Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives

: The main aim of the paper is to present an algorithm to solve approximately initial value problems for a scalar non-linear fractional differential equation with generalized proportional fractional derivative on a ﬁnite interval. The main condition is connected with the one sided Lipschitz condition of the right hand side part of the given equation. An iterative scheme, based on appropriately deﬁned mild lower and mild upper solutions, is provided. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. The approximate method is based on the application of the method of lower and upper solutions combined with the monotone-iterative technique.


Introduction
Fractional differential equations are effective in both theoretical and applied mathematics and arise in models of medicine, engineering, biochemistry, thermal and mechanical systems, acoustics and modeling of materials, etc. There are different forms of fractional derivatives and consequently numerous fractional derivatives have appeared (see, for example, [1][2][3][4][5][6] and the references cited therein). Jarad et al. [7] introduced a new generalized proportional derivative which is well-behaved and has several advantages over classical derivatives and generalizes known derivatives in the literature. For recent contributions relevant to fractional differential equations via generalized proportional derivatives, see e.g., [8][9][10][11][12]. We note that initial value problems for Riemann-Liouville fractional differential equations differ from the Caputo fractional ones and requires a separate study.
The theory of impulsive differential equations has undergone rapid development over the years (see, for example, the monographs by Benchohra et al. [13], Lakshmikantham et al. [14], Samoilenko and Perestyuk [15], and the references therein). Impulses were also considered for fractional-order differential systems, and the theory of impulsive fractional differential systems was presented in the literature, mainly for fractional derivatives of Caputo type (see for example, [16][17][18]).
Note that most fractional differential equations have no explicit solutions, so developing approximate methods is usually required. In this paper, a new algorithm for approximate solving an initial value problem for scalar non-linear fractional differential equations with generalized proportional fractional derivative is proposed. This method is based on the application of the method of lower and upper solutions and the monotone-iterative technique. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem.
Consider the following fractional differential equation with the generalized proportional fractional derivative with fractional initial and impulsive conditions (PIVP): ( R t i D α,ρ u)(t) = ψ(t, u(t)), t ∈ (t i , t i+1 ], i = 0, 1, 2, . . . , m, where u : [0, T] → R is a function, ρ ∈ (0, 1], α ∈ (0, 1) are two reals, u 0 is a real constant, and ψ : [0, T] × R → R and Ψ i : R → R, i = 1, 2, . . . , m are two functions. We recall that the generalized proportional fractional integral and the generalized proportional fractional derivative of a function υ : [a, b] → R are defined, respectively, by (see [7]) Remark 1. Note that the generalized proportional fractional derivative of Riemann-Liouville fractional type leads to an appropriate definition of the impulsive conditions similar to the initial condition (see the last two equations in problem (1). Additionally, we consider the case when the lower limit of the fractional derivative is changed at any impulsive point.
Observe that a solution of the PIVP (1) can have singularities at the points t i , for i = 0, 1, 2, . . . , m. Let equipped with the norms

Mild Lower/Upper Solutions
. . , m be positive constants (to be determined later). Then PIVP (1) can be equivalently written in the form where and The solution where for t ∈ (t k , t k+1 ] and k = 0, 1, 2, . . . , m, and Remark 3. According to Lemma 1, the solution Based on (15) we will define mild lower/upper solutions of (1).

According to the claims [c]-[e], the inequalities
hold. Take the limit as n → ∞ in (40), use the continuity of the function ψ, the definition (14) of the function G k , and we obtain the Volterra fractional integral equatioñ and lim t→t k +Ṽ Therefore, the equalities and hold. Define the function is a mild solution of the PIVP (1) on [0, T], i.e., the functions V(·) and W(·) satisfy the initial value problem in (1).

Conclusions
Recently many different types of fractional derivatives are defined and applied to model more adequate real world phenomena. One of the last introduced fractional derivatives is the so called generalized proportional fractional derivative, which is a generalization of the classical Caputo and Riemann-Liouville fractional ones. The main difficulties in the application of these derivatives to differential equations is that it is very difficult to obtain exact solutions even in the scalar case. As a result we require some algorithm to solve the corresponding initial value problems approximately. In this paper an approximate method for solving an initial value problem for a scalar non-linear fractional differential equation with generalized proportional fractional derivative of Riemann-Liouville type on a finite interval is proposed. We study the case when some impulsive perturbations with negligible small action time are applied to the equation. In connection with these impulses we set up in appropriate way both the impulsive and the initial conditions. Additionally, we consider the case when the lower limit of the fractional derivative is hanged at any impulsive time. The suggested approximate scheme is based on the method of lower and upper solutions combined with the monotone-iterative technique. Mild lower and mild upper solutions are defined in an appropriate way. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. To the best of our knowledge it is the first approximate scheme suggested to the initial value problem of this type of fractional differential equation.