A unified integral operator and further its consequences

Abstract: The aim of this paper is to construct left sided and right sided integral operators in a unified form. These integral operators produce various well known integral operators in the theory of fractional calculus. Formulated integral operators of this study include generalized fractional integral operators of Riemann-Liouville type and operators containing Mittag-Leffler functions in their kernels. Also boundedness of all these fractional integral operators is derived from the boundedness of unified integral operators. The existence of new integral operators may have useful consequences in applied sciences besides in fractional calculus.


Introduction
W e start with the following compact form of fractional integrals: Definition 1. [1] Let f : [a, b] → R be an integrable function. Also let g be an increasing and positive function on (a, b], having a continuous derivative g on (a, b). The left-sided and right-sided fractional integrals of a function f with respect to another function g on [a, b] of order µ ∈ C (R(µ) > 0) are defined as: and where Γ(.) is the gamma function.
A k-fractional analogue of above definition is given as follows: [2] Let f : [a, b] → R be an integrable function. Also let g be an increasing and positive function on (a, b], having a continuous derivative g on (a, b). The left-sided and right-sided k-fractional integral operators, k > 0 of a function f with respect to another function g on [a, b] of order µ ∈ C (R(µ) > 0) are defined as: and where is the k-gamma function.
Fractional integral operators which have been studied in recent decades are special cases of generalized classical integral operators of Riemann-Liouville type which are defined in (1) and (2). Instead of studying the derivations of recently defined fractional integral operators from (3) and (4), the authors of these decades have derived them independently, see [1,[3][4][5][6][7][8]. Remark 1 provides derivations of these fractional integrals from (1) and (2) and along with their k-analogues (3) and (4). A detail study of fractional integrals associated with (3) and (4), which have been investigated by the authors of recent years, is summarized in the following remark: (3) and (4) particularly produce several known fractional integrals corresponding to different settings of k and g.
The extended generalized Mittag-Leffler function E γ,δ,ν,c µ,α,l (t; p) is absolutely convergent for ν < δ + (µ), (see [9]). If S is the sum of series of absolute terms of the Mittag-Leffler function E γ,δ,ν,c µ,α,l (t; p), then one can obtain E γ,δ,ν,c µ,α,l (t; p) ≤ S. This property of absolutely convergence of the Mittag-Leffler function is used in establishing Theorems 6, 7, 8. The rest of the paper is organized as follows: In Section 2, we establish the existence of integral operators in a unified form. The bounds of these unified integral operators are also obtained. It is important to note that the unified integral operators produce almost all Riemann-Liouville type fractional integral operators as well as fractional integral operators containing the Mittag-Leffler function in their kernels. Furthermore, the bounds of all these fractional integral operators are obtained in Section 2 and Section 3, from the bounds which have been established for unified integral operators.

Existence of new unified integral operators
The first result provides the existence of new integral operators with upper bounds in variable form.
From above theorem, we are motivated to give the definition of a new unified two-sided integral operator as follows: be the functions such that f be positive and f ∈ L 1 [a, b], and g be differentiable and strictly increasing. Also let φ x be an increasing function on [a, ∞) and ω, α, l, γ, c ∈ C, (α), (l) > 0, (c) > (γ) > 0 with p ≥ 0, µ, δ > 0 and 0 < ν ≤ δ + µ. Then for x ∈ [a, b] the left and right integral operators are defined by The upcoming theorem provides the boundedness of the unified integral operators.

Theorem 6.
Under the assumption of Theorem 4, the following bounds hold for integral operators (15) and (16): and where S is the sum of absolute terms of series in (7) and Proof. From (11), it can be obtained x a g (t) f (t) dt by simplifying the above inequality we get x] (g(x) − g(a)).
and hence (17) can be achieved. Similarly from (14), one can obtain and hence (18) can be achieved. By using (17) and (18), the inequality in (19) can be obtained.
Integral operators defined in (15) and (16) have connection with fractional integral operators given in Section 1. The upcoming section describes the connection of these integral operators with fractional integral operators (5) and (6) and their bounds are computed.

Bounds of fractional integral operators containing Mittag-Leffler functions
In this whole section, we set the function φ(x) = x α , α > 0 and the function g(x) = I(x), where I denotes the identity function. In this case bounds of fractional integral operators containing Mittag-Leffler functions defined in [1,[3][4][5][6][7][8] can be obtained at once from bounds of unified integral operators (15) and (16). As an example the bounds of fractional integral operators defined by Andrić et al. in [9] are obtained. Computation of rest of the bounds of related fractional integrals described in Remark 2 are left for the reader.  (5) and (6) are bounded for α > 1, further the following inequality holds: Proof. Let g(x) = x and φ(x) = x α . Then φ x is increasing for α > 1. Therefore and γ,δ,ν,c Hence boundedness of fractional operators (5) and (6) is followed and (21) can be achieved. (22) and (23), the boundedness of all fractional integrals containing Mittag-Leffler functions compiled in Remark 2 can be obtained.

Bounds of fractional integral operators associated with generalized k-fractional integrals
In this whole section, we set the function φ(x) = x β k kΓ k (β) , β, k > 0. In this case bounds of fractional integral operators defined in [10][11][12][13] can be obtained at once from bounds of unified integral operators (15) and (16). As an example the bounds of fractional integral operators (3) and (4) (3) and (4) are bounded for β > k, further the following inequality holds: Proof. Let φ(x) = x β k kΓ k (β) , β, k > 0 and p = ω = 0. Then φ x is increasing for β > k. Therefore and Hence boundedness of fractional operators (3) and (4) is followed and (24) can be achieved. (25) and (26), the boundedness of all fractional integrals compiled in Remark 1 can be obtained. Also by setting g(x) = I(x) in (22) and (23) integral operators defined in [14] can be obtained and using Theorem 6 bounds of these integral operators can be achieved.

Concluding remarks
The aim of this study is to develop unified integral operators which provide the fractional integral operators of Riemann-Liouville type as well as fractional integral operators containing Mittag-Leffler functions in their kernels. The bounds of these new integral operators are computed which simultaneously provide the bounds of all fractional integral operators defined in [1,[3][4][5][6][7][8][10][11][12][13]. The existence of new generalized integral operators may be useful in applied sciences along with the theory and applications of fractional calculus.