
\infty \sum
n=0
(\gamma )(\gamma + 1)qn
\Gamma (\alpha n+ \beta )
zn
n!
= \gamma E\gamma +1,q
\alpha ,\beta (z).
Remark 6.1. On setting q = 1 in (23), this gives result due to Prabhakar [6] and for q = 1,
\gamma = 1 yields relations proved by Wiman [16].
7. Conclusion. Since long the integral equations with special functions are not studied, in this
paper, we obtained some properties of integral equation through generalized Mittag-Leffler function.
This work may be useful in the study on integral equation and special functions.
Acknowledgment. The authors are thankful to Prof. P. K. Banerji, JNVU, Jodhpur, India, for his
valuable suggestions.
References
1. R. Desai, I. A. Salehbhai, A. K. Shukla, Note on generalized Mittag-Leffler function, SpringerPlus, 5 (2016).
2. A. A. Kilbas, M. Saigo, On Mittag-Leffler type function, fractional calculas operators and solutions of integral
equations, Integral Transforms and Spec. Functions, 4, № 4, 355 – 370 (1996).
3. A. A. Kilbas, M. Saigo, R. K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus
operators, Integral Transforms and Spec. Functions, 15, № 1, 31 – 49 (2004).
4. K. S. Miller, B. Ross, An introduction to fractional calculus and fractional differential equations, Wiley, New York
(1993).
5. G. Mittag-Leffler, Sur la nouvelle fonction E\alpha (x), C. R. Acad. Sci. Paris, 137, 554 – 558 (1903).
6. T. R. Prabhakar, A singular equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J.,
19, 7 – 15 (1971).
7. S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives — theory and applications, Gordon
and Breach, Switzerland (1993).
8. R. K. Saxena, J. P. Chauhan, R. K. Jana, A. K. Shukla, Further results on the generalized Mittag-Leffler function
operator, J. Inequal. and Appl., 2015, № 75 (2015).
9. A. K. Shukla, J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. and
Appl., 336, № 2, 797 – 811 (2007).
10. A. K. Shukla, J. C. Prajapati, On a generalization of Mittag-Leffler function and generalized integral operator, Math.
Sci. Rec. J., 12, № 12, 283 – 290 (2008).
11. K. N. Srivastava, A class of integral equations involving Laguree polynomials as kernel, Proc. Edinburgh Math. Soc.,
15, 33 – 36 (1966).
12. K. N. Srivastava, On integral equations involving Whittaker’s function, Proc. Glasgow Math. Assoc., 7, 125 – 127
(1966).
13. H. M. Srivastava, R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. and Comput.,
118, 1 – 52 (2001).
14. H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler
function in the kernel, Appl. Math. and Comput., 211, № 1, 198 – 210 (2009).
15. Z. Tomovski, R. Hilfer, H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative
operators and Mittag-Leffler type functions, Integral Transforms and Spec. Functions, 21, № 11, 797 – 814 (2010).
16. A. Wiman, Über den fundamentalsatz in der theorie der funktionen E(x), Acta Math., 29, 191 – 201 (1905).
17. J. Wimp, Two integral transform pairs involving hypergeometric functions, Proc. Glasgow Math. Assoc., 7, 42 – 44
(1965).
Received 01.04.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
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| id | umjimathkievua-article-6014 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:25:15Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d6/4068a739bb35756bc3f8104c093095d6.pdf |
| spelling | umjimathkievua-article-60142022-03-26T11:01:44Z Integral equations involving generalized Mittag-Leffler function Integral equations involving generalized Mittag-Leffler function Integral equations involving generalized Mittag-Leffler function Desai, Rachana Salehbhai, I. A. Shukla, A. K. Desai, Rachana Salehbhai, I. A. Shukla, A. K. Desai, Rachana Salehbhai, I. A. Shukla, A. K. Integral equations; Integral Operators; Generalized Mittag-Leffler function; Fractional calculus UDC 517.5 The paper deals with solving the integral equation with a generalized Mittag-Leffler function $E_{\alpha ,\beta }^{\gamma ,q}(z)$ that defines a kernel using a fractional integral operator. The existence of the solution is justified and necessary conditions on the integral equation admiting a solution are discussed. Also, the solution of the integral equation is derived. Розглядається розв'язність інтегрального рівняння з узагальненою функцією Міттаг-Леффлера $E_{\alpha,\beta}^{\gamma,q}(z),$ що визначає ядро з використанням дробового інтегрального оператора. Існування розв'язку обгрунтовано, обговорюються необхідні умови для інтегрального рівняння, що допускають розв’язок. Також наведено розв’язок такого інтегрального рівняння. Institute of Mathematics, NAS of Ukraine 2020-06-05 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6014 10.37863/umzh.v72i5.6014 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 5 (2020); 620–627 Український математичний журнал; Том 72 № 5 (2020); 620–627 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/6014/8721 |
| spellingShingle | Desai, Rachana Salehbhai, I. A. Shukla, A. K. Desai, Rachana Salehbhai, I. A. Shukla, A. K. Desai, Rachana Salehbhai, I. A. Shukla, A. K. Integral equations involving generalized Mittag-Leffler function |
| title | Integral equations involving generalized Mittag-Leffler function |
| title_alt | Integral equations involving generalized Mittag-Leffler function Integral equations involving generalized Mittag-Leffler function |
| title_full | Integral equations involving generalized Mittag-Leffler function |
| title_fullStr | Integral equations involving generalized Mittag-Leffler function |
| title_full_unstemmed | Integral equations involving generalized Mittag-Leffler function |
| title_short | Integral equations involving generalized Mittag-Leffler function |
| title_sort | integral equations involving generalized mittag-leffler function |
| topic_facet | Integral equations Integral Operators Generalized Mittag-Leffler function Fractional calculus |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6014 |
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