On Fractional Volterra integrodifferential equations with fractional integrable impulses

We consider a class of nonlinear fractional Volterra integrodifferential equation with fractional integrable impulses and investigate the existence and uniqueness results in the Bielecki's normed Banach spaces. Further, Bielecki--Ulam type stabilities have been demonstrated on a compact interval. A concrete example is provided to illustrate the outcomes we acquired.


Introduction
Famous "Ulam stability problem" of functional differential equation raised by Ulam [1] have been extended to different kinds of equations. Wang et al. [2] are the first mathematicians who investigated the Ulam stability and data dependence for fractional differential equations. Thereafter, several interesting works on different Ulam type stabilities of fractional differential and integral equations have been reported (see for instance [3], [4], [5], [6], [7]).
An overview pertaining to impulsive differential equations with instantaneous impulses and its applicability in the practical dynamical systems have provided in the monograph [8,9,10]. The impulsive differential equations with time variable impulses dealt in an interesting papers [11,12,13]. Wang et al. [14,15] studied existence and uniqueness of solutions and established generalized β-Ulam-Hyers-Rassias stability to differential equations with not instantaneous impulses in a P β-normed Banach space.
Recently, Wang and Zang [16] introduced a new class of impulsive differential equations of the form x ′ (t) = f (t, x(t)) , t ∈ (s i , t i+1 ], i = 0, 1, · · · , m, x(t) = I β t i ,t h i (t, x(t)), t ∈ (t i , s i ], i = 1, 2, · · · , m, β ∈ (0, 1). and examined the existence and uniqueness of solutions in Bielecki's normed Banach spaces. Further, demonstrated that the corresponding equations are Bielecki-Ulam's-Hyer stable. It is seen that such sort of formulations are adequate to depict the memory procedures of the drugs in the circulation system and the subsequent absorption for the body.
We comment that within this scope, the class of equations considered in the present paper is more broad and the outcomes acquired are the generalization of the fundamental results obtained by Wang and Zang [16]. We support our main results with the examples.
In section 2, we introduce some preliminaries and auxiliary lemmas related to fractional calculus. In Section 3, we prove existence and uniqueness results for (1.1) by using Banach contraction principle via Bielecki norm. In Section 4, adopting the idea of Wang and Zang [16] we examine different BieleckiUlams type stability for the problem (1.1). Finally, an example has been provided to illustrate the results we obtained.

Preliminaries
x is continuous function} be the Banach space endowed with a Bielecki norm where θ > 0 is a fixed real number. Let If P C(J, R) is endowed with the norm with the norm x P B ′ = max { x P B , x ′ P B } then P C 1 (J, R), · P B ′ is a Banach space.
Next, we use definitions and the results listed bellow from fractional calculus. For more details, we refer the readers to the monographs [17] Definition 2.1 Let g ∈ C[a, T ] with T > a ≥ 0 and β ≥ 0, then the Riemann-Liouville fractional integral I β a,τ of order β of a function g is defined as provided the integral exists.
Following lemma plays an important role to obtain our results.

Existence and Uniqueness Results
Proof: For i = 0, on operating Riemann-Liouville fractional integral operator I α σ i ,τ on both sides of fractional differential equation (3.1), we get As 0 < α < 1, in view of lemma 2.1, we obtain Similarly, for each i (i = 1, 2, · · · m), operating I α σ i ,τ on both sides of (3.1), we get But from equation (3.1), we have ✷ We list the following hypotheses in order to establish our main results.
(H1) The function f ∈ C(J × R × R, R) satisfies the Lipschitz condition (H2) The function h ∈ C(J × R, R) satisfies the Lipschitz condition where K h > 0.

Bielecki-Ulam-Hyers Stability
We adopt the idea of Wang and Zang [16] to investigate the concepts of Bielecki-Ulams type stability for the class of nonlinear fractional order Volterra integrodifferential equation (1.1).

Definition 4.4 The equation (1.1) is generalized Bielecki-Ulam-Hyers-Rassias stable with respect to
Lemma 4.1 If y ∈ P C 1 (J, R) is a solution of inequality (4.3) then y satisfies the following integral inequalities (4.4) Proof: If y ∈ P C 1 (J, R) is a solution of the inequality (4.3) then there exists H ∈ P C(J, R) and constants H i , i = 1, 2, · · · m (which depend on y) such that In the view of Lemma 3.1, above equation is equivalent to the integral equations For any τ ∈ (σ i , τ i+1 ], i = 1, 2, · · · m, For τ ∈ (0, τ 1 ], The last three inequalities are the required equivalent integral inequalities in (4.4). ✷ Following additional assumption is needed to prove the Bielecki-Ulam-Hyers-Rassias stability of equation (1.1).