On certain triple q-integral equations involving the third Jackson q ...

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Mansour and Al-Towailb Advances in Difference Equations (2016) 2016:81 DOI 10.1186/s13662-016-0809-3 RESEARCH Open Access On certain triple q-integral equations involving the third Jackson q-Bessel functions as kernel Zeinab S Mansour 1,2 and Maryam A Al-Towailb 3* * Correspondence: [email protected] 3 Department of Natural and Engineering Science, Faculty of Applied Studies and Community Service, King Saud University, Riyadh, Kingdom of Saudi Arabia Full list of author information is available at the end of the article Abstract In this paper, we consider two different systems of triple q-integral equations, where the kernel is a third Jackson q-Bessel function. The solution of the first system is reduced to two simultaneous Fredholm q-integral equations of the second kind. We solve the second system by using the solution of specific dual q-integral equations. Some examples are included. MSC: Primary 45F10; secondary 31B10; 26A33; 33D45 Keywords: third Jackson q-Bessel functions; fractional q-integral operators; triple q-integral equations 1 Introduction In the past years, several authors have described various methods to solve triple integral equations especially of the form A(u)K (u, x) du = f (x), < x < a, w(u)A(u)K (u, x) du = g (x), a < x < b, A(u)K (u, x) du = h(x), b < x < , where w(u) is the weight function, K (u, x) is the kernel function (see for example [–]). Some mixed boundary value problems of the mathematical theory of elasticity are solved by reducing them to multiple integral equations. For example, the axisymmetric problem of a torsion of an elastic space, weakened by a conical crack, under the assumption that on the boundaries of the crack, the tangential displacements of the shear stress are pre- scribed, is solved by the application of dual integral equations (see []). Harmonic shear oscillations of a rigid stamp with a plane base coupled to an elastic half-space were stud- ied in [] and reduced to dual integral equations. For more applications see [, ]. This type of equations can be solved by a Fredholm integral equation of the second kind by us- ing the modified Hankel transform operator and the Erdélyi-Kober fractional integration operator. © 2016 Mansour and Al-Towailb. This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Transcript of On certain triple q-integral equations involving the third Jackson q ...

Page 1: On certain triple q-integral equations involving the third Jackson q ...

Mansour and Al-Towailb Advances in Difference Equations (2016) 2016:81 DOI 10.1186/s13662-016-0809-3

R E S E A R C H Open Access

On certain triple q-integral equationsinvolving the third Jackson q-Bessel functionsas kernelZeinab S Mansour1,2 and Maryam A Al-Towailb3*

*Correspondence:[email protected] of Natural andEngineering Science, Faculty ofApplied Studies and CommunityService, King Saud University,Riyadh, Kingdom of Saudi ArabiaFull list of author information isavailable at the end of the article

AbstractIn this paper, we consider two different systems of triple q-integral equations, wherethe kernel is a third Jackson q-Bessel function. The solution of the first system isreduced to two simultaneous Fredholm q-integral equations of the second kind. Wesolve the second system by using the solution of specific dual q-integral equations.Some examples are included.

MSC: Primary 45F10; secondary 31B10; 26A33; 33D45

Keywords: third Jackson q-Bessel functions; fractional q-integral operators; tripleq-integral equations

1 IntroductionIn the past years, several authors have described various methods to solve triple integralequations especially of the form

∫ ∞

A(u)K(u, x) du = f (x), < x < a,

∫ ∞

w(u)A(u)K(u, x) du = g(x), a < x < b,

∫ ∞

A(u)K(u, x) du = h(x), b < x < ∞,

where w(u) is the weight function, K(u, x) is the kernel function (see for example [–]).Some mixed boundary value problems of the mathematical theory of elasticity are solved

by reducing them to multiple integral equations. For example, the axisymmetric problemof a torsion of an elastic space, weakened by a conical crack, under the assumption thaton the boundaries of the crack, the tangential displacements of the shear stress are pre-scribed, is solved by the application of dual integral equations (see []). Harmonic shearoscillations of a rigid stamp with a plane base coupled to an elastic half-space were stud-ied in [] and reduced to dual integral equations. For more applications see [, ]. Thistype of equations can be solved by a Fredholm integral equation of the second kind by us-ing the modified Hankel transform operator and the Erdélyi-Kober fractional integrationoperator.

© 2016 Mansour and Al-Towailb. This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commonslicense, and indicate if changes were made.

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In this paper, we consider triple q-integral equation where the kernel is the third Jacksonq-Bessel function and the q-integral is a Jackson q-integral. It is worth mentioning thatdifferent approaches for solving a dual q-integral equation are in []. Also, solutions fordual and triple sequence involving q-orthogonal polynomials are in [].

The paper is organized in the following manner. The next section includes the main no-tations and some results we need in our investigations. In Section , we solve the tripleq-integral equations by reducing the system to two simultaneous Fredholm q-integralequation of the second kind, we shall use a method due to Singh et al. []. The approachdepends on fractional q-calculus. Furthermore, we will conclude the solutions of two dualq-integral equations to be special cases of the solution of the triple q-integral equation,and we show that this coincides with the results in []. In the last section, we use a resultfrom [] for a solution of dual q-integral equations to solve triple q-integral equations.The result of this section is a q-analog of the results introduced by Cooke in [].

2 q-Notations and resultsThroughout this paper, we will assume that q is a positive number less than one andwe follow Gasper and Rahman [] for the definitions of the q-shifted factorial, multi-ple q-shifted factorials, basic hypergeometric series, Jackson q-integrals, the q-gamma,and beta functions. We also follow Annaby and Mansour [] for the definition of theq-derivative at zero.

For t > , let Aq,t , Bq,t , and Rq,t,+ be the sets defined by

Aq,t :={

tqn : n ∈N}

, Bq,t :={

tq–n : n ∈N}

,

Rq,t,+ :={

tqk : k ∈ Z}

,

where N := {, , , . . .}, and N := {, , . . .}. Notice, if t = we write Aq, Bq, and Rq,+, andwe define the following spaces:

Lq,η(Rq,+) :={

f : ‖f ‖q,η :=∫ ∞

∣∣tηf (t)∣∣dqt < ∞

},

Lq,η(Aq) :={

f : ‖f ‖Aq ,η :=∫

∣∣tηf (t)∣∣dqt < ∞

},

Lq,η(Bq) :={

f : ‖f ‖Bq ,η :=∫ ∞

∣∣tηf (t)∣∣dqt < ∞

},

where η ∈C and Lq,η(Rq,+) = Lq,η(Aq) ∩ Lq,η(Bq).Koornwinder and Swarttouw [] introduced the following inverse pair of q-Hankel in-

tegral transforms under the side condition f , g ∈ Lq(Rq,+):

g(λ) =∫ ∞

xf (x)Jν

(λx; q)dqx; f (x) =

∫ ∞

λg(λ)Jν

(λx; q)dqλ, (.)

where λ, x ∈Rq,+.Now we recall some definitions and results which will be needed in the sequel. Let α ∈C,

the q-binomial coefficient is defined by

k

]

q

=

{, k = ,(–qα )(–qα–)···(–qα–k+)

(q;q)k, k ∈N.

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The third Jackson q-Bessel function J ()ν (z; q), see [] and [], is defined by

Jν(z; q) = J ()ν (z; q) :=

(qν+; q)∞(q; q)∞

∞∑n=

(–)n qn(n+)/zn+ν

(q; q)n(qν+; q)n, z ∈C, (.)

and it satisfies the following relations (see []):

Dq[(·)–νJν

(·; q)](z) = –q–νz–ν

– qJν+

(qz; q), (.)

Dq[(·)νJν

(·; q)](z) =zν

– qJν–

(z; q). (.)

Also, for �(ν) > –, the q-Bessel function Jν(·; q) satisfies (see [])

∣∣Jν(qn; q)∣∣ ≤ (–q; q)∞(–qν+; q)∞(q; q)∞

{qnν , n ∈N,qn–(ν+)n, n ∈N.

(.)

The functions cos(z; q) and sin(z; q) are defined by

cos(z; q) :=(q; q)∞(q; q)∞

(zq–

( – q))

J–

(z( – q)/

√q; q),

sin(z; q) :=(q; q)∞(q; q)∞

(z( – q)

) J

(z( – q); q), z ∈C.

We need the following results from [].

Proposition . Let α,β ∈C, ρ, t ∈ Rq,+. Then, for R(β) > R(α) > –,

∫ ∞

tα–β+Jα

(ξ t; q)Jβ

(ρt; q)dqt

=

⎧⎨⎩

, ξ > ρ,(–q)(–q)–β+α

q (β–α) ξαρ(β–α–)(qξ /ρ; q)β–α–, ξ ≤ ρ.

Proposition . Let ν and α be complex numbers such that �(ν) > –. Then, for ρ, u ∈Rq,+,

∫ ∞

ρ

xα–ν–(ρ/x; q)α–Jν

(ux; q)dqx =

( – q)q (α)( – q)–α

ρα–νu–αqαJν–α

(uρ/q; q).

Proposition . Let x, ν , and γ be complex numbers and u ∈ Rq,+. Then, for �(γ ) > –and �(ν) > –, the following identity holds:

∫ x

ρν+(qρ/x; q)

γJν

(uρ; q)dqρ

= xν–γ +u–γ –( – q)( – q)γ

q (γ + )Jγ +ν+(ux; q). (.)

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Moreover, if �(γ ) > and �(ν) > –, then

∫ ∞

xργ –ν–(x/ρ; q)

γ –Jν(uρ; q)dqρ

= xγ –νu–γ ( – q)qγ (q; q)∞(qγ ; q)∞

Jν–γ

(uxq

; q)

. (.)

The following are consequences of the above results.

Lemma . Let x, u, and α be complex numbers such that u ∈Rq,+, �(α) > –, and �(ν) >–. Then

uαJν–α

(ux; q)

=( – q)α

q ( – α)xα–ν–Dq,x

[x–α

∫ x

ρν+(qρ/x; q)

–αJν

(uρ; q)dqρ

]. (.)

Proof Applying (.) with γ = –α, we have

∫ x

ρν+(qρ/x; q)

–αJν

(uρ; q)dqρ

= xν+α+uα–( – q)( – q)–α

q ( – α)Jν–α+(ux; q). (.)

Multiplying both sides of equation (.) by x–α , and then calculating the q-derivative ofthe two sides with respect to x and using (.), we get the required result. �

Similarly, by using (.) we obtain the following result.

Lemma . Let x, u, and α be complex numbers such that u ∈ Rq,+, �(α) > , and �(ν) >–. Then

uαJν+α

(ux; q)

= –( – q)αqα+ν–xα+ν–

q ( – α)Dq,x

∫ ∞

xρ–α–ν+(x/ρ; q)

–αJν

(uρ; q)dqρ. (.)

We end this section by introducing some q-fractional operators that we use in solvingthe triple q-integral equations under consideration. The technique of using fractional op-erators in solving dual and triple integral equations is not new. See for example [, , ].

A q-analog of the Riemann-Liouville fractional integral operator is introduced in []by Al-Salam through

Iαq f (x) :=

xα–

q(α)

∫ x

(qt/x; q)α–f (t) dqt, α /∈ {–, –, . . .}.

In [], Agarwal defined the q-fractional derivative to be

Dαq f (x) := I–α

q f (x) =x–α–

q(–α)

∫ x

(qt/x; q)–α–f (t) dqt.

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Also, we have (see [], Lemma .)

Iαq Dα

q f (x) = f (x) – I–αq f ()

xα–

q(α), < α < . (.)

In [], Al-Salam defined a two parameter q-fractional operator by

Kη,αq φ(x) :=

q–ηxη

q(α)

∫ ∞

x(x/t; q)α–t–η–φ

(tq–α

)dqt,

α = –, –, . . . . This is a q-analog of the Erdélyi and Sneddon fractional operator, cf. [,],

Kη,αf (x) =xη

(α)

∫ ∞

x(t – x)α–t–η–f (t) dt.

In [], the authors introduced a slight modification of the operator Kη,αq . This operator is

denoted by Kη,αq and defined by

Kη,αq φ(x) :=

q–ηxη

q(α)

∫ ∞

x(x/t; q)α–t–η–φ(qt) dqt, (.)

where α = –, –, . . . . In case of η = –α, we set

Kαq f (x) := q–αxαqα(α–)/K–α,α

q f (x)

=q–α(α–)/

q(α)

∫ ∞

xtα–(x/t; q)α–f (qt) dqt. (.)

Note that this operator satisfies the following semigroup identity:

KαqKβ

q φ(x) = Kα+βq φ(x) for all α and β . (.)

The proof of (.) is completely similar to the proof of Theorem . in [] and is omit-ted.

Lemma . Let α ∈ C, x ∈ Bq. If � ∈ Lq,α–(Bq) and G(x) = Dq,xKαq �(x), then

�(x) = –qα–K–αq G

(xq

).

Proof According to (.), we have

G(x) =q–α(α–)/

q(α)Dq,x

∫ ∞

xtα–(x/t; q)α–�(qt) dqt

=q–α(α–)/

q(α)

[∫ ∞

xtα–(Dq,x(x/t; q)α–

)�(qt) dqt – xα–(q; q)α–�(qx)

]. (.)

Note that

Dq,x(x/t; q)α– = –( – qα–)t( – q)

(qx/t; q)α– = –t

[α – ](qx/t; q)α–

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and∫ ∞

qxg(t) dqt =

∫ ∞

xg(t) dqt + x( – q)g(x).

Hence,

G(x) = –q–α(α–)/

q(α)[α – ]

[∫ ∞

xtα–(qx/t; q)α–�(qt) dqt – xα–(q; q)α–�(qx)

]

= –q–α(α–)/

q(α)[α – ]

∫ ∞

qxtα–(qx/t; q)α–�(qt) dqt

= –q–α(α–)/

q(α – )

∫ ∞

qxtα–(qx/t; q)α–�(qt) dqt = –q–αK(α–)

q �(qx).

This implies

K(α–)q �(x) = –qα–G(x/q).

Using (.), we obtain the result and completes the proof. �

3 A system of triple q-integral equationsThe goal of this section is to solve the following triple q-integral equations:

∫ ∞

ψ(u)Jν

(uρ; q)dqu = f(ρ), ρ ∈ Aq,a, (.)

∫ ∞

u–αψ(u)

[ + w(u)

]Jν

(uρ; q)dqu = f(ρ), ρ ∈ Aq,b ∩ Bq,a, (.)

∫ ∞

ψ(u)Jν

(uρ; q)dqu = f(ρ), ρ ∈ Bq,b, (.)

where < a < b < ∞, and α, ν are complex numbers satisfying

�(ν) > – and < �(α) < .

ψ is an unknown function to be determined, fi (i = , , ) are known functions, and w is anon-negative bounded function defined on Rq,+.

Clearly from (.), a sufficient condition for the convergence of the q-integrals on theleft-hand side of (.)-(.) is that

ψ ∈ Lq,ν(Rq,+) ∩ Lq,ν–α(Rq,+). (.)

For getting the solution of the triple q-integral equations (.)-(.), we define a functionC by

C(u) := u–αψ(u)[ + w(u)

], u ∈ Rq,+.

This implies

ψ(u) = uαC(u) – uαC(u)[

w(u) + w(u)

],

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and the triple q-integral equations (.)-(.) can be represented as

∫ ∞

uαC(u)Jν

(uρ; q)dqu –

∫ ∞

uαC(u)

[w(u)

+ w(u)

]Jν

(uρ; q)dqu

= f(ρ), ρ ∈ Aq,a, (.)∫ ∞

C(u)Jν

(uρ; q)dqu = f(ρ), ρ ∈ Aq,b ∩ Bq,a, (.)

∫ ∞

uαC(u)Jν

(uρ; q)dqu –

∫ ∞

w(u) + w(u)

uαC(u)Jν(uρ; q)dqu

= f(ρ), ρ ∈ Bq,b. (.)

Since equation (.) is linear in C, we may assume that C := C + C and

f = g + g, on Aq,b ∩ Bq,a,

where g defined on Aq,b and g defined on Bq,a. Therefore,

∫ ∞

C(u)Jν

(uρ; q)dqu = g(ρ), ρ ∈ Aq,b,

∫ ∞

C(u)Jν

(uρ; q)dqu = g(ρ), ρ ∈ Bq,a.

So, the triple q-integral equations (.)-(.) can be rewritten in the following form:

∫ ∞

[C(u) + C(u)

]Jν

(uρ; q)dqu

–∫ ∞

[C(u) + C(u)

] w(u) + w(u)

Jν(uρ; q)dqu

= f(ρ), ρ ∈ Aq,a, (.)∫ ∞

C(u)Jν

(uρ; q)dqu = g(ρ), ρ ∈ Aq,b, (.)

∫ ∞

C(u)Jν

(uρ; q)dqu = g(ρ), ρ ∈ Bq,a, (.)

∫ ∞

[C(u) + C(u)

]Jν

(uρ; q)dqu

–∫ ∞

w(u) + w(u)

uα[C(u) + C(u)

]Jν

(uρ; q)dqu

= f(ρ), ρ ∈ Bq,b. (.)

Proposition . Let ψ, ψ be the functions defined by

ψ(x) :=∫ ∞

uαC(u)Jν–α

(ux; q)dqu, x ∈ Bq,b, (.)

ψ(x) :=∫ ∞

uαC(u)Jν+α

(ux; q)dqu, x ∈ Aq,a, (.)

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provided that < �(α) < , �(ν) > –, �(ν + α) > , and C ∈ Lq,ν(Rq,+), C ∈ Lq,–t(Rq,+)where

�(ν) + > �(t) > –�(ν) + �( – α).

Then, for u ∈Rq,+, we have

C(u) = u–α

[∫ b

x�(x)Jν–α

(ux; q)dqx +

∫ ∞

bxψ(x)Jν–α

(ux; q)dqx

], (.)

C(u) = u–α

[∫ a

xψ(x)Jν+α

(ux; q)dqx +

∫ ∞

ax�(x)Jν+α

(ux; q)dqx

], (.)

where

�(x) =( – q)αxα–ν–

q ( – α)Dq,x

[x–α

∫ x

g(ρ)ρν+(qρ/x; q)

–αdqρ

]

=( – q)αxα–ν–Dα

q,x

((·)ν/g(

√·))(x), x ∈ Aq,b, (.)

�(x) = –( – q)αqα+ν–xα+ν–

q ( – α)Dq,x

∫ ∞

xg(ρ)ρ–α–ν

(x/ρ; q)

–αdqρ

= –qα(–α)

( – q)αxα+ν–Dq,xK(–α)

q

[(·)–ν/g(

√·)](

xq

), x ∈ Bq,a. (.)

Proof We start with proving (.). Let x ∈ Aq,b. Multiplying both sides of (.) byx–αρν+(qρ/x; q)–α and then integrating with respect to ρ from to x, we get

∫ x

x–αρν+(qρ/x; q)

–α

∫ ∞

C(u)Jν

(uρ; q)dqu dqρ

=∫ x

g(ρ)x–αρν+(qρ/x; q)

–αdqρ. (.)

Notice that the double q-integral on the left-hand side of (.) is absolutely convergentfor < �(α) < and for �(ν) > – provided that C ∈ Lq,ν(Rq,+). So, we can interchange theorder of the q-integrations to obtain

∫ ∞

C(u)x–α

∫ x

ρν+

(qρ

x ; q)

–α

Jν(uρ; q)dqρ dqu

=∫ x

g(ρ)x–αρν+

(qρ

x ; q)

–α

dqρ. (.)

By calculating the q-derivative of the two sides of (.) with respect to x and using (.),we get

∫ ∞

uαC(u)Jν–α

(ux; q)dqu = �(x), x ∈ Aq,b, (.)

where

�(x) =( – q)αxα–ν–

q ( – α)Dq,x

[x–α

∫ x

g(ρ)x–αρν+

(qρ

x ; q)

–α

dqρ

].

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To prove (.), let x ∈ Bq,a. Multiplying both sides of (.) by ρ–α–ν+(x/ρ; q)–α andq-integrating with respect to ρ from x to ∞, we get

∫ ∞

xρ–α–ν

(x/ρ; q)

–α

∫ ∞

C(u)Jν

(uρ; q)dqu dqρ

=∫ ∞

xg(ρ)ρ–α–ν+(x/ρ; q)

–αdqρ. (.)

From (.), we can prove that utJν(u; q) is bounded on Rq,+ provided that �(t + ν) > –.So, if we take t such that �(ν) + > �(t) > –�(ν) + �( – α), we can prove that the doubleq-integral

∫ ∞

xρ–α–ν

(x/ρ; q)

–α

∫ ∞

C(u)Jν

(uρ; q)dqu dqρ

is absolutely convergent and then we can interchange the order of the q-integration toobtain

∫ ∞

C(u)

∫ ∞

xρ–α–ν

(x/ρ; q)

–αJν

(uρ; q)dqρ dqu

=∫ ∞

xg(ρ)ρ–α–ν+(x/ρ; q)

–αdqρ. (.)

Calculating the q-derivative of the two sides of (.) with respect to x and using (.)yields

∫ ∞

uαC(u)Jν+α

(ux; q)dqu = �(x), x ∈ Bq,a, (.)

where

�(x) = –( – q)αqα+ν–xα+ν–

q ( – α)Dq,x

∫ ∞

xg(ρ)ρ–α–ν

(x/ρ; q)

–αdqρ.

By the above argument, if we assume that ψ and ψ are given by (.) and (.), then

∫ ∞

uαC(u)Jν–α

(ux; q)dqx =

{φ(x), x ∈ Aq,b,ψ(x), x ∈ Bq,b

(.)

and

∫ ∞

uαC(u)Jν+α

(ux; q)dqx =

{φ(x), x ∈ Bq,a,ψ(x), x ∈ Aq,a.

(.)

Hence, (.) and (.) follow by applying the inverse pair of q-Hankel transforms (.)on (.) and (.). This completes the proof. �

Remark . From the definitions of ψi and φi, i = , , in Proposition ., one can verifythat x–ν–αφ is a bounded function in Bq,a and x–ν–αψ is bounded in Aq,a. Also, x–ν+αφ isbounded in Aq,b and x–ν+αψ is bounded in Bq,b.

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Proposition . For ρ ∈ Bq,b, ψ(ρ) satisfies the Fredholm q-integral equation of the form

ψ(ρ) = F̃(ρ) +q–α–α+ν

( – q)

∫ ∞

bxψ(x)K(ρ, x) dqx, (.)

where

K(ρ, x) =∫ ∞

uw(u) + w(u)

Jν–α

(ux; q)Jν–α

(uρ; q)dqu,

F̃(ρ) = F(ρ) –q–α–α+ν

( – q)

∫ a

xψ(x)

∫ ∞

u + w(u)

Jν+α

(ux; q)Jν–α

(uρ; q)dqu dqx,

and

F(ρ) = ρν–α q–α–α+ν( + q)( – q)–α

( – q)q (α)

∫ ∞

ρ

xα–ν–f(qx)(ρ/x; q)

α– dqx

–q–α–α+ν

( – q)

[∫ ∞

ax�(x)

∫ ∞

u + w(u)

Jν+α

(ux; q)Jν–α

(uρ; q)dqu dqx

+∫ b

x�(x)

∫ ∞

uw(u) + w(u)

Jν–α

(ux; q)Jν–α

(uρ; q)dqu dqx

].

Proof Equation (.) can be written in the following form:

∫ ∞

uαC(u)Jν

(uρ; q)dqu = G(ρ), ρ ∈ Bq,b, (.)

where

G(ρ) = f(ρ) –∫ ∞

uαC(u)

+ w(u)

Jν(uρ; q)dqu

+∫ ∞

uαC(u)

w(u) + w(u)

Jν(uρ; q)dqu. (.)

By using equations (.) and (.), we get

G(ρ) = –( – q)ρν–qν–Dq,ρρ–ν

∫ ∞

uα–C(u)Jν–

(uρq–; q)dqu. (.)

Substituting the value of C(u) from (.) into (.), we obtain

Dq,ρρ–ν

∫ ∞

[∫ b

x�(x)Jν–α

(ux; q)dqx +

∫ ∞

bxψ(x)Jν–α

(ux; q)dqx

]

× Jν–(uρq–; q)dqu = –

ρ–νq–νG(ρ)( – q)

, ρ ∈ Bq,b. (.)

From (.), there exists M > such that

∣∣Jν–α

(ux; q)∣∣ ≤ M(ux)�(ν–α) for all u, x ∈ Rq,b,+.

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Hence, from Remark ., the double q-integration is absolutely convergent and we caninterchange the order of the q-integrations to obtain

G(ρ) = –( – q)ρν–qν–[∫ b

x�(x) dqx +

∫ ∞

bxψ(x) dqx

]

× Dq,ρρ–ν

∫ ∞

uαJν–

(uρq–; q)Jν–α

(ux; q)dqu, ρ ∈ Bq,b. (.)

Therefore, applying Proposition . with �(ν – α) > �(ν – ) > – we obtain

G(ρ) =–( – q)( – q)α

q ( – α)ρν–Dq,ρ

∫ ∞

ρ

x–ν–αψ(x)(ρ/x; q)

–αdqx. (.)

By using

∫ ∞

xf (t) dqt =

+ q

∫ ∞

x

f (√

t)√t

dq t, Dq,ρ(f(ρ)) = ρ( + q)(Dq f )

(ρ), (.)

we obtain

G(ρ) =–( – q)( – q)α

q ( – α)ρνDq,ρ

∫ ∞

ρx

–(ν+α) ψ(

√x)

(ρ/x; q)

–αdq x

= –( – q)( – q)αqα–α–νρν(DqK–α

q((·)– ν+α

ψ(·)))(ρ/q).

Replacing ρ by qρ yields

–q–α+α( – q)–α( – q)–[(·)–ν/G(q

√·)](ρ)

= Dq,ρK–αq

[(√·)(α–ν)ψ(

√·)](ρ). (.)

Thus, applying Proposition . yields

ρα–νψ(ρ) = q–α( – q)–( – q)–αKα

q[(·)–ν/G(q

√·)](ρ/q)

=q–α–α+ν( – q)–α( – q)–

q (α)

∫ ∞

ρx– ν

+α–G(q√

x)(ρ/x; q)

α– dq x.

Using∫ ∞

x f (t) dq t = ( + q)∫ ∞

x tf (t) dqt, we obtain

ρα–νψ(ρ) =q–α–α+ν( – q)–α( + q)

( – q)q (α)

∫ ∞

ρ

xα–ν–G(qx)(ρ/x; q)

α– dqx.

From (.), we can write the last equation in the following form:

ψ(ρ) + ρν–α q–α–α+ν( – q)–α( + q)( – q)q (α)

∫ ∞

ρ

xα–ν–(ρ/x; q)α–

×[∫ ∞

+ w(u)C(u)Jν

(qux; q)dqu

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–∫ ∞

w(u) + w(u)

uαC(u)Jν(qux; q)dqu

]dqx

= ρν–α q–α–α+ν( – q)–α( + q)( – q)q (α)

×∫ ∞

ρ

xα–ν–f(qx)(ρ/x; q)

α– dqx, ρ ∈ Bq,b. (.)

From the condition on the function C, we can prove that the double q-integration∫ ∞

ρ

xα–ν–(ρ/x; q)α–

∫ ∞

C(u)

+ w(u)Jν

(qux; q)dqu dqx

is absolutely convergent. Therefore, we can interchange the order of the q-integrationsand use Proposition . to obtain

ψ(ρ) +q–α–α+ν

( – q)

[∫ ∞

+ w(u)C(u)Jν–α

(uρ; q)dqu

–∫ ∞

uαw(u) + w(u)

C(u)Jν–α

(uρ; q)dqu

]

= ρν–α q–α–α+ν( – q)–α( + q)( – q)q (α)

×∫ ∞

ρ

xα–ν–f(qx)(ρ/x; q)

α– dqx, ρ ∈ Bq,b. (.)

Substituting the value of C(u) and C(u) from equations (.) and (.) into equation(.), and then interchanging the order of the q-integrations we get

ψ(ρ) +qν–α

( – q)

[∫ a

xψ(x)

∫ ∞

u + w(u)

Jν+α

(ux; q)Jν–α

(uρ; q)dqu dqx

–∫ ∞

bxψ(x)

∫ ∞

uw(u) + w(u)

Jν–α

(ux; q)Jν–α

(uρ; q)dqu dqx

]

= F(ρ), ρ ∈ Bq,b, (.)

where

F(ρ) = ρν+α qν–α( + q)( – q)–α

( – q)q (α)

∫ ∞

ρ

xα–ν–f(qx)(ρ/x; q)

α– dqx

–qν–α

( – q)

[∫ ∞

ax�(x)

∫ ∞

u + w(u)

Jν+α

(ux; q)Jν–α

(uρ; q)dqu dqx

+∫ b

x�(x)

∫ ∞

uw(u) + w(u)

Jν–α

(ux; q)Jν–α

(uρ; q)dqu dqx

].

Equation (.) is nothing else but the Fredholm q-integral equation of the second kind(.). This completes the proof. �

Proposition . For ρ ∈ Aq,a, ψ(ρ) satisfies the Fredholm q-integral equation of the form

ψ(ρ) = F̃(ρ) +

( – q)

∫ a

xK(ρ, x)ψ(x) dqx, (.)

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where

K(ρ, x) =∫ ∞

uw(u) + w(u)

Jν+α

(ux; q)Jν+α

(uρ; q)dqu,

F̃(ρ) = F(ρ) –

( – q)

∫ ∞

bxψ(x)

∫ ∞

u + w(u)

Jν–α

(ux; q)Jν+α

(uρ; q)dqu dqx,

and

F(ρ) =( – q)–α( + q)ρα–ν–

( – q)q (α)

∫ ρ

(qx/ρ; q)

α–xν+f(x) dqx

+

( – q)

∫ ∞

ax�(x)

∫ ∞

uw(u) + w(u)

Jν+α

(ux; q)Jν+α

(uρ; q)dqu dqx

( – q)

∫ b

x�(x)

∫ ∞

u + w(u)

Jν–α

(ux; q)Jν+α

(uρ; q)dqu dqx.

Proof The proof is similar to the proof of Proposition . and is omitted. �

Theorem . The solution of (.)-(.) is given by

ψ(u) =uα

+ w(u)(C(u) + C(u)

).

The functions C, C, φ, and φ are given by Proposition ., and ψ, ψ satisfies the Fred-holm q-integral equations (.) and (.) of second kind.

Example . Take b = aq–m and assume that m → ∞. If we assume that f = f , f = f , andw = . Then the system (.)-(.) is reduced to the dual q-integral equations

∫ ∞

ψ(u)Jν

(uρ; q)dqu = f (ρ), ρ ∈ Aq,a, (.)

∫ ∞

u–αψ(u)Jν

(uρ; q)dqu = , ρ ∈ Bq,a. (.)

Hence, from Theorem .,

ψ(u) = u+α

∫ ∞

xψ(x)Jν+α

(ux; q)dqx, u ∈Rq,+,

ψ(ρ) =( – q)–α( + q)ρα–ν–

( – q)q (α)

∫ ρ

(qx/ρ; q)

α–xν+f (x) dqx

= ρ–α–ν ( – q)–α

( – q) Iαq

(tν/f (

√t)

)(ρ).

Hence,

ψ(u) = u+α ( – q)–α

( – q)

∫ ∞

x–α–νIα

q(tν/f (

√t)

)(x)Jν+α

(ux; q)dqx.

This coincides with the result in [], Theorem ., for solutions of double q-integral equa-tions.

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. Let a = qm and assume that m → ∞. If we assume that f = , and f = f , we obtainthe dual q-integral system of equations

∫ ∞

u–αψ(u)Jν

(uρ; q)dqu = , ρ ∈ Aq,b, (.)

∫ ∞

ψ(u)Jν

(uρ; q)dqu = f , ρ ∈ Bq,b. (.)

Hence, from Theorem .,

ψ(u) = u+α

∫ ∞

bxψ(x)Jν–α

(ux; q)dqx, u ∈ Rq,+,

ψ(ρ) = –( – q)–αq–αρα+ν

( – q)q (α)

∫ ∞

ρ

(ρ/x; q)

α–xα–ν–f (x) dqx.

This is a special case of Theorem . in [].

Example We consider the triple q-integral equations

∫ ∞

ψ(u)J

(uρ; q)dqu = , ρ ∈ Aq,a, (.)

∫ ∞

u–ψ(u)J

(uρ; q)dqu = , ρ ∈ Aq,b ∩ Bq,a, (.)

∫ ∞

ψ(u)J

(uρ; q)dqu = , ρ ∈ Bq,b. (.)

Hence, we have ν = , g = , g = , f = f = , w = , and α = .

From Theorem .,

ψ(u) = u(C(u) + C(u)

),

where

C(u) =( – q)( – q)

q (/)

sin( bu–q ; q)u

+√

– q

q (/)

∫ ∞

b

√xψ(x) cos

(xu√q – q

; q)

dqx,

C(u) =√

– q

q (/)

∫ a

√xψ(x) sin

(xu

– q; q

)dqx,

ψ(ρ) =√

ρ( + q)q( – q)

q (/)

∫ a

x/ ψ(x)

qρ – x dqx, ρ ∈ Bq,b, (.)

ψ(ρ) = –( + q)√ρ

( – q)q (/)

∫ ∞

b

√xψ(x)

qx – ρ dqx

+( + q)/

√ – q

q (/)√

ρ

∫ b

ρ/q

dqxqx – ρ . (.)

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We used [], pp.- or Proposition . of [] to calculate ψ and ψ in equations(.) and (.), respectively. Substituting from (.) into (.), we obtain the secondorder Fredholm q-integral equation

ψ(ρ) = –q–√ρ( + q)

( – q)q (/)

∫ a

t/ψ(t)K(ρ, t) dqt

+( + q)/

√ – q

q (/)√

ρ

∫ b

ρ/q

dqxqx – ρ , (.)

where ρ ∈ Aq,a and

K(ρ, t) =∫ ∞

b

x(t – qx)(ρ – qx)

dqt.

4 Solving system of triple q2-integral equations by using solutions of dualq-integral equations

In [], Cooke solved certain triple integral equations involving Bessel functions by using aresult for Noble [] for solutions for dual integral equations with Bessel functions as ker-nel. In this section, we use the result, Theorem A, introduced in [] to solve the followingtriple q-integral equations:

ξ–γ

∫ ∞

ρ–γ ψ(ρ)Jκ

(√ρξ ; q)dqρ = f (ξ ), ξ ∈ Aq , (.)

ξ–α

∫ ∞

ρ–αψ(ρ)Jμ

(√ρξ ; q)dqρ = g(ξ ), ξ ∈ Aq ∩ Bq , (.)

ξ–β

∫ ∞

ρ–βψ(ρ)Jν

(√ρξ ; q)dqρ = h(ξ ), ξ ∈ Bq , (.)

where a, α, β , γ , μ, ν , and κ are complex numbers such that

�(ν) > –, �(μ) > –, �(κ) > –, and < a < ,

the functions f (ρ), g(ρ), and h(ρ) are known functions, and ψ(u) is the solution functionto be determined.

The following is a result from [] that we shall use to solve the system (.)-(.).

Theorem A Let α, β , μ, and ν be complex numbers and let λ := (μ + ν) – (α – β) > –.

Assume that

�(ν) > –, �(μ) > –, �(λ) > –, and �(λ – μ – α) > .

Let f ∈ Lq, μ +α(Aq ) and g ∈ Lq, –μ +α–(Bq ). Then the dual q-integral equations

ξ–α

∫ ∞

ρ–αψ(ρ)Jμ

(√ρξ ; q)dqρ = f (ξ ), ξ ∈ Aq , (.)

ξ–β

∫ ∞

ρ–βψ(ρ)Jν

(√ρξ ; q)dqρ = g(ξ ), ξ ∈ Bq , (.)

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has a solution of the form

ψ(ξ ) =( – q)λ–ν+α–

ξλ/–μ/+α

(√ρξ ; q)Iμ/+α,λ–μ

q f (ρ) dqρ

+( – q)λ–ν–

ξλ/–μ/+α

∫ ∞

(√ρξ ; q)Kλ/–ν/–β ,ν–λ

q g(ρ) dqρ,

in Lq, μ –α(Rq,+) ∩ Lq, ν –β (Rq,+) ∩ Lq, ν –β–γ (Rq,+), for γ satisfying

+ �(ν) > �(γ ) > max{

,�(ν – λ)}

.

Now we shall solve the system of triple q-integral equations (.)-(.). Since the func-tion g(ρ) is only defined in Aq ∩ Bq , we can write

g(ξ ) = g(ξ ) + g(ξ ),

g and g defined in Aq and Bq , respectively. So, we may assume that

ψ = A + A,

and we solve the equations in the form

ξ–γ

∫ ∞

ρ–γ

[A(ρ) + A(ρ)

]Jκ

(√ρξ ; q)dqρ = f (ξ ), ξ ∈ Aq , (.)

ξ–α

∫ ∞

ρ–αA(ρ)Jμ

(√ρξ ; q)dqρ = g(ξ ), ξ ∈ Aq , (.)

ξ–α

∫ ∞

ρ–αA(ρ)Jμ

(√ρξ ; q)dqρ = g(ξ ), ξ ∈ Bq , (.)

ξ–β

∫ ∞

ρ–β

[A(ρ) + A(ρ)

]Jν

(√ρξ ; q)dqρ = h(ξ ), ξ ∈ Bq . (.)

We rewrite the equations as two pairs of dual q-integral equations, namely

{ξ–α

∫ ∞ ρ–αA(ρ)Jμ(

√ρξ ; q) dqρ = g(ξ ), ξ ∈ Aq ,

ξ–β∫ ∞

ρ–βA(ρ)Jν(√

ρξ ; q) dqρ = h(ξ ) – f(ξ ), ξ ∈ Bq ,(.)

{ξ–α

∫ ∞ ρ–αA(ρ)Jμ(

√ρξ ; q) dqρ = g(ξ ), ξ ∈ Bq ,

ξ–γ∫ ∞

ρ–γ A(ρ)Jκ (√

ρξ ; q) dqρ = f (ξ ) – f(ξ ), ξ ∈ Aq ,(.)

where

ξ–γ

∫ ∞

ρ–γ A(ρ)Jκ

(√ρξ ; q)dqρ = f(ξ ), ξ ∈ Aq ,

ξ–β

∫ ∞

ρ–βA(ρ)Jν

(√ρξ ; q)dqρ = f(ξ ), ξ ∈ Bq .

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Then we can solve the first and second pairs by Theorem A. For the first pairs

A(ξ ) =( – q)λ–ν+α–

ξλ/–μ/+α

(√ρξ ; q)Iμ/+α,λ–μ

q g(ρ) dqρ

+( – q)λ–ν–

ξλ/–μ/+α

∫ ∞

(√ρξ ; q)Kλ/–ν/–β ,ν–λ

q

[h(ρ) – f(ρ)

]dqρ,

where λ := (μ + ν) – (α – β) > –.

The solution of the second pair has the form

A(ξ ) =( – q)λ–μ+γ –

ξλ/–κ/+γ

∫ a

(√ρξ ; q)Iκ/+γ ,λ–κ

q

[f (ρ) – f(ρ)

]dqρ

+( – q)λ–μ–

ξλ/–κ/+γ

∫ ∞

aJλ

(√ρξ ; q)Kλ/–μ/–α,μ–λ

q g(ρ) dqρ,

where λ := (μ + κ) – (γ – α) > –.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsThe authors have made equal and significant contributions in writing this paper. They read and approved the finalmanuscript.

Author details1Present address:Department of Mathematics, Faculty of Science, King Saud University, Riyadh, Kingdom of Saudi Arabia.2Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt. 3Department of Natural and EngineeringScience, Faculty of Applied Studies and Community Service, King Saud University, Riyadh, Kingdom of Saudi Arabia.

AcknowledgementsThe authors very grateful to the reviewers and editors for their suggestions and comments which improved final versionof this paper. This research is supported by the DSFP program of the King Saud University in Riyadh through grantDSFP/MATH 01.

Received: 7 January 2016 Accepted: 13 March 2016

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