An estimate of Sumudu transforms for Boehmians

  • Shrideh Khalaf Qasem Al-Omari1 and

    Affiliated with

    • Adem Kılıçman2Email author

      Affiliated with

      Advances in Difference Equations20132013:77

      DOI: 10.1186/1687-1847-2013-77

      Received: 14 January 2013

      Accepted: 5 March 2013

      Published: 26 March 2013

      Abstract

      The space of Boehmians is constructed using an algebraic approach that utilizes convolution and approximate identities or delta sequences. A proper subspace can be identified with the space of distributions. In this paper, we first construct a suitable Boehmian space on which the Sumudu transform can be defined and the function space S can be embedded. In addition to this, our definition extends the Sumudu transform to more general spaces and the definition remains consistent for S elements. We also discuss the operational properties of the Sumudu transform on Boehmians and finally end with certain theorems for continuity conditions of the extended Sumudu transform and its inverse with respect to δ- and Δ-convergence.

      MSC:54C40, 14E20, 46E25, 20C20.

      Keywords

      Sumudu transforms Boehmian spaces the space H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq1_HTML.gif the space H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq2_HTML.gif distributions

      1 Introduction

      The Sumudu transform of one variable function f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq3_HTML.gif is introduced as a new integral transform by Watugala in [1] and is given by
      S f ( t ) ( y ) = 1 y R + f ( t ) exp ( t y ) d t , y ( τ 1 , τ 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equa_HTML.gif
      over the set of functions
      A = { f ( t ) : M , τ 1 , τ 2 > 0 , | f ( t ) | < M e t τ j , t ( 1 ) j × ( 0 , ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equb_HTML.gif

      where f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq4_HTML.gif is a function that can be expressed as a convergent series [2, 3]. The Sumudu transform was applied to solve the ordinary differential equations in control engineering problems; see [3].

      The Sumudu transform of the convolution product of f and u is given by
      S ( f u ) ( y ) = y f s ( y ) u s ( y ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equc_HTML.gif

      where f s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq5_HTML.gif and u s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq6_HTML.gif are the Sumudu transforms of f and u, respectively.

      Some of the properties were established by Weerakoon in [4, 5]. In [6], further fundamental properties of this transform were also established by Asiru. Similarly, this transform was applied to a one-dimensional neutron transport equation in [7] by Kadem.

      In [8], the Sumudu transform was extended to the distributions and some of their properties were also studied. Recently, this transform has been applied to solve the system of differential equations; see Kılıçman et al. in [9].

      Note that a very interesting fact about Sumudu transform is that the original function and its Sumudu transform have the same Taylor coefficients except the factor n; see Zhang [10]. Similarly, the Sumudu transform sends combinations C ( m , n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq7_HTML.gif into permutations P ( m , n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq8_HTML.gif, and hence it will be useful in the discrete systems.

      The following are the general properties of the Sumudu transform which are auxiliary from the substitution method and the properties of integral operators.
      1. (i)
        If k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq9_HTML.gif and k 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq10_HTML.gif are non-negative integers and S 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq11_HTML.gif and S 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq12_HTML.gif are the corresponding Sumudu transforms of f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq13_HTML.gif and f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq14_HTML.gif, respectively, then
        S ( k 1 f 1 + k 2 f 2 ) ( y ) = k 1 S 1 ( y ) + k 2 S 2 ( y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equd_HTML.gif
         
      2. (ii)

        S f ( k t ) ( y ) = S ( k y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq15_HTML.gif, k R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq16_HTML.gif.

         
      3. (iii)

        lim t 0 f ( t ) = lim u 0 S ( y ) = f ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq17_HTML.gif, where S ( y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq18_HTML.gif is the Sumudu transform of f.

         

      More properties of the Sumudu transforms a long with a some of applications were given in [11] and [12].

      2 Boehmian space

      Boehmians were first constructed as a generalization of regular Mikusinski operators [13]. The minimal structure necessary for the construction of Boehmians consists of the following elements:
      1. (i)

        a nonempty set A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq19_HTML.gif;

         
      2. (ii)

        a commutative semigroup ( B , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq20_HTML.gif;

         
      3. (iii)

        an operation : A × B A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq21_HTML.gif such that for each x A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq22_HTML.gif and s 1 , s 2 , B http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq23_HTML.gif, x ( s 1 s 2 ) = ( x s 1 ) s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq24_HTML.gif;

         
      4. (iv)

        a collection Δ B N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq25_HTML.gif such that

         
      5. (a)

        If x , y A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq26_HTML.gif, ( s n ) Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq27_HTML.gif, x s n = y s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq28_HTML.gif for all n, then x = y http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq29_HTML.gif;

         
      6. (b)

        If ( s n ) , ( t n ) Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq30_HTML.gif, then ( s n t n ) Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq31_HTML.gif.

         
      Elements of Δ are called delta sequences. Consider
      Q = { ( x n , s n ) : x n A , ( s n ) Δ , x n s m = x m s n , m , n N } . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Eque_HTML.gif

      Now if ( x n , s n ) , ( y n , t n ) Q http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq32_HTML.gif, x n t m = y m s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq33_HTML.gif, m , n N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq34_HTML.gif, then we say ( x n , s n ) ( y n , t n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq35_HTML.gif. The relation ∼ is an equivalence relation in ℚ. The space of equivalence classes in ℚ is denoted by β. Elements of β are called Boehmians.

      We note that between A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq19_HTML.gif and β there is a canonical embedding expressed as x x s n s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq36_HTML.gif. The operation ⊙ can also be extended to β × A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq37_HTML.gif by x n s n t = x n t s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq38_HTML.gif. The relationship between the notion of convergence and the product ⊙ is given by:
      1. (i)

        If f n f http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq39_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif in A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq19_HTML.gif and ϕ B http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq41_HTML.gif is any fixed element, then f n ϕ f ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq42_HTML.gif in A (as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq43_HTML.gif);

         
      2. (ii)

        If f n f http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq39_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif in A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq19_HTML.gif and ( δ n ) Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq44_HTML.gif, then f n δ n f http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq45_HTML.gif in A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq19_HTML.gif (as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif).

         
      The operation ⊙ can be extended to β × B http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq46_HTML.gif as follows: If [ f n s n ] β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq47_HTML.gif and ϕ B http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq48_HTML.gif, then [ f n s n ] ϕ = [ f n ϕ s n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq49_HTML.gif. In β, there are two types of convergence as follows.
      1. (1)

        A sequence ( h n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq50_HTML.gif in β is said to be δ-convergent to h in β, denoted by h n δ h http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq51_HTML.gif, if there exists ( s n ) Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq52_HTML.gif such that ( h n s n ) , ( h s n ) A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq53_HTML.gif, k , n N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq54_HTML.gif, and ( h n s k ) ( h s k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq55_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq56_HTML.gif in A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq19_HTML.gif for every k N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq57_HTML.gif.

         
      2. (2)

        A sequence ( h n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq50_HTML.gif in β is said to be Δ-convergent to h in β, denoted by h n Δ h http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq58_HTML.gif, if there exists a ( s n ) Δ http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq59_HTML.gif such that ( h n h ) s n A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq60_HTML.gif, n N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq61_HTML.gif, and ( h n h ) s n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq62_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif in A http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq19_HTML.gif.

         

      For further discussion, see [1416].

      3 The Boehmian space H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq63_HTML.gif

      Denote by S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq64_HTML.gif and D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq65_HTML.gif the space of all rapidly decreasing functions over R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq66_HTML.gif ( R + = ( 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq67_HTML.gif) and the space of all test functions of compact support, respectively. In what follows, we obtain preliminary results required to construct the Boehmian space H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq68_HTML.gif, where Y = ( S + , D + , Δ + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq69_HTML.gif.

      Lemma 3.1
      1. (1)

        If u 1 , u 2 D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq70_HTML.gif, then u 1 u 2 D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq71_HTML.gif.

         
      2. (2)

        If f 1 , f 2 S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq72_HTML.gif and u 1 D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq73_HTML.gif, then ( f 1 + f 2 ) u 1 = f 1 u 1 + f 2 u 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq74_HTML.gif.

         
      3. (3)

        u 1 u 2 = u 2 u 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq75_HTML.gif, u 1 , u 2 D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq76_HTML.gif.

         
      4. (4)

        If f S + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq77_HTML.gif, u 1 , u 2 D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq70_HTML.gif, then ( f u 1 ) u 2 = f ( u 1 u 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq78_HTML.gif.

         

      Proofs are analogous to those of classical cases and details are omitted.

      Definition 3.2 A sequence ( s n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq79_HTML.gif of functions from D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq80_HTML.gif is said to be in Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq81_HTML.gif if and only if
      Δ + 1 R + s n ( x ) d x = 1 ; Δ + 2 R + | s n ( x ) | d x M , M  is a positive integer ; Δ + 3 supp s n ( x ) ( 0 , ε n ) , ε n 0  as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equf_HTML.gif

      This means that ( s n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq82_HTML.gif shrinks to zero as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif. Each member of Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq83_HTML.gif is called a delta sequence or an approximate identity or, sometimes, a summability kernel. Delta sequences, in general, appear in many branches of mathematics, but probably the most important applications are those in the theory of generalized functions. The basic use of delta sequences is the regularization of generalized functions, and further, they can be used to define the convolution product and the product of generalized functions.

      Lemma 3.3 If ( s n ) , ( t n ) Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq84_HTML.gif, then supp ( s n t n ) supp s n + supp t n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq85_HTML.gif.

      Lemma 3.4 If u 1 , u 2 D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq86_HTML.gif, then so is u 1 u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq87_HTML.gif and
      R + | u 1 u 2 | R + | u 1 | R + | u 2 | . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equg_HTML.gif

      Theorem 3.5 Let f 1 , f 2 S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq88_HTML.gif and ( s n ) Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq89_HTML.gif such that f 1 s n = f 2 s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq90_HTML.gif, n = 1 , 2 , 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq91_HTML.gif , then f 1 = f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq92_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq93_HTML.gif.

      Proof We show that f 1 s n = f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq94_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq95_HTML.gif. Let K be a compact set containing the supp s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq96_HTML.gif for every n N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq97_HTML.gif. Using Δ + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq98_HTML.gif, we write
      | x k D m ( f 1 s n f 1 ) ( x ) | K | s n ( t ) | | x k D m ( f 1 ( x t ) f 1 ( x ) ) | d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equ1_HTML.gif
      (3.1)
      The mapping t f 1 t http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq99_HTML.gif, where f 1 t ( x ) = f 1 ( x t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq100_HTML.gif, is uniformly continuous from R + R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq101_HTML.gif. From the hypothesis that supp s n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq102_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif (by Δ + 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq103_HTML.gif), we choose r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq104_HTML.gif such that supp s n [ 0 , r ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq105_HTML.gif for large n and t < r http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq106_HTML.gif. This implies
      | f 1 ( x t ) f 1 ( x ) | = | f 1 t f 1 | < ε n M . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equ2_HTML.gif
      (3.2)
      Hence using Δ + 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq107_HTML.gif and (3.2), (3.1) becomes
      | x k D m ( f 1 s n f 1 ) ( x ) | < ε n 0 as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equh_HTML.gif

      Thus f 1 s n f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq108_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq109_HTML.gif. Similarly, we show that f 2 s n f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq110_HTML.gif. This completes the proof of the theorem. □

      Theorem 3.6 If lim n f n = f http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq111_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq112_HTML.gif and u D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq113_HTML.gif, then
      lim n f n u = f u . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equi_HTML.gif

      Proof

      In view of the hypothesis of the theorem, we write
      | x k D m ( f n u f u ) ( x ) | = | x k ( D m ( f n f ) u ) ( x ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equ3_HTML.gif
      (3.3)
      The last equation follows from the fact that [17]
      D m f u = D m f u = f D m u . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equj_HTML.gif
      Hence, for each u D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq114_HTML.gif, we have
      | x k D m ( f n u f u ) ( x ) | K x k | D m ( f n f ) ( x t ) | | u ( t ) | d t M γ k ( f n f ) for some constant  M 0 as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equk_HTML.gif

      The proof of the theorem is completed. □

      Theorem 3.7 If lim n f n = f http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq111_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq112_HTML.gif and ( s n ) Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq115_HTML.gif, then lim n f n s n = f http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq116_HTML.gif.

      Proof

      In view of the analysis employed for Theorem 3.5, we get
      lim n f n s n = f n f as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equl_HTML.gif
      Hence
      lim n f n s n = f as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equm_HTML.gif

      This completes the proof. The Boehmian space H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq117_HTML.gif is therefore constructed. □

      The canonical embedding between S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq118_HTML.gif and H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq119_HTML.gif is expressed as x [ x s n s n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq120_HTML.gif. The extension of ⋆ to H ( Y ) × S + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq121_HTML.gif is given by [ x n s n ] t = [ x n t s n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq122_HTML.gif. Convergence in H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq119_HTML.gif is defined in a natural way:

      δ-convergence: A sequence ( h n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq123_HTML.gif in H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq119_HTML.gif is said to be δ-convergent to h in H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq124_HTML.gif, denoted by h n δ h http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq125_HTML.gif, if there exists a delta sequence ( s n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq126_HTML.gif such that ( h n s n ) , ( h s n ) S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq127_HTML.gif, k , n N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq128_HTML.gif, and ( h n s k ) ( h s k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq129_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq118_HTML.gif for every k N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq57_HTML.gif.

      Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq83_HTML.gif-convergence: A sequence ( h n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq130_HTML.gif in H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq131_HTML.gif is said to be Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq83_HTML.gif-convergent to h in H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq119_HTML.gif, denoted by h n Δ h http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq132_HTML.gif, if there exists a ( s n ) Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq133_HTML.gif such that ( h n h ) s n S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq134_HTML.gif, n N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq135_HTML.gif, and ( h n h ) s n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq136_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq43_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq64_HTML.gif.

      Theorem 3.8 The mapping f [ f s n s n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq137_HTML.gif is a continuous embedding of S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq64_HTML.gif into H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq124_HTML.gif.

      Proof The mapping is one-to-one. For detailed proof, let [ f 1 s n s n ] = [ f 2 t n t n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq138_HTML.gif, then ( f 1 s n ) t m = ( f 2 t m ) s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq139_HTML.gif. Then since ( s n ) , ( t n ) Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq140_HTML.gif, f 1 ( s m t n ) = f 2 ( t n s m ) = f 2 ( s m t n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq141_HTML.gif. Using Theorem 3.5, we get f 1 = f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq92_HTML.gif. To show the mapping is continuous, let f n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq142_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq118_HTML.gif. Then we show that
      [ f n s m s m ] δ 0 as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equn_HTML.gif

      From Theorem 3.5, [ f n s m s m ] s m = f n s m 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq143_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq56_HTML.gif. This completes the proof of the theorem. □

      Theorem 3.9 Let f S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq144_HTML.gif and u D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq145_HTML.gif, then
      S ( f u ) ( y ) = y f s ( y ) u s ( y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equo_HTML.gif

      4 The Boehmian space H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq146_HTML.gif

      We describe another Boehmian space as follows. Let S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq64_HTML.gif be the space of rapidly decreasing functions [17]. Define
      D + s ( R ) = { u s :  for all  u D + ( R ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equ4_HTML.gif
      (4.1)
      where u s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq6_HTML.gif denotes the Sumudu transform of u. We also define f u s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq147_HTML.gif by
      ( f u s ) ( y ) = y f ( y ) u s ( y ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equ5_HTML.gif
      (4.2)

      Lemma 4.1 Let f S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq148_HTML.gif and u s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq149_HTML.gif, then f u s S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq150_HTML.gif.

      Proof If f S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq151_HTML.gif and u s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq152_HTML.gif, then using the topology of S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq64_HTML.gif and Leibnitz’ theorem, we get
      | x k D x m ( f u s ) ( x ) | | x k j = 1 m D m j ( x f ( x ) ) D j u s ( x ) | j = 1 m | x k D m j ( x f ( x ) ) | | D j u s ( x ) | = j = 1 m | x k D m j f 1 ( x ) | | K u ( t ) D x j e t x x d t | , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equp_HTML.gif
      where f 1 ( x ) = x f ( x ) S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq153_HTML.gif and K is a compact subset containing the supp u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq154_HTML.gif. Hence
      | x k D x m ( f u s ) ( x ) | M γ k , m j ( f 1 ) < http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equq_HTML.gif

      for some positive constant M. This completes the proof of the lemma. □

      Lemma 4.2 The mapping
      S + × D + s S + , ( f , u s ) f u s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equr_HTML.gif
      satisfies the following properties:
      1. (1)

        If u 1 s , u 2 s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq155_HTML.gif, then u 1 s u 2 s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq156_HTML.gif.

         
      2. (2)

        If f 1 , f 2 S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq157_HTML.gif, u s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq158_HTML.gif, then ( f 1 + f 2 ) u s = f 1 u s + f 2 u s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq159_HTML.gif.

         
      3. (3)

        For u 1 s , u 2 s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq160_HTML.gif, u 1 s u 2 s = u 2 s u 1 s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq161_HTML.gif.

         
      4. (4)

        For f S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq162_HTML.gif, u 1 s , u 2 s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq163_HTML.gif, then ( f u 1 s ) u 2 s = f ( u 1 s u 2 s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq164_HTML.gif.

         

      Proof The proof of the above lemma is straightforward. Detailed proof is as follows.

      Proof of (1). Let u 1 , u 2 D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq165_HTML.gif, then u 1 u 2 D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq166_HTML.gif. Hence ( u 1 u 2 ) s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq167_HTML.gif by (4.1). Theorem 3.9 implies u 1 s u 2 s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq156_HTML.gif.

      Proof of (2) is obvious.

      Proof of (3). We have
      ( u 1 s u 2 s ) ( x ) = x u 1 s ( x ) u 2 s ( x ) = x u 2 s ( x ) u 1 s ( x ) = ( u 2 s u 1 s ) ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equs_HTML.gif

      Hence u 1 s u 2 s = u 2 s u 1 s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq161_HTML.gif.

      Proof of (4). Let f S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq168_HTML.gif, u 1 s , u 2 s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq169_HTML.gif, then
      ( ( f u 1 s ) u 2 s ) ( x ) = x ( f u 1 s ) ( x ) u 2 s = x x f ( x ) u 1 s ( x ) u 2 s ( x ) = x f ( x ) x u 1 s ( x ) u 2 s ( x ) = x f ( x ) ( u 1 s u 2 s ) ( x ) = f ( u 1 s u 2 s ) ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equt_HTML.gif
      that is,
      ( f u 1 s ) u 2 s = f ( u 1 s u 2 s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equu_HTML.gif

      This completes the proof of the theorem. □

      Denote by Δ + s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq170_HTML.gif the set of all Sumudu transforms of delta sequences from Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq83_HTML.gif. That is,
      Δ + s = { ( s n s ) : ( s n ) Δ + , n N } . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equ6_HTML.gif
      (4.3)

      Lemma 4.3 Let f 1 , f 2 S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq171_HTML.gif, ( s n s ) Δ + s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq172_HTML.gif be such that f 1 s n s = f 2 s n s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq173_HTML.gif, ∀n, then f 1 = f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq92_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq93_HTML.gif.

      Proof Let f 1 , f 2 S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq174_HTML.gif and ( s n s ) Δ + s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq175_HTML.gif. Since f 1 s n s = f 2 s n s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq173_HTML.gif, (4.2) implies x f 1 ( x ) s n s ( x ) = x f 2 ( x ) s n s ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq176_HTML.gif. Hence f 1 ( x ) = f 2 ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq177_HTML.gif for all x. The proof is completed. □

      Lemma 4.4 For each ( s n ) , ( t n ) Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq84_HTML.gif, ( s n s t n s ) Δ + s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq178_HTML.gif.

      Proof Since ( s n ) , ( t n ) Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq140_HTML.gif, s n t n Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq179_HTML.gif for all n. Hence, from Theorem 3.9, we get S ( s n t n ) ( x ) = x s n s ( x ) t n s ( x ) = s n s t n s Δ + s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq180_HTML.gif for every n. This completes the proof of the lemma. □

      By aid of Lemma 4.3. and Lemma 4.4, Δ + s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq170_HTML.gif can be regarded as a delta sequence.

      Lemma 4.5 Let f n f http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq39_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq64_HTML.gif, u s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq181_HTML.gif, then f n u s f u s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq182_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq183_HTML.gif.

      Proof It is clear that u s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq6_HTML.gif is bounded in D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq184_HTML.gif. Further,
      ( f n u s ) ( x ) x f ( x ) u s ( x ) ( f n u s ) ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equv_HTML.gif

      Hence ( f n u s ) f u s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq185_HTML.gif. □

      Lemma 4.6 Let f n f http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq39_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq64_HTML.gif, ( s n s ) Δ + s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq175_HTML.gif, then f n s n s f http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq186_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq64_HTML.gif.

      Proof Let ( s n ) Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq115_HTML.gif, then s n s ( x ) 1 x http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq187_HTML.gif uniformly on compact subsets of R + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq66_HTML.gif. Hence
      | x k D x m ( f n s n s f ) ( x ) | = | x k D x m ( x f n ( x ) s n s ( x ) f ( x ) ) | | x k D x m ( f n f ) ( x ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equw_HTML.gif

      as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif. Thus | x k D x m ( f n s n s f ) ( x ) | 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq188_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif. This yields f n s n s f http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq186_HTML.gif in the topology of S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq118_HTML.gif. The proof is therefore completed. The space H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq189_HTML.gif can be regarded as a Boehmian space, where Y s = ( S + , D + s , Δ + s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq190_HTML.gif. □

      Lemma 4.7 The mapping
      f [ f s n s s n s ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equ7_HTML.gif
      (4.4)

      is a continuous embedding of S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq191_HTML.gif into H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq192_HTML.gif.

      Proof For f S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq193_HTML.gif, s n s Δ + s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq194_HTML.gif, f s n s s n s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq195_HTML.gif is a quotient of sequences in the sense that ( f s n s ) s m s = f ( s m s s n s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq196_HTML.gif. We show that the map (4.4) is one-to-one. Let [ f 1 s n s s n s ] = [ f 2 t n s t n s ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq197_HTML.gif, then ( f 1 s n s ) t m s = ( f 2 t m s ) s n s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq198_HTML.gif, m , n N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq199_HTML.gif. Using Lemma 4.2 and Lemma 4.3, we conclude f 1 = f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq92_HTML.gif. □

      To establish the continuity of (4.4), let f n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq142_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq112_HTML.gif. Then f n s n s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq200_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif by Lemma 4.6, and hence
      [ f n s n s s n s ] 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equx_HTML.gif

      as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif in H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq192_HTML.gif. This completes the proof of the lemma.

      5 The Sumudu transform of Boehmians

      Let β = [ f n s n ] H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq201_HTML.gif, then we define the Sumudu transform of β by the relation
      β 1 s = [ f n s s n s ] in  H ( Y s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equ8_HTML.gif
      (5.1)

      Theorem 5.1 β 1 s : H ( Y ) H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq202_HTML.gif is well defined.

      Proof Let β 1 = β 2 H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq203_HTML.gif, where β 1 = [ f n s n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq204_HTML.gif, β 2 = [ g n t n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq205_HTML.gif. Then the concept of quotients yields f n t m = g m s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq206_HTML.gif. Employing Theorem 3.9, we get x f n s ( x ) t m s ( x ) = x g m s ( x ) s n s ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq207_HTML.gif, i.e., f n s t m s = g m s s n s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq208_HTML.gif. Equivalently, f n s s n s g n s t n s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq209_HTML.gif. Thus β 1 s = β 2 s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq210_HTML.gif. This completes the proof of the theorem. □

      Theorem 5.2 β s : H ( Y ) H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq211_HTML.gif is continuous with respect to δ-convergence.

      Proof Let β n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq212_HTML.gif in H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq117_HTML.gif, then by [14], β n = [ f n , k s k ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq213_HTML.gif and f n , k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq214_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif in S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq64_HTML.gif. Applying the Sumudu transform to both sides yields f n , k s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq215_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif. Hence
      β n s = [ f n , k s s k s ] 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equy_HTML.gif

      as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif in H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq192_HTML.gif. This proves the theorem. □

      Theorem 5.3 β s : H ( Y ) H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq216_HTML.gif is a one-to-one mapping.

      Proof Assume β 1 s = [ f n s s n s ] = [ g n s t n s ] = β 2 s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq217_HTML.gif, then f n s t m s = g m s s n s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq208_HTML.gif. Hence
      ( f n t m ) s = ( g m s n ) s . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equz_HTML.gif
      Since the Sumudu transform is one-to-one, we get f n t m = g m s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq206_HTML.gif. Thus
      f n s n g n t n . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equaa_HTML.gif
      Hence
      [ f n s n ] = β 1 = [ g n t n ] = β 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equab_HTML.gif

      This completes the proof of the theorem. □

      Theorem 5.4 Let β 1 , β 2 H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq218_HTML.gif, then
      1. (1)

        ( β 1 + β 2 ) s = β 1 s + β 2 s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq219_HTML.gif;

         
      2. (2)

        ( k β ) s = k β s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq220_HTML.gif, λ C http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq221_HTML.gif.

         

      Proof is immediate from the definitions.

      Theorem 5.5 β s : H ( Y ) H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq216_HTML.gif is continuous with respect to Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq83_HTML.gif-convergence.

      Proof Let β n Δ β http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq222_HTML.gif in H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq119_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif. Then there exist f n S + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq223_HTML.gif and ( s n ) Δ + http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq115_HTML.gif such that ( β n β ) s n = [ f n s k s k ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq224_HTML.gif and f n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq142_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif. Employing Eq. (5.1), we get
      S ( ( β n β ) s n ) = [ S ( f n s k ) s k s ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equac_HTML.gif
      Hence, we have S ( ( β n β ) s n ) = [ y f n s s k s s k s ] 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq225_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif in H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq189_HTML.gif. Therefore
      S ( ( β n β ) s n ) = y ( β n s β s ) s n s 0 as  n . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equad_HTML.gif

      Hence, β n s Δ β s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq226_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq40_HTML.gif. □

      Theorem 5.6 β s : H ( Y ) H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq227_HTML.gif is onto.

      Proof Let [ f n s s n s ] H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq228_HTML.gif be arbitrary, then f n s s m s = f m s s n s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq229_HTML.gif for every m , n N http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq199_HTML.gif. Then f n s m = f m s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq230_HTML.gif. That is, f n s n http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq231_HTML.gif is the corresponding quotient of sequences of f n s s n s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq232_HTML.gif. Thus [ f n s n ] H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq233_HTML.gif is such that S [ f n s n ] = [ f n s s n s ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq234_HTML.gif in H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq146_HTML.gif. This completes the proof of the lemma.

      Let β s = [ f n s s n s ] H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq235_HTML.gif, then we define the inverse Sumudu transform of β s http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq236_HTML.gif by
      β s 1 = [ f n s n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equae_HTML.gif

      in the space H ( Y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq63_HTML.gif. □

      Theorem 5.7 Let [ f n s s n s ] H ( Y s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq237_HTML.gif and u D + ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq113_HTML.gif, u s D + s ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_IEq238_HTML.gif
      β ( [ f n s n ] u ) = [ f n s s n s ] u and β s 1 ( [ f n s s n s ] u s ) = [ f n s n ] u . http://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-77/MediaObjects/13662_2013_Article_424_Equaf_HTML.gif

      Proof is immediate from the definitions.

      Declarations

      Acknowledgements

      Dedicated to Professor Hari M Srivastava.

      The authors express their sincere thanks to the referee(s) for the careful and detailed reading of the manuscript and very helpful suggestions that improved the manuscript substantially.

      Authors’ Affiliations

      (1)
      Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University
      (2)
      Department of Mathematics and Institute of Mathematical Research, Universiti Putra Malaysia (UPM)

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      © Al-Omari and Kılıçman; licensee Springer. 2013

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