Certain triple q-integral equations involving third Jackson $q$-Bessel functions as kernel

In this paper, we employ the fractional $q$-calculus in solving a triple system of $q$-Integral equations, where the kernel is the third Jackson $q$-Bessel functions. The solution is reduced to two simultaneous Fredholm $q$-integral equation of the second kind. Examples are included. We also apply a result in~[Pacific J. Math. \textbf{275}(1) (2015)63--102] for solutions of dual $q^2$-integral equations to solve certain triple integral equations.


Introduction
Some three-parts mixed boundary value problems of the mathematical theory of elasticity are solved by reducing them to triple integral equations. Many of the triple integral equations are of the form where w(u) is the weight function, K(u, x) is the kernel function. Several authors have described various methods to solve dual and triple integral equations especially when the kernel is a Bessel function. For the dual integral equations, see for example [13,15,21,24,26,27,29,30]. For the triple integral equations, see for example [9-12, 14, 25, 31, 32, 34, 35]. In this paper, we consider triple q-integral equation where the kernel is the third Jackson q-Bessel function and the q-integral is Jackson q-integral. It is worth mentioning that different approaches for solving dual q-integral equation is in [6]. Also, solutions for dual and triple sequence involving q-orthogonal polynomials is in [7]. This paper is organized as follows. The next section is introductory section includes the main notions and notations we need in our investigations. In Section 3, we solve the triple q-integral equations by reducing the system to two simultaneous Fredholm q-integral equation of the second kind, we shall use a method due to Singh, Rokne and Dhaliwal [31]. The approach depends on fractional q-calculus. We include solutions of two dual q-integral equations as special cases of the solution of the triple q-integral equation included in this section, and we show that this coincides with the results in [6]. In the last section, we use a result from [6] for a solution of dual q 2 -integral equations to solve triple q 2 -integral equations. The result of this section is a q-analogue of the results introduced by cooke in [11].

q-Notations and Results
In this paper, we assume that q is a positive number less than one. We introduce some of the needed q-notations and results (see [5]).
Let t > 0, A q,t , B q,t and R q,t,+ be the sets defined by (Note that if t = 1, we write A q , B q and R q,+ ). We follow Gasper and Rahman [17] for the definitions of the qshifted factorial, multiple q-shifted factorials, basic hypergeometric series, Jackson q-integrals, the q-gamma and beta functions. We also follow Annaby and Mansour [5] for the definition of the q-derivative at zero.
Let α ∈ C, we will use the following notation For η ∈ C and a function f defined on R q,+ , we define the following spaces The third Jackson q-Bessel function J ν (z; q), see [18] and [19], is defined by and satisfies the following relations (see [33]): Also, for ℜ(ν) > −1, the q-Bessel function J ν (.; q 2 ) satisfies (see [22]): We recall that the functions cos(z; q) and sin(z; q) are defined for z ∈ C by . We need the following results from [5]: Proposition 2.2. Let ν and α be complex numbers such that ℜ(ν) > −1.
Corollary 2.5. Let x, u and α be complex numbers such that u ∈ R q,+ , ℜ(α) > 0 and ℜ(ν) > −1. Then Proof. The proof is similar to the proof of Corollary 2.4 and is omitted.
Koornwinder and Swarttouw [22] introduced the following inverse pair of q-Hankel integral transforms under the side condition f, g ∈ L 2 q (R q,+ ): where λ, x ∈ R q,+ . In the following, we introduce A q-analogue of the Riemann-Liouville fractional integral operator is introduced in [3] by Al-Salam through In [1], Agarwal defined the q-fractional derivative to be We shall also use that see [5,Lemma 4.17].
In the following, we introduce some q-fractional operators that we use in solving the triple q-integral equations under consideration. The technique of using fractional operators in solving dual and triple integral equations is not new. See for example [2,6,30]. In [3], Al-Salam defined a two parameter q-fractional operator by .. This is a q-analogue of the Erdélyi and Sneddon fractional operator, cf. [15,16], In [6], the authors introduced a slight modification of the operator K η,α q . This operator is denoted by K η,α q and defined by This is a slight modification of the operator K α f (x; q) introduced in [18, (19.4.8)] and by Al-salam in [3]. Note that this operator satisfies the following semigroup identity for all α and β.
The proof of (2.15) is completely similar to the proof of [5, Theorem 5.13] and is omitted.
Proof. First, we show that According to (2.14), we have so, (2.16) can be written as Hence, This implies, , and by using (2.15), we obtain the result and completes the proof.

A system of triple q-Integral Equations
The goal of this section is to solve the following triple q-integral equations: where 0 < a < b < ∞, and α, ν are complex numbers satisfying ψ is an unknown function to be determined, and f i (i = 1, 2, 3) are known functions, and w is a non-negative bounded function defined on R q,+ .
Clearly from (2.5), a sufficient condition for the convergence of the q-integrals on the left hand side of (3.1)-(3.2) is that For getting the solution of the triple q-integral Hence, , and the triple q-integral equation (3.1)-(3.3) can be represented as: Now assume that C : where So, the triple q-integral equations (3.5)-(3.7) can be rewritten in the following form: Proposition 3.1. Let the functions ψ 1 , ψ 2 be defined by Then for u ∈ R q,+ , (3.14) Proof. We start with proving (3.16). Let x ∈ A q,b . Multiply both sides of (3.9) by x −2α ρ ν+1 (q 2 ρ 2 /x 2 ; q 2 ) −α and integrate with respect to ρ from 0 to x, we get We can prove that the double q-integral on the left hand side of (3.18) is absolutely convergent for 0 < ℜ(α) < 1 and for ℜ(ν) > −1 provided that C 1 ∈ L q,ν (R q,+ ). So, we can interchange the order of the q-integrations to obtain Calculate the q-derivative of the two sides of (3.19) with respect to x and using (2.8), we get Now, we prove (3.17). Let x ∈ B q,a . Multiply both sides of (3.10) by ρ −2α−ν+1 (x 2 /ρ 2 ; q 2 ) −α and q-integrate with respect to ρ from x to ∞, we get From (2.5), we can prove that u t J ν (u; q 2 ) is bounded on R q,+ provided that ℜ(t + ν) > −1. So, if we take t such that ℜν + 2 > ℜt > −ℜν + 2ℜ(1 − α), we can prove that the double q-integral is absolutely convergent and we can interchange the order of the q-integration to obtain Calculating the q-derivative of the two sides of (3.22) with respect to x and using (2.10) yields By the above argument, If we assume that ψ 1 and ψ 2 are given by (3.12) and (3.13), then Hence, (3.14) and (3.15) follow by applying the inverse pair of q-Hankel transforms (2.11) on (3.24) and (3.25). This completes the proof.
Remark 3.2. From the definitions of ψ i and φ i , i = 1, 2, in Proposition 3.1, one can verify that x −ν−α φ 2 is bounded function in B q,a and and Proof. Equation (3.11) can be written in the following form: By using equations (2.3) and (3.27), we get Substituting the value of C 1 (u) from (3.14) into (3.29), we obtain (3.30) From (2.5), there exists M > 0 such that Hence, the double q-integration is absolutely convergent and we can interchange the order of the q-integrations to obtain Therefore, applying Proposition 2.1 with ℜ(ν − α) > ℜ(ν − 1) > −1, we obtain By using we obtain Replacing ρ by qρ yields Thus, applying Proposition 3.3 yields From (3.28), we can write the last equation in the following form (3.34) From the condition on the function C 2 , we can prove that the double q-integration is absolutely convergent. Therefore, we can interchange the order of the q-integrations and use Proposition 2.2 to obtain (3.35) Substitute the value of C 1 (u) and C 2 (u) from equations (3.15) and (3.14) into equation (3.35), and then interchange the order of the q-integrations we get (3.36) where Equation (3.36) is nothing else but the Fredholm q-integral equation of the second kind (3.26). This completes the proof.
Proposition 3.4. For ρ ∈ A q,a , Ψ 2 (ρ) satisfies the Fredholm q-integral equation of the form Proof. The proof is similar to the proof of Proposition 3.3 and is omitted.
Hence, from Theorem 3.5 Hence, This coincides with the result in [6,Theorem 4.1] for solutions of double q-integral equations.
Let a = q m and assume that m → ∞. If we assume that f 2 = 0, and f 3 = f , we obtain the dual q-integral system of equations Hence, from Theorem 3.5 This is a special case of [6, Theorem 5.1].

4.
Solving system of triple q 2 -Integral Equations by using solutions of dual q-integral equations In [11], Cooke solved certain triple integral equations involving Bessel functions by using a result for Noble [28] for solutions for dual integral equations with Bessel functions as kernel. In this section, we use the result, Theorem A , which introduced in [6] to solve the following triple q-integral equations: where a, α, β, γ, µ, ν and κ are complex numbers such that ℜ(ν) > −1, ℜ(µ) > −1, ℜ(κ) > −1, and 0 < a < 1, , the functions f (ρ), g(ρ) and h(ρ) are known functions, and ψ(u) is the solution function to be determined.
The following is a result from [6] that we shall use to solve the system (4.1)-(4.3).