1 Preliminaries

The subject of fractional calculus has attained an eminent concern during the past decades due to various applications of this subject in various fields of science and engineering. Recently, an increase of interest in this area has duly been implemented and utilized in the theory of ordinary fractional calculus, optimal control problems, q-transform analysis, statistics, mathematical physics, q-difference equations, and q-integral equations (see, e.g., [1, 2]). By fixing a real number q such that \(0< q<1\), the q-derivative of a differentiable function ϑ is defined by

$$ D_{q}\vartheta ( x ) = \frac{\vartheta ( x ) -\vartheta ( qx ) }{ ( 1-q ) x} \quad ( x\neq 0 ) . $$

The q-integrals from 0 to y and from 0 to ∞ have been respectively defined by Jackson [3] as follows:

$$ I_{q}\vartheta ( y ) = \int _{0}^{y}\vartheta ( x ) \,d_{q}x= ( 1-q ) y\sum_{n=0}^{\infty } \vartheta \bigl( yq^{n} \bigr) q^{n} $$

and

$$ \int _{0}^{\infty }\vartheta ( x ) \,d_{q}x= ( 1-q ) \sum_{n=-\infty }^{\infty } \vartheta \bigl( q^{n} \bigr) q^{n}, $$

provided the two sums converge absolutely. The q-integration by parts has been defined by

$$ \int _{a}^{b}g ( x ) D_{q}\vartheta ( x ) \,d_{q}x=\vartheta ( b ) g ( b ) - \vartheta ( a ) g ( a ) - \int _{a}^{b} \vartheta ( qx ) D_{q}g ( x ) \,d_{q}x. $$

Ever after Jackson [3] presented the q-integral concept, various q-analogues of various types of integral transforms were given in a classical way (see, e.g., [4–19]). Consisting of the notion of regular operators [20], the theory of Boehmians was first introduced by [21] to generalize distributions and regular operators [22] (see, e.g., [21, 23–31]). Boehmians with their abstract nature are equivalence classes of quotients of sequences obtained from an integral domain, where the operations are addition and convolution. Complying with the q-calculus concept, we introduce the q-Boehmian concept as follows: Let A be a complex linear space and B be a subspace of A. Let \(\bullet ^{q}:A\times B\rightarrow A\) be a binary operation such that, for all \(\alpha \in \mathbb{C}\), \(\hat{\delta },\check{\delta },\tilde{\delta }\in A\) and \(\gamma _{1},\gamma _{2},\gamma \in B\), we have \(( \hat{\delta }+\check{\delta } ) \bullet ^{q}\gamma =\hat{\delta }\bullet ^{q}\gamma + \check{\delta }\bullet ^{q}\gamma \), \(( \alpha \hat{\delta } ) \bullet ^{q} \gamma =\alpha ( \hat{\delta }\bullet ^{q}\gamma ) \), \(\hat{\delta }\bullet ^{q} ( \gamma _{1}\bullet ^{q}\gamma _{2} ) = ( \hat{\delta }\bullet ^{q}\gamma _{1} ) \bullet ^{q}\gamma _{2}\), \(\tilde{\delta }_{n}\bullet ^{q}\gamma \rightarrow \tilde{\delta }\bullet ^{q}\gamma \) as \(\tilde{\delta }_{n}\rightarrow \tilde{\delta }\) in A as \(n\rightarrow \infty \) and, for all \(( x_{n} ) , ( y_{n} ) \in \Delta _{q}\), we have \(x_{n}\bullet ^{q}y_{n}\in \Delta _{q}\), where \(\Delta _{q}\) is a collection of sequences in B and \(x_{n}\bullet ^{q}\tilde{\delta }_{n}\rightarrow \tilde{\delta }\) as \(n\rightarrow \infty \) provided \(\tilde{\delta }_{n}\rightarrow \tilde{\delta }\) in A as \(n\rightarrow \infty \). The name of q-Boehmian is proposed to mean the equivalence class \(\frac{\tilde{\delta }_{n}}{x_{n}}\) obtained from the equivalence relation

$$ \tilde{\delta }_{n}\bullet ^{q}x_{n}=\tilde{ \delta }_{m}\bullet ^{q}x_{n} \quad ( m,n\in \mathbb{N} ), $$

where \(( \tilde{\delta }_{n} ) \in A\) and \(( x_{n} ) \in \Delta _{q}\). The collection of all q-Boehmians is claimed to form a q-Boehmian space denoted by \(B_{q}\). The linear space A is identified as a subspace of the q-Boehmian space \(B_{q}\) justified by the identification formula

$$ \check{\delta }\rightarrow \frac{\check{\delta }\bullet ^{q}x_{n}}{x_{n}},\quad \text{where } \check{ \delta }\in A\text{ and } ( x_{n} ) \in \Delta _{q}. $$

Two q-Boehmians \(\frac{\hat{\delta }_{n}}{x_{n}}\) and \(\frac{\tilde{\delta }_{n}}{y_{n}}\) are said to be equal in \(B_{q}\) if \(\hat{\delta }_{n}\bullet ^{q}x_{m}=\tilde{\delta }_{m}\bullet ^{q}x_{m}\), \(\forall m,n\in \mathbb{N} \). The addition is defined in \(B_{q}\) as

$$ \frac{\hat{\delta }_{n}}{x_{n}}+\frac{\tilde{\delta }_{n}}{y_{n}}= \frac{\hat{\delta }_{n}\bullet ^{q}y_{n}+\tilde{\delta }_{n}\bullet ^{q}x_{n}}{x_{n}\bullet ^{q}y_{n}}. $$

The scalar multiplication is defined in \(B_{q}\) as

$$ \alpha \frac{\tilde{\delta }_{n}}{x_{n}}= \frac{\alpha \tilde{\delta }_{n}}{x_{n}}\quad \text{for all }\alpha \in \mathbb{C}. $$

For every \(( x_{n} ) \in \Delta _{q}\), convergence of type \(\delta _{q}\), \(\beta _{n}\overset{\delta _{q}}{\rightarrow }\beta \), is defined in \(B_{q}\) when

$$ \beta _{n}\bullet ^{q}x_{k}\in A,\forall k,n\in \mathbb{N} , \beta \bullet ^{q}x_{k} \in A,\forall k\in \mathbb{N}, $$

and for each \(k\in \mathbb{N} \), \(\beta _{n}\bullet ^{q}x_{k}\rightarrow \beta \bullet ^{q}x_{k}\) as \(n\rightarrow \infty \) in A, whereas, convergence of type \(\Delta _{q}\), \(\beta _{n}\overset{\Delta _{q}}{\rightarrow }\beta \), is defined in \(B_{q}\) if for some \(( x_{n} ) \in \Delta _{q}\), \(( \beta _{n}-\beta ) \bullet ^{q}x_{n}\in A\), \(\forall n\in \mathbb{N} \) and

$$ ( \beta _{n}-\beta ) \bullet ^{q}x_{n} \rightarrow 0 \quad \text{as }n\rightarrow \infty \text{ in }A. $$

The space of q-Boehmians furnished with the current convergence notions emerges to be a complete quasi-normed space. Over the set of functions

$$ A= \biggl\{ \check{\delta } ( t ) :\exists M,\tau _{1},\tau _{2}>0, \bigl\vert \check{\delta } ( t ) \bigr\vert < ME_{q} \biggl( \frac{ \vert t \vert }{\tau _{j}} \biggr) , t\in ( -1 ) ^{j}\times [ 0,\infty ) \biggr\} , $$

the q-analogue of the Sumudu transform of the first type was latterly defined by [8, (1.18)] as follows:

$$ S_{q} \bigl\{ \check{\delta } ( t ) ;s \bigr\} = \frac{1}{ ( 1-q ) s} \int _{0}^{s}E_{q} \biggl( \frac{q}{s}t \biggr) \check{\delta } ( t ) \,d_{q}t,\quad s\in ( -\tau _{1}, \tau _{2} ) . $$
(1)

The properties of q-analogue \(S_{q} \) of the Sumudu transform including the convergence conditions and its relation with the q-Laplace integral transform have been derived by Albayrak et al. [8]. Over and above, the authors investigated certain fundamental aspects of the cited integral enfolding linearity, shifting theorems, differentiation, integration, etc. Also an attempt has been made to obtain the convolution theorem in a convergent series type. On the other hand, the authors in [6] provided some applications of the q-Sumudu transform to q-polynomials, q-functions, and q-Fox’s H-functions as well.

The aimed goal of this paper is to discuss the generalized q-theory of the q-integrable functions in the space \(L_{q}^{1}\) and to investigate fundamental properties of the q-analogue (1) in the generalized q-theory. In Sect. 2, we introduce a concept of q-Boehmians and a concept of q-delta sequences. We also establish the space \(B_{q}^{1}\) of q-Boehmians. In Sect. 3, we establish the second space \(B_{q}^{2}\) of q-Boehmians. In Sect. 4, we introduce the generalized q-Sumudu transform and discuss several properties. In Sect. 5, we provide a conclusion part.

2 The space \(B_{q}^{1}\)

Denote by \(L_{q}^{1}\) the space of all q-integrable functions δ̌ on \(\mathbb{R} _{+}\) defined by

$$ \Vert \check{\delta } \Vert _{L_{q}^{1} ( \mathbb{R} _{+} ) }=\frac{1}{1-q} \int _{0}^{\infty } \bigl\vert \check{\delta } ( x ) \bigr\vert \,d_{q}x< \infty , $$
(2)

whose comparable definition in a series expression formula is given as \(\sum_{-\infty }^{\infty }q^{n}\check{\delta } ( q^{n} ) \), provided the series converges absolutely. Denote by \(D_{q}\) the q-space of all test functions of compact supports on \(\mathbb{R} _{+}\), i.e.,

$$ D_{q}= \bigl\{ \check{\delta }\in C^{\infty } ( \mathbb{R} ) : \underset{0< x< \infty }{\sup } \bigl\vert D_{q} \check{\delta } ( x ) \bigr\vert < \infty \bigr\} . $$

Denote by \(\Delta _{q}\) the set of all sequences from \(D_{q}\) such that \(\Delta _{q}^{1}-\Delta _{q}^{3}\) satisfy

$$\begin{aligned}& \Delta _{q}^{1}: \int _{0}^{\infty } \bigl\vert x_{n} ( x ) \bigr\vert \,d_{q}x=1,\quad \forall n\in \mathbb{N}. \\& \Delta _{q}^{2}: \bigl\vert x_{n} ( x ) \bigr\vert < M,\quad M>0,M \in \mathbb{R} _{+}. \\& \Delta _{q}^{3}:\operatorname{supp}(x_{n}) \subseteq ( 0,b_{n} ) ,b_{n} \rightarrow 0\quad \mbox{as }n\rightarrow \infty , \forall n\in \mathbb{N}. \end{aligned}$$

Between two integrable functions δ̌ and δ̂ in \(L_{q}^{1}\), we denote by \(\ast ^{q}\) the q-convolution product defined by

$$ \bigl( \check{\delta }\ast ^{q}\hat{\delta } \bigr) ( x ) = \int _{0}^{x}\check{\delta } ( x-t ) \hat{ \delta } ( t ) \,d_{q}t, $$
(3)

provided the right-hand side integral exists for every real number \(x>0\). It is clear from the context that \(\check{\delta }\ast ^{q}\hat{\delta }\in L_{q}^{1}\) for all δ̌ and δ̂ in \(L_{q}^{1}\). Following Belgacem [32], the q-convolution theorem of the q-Sumudu transform of the convolution \(\check{\delta }\ast ^{q}\hat{\delta }\) can be easily established by using [21, (2.1)] (see, also [11]) as follows:

$$ S_{q} \bigl( \check{\delta }\ast ^{q}\hat{\delta } \bigr) ( x ) =xS_{q}\check{\delta } ( x ) S_{q}\hat{\delta } ( x ) . $$
(4)

An imperative result for categorizing the q-delta sequences may be introduced as follows.

Lemma 1

Let \(( x_{n} ) \) and \(( y_{n} ) \) be arbitrary in \(\Delta _{q}\). Then the sequence \(( x_{n}\ast ^{q}y_{n} ) \) is a q-delta sequence in \(\Delta _{q}\).

Proof

To establish this lemma, we examine that \(\Delta _{q}^{1}\), \(\Delta _{q}^{2}\), and \(\Delta _{q}^{3}\) satisfy for \(( x_{n}\ast ^{q}y_{n} ) \). To examine the correctness of \(\Delta _{q}^{1}\), we use integral Eq. (3) to write

$$ \int _{0}^{\infty } \bigl( x_{n}\ast ^{q}y_{n} \bigr) ( x ) \,d_{q}x= \int _{0}^{\infty } \biggl( \int _{0}^{x}x_{n} ( x-t ) y_{n} ( t ) \,d_{q}t \biggr) \,d_{q}x. $$

Now we change the order of integration to have

$$ \int _{0}^{\infty } \bigl( x_{n}\ast ^{q}y_{n} \bigr) ( x ) \,d_{q}x= \int _{0}^{\infty } \int _{t}^{ \infty }x_{n} ( x-t ) y_{n} ( t ) \,d_{q}t\,d_{q}x. $$

By changing the variables in the inner integral, i.e., substituting the change of variables \(x-t=y\), hence \(d_{q}x=d_{q}y\), the above equation reveals

$$ \int _{0}^{\infty } \bigl( x_{n}\ast ^{q}y_{n} \bigr) ( x ) \,d_{q}x= \int _{0}^{\infty }y_{n} ( t ) x_{n} ( y ) \,d_{q}t\,d_{q}y= \int _{0}^{ \infty }y_{n} ( t ) \,d_{q}t \int _{0}^{\infty }x_{n} ( y ) \,d_{q}y=1. $$

This proves \(\Delta _{q}^{1}\). The proof of \(\Delta _{q}^{2}\) follows from the fact that \(\vert x_{n}\ast ^{q}y_{n} \vert \leq \vert x_{n} \vert \vert y_{n} \vert \). The proof of \(\Delta _{q}^{3} \) follows from the fact that \(\operatorname{supp} ( x_{n}\ast ^{q}y_{n} ) \subset \operatorname{supp}(x_{n})+\operatorname{supp}(y_{n})\) for \(( x_{n} ) , ( y_{n} ) \in \Delta _{q}\). This completes the proof of the lemma. □

Hence, Lemma 1 shows that every sequence in \(\Delta _{q}\) forms an appropriate q-delta sequence.

Lemma 2

Let δ̌ and δ̂ be in \(L_{q}^{1}\). Then, for every \(\gamma ,\hat{\gamma }\in D_{q}\) and \(\alpha \in \mathbb{C}\), the following assertions are valid:

$$\begin{aligned} \textstyle\begin{array}{l@{\qquad }l} \mathrm{(i)}\quad \gamma \ast ^{q}\hat{\gamma }=\hat{\gamma }\ast ^{q} \gamma ,& \mathrm{(ii)}\quad ( \check{\delta }+\hat{\delta } ) \ast ^{q}\gamma =\check{\delta }\ast ^{q}\gamma +\hat{\delta }\ast \gamma , \\ \mathrm{(iii)}\quad ( \alpha \check{\delta } ) \ast ^{q} \gamma = \alpha \bigl( \check{\delta }\ast ^{q}\gamma \bigr) , & \mathrm{(iv)} \quad \check{\delta }\ast ^{q} \bigl( \gamma \ast ^{q} \hat{\gamma } \bigr) = \bigl( \check{\delta }\ast ^{q}\gamma \bigr) \ast ^{q}\hat{\gamma }.\end{array}\displaystyle \end{aligned}$$

Proof

(i) By using the definition of the convolution product \(\ast ^{q}\) given by (3) and applying the change of variables \(y=x-t\), hence \(d_{q}y=-d_{q}t\), we get

$$\begin{aligned} \bigl( \gamma \ast ^{q}\hat{\psi } \bigr) ( x ) =& \int _{0}^{x}\gamma ( x-t ) \hat{\gamma } ( t ) \,d_{q}t \\ =& \int _{x}^{0}\gamma ( y ) \hat{\gamma } ( x-y ) ( -1 ) \,d_{q}y \\ =& \int _{0}^{x}\hat{\gamma } ( x-y ) \gamma ( y ) \,d_{q}y . \end{aligned}$$

That is,

$$ \bigl( \gamma \ast ^{q}\hat{\psi } \bigr) ( x )= \bigl( \hat{\gamma }\ast ^{q}\gamma \bigr) ( x ) . $$
(5)

Proof of (ii) and (iii) follows from simple integral calculus. The proof of (iv) follows from a similar argument to that of (i). This completes the proof of the lemma. □

Lemma 3

Let δ̌ and \(( \check{\delta }_{n} ) \) be sequences of integrable functions in the space \(L_{q}^{1}\) such that \(\check{\delta }_{n}\rightarrow \check{\delta }\) as \(n\rightarrow \infty \). Then we have

$$ \check{\delta }_{n}\ast ^{q}\gamma \rightarrow \check{ \delta }\ast ^{q} \gamma \quad \textit{as }n\rightarrow \infty $$
(6)

for every \(\gamma \in D_{q}\).

The proof of this lemma follows from simple integration. Hence, we delete the details.

Finally, we have to establish the following lemma.

Lemma 4

Let δ̌ and δ̂ be arbitrary functions in \(L_{q}^{1}\) and \(( x_{n} ) \) be in \(\Delta _{q}\) such that \(\check{\delta }\ast ^{q}x_{n}=\hat{\delta }\ast ^{q}x_{n}\). Then we have \(\check{\delta }=\hat{\delta }\) in \(L_{q}^{1}\) for every \(n\in \mathbb{N}\).

Proof

The proof of this lemma follows from Eq. (2) and Lemma 3. Thus, we omit the details. □

Lemma 5

Let δ̌ be an integrable function in \(L_{q}^{1}\), then, for every \(( x_{n} ) \) in \(\Delta _{q}\), we have \(\check{\delta }\ast ^{q}x_{n}\rightarrow \check{\delta }\) as \(n\rightarrow \infty \).

Proof

By the assumption that \(\Delta _{q}^{1}\) and \(\Delta _{q}^{3}\) hold for \(( x_{n} ) \) and allowing the support of \(x_{n}\) to be included in the interval \(( 0,b_{n} ) \), for some real numbers \(b_{n}\), \(n\in \mathbb{N}\), we obtain

$$\begin{aligned} \bigl\Vert \check{\delta }\ast ^{q}x_{n}-\check{\delta } \bigr\Vert _{L_{q}^{1} ( \mathbb{\mathbb{R} }_{+} ) } =&\frac{1}{1-q} \int _{0}^{\infty } \bigl\vert \bigl( \check{\delta }\ast ^{q}x_{n} \bigr) ( x ) - \check{\delta } ( x ) \bigr\vert \,d_{q}x \\ \leq &\frac{1}{1-q} \int _{0}^{\infty } \int _{0}^{x} \bigl\vert \check{\delta } ( x-t ) -\check{\delta } ( x ) \bigr\vert \bigl\vert x_{n} ( t ) \bigr\vert \,d_{q}t\,d_{q}x \\ =&\frac{1}{1-q} \int _{0}^{\infty } \int _{0}^{b_{n}} \bigl\vert \check{\delta } ( x-t ) -\check{\delta } ( x ) \bigr\vert \bigl\vert x_{n} ( t ) \bigr\vert \,d_{q}t\,d_{q}x. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \bigl\Vert \check{\delta }\ast ^{q}x_{n}-\check{\delta } \bigr\Vert _{L_{q}^{1} ( \mathbb{R}_{+} ) } \leq & \frac{1}{1-q} \int _{0}^{\infty } \int _{0}^{b_{n}} \bigl\vert \check{\delta } ( x-t ) \bigr\vert \bigl\vert x_{n} ( t ) \bigr\vert \,d_{q}t\,d_{q}x \\ &{}+\frac{1}{1-q} \int _{0}^{\infty } \int _{0}^{b_{n}} \bigl\vert \check{\delta } ( x ) \bigr\vert \bigl\vert x_{n} ( t ) \bigr\vert \,d_{q}t\,d_{q}x. \end{aligned}$$

Since \(\check{\delta }\in L_{q}^{1}\), the above equation can be developed to give

$$ \bigl\Vert \check{\delta }\ast ^{q}x_{n}-\check{\delta } \bigr\Vert _{L_{q}^{1} ( \mathbb{R}_{+} ) }\leq A\frac{1}{1-q} \int _{0}^{b_{n}} \bigl\vert x_{n} ( t ) \bigr\vert \,d_{q}t+A \frac{1}{1-q} \int _{0}^{b_{n}} \bigl\vert x_{n} ( t ) \bigr\vert \,d_{q}t $$
(7)

for some constant A. Hence, by applying \(\Delta _{q}^{2}\) and integrating, from Eq. (7) we get

$$ \bigl\Vert \check{\delta }\ast ^{q}x_{n}-\check{\delta } \bigr\Vert _{L_{q}^{1} ( \mathbb{R}_{+} ) }\leq AMb_{n}+BMb_{n} \rightarrow 0 $$

as \(n\rightarrow \infty \), where M is a certain positive constant. Hence the proof of this lemma is completed. □

Therefore, the space \(B_{q}^{1}\) is defined with the sets \(( L_{q}^{1},\ast ^{q} )\), \(( D_{q},\ast ^{q} )\), and \(\Delta _{q}\). The canonical q-embedding of the space \(L_{q}^{1}\) in the space \(B_{q}^{1}\) is given as

$$ \check{\delta }\rightarrow \frac{\check{\delta }\ast ^{q}x_{n}}{x_{n}}. $$
(8)

Therefore, every δ̌ in \(L_{q}^{1}\) can be identified in \(B_{q}^{1}\) as \(\frac{\check{\delta }\ast ^{q}x_{n}}{x_{n}}\). In addition, scalar multiplication, differentiation, \(\Delta _{q}\) convergence, and \(\delta _{q}\) convergence are defined in a natural way. If \(( \check{\delta }_{n} ) \in L_{q}^{1}\) and \(( x_{n} ) \in \Delta _{q}\), then the pair \(( \check{\delta }_{n},x_{n} ) \) \(( \text{or }\frac{\check{\delta }_{n}}{x_{n}} ) \) is said to be a quotient of sequences if \(\check{\delta }_{n}\ast ^{q}x_{m}=\check{\delta }_{m}\ast ^{q}x_{n}\), \(\forall n,m\in \mathbb{N}\). Therefore, if \(\frac{\check{\delta }_{n}}{x_{n}}\) and \(\frac{\hat{\delta }_{n}}{y_{n}}\) are quotients of sequences and \(\check{\delta }\in L_{q}^{1}\), then it is easy to see that

$$ \frac{\check{\delta }\ast ^{q}x_{n}}{x_{n}}\quad \text{and}\quad \frac{\check{\delta }_{n}\ast ^{q}x_{n}+\hat{\delta }_{n}\ast ^{q}x_{n}}{x_{n}\ast ^{q}y_{n}} $$

are also quotients of sequences. Further, the following equivalence relations can be easily checked:

$$ \frac{\check{\delta }_{n}}{x_{n}\ast ^{q}\check{\delta }}\sim \frac{\check{\delta }_{n}\ast ^{q}\check{\delta }}{x_{n}}\quad \text{and}\quad \frac{\check{\delta }_{n}}{x_{n}\ast ^{q}\hat{\delta }_{n}}\sim \frac{\check{\delta }_{n}\ast ^{q}\hat{\delta }_{n}}{x_{n}}. $$

Two quotients of sequences \(\frac{\check{\delta }_{n}}{x_{n}}\) and \(\frac{\hat{\delta }_{n}}{y_{n}}\) are said to be equivalent if \(\check{\delta }_{n}\ast ^{q}y_{m}=\hat{\delta }_{m}\ast ^{q}x_{n}\), \(\forall n,m\in \mathbb{N}\). The equivalent class \(\breve{w}_{B}= ( \frac{\check{\delta }_{n}}{x_{n}} ) \) of quotients of sequences containing \(\frac{\check{\delta }_{n}}{x_{n}}\) is said to be a q-Boehmian. The space of such q-Boehmians is denoted by \(B_{q}^{1}\).

For two q-Boehmians \(\breve{w}_{B}=\frac{\check{\delta }_{n}}{x_{n}}\) and \(\breve{z}_{B}=\frac{\hat{\delta }_{n}}{y_{n}}\) in \(B_{q}^{1}\), the following are well defined on \(B_{q}^{1}\):

$$\begin{aligned}& \mathrm{( i )}\quad \breve{w}_{B}+\breve{z}_{B}= \frac{\check{\delta }_{n}\ast ^{q}x_{n}+\hat{\delta }_{n}\ast ^{q}\delta _{n}}{x_{n}\ast ^{q}y_{n}}, \qquad \mathrm{( ii )}\quad \beta \breve{w}_{B}= \frac{\beta \check{\delta }_{n}}{x_{n}},\\& \mathrm{( iii )}\quad \breve{w}_{B} \ast ^{q} \breve{z}_{B}= \frac{\check{\delta }_{n}\ast ^{q}\hat{\delta }_{n}}{x_{n}\ast ^{q}y_{n}}, \qquad \mathrm{( iv )}\quad D^{k}\breve{w}_{B}= \frac{D^{k}\check{\delta }_{n}}{x_{n}},\quad \text{and} \quad \mathrm{( v )}\quad \breve{w}_{B}\ast ^{q}\check{\delta }= \frac{\check{\delta }_{n}\ast ^{q}\check{\delta }}{x_{n}}, \end{aligned}$$

where \(k\in \mathbb{R}\), \(\beta \in \mathbb{C}\), \(D^{k}\breve{w}_{B}\) is the kth derivative of \(\breve{w}_{B}\) and \(\check{\delta }\in L_{q}^{1}\).

Definition 6

For \(n=1,2,3,\ldots \) and \(\breve{w}_{B,n},\breve{w}_{B}\in B_{q}^{1}\), the sequence \(( \breve{w}_{B,n} ) \) is said to be \(\delta _{q}\)-convergent to \(\breve{w}_{B}\), denoted by \(\delta _{q}-\lim_{n\rightarrow \infty }\breve{w}_{B,n}=\breve{w}_{B}\), provided there can be found a delta sequence \(( x_{n} ) \) such that

  1. (a)

    \(( \breve{w}_{B,n}\ast ^{q}x_{k} ) \), \(( \breve{w}_{B}\ast ^{q}x_{k} )\) in \(L_{q}^{1}\) for all \(n,k\in \mathbb{N}\);

  2. (b)

    \(\lim_{n\rightarrow \infty }\breve{w}_{B,n}\ast ^{q}x_{k}=\breve{w}_{B}\ast ^{q}x_{k}\) in \(L_{q}^{1}\) for every \(k\in \mathbb{N}\).

Definition 7

For \(n=1,2,3,\ldots \) and \(\breve{w}_{B,n},\breve{w}_{B}\in B_{q}^{1}\), the sequence \(( \breve{w}_{B,n} ) \) is said to be \(\Delta _{q}\)-convergent to \(\breve{w}_{B}\), denoted by \(\Delta _{q}\)-\(\lim_{n\rightarrow \infty }\breve{w}_{B,n}=\breve{w}_{B}\), provided there can be found a delta sequence \(( x_{n} ) \) such that

  1. (a)

    \(( \breve{w}_{B,n}-\breve{w}_{B} ) \ast ^{q}x_{n} \in L_{q}^{1} ( \forall n\in \mathbb{N} )\);

  2. (b)

    \(\lim_{n\rightarrow \infty } ( \breve{w}_{B,n}- \breve{w}_{B} ) \ast ^{q}x_{n}=0\) in \(L_{q}^{1}\).

Remark 8

Let \(\check{\delta }\in L_{q}^{1}\) and \(( x_{n} ) \in \Delta _{q}\) be fixed. Then the mapping

$$ \check{\delta }\rightarrow \breve{w}_{B}, $$

where \(\breve{w}_{B}=\frac{\check{\delta }\ast ^{q}x_{n}}{x_{n}}\) is an injective mapping from \(L_{q}^{1}\) into \(B_{q}^{1}\).

Therefore, it can be easily checked that \(L_{q}^{1}\) may be identified as a subspace of \(B_{q}^{1}\).

Remark 9

Let \(( x_{n} ) \in \Delta _{q}\). Then, if \(\check{\delta }_{n}\rightarrow \check{\delta }\) in \(L_{q}^{1}\) as \(n\rightarrow \infty \), then for all \(k\in \mathbb{N}\),

$$ \check{\delta }_{n}\ast ^{q}x_{k} \rightarrow \check{\delta }\ast ^{q}x_{k} $$

as \(n\rightarrow \infty \). That is, \(\breve{w}_{B,n}\rightarrow \breve{w}_{B} \) in \(B_{q}^{1}\) as \(n\rightarrow \infty \).

The above remark states the following.

Theorem 10

The mapping \(\check{\delta }\rightarrow \breve{w}_{B}\), \(\breve{w}_{B}=\frac{\check{\delta }\ast ^{q}x_{n}}{x_{n}}\), is a continuous q-embedding of the space \(L_{q}^{1}\) into the space \(B_{q}^{1}\).

3 The space \(B_{q}^{2}\)

In this section we provide the basic essentials that we need in defining the space \(B_{q}^{2}\) of q-Boehmians, with the set \(( L_{q}^{1},\bullet ^{q} ) \), the subset \(( D_{q}^{S_{q}},\ast ^{q} ) \), the set of delta sequences \(( \Delta _{q}^{S_{q}},\ast ^{q} ) \), the convolution products \(\ast ^{q}\), and the operation \(\bullet ^{q}\); see Eq. (9). To establish the space \(B_{q}^{2}\), let us define the following convolution product.

Definition 11

Let δ̌ and δ̂ be integrable functions in \(L_{q}^{1}\). Between δ̌ and δ̂ we define the q-product \(\bullet ^{q}\) as follows:

$$ \bigl( \check{\delta }\bullet ^{q}\hat{\delta } \bigr) ( x ) =xS_{q}\check{\delta } ( x ) \hat{\delta } ( x ) , $$
(9)

where \(S_{q}\check{\delta }\) is the q-Sumudu transform of δ̌.

We have the following assertion.

Theorem 12

Let δ̌ be an integrable function in \(L_{q}^{1}\), then \(\check{\delta }\bullet ^{q}\psi \in L_{q}^{1}\) for all \(\psi \in D_{q}^{S_{q}}\).

Proof

Let \(\check{\delta }\in L_{q}^{1}\) and \(\psi =S_{q}\gamma \) for some \(\gamma \in D_{q}\). Then, by the definition of the space \(L_{q}^{1}\) and the product \(\bullet ^{q}\), we write

$$ \bigl\Vert \check{\delta }\bullet ^{q}\psi \bigr\Vert _{L_{q}^{1} ( \mathbb{R} _{+} ) } =\frac{1}{1-q} \int _{0}^{ \infty } \bigl\vert \bigl( \check{\delta }\bullet ^{q}\psi \bigr) ( x ) \bigr\vert \,d_{q}x =\frac{1}{1-q} \int _{0}^{\infty }xS_{q}\check{\delta } ( x ) \psi ( x ) \,d_{q}x. $$

Hence, we have

$$ \bigl\Vert \check{\delta }\bullet ^{q}\psi \bigr\Vert _{L_{q}^{1} ( \mathbb{R} _{+} ) }\leq \frac{1}{1-q} \int _{0}^{ \infty }xS_{q}\check{\delta } ( x ) S_{q} ( \gamma ) ( x ) \,d_{q}x. $$
(10)

Therefore, if \([ a,b ] \) denotes a closed interval containing the support of ψ, then the hypothesis that \(\check{\delta }\in L_{q}^{1}\) yields

$$\begin{aligned} \bigl\Vert \check{\delta }\bullet ^{q}\psi \bigr\Vert _{L_{q}^{1} ( \mathbb{R}_{+} ) } =&\frac{1}{1-q} \int _{0}^{ \infty } \bigl\vert \bigl( \check{\delta }\bullet ^{q}\psi \bigr) ( x ) \bigr\vert \,d_{q}x \\ \leq &M\frac{1}{1-q} \int _{a}^{b} \bigl\vert xS_{q} \gamma ( x ) \bigr\vert \,d_{q}x \\ =&M\frac{1}{1-q} \int _{a}^{b} \bigl\vert x\psi ( x ) \bigr\vert \,d_{q}x \\ < &\infty \end{aligned}$$

for every \(\psi \in D_{q}^{S_{q}}\), M being a positive constant.

This establishes the theorem. □

Now, we reciprocate the product \(\bullet ^{q}\) as follows.

Theorem 13

Let δ̌ be an integrable function in \(L_{q}^{1}\), then \(\check{\delta }\bullet ^{q} ( \psi \bullet ^{q}\hat{\psi } ) = ( \check{\delta }\bullet ^{q}\psi ) \bullet ^{q}\hat{\psi }\) for all \(\psi ,\hat{\psi }\in D_{q}^{S_{q}}\).

Proof

Let \(\psi =S_{q}\gamma \), \(\hat{\psi }=S_{q}\hat{\gamma }\), \(\gamma ,\hat{\gamma }\in D_{q}\). Then, by the concept of q-convolution \(\bullet ^{q}\) and Eq. (3), we get

$$ \bigl( \check{\delta }\bullet ^{q} \bigl( \psi \bullet ^{q}\hat{\psi } \bigr) \bigr) ( x ) =x ( S_{q}\check{ \delta } ) ( x ) \bigl( S_{q}\gamma \bullet ^{q}S_{q} \hat{\gamma } \bigr) ( x ) =xS_{q}\check{\delta } ( x ) xS_{q}\gamma ( x ) S_{q}\hat{\gamma } ( x ) . $$
(11)

Rearranging Eq. (11) and again the concept of q-convolution \(\bullet ^{q}\) gives

$$\begin{aligned} \bigl( \check{\delta }\bullet ^{q} \bigl( \psi \bullet ^{q}\hat{\psi } \bigr) \bigr) ( x ) =&x \bigl( xS_{q} \check{\delta } ( x ) xS_{q}\gamma ( x ) \bigr) \bigl( S_{q} \hat{\gamma } ( x ) \bigr) \\ =&x \bigl( \check{\delta }\bullet ^{q}\psi \bigr) S_{q} ( \hat{\gamma } ) ( x ) \\ =& \bigl( \bigl( \check{\delta }\bullet ^{q}\psi \bigr) \bullet ^{q} \hat{\psi } \bigr) ( x ) . \end{aligned}$$

Hence the proof of the theorem follows. □

Theorem 14

(i) Let δ̌ and δ̂ be integrable functions in \(L_{q}^{1}\), then \(( \check{\delta }+\hat{\delta } ) \bullet ^{q}\psi =\check{\delta }\bullet ^{q} \psi +\hat{\delta }\bullet ^{q}\psi \) for all \(\psi \in D_{q}^{S_{q}}\).

(ii) Let δ̌ be an integrable function in \(L_{q}^{1}\), then \(( \alpha \check{\delta }\bullet ^{q}\psi ) =\alpha ( \check{\delta }\bullet ^{q}\psi ) \) for all \(\psi \in D_{q}^{S_{q}}\), \(\alpha \in \mathbb{C}\).

Proof

(i) Let \(\psi \in D_{q}^{S_{q}}\), \(\psi \in D_{q}^{S_{q}}\), \(\psi =S_{q}\gamma \), \(\gamma \in D_{q}\). Then, by the definition of \(\bullet ^{q}\) and Eq. (11), we write

$$\begin{aligned} \bigl( ( \check{\delta }+\hat{\delta } ) \bullet ^{q}\psi \bigr) ( x ) =&xS_{q} ( \check{\delta }+ \hat{\delta } ) ( x ) \psi ( x ) \\ =&xS_{q}\check{\delta } ( x ) S_{q}\gamma ( x ) +xS_{q}\hat{\delta } ( x ) S_{q}\gamma ( x ) \\ =& \bigl( \check{\delta }\bullet ^{q}\psi \bigr) ( x ) + \bigl( \hat{\delta }\bullet ^{q}\psi \bigr) ( x ) . \end{aligned}$$

The proof of the second part is similar. The proof of this theorem is therefore completed. □

Theorem 15

Let δ̌, δ̂ and \(( \hat{\delta }_{n} ) \) be integrable in \(L_{q}^{1}\). Then the following hold.

(i) If \(\hat{\delta }_{n}\rightarrow \check{\delta }\) in \(L_{q}^{1}\) as \(n\rightarrow \infty \), then \(\hat{\delta }_{n}\bullet ^{q}\psi \rightarrow \check{\delta }\bullet ^{q} \psi \) for all \(\psi \in D_{q}^{S_{q}}\), \(\psi =S_{q}\gamma \), \(\gamma \in D_{q}\) as \(n\rightarrow \infty \).

(ii) If \(\check{\delta }\bullet ^{q}x_{n}=\hat{\delta }\bullet ^{q}z_{n}\), then \(\check{\delta }=\hat{\delta }\) in \(L_{q}^{1}\) for all \(( z_{n} ) \in \Delta _{q}^{S_{q}}\), \(z_{n}=S_{q}x_{n}\), \(n\in \mathbb{N} \).

(iii) \(\check{\delta }\bullet ^{q}x_{n}\rightarrow 0\) in \(L_{q}^{1}\) for all \(( x_{n} ) \in \Delta _{q}^{S_{q}}\) as \(n\rightarrow \infty \).

Proof

(i) Let δ̌, δ̂ and \(( \hat{\delta }_{n} ) \) be integrable in \(L_{q}^{1}\) and \(\psi \in D_{q}^{S_{q}}\), \(\psi =S_{q}\gamma \), \(\gamma \in D_{q}\), such that \(\hat{\delta }_{n}\rightarrow \hat{\delta }\) in \(L_{q}^{1}\) as \(n\rightarrow \infty \). Then we have

$$ \bigl( \hat{\delta }_{n}\bullet ^{q}\psi \bigr) ( x ) =xS_{q} \hat{\delta }_{n} ( x ) S_{q}\gamma ( x ) \rightarrow xS_{q}\hat{ \delta } ( x ) S_{q}\gamma ( x ) =xS_{q} \hat{\delta } ( x ) \psi ( x ) =\check{\delta }\bullet ^{q} \psi \text{ as }n\rightarrow \infty . $$

Proof (ii) Let δ̌ and δ̂ be integrable functions in \(L_{q}^{1}\) and \(( z_{n} ) \in \Delta _{q}^{S_{q}}\), \(z_{n}=S_{q}x_{n}\), \(n\in \mathbb{N}\) such that \(\check{\delta }\bullet ^{q}z_{n}=\hat{\delta }\bullet ^{q}z_{n}\). Then \(xS_{q}\check{\delta } ( x ) S_{q}x_{n} ( x ) -xS_{q}\hat{\delta } ( x ) S_{q} ( x_{n} ) ( x ) =0\) for all \(x\in L_{q}^{1}\). Hence, we have

$$ xS_{q}x_{n} ( x ) S_{q} ( \check{\delta }- \hat{\delta } ) ( x ) =0 $$

for all \(x\in L_{q}^{1}\). Therefore, e infers that \(\check{\delta } ( x ) =\hat{\delta } ( x ) \) for all \(x\in L_{q}^{1}\). The proof of (iii) is analogous. Hence the proof is completed. □

If \(( \check{\delta }_{n} ) \in L_{q}^{1}\) and \(( z_{n} ) \in \Delta _{q}^{S_{q}}\), \(z_{n}=S_{q}x_{n}\), \(n\in \mathbb{N}\), then the pair \(( \check{\delta }_{n},z_{n} ) \) (or \(\frac{\check{\delta }_{n}}{z_{n}}\)) is said to be a quotient of sequences if \(\check{\delta }_{n}\bullet ^{q}z_{m}=\check{\delta }_{m}\bullet ^{q}z_{n}\), \(\forall n,m\in \mathbb{N}\). Therefore, if \(\frac{\check{\delta }_{n}}{z_{n}}\) and \(\frac{\hat{\delta }_{n}}{y_{n}}\) are quotients of sequences and \(\check{\delta }\in L_{q}^{1}\), then it is easy to see that

$$ \frac{\check{\delta }\bullet ^{q}z_{n}}{z_{n}}\quad \text{and} \quad \frac{\check{\delta }_{n}\bullet ^{q}z_{n}+\hat{\delta }_{n}\bullet ^{q}z_{n}}{z_{n}\bullet ^{q}y_{n}} $$

are quotients of sequences. Further, we can easily check the following equivalence relations:

$$ \frac{\check{\delta }_{n}}{z_{n}\bullet ^{q}\check{\delta }}\sim \frac{\check{\delta }_{n}\bullet ^{q}\check{\delta }}{z_{n}}\quad \text{and}\quad \frac{\check{\delta }_{n}}{z_{n}\bullet ^{q}\hat{\delta }_{n}}\sim \frac{\check{\delta }_{n}\bullet ^{q}\hat{\delta }_{n}}{z_{n}}. $$

Two quotients of sequences \(\frac{\check{\delta }_{n}}{z_{n}}\) and \(\frac{\hat{\delta }_{n}}{y_{n}}\) are said to be equivalent if \(\check{\delta }_{n}\bullet ^{q}y_{m}=\hat{\delta }_{m}\bullet ^{q}z_{n}\), \(\forall n,m \in \mathbb{N}\). The equivalent class \(\breve{w}_{B}=\frac{\check{\delta }_{n}}{z_{n}}\) of quotients of sequences containing \(\frac{\check{\delta }_{n}}{z_{n}}\) is said to be a q-Boehmian. The space of such q-Boehmians is denoted by \(B_{q}^{2}\). For two q-Boehmians \(\breve{w}_{B}=\frac{\check{\delta }_{n}}{z_{n}}\) and \(\breve{z}_{B}=\frac{\hat{\delta }_{n}}{y_{n}}\) in \(B_{q}^{2}\), the following are well defined on \(B_{q}^{2}\):

$$\begin{aligned} \textstyle\begin{array}{l@{\qquad }l} \displaystyle\mathrm{(i)} \quad \breve{w}_{B}+\breve{z}_{B}= \frac{\check{\delta }_{n}\bullet ^{q}z_{n}+\hat{\delta }_{n}\bullet ^{q}z_{n}}{z_{n}\bullet ^{q}y_{n}},& \mathrm{(ii)}\quad \beta \breve{w}_{B}= \frac{\beta \check{\delta }_{n}}{z_{n}}, \\ \displaystyle\mathrm{(iii)}\quad \breve{w}_{B}\bullet ^{q} \breve{z}_{B}= \frac{\check{\delta }_{n}\bullet ^{q}\hat{\delta }_{n}}{z_{n}\bullet ^{q}y_{n}},& \mathrm{(iv)}\quad D^{k}\breve{w}_{B}=\frac{D^{k}\check{\delta }_{n}}{z_{n}},\quad \text{and} \\ \displaystyle\mathrm{(v)}\quad \breve{w}_{B}\bullet ^{q}\check{ \delta }= \frac{\check{\delta }_{n}\bullet ^{q}\check{\delta }}{z_{n}}, \end{array}\displaystyle \end{aligned}$$

where \(k\in \mathbb{R}\), \(\beta \in \mathbb{C}\), and \(D^{k}\breve{w}_{B}\) is the kth derivative of \(\breve{w}_{B}\) and \(\check{\delta }\in L_{q}^{1}\).

Definition 16

For \(n=1,2,3,\ldots \) and \(\breve{w}_{B,n},\breve{w}_{B}\in B_{q}^{2}\), the sequence \(( \breve{w}_{B,n} ) \) is said to be \(\delta _{q}\)-convergent to \(\breve{w}_{B}\), denoted by \(\delta _{q}-\lim_{n\rightarrow \infty }\breve{w}_{B,n}=\breve{w}_{B}\), provided there can be found a delta sequence \(( z_{n} ) \), \(z_{n}=S_{q}x_{n}\), \(n\in \mathbb{N}\), such that

  1. (a)

    \(( \breve{w}_{B,n}\bullet ^{q}z_{k} ) \), \(( \breve{w}_{B}\bullet ^{q}z_{k} )\) in \(L_{q}^{1}\) for all \(n,k\in \mathbb{N}\);

  2. (b)

    \(\lim_{n\rightarrow \infty }\breve{w}_{B,n}\bullet ^{q}z_{k}=\breve{w}_{B}\bullet ^{q}z_{k}\) in \(L_{q}^{1}\) for every \(k \in \mathbb{N}\).

Definition 17

For \(n=1,2,3,\ldots \) and \(\breve{w}_{B,n},\breve{w}_{B}\in B_{q}^{2}\), the sequence \(( \breve{w}_{B,n} ) \) is said to be \(\Delta _{q}^{S_{q}}\)-convergent to \(\breve{w}_{B}\), denoted by \(\Delta _{q}\)-\(\lim_{n\rightarrow \infty } \breve{w}_{B,n}=\breve{w}_{B}\), provided there can be found a delta sequence \(( z_{n} ) \) such that

  1. (a)

    \(( \breve{w}_{B,n}-\breve{w}_{B} ) \bullet ^{q}z_{n}\in L_{q}^{1}\) (\(\forall n\in \mathbb{N} \));

  2. (b)

    \(\lim_{n\rightarrow \infty } ( \breve{w}_{B,n}- \breve{w}_{B} ) \bullet ^{q}z_{n}=0\) in \(L_{q}^{1}\).

Remark 18

Let \(\check{\delta }\in L_{q}^{1}\) and \(( z_{n} ) \in \Delta _{q}^{S_{q}}\), \(z_{n}=S_{q}x_{n}\), \(n\in \mathbb{N}\), be fixed. Then the mapping

$$ \check{\delta }\rightarrow \breve{w}_{B}, $$

where \(\breve{w}_{B}=\frac{\check{\delta }\bullet ^{q}z_{n}}{z_{n}}\) is an injective mapping from \(L_{q}^{1}\) into \(B_{q}^{2}\).

Therefore, it can be easily checked that \(L_{q}^{1}\) may be identified as a subspace of \(B_{q}^{2}\).

Remark 19

Let \(( z_{n} ) \in \Delta _{q}^{S_{q}}\), \(z_{n}=S_{q}x_{n}\), \(n\in \mathbb{N}\). Then if \(\check{\delta }_{n}\rightarrow \check{\delta }\) in \(L_{q}^{1}\) as \(n\rightarrow \infty \), then, for all \(k\in \mathbb{N}\),

$$ \check{\delta }_{n}\bullet ^{q}z_{k} \rightarrow \check{\delta }\bullet ^{q}z_{k} $$

as \(n\rightarrow \infty \). That is, \(\breve{w}_{B,n}\rightarrow \breve{w}_{B} \) in \(B_{q}^{2}\) as \(n\rightarrow \infty \).

The above remark states the following.

Theorem 20

The mapping \(\psi \rightarrow \breve{w}_{B}\), \(\breve{w}_{B}=\frac{\check{\delta }\bullet ^{q}z_{n}}{z_{n}}\), is a continuous q-embedding of the space \(L_{q}^{1}\) into the space \(B_{q}^{2}\).

4 The q-Sumudu transform of generalized q-theory

Definition 21

Let \(\frac{\check{\delta }_{n}}{x_{n}}\) be a q-Boehmian in the space \(B_{q}^{1}\). Then we define the q-Sumudu transform of \(\frac{\check{\delta }_{n}}{x_{n}}\) as follows:

$$ S_{q}^{1}\frac{\check{\delta }_{n}}{x_{n}}= \frac{S_{q}\check{\delta }_{n}}{S_{q}x_{n}}\quad \text{for all } ( \check{\delta }_{n} ) \in L_{q}^{1}\text{ and } ( x_{n} ) \in \Delta _{q}. $$

It is clear that \(S_{q}^{1}\frac{\check{\delta }_{n}}{x_{n}}\) belongs to \(B_{q}^{2}\) as \(( S_{q}\check{\delta }_{n} ) \) and \(( S_{q}x_{n} ) \) are elements of the spaces \(L_{q}^{1}\) and \(\Delta _{q}^{S_{q}}\), respectively. The linearity of \(S_{q}^{1}\) follows by easy techniques.

Theorem 22

The operator \(S_{q}^{1}:B_{q}^{1}\rightarrow B_{q}^{2}\) is q-sequentially continuous, i.e., if \(\Delta _{q}-\lim_{n\rightarrow \infty }\breve{w}_{B,n}=\breve{w}_{B}\) in \(B_{q}^{1}\), then

$$ \Delta _{q}-\lim_{n\rightarrow \infty }S_{q}^{1} \breve{w}_{B,n}=S_{q}^{1} \breve{w}_{B}\quad \textit{in }B_{q}^{2}. $$

Proof

Let \(\Delta _{q}-\lim_{n\rightarrow \infty }\breve{w}_{B,n}=\breve{w}_{B}\) in \(B_{q}^{1}\), then there is \(( x_{n} ) \) in \(\Delta _{q}\) such that \(\Delta _{q}- \lim_{n\rightarrow \infty } ( \breve{w}_{B,n}- \breve{w}_{B} ) \ast ^{q}x_{n}=0\) in \(B_{q}^{1}\). The continuity of the integral operator gives

$$ \Delta _{q}^{S_{q}}-\lim_{n\rightarrow \infty }S_{q}^{1} \bigl( ( \breve{w}_{B,n}-\breve{w}_{B} ) \ast ^{q}x_{n} \bigr) = \Delta _{q}^{S_{q}}- \lim_{n\rightarrow \infty } \bigl( \bigl( S_{q}^{1} \breve{w}_{B,n}-S_{q}^{1} \breve{w}_{B} \bigr) \bullet ^{q}S_{q}x_{n} \bigr) =0. $$

Thus, we have \(\Delta _{q}^{S_{q}}-\lim_{n\rightarrow \infty }S_{q}^{1}\breve{w}_{B,n}=S_{q}^{1}\breve{w}_{B}\) in \(B_{q}^{2}\).

This finishes the proof of the theorem. □

Theorem 23

\(S_{q}^{1}:B_{q}^{1}\rightarrow B_{q}^{2}\) is one-one, onto, continuous with respect to \(\delta _{q}\) and \(\Delta _{q}\)-convergence and consists with the classical operator \(S_{q}\).

Proof

Proofs of the parts that \(S_{q}^{1}\) is one-one, onto, continuous with respect to \(\delta _{q}\) and \(\Delta _{q}\)-convergence are analogous to those given in the literature. To prove that \(S_{q}^{1}\) consists with the classical operator \(S_{q}^{1}\), let \(\hat{\delta }\in L_{q}^{1}\) and let \(\frac{\hat{\delta }\ast ^{q}z_{n}}{z_{n}}\) be its representative in \(B_{q}^{2}\) for all \(z_{n}=S_{q}x_{n}\), \(( x_{n} ) \in \Delta _{q}\). Clearly, for all \(n\in \mathbb{N}{\small ,}\) \(( z_{n} ) \) is independent of the representative. Hence, by the convolution theorem we get

$$ S_{q}^{1}\frac{\hat{\delta }\ast ^{q}z_{n}}{z_{n}}= \frac{S_{q}\hat{\delta }\bullet ^{q}S_{q}x_{n}}{S_{q}x_{n}}=S_{q} \hat{\delta }\bullet ^{q}S_{q}^{1}\frac{x_{n}}{x_{n}}. $$

That is, \(\frac{S_{q}\hat{\delta }\bullet ^{q}S_{q}x_{n}}{S_{q}x_{n}}= \frac{S_{q}\hat{\delta }\bullet ^{q}z_{n}}{z_{n}}\) is the representative of \(S_{q}\hat{\delta }\) in the space \(L_{q}^{1}\). The proof is, therefore, finished.

We introduce the transform inversion formula as follows. □

Definition 24

We define the inverse integral operator of \(S_{q}^{1}\) of a q-Boehmian \(\frac{S_{q}\check{\delta }_{n}}{z_{n}}\) in \(B_{q}^{2}\) as a q-Boehmian in \(B_{q}^{1}\) defined by

$$ S_{q}^{-1}\frac{S_{q}\check{\delta }_{n}}{z_{n}}= \frac{\check{\delta }_{n}}{x_{n}}, $$

where \(z_{n}=S_{q}x_{n}\), \(( x_{n} ) \) is a delta sequence in \(\Delta _{q}\), and \(( \check{\delta }_{n} ) \) is a sequence of integrable functions in \(L_{q}^{1}\).

Theorem 25

Let \(\frac{S_{q}\check{\delta }_{n}}{z_{n}}\) be a q-Boehmian in \(B_{q}^{2}\), \(z_{n}=S_{q}x_{n}\), \(( x_{n} ) \in \Delta _{q}\), and \(\hat{\delta }\in L_{q}^{1}\). Then we have

$$ S_{q}^{1} \biggl( \frac{\check{\delta }_{n}}{x_{n}}\ast ^{q} \hat{\delta } \biggr) =\frac{S_{q}\check{\delta }_{n}}{z_{n}}\bullet ^{q} \hat{\delta }\quad \textit{and}\quad S_{q}^{-1} \biggl( \frac{S_{q}\check{\delta }_{n}}{z_{n}}\bullet ^{q}\hat{\delta } \biggr) =\frac{\check{\delta }_{n}}{x_{n}}\ast ^{q} \hat{\delta }. $$

Proof

Assume that \(\frac{S_{q}\check{\delta }_{n}}{z_{n}}\) is a q-Boehmian in the space \(B_{q}^{2},z_{n}=S_{q}x_{n}\), \(( x_{n} ) \in \Delta _{q}\), and \(\hat{\delta }\in L_{q}^{1}\). Then, by using the convolution theorem, Definition 21, and Eq. (9), we have

$$\begin{aligned} S_{q}^{1} \biggl( \frac{\check{\delta }_{n}}{x_{n}}\ast ^{q} \hat{\delta } \biggr) =&S_{q}^{1} \biggl( \frac{\check{\delta }_{n}\ast ^{q}\hat{\delta }}{x_{n}} \biggr) \\ =&\frac{S_{q} ( \check{\delta }_{n}\ast ^{q}\hat{\delta } ) }{S_{q}x_{n}} \\ =&\frac{xS_{q}\check{\delta }_{n} ( x ) S_{q}\hat{\delta } ( x ) }{S_{q}x_{n}} \\ =&\frac{S_{q}\check{\delta }_{n}\bullet ^{q}\hat{\delta }}{S_{q}x_{n}} \\ =&\frac{S_{q}\check{\delta }_{n}}{S_{q}x_{n}}\bullet ^{q}\hat{\delta } \\ =&\frac{S_{q}\check{\delta }_{n}}{z_{n}}\bullet ^{q}\hat{\delta }. \end{aligned}$$

Similarly, by using the convolution theorem, Definition 24, and Eq. (9), we obtain

$$\begin{aligned} S_{q}^{-1} \biggl( \frac{S_{q}\check{\delta }_{n}}{z_{n}}\bullet ^{q} \hat{\delta } \biggr) =&S_{q}^{-1} \frac{S_{q}\check{\delta }_{n}\bullet ^{q}\hat{\delta }}{z_{n}} \\ =&S_{q}^{-1} \frac{xS_{q}\check{\delta }_{n} ( x ) S_{q}\hat{\delta } ( x ) }{z_{n}} \\ =&S_{q}^{-1} \frac{S_{q} ( \check{\delta }_{n}\ast ^{q}\hat{\delta } ) }{S_{q}x_{n}} \\ =&\frac{\check{\delta }_{n}\ast ^{q}\hat{\delta }}{x_{n}} \\ =&\frac{\check{\delta }_{n}}{x_{n}}\ast ^{q}\hat{\delta }. \end{aligned}$$

This completely finishes the proof of the theorem. □

5 Conclusion

This paper could be an evolution of idea. It gives an extension to a set of q-integrable functions to a set of q-integrable q-generalized functions. It verifies that the q-analysis of this paper generalizes the q-analysis followed by Albayrak et al. 2013. Moreover, this paper has also shown that the generalized q-Sumudu transform and its q-inversion formula are well-defined mappings possessing properties alike to the classical properties of their corresponding classical versions.