Existence of solutions of multi-term fractional differential equations with impulse effects on a half line

Abstract. A class of boundary value problem for impulsive fractional differential equation on a half line is proposed. Some results on existence of solutions of this kind of boundary value problem for impulsive multi-term fractional differential equation on a half line are established by constructing a weighted Banach space, a completely continuous operator and using a fixed point theorem in the Banach space. Some unsuitable lemmas in recent published papers are pointed out. An example is given to illustrate the efficiency of the main theorems.


Introduction
Fractional differential equation is a generalization of ordinary differential equation to arbitrary noninteger orders.Recent investigations have shown that many physical systems can be represented more accurately through fractional derivative formulation [10,13].There have been many excellent books and monographs available on this field [7,11,12].
Many authors have studied the existence and uniqueness of solutions of the impulsive fractional differential equations involving the Caputo fractional derivatives.Impulsive fractional differential equations is an important area of study [1].There have been many questions needed be studied.For example, authors in papers [2,3,6] studied the existence of solutions of the different initial value problems for the impulsive fractional differential equations.
In the literature, D α 0 + u(t) + f (t, u(t)) = 0 is known as a single term equation.In certain cases, we find equations containing more than one differential terms.A classical example is the so-called Bagley-Torvik equation where A, B, C are constants, and f is a given function.This equation arises from example for the modelling of motion of a rigid plate immersed in a Newtonian fluid.It was originally proposed in [14].Another example for an application of equations with more than one fractional derivatives is the Basset equation where 0 < n < 1.This equation is most frequently, but not exclusively, used with n = 1/2.It describes the forces that occur when a spherical object sinks in a (relatively dense) incompressible viscous fluid; see [5,10].
In recent paper [8], Liu studied existence of positive solutions for the following boundary value problems (BVP) for fractional impulsive differential equations: where D α 0 + is the Riemann-Liouville fractional derivative of order α ∈ (1, 2) with the base point 0, m is a positive integer, c k ∈ (0, 1/2), f : [0, 1] × [0, +∞) → [0, +∞) is a given continuous function, u(t + k ) and u(t − k ) denote the right limit and left limit of u at t k and u(t + k ) = u(t k ), i.e., u is right continuous at t k .In [15], Zhao and Ge studied the following fractional impulsive boundary value problem on infinite intervals: where α ∈ (1, 2], D * 0 + is the Riemann-Liouville fractional derivatives of orders * > 0, +∞), and u → I k (u) is nonnegative, continuous, and bounded.Existence, uniqueness, and computational method of unbounded positive solutions were established.
We note that Lemma 3.1 in [8] and Lemma 3.1 in [15] are unsuitable; see Remarks 2 and 3 in Section 3.This motivates us to establish results on solutions of impulsive fractional differential equations with order α ∈ (1, 2).In this paper, we discuss the following boundary value problems for nonlinear impulsive fractional differential equations: are finite, and x satisfies all equations in (3).We obtain the results on existence of solutions for BVP (3).An example is given to illustrate the efficiency of the main theorem.
The remainder of this paper is as follows: in Section 2, we present preliminary results.In Section 3, the main theorem on the existence of solutions of (3) are presented.In Section 4, an example is given to illustrate the main results.

Preliminary results
For the convenience of the readers, we present the necessary definitions from the fractional calculus theory.These definitions and results can be found in the monograph [11].
Let the gamma and beta functions Γ(α), B(p, q) be defined by ds, provided that the right-hand side exists [11].
For ease of expression, denote Definition 3. Let σ > 1 + k.We call K is called a Carathéodory function if it satisfies the followings: for almost all t ∈ (0, ∞); (iii) for each r > 0, there exists a constant A r > 0 such that For x ∈ X, define the norm by By standard method, we can show that X is a real Banach space.
Lemma 1. Suppose that x ∈ X, (a)-(d) hold, and Then u ∈ X is a solution of https://www.mii.vu.lt/NA if and only if Proof.From x ∈ X, there exists r > 0 such that Since f 1 is a Carathéodory function, by Definition 3, there exists A r > 0 such that We divide the whole proof into two steps.
Lemma 2. Suppose that (a)-(d) hold, and Π = 0 defined in Lemma 1. Then T : X → X is well defined and completely continuous.
Proof.The proof is standard and is omitted.
In this section, we shall establish existence result of at least one solution of (3).For easy referencing, we list the necessary conditions as follows: (A σ ) There exist positive numbers σ 1 > σ 2 > • • • > σ n > 0, bounded functions ψ : (0, ∞) → R, and nonnegative numbers a j , b j (j = 1, 2, . . ., n) such that Set Theorem 1. Suppose that (a)-(d) and (A σ ) hold, Π = 0 defined in Lemma 1. Then system (3) has at least one solution if one of the following items is satisfied: Nonlinear Anal.Model.Control, 22(5):679-701 Proof.Let the Banach space X and its norm be defined as in Section 2. Define the nonlinear operator T by (14).Then (i) T : X → X is well defined; (ii) x is a bounded positive solution of BVP (3) if and only if x is a solution of the operator equation x = T x in X; (iii) T : X → X is completely continuous.
It is easy to show that Ψ ∈ X.Let r > 0, and define M r = {x ∈ X: x − Ψ r}.For x ∈ M r , we find x x − Ψ + Ψ and By definition of T and Ψ and (15), we have for t ∈ (t i , t i+1 ] that Similarly, we get https://www.mii.vu.lt/NAwhere then u is a solution of We find that this result is wrong.In fact, ( 16) can be rewritten by Hence, we have for t ∈ (t k , t k+1 ] (k = 1, 2, . . ., m) that Nonlinear Anal.Model.Control, 22 (5):679-701 By changing the order of sum and integral, we get ds y(u) du .
It was proved that u is a solution of BVP (1) (x is continuous at each point t = t i , right continuous at t i , the left limit lim t→t − i x(t) is finite and satisfies (1)).Here We find that Lemma of [8] is wrong.In fact, if u is a fixed point of A, then we get It is rewritten by One can easily verifies that D α 0 + x(t) = f (t, x(t)), t ∈ (t i , t i+1 ], i = 1, . . ., m, similarly to above discussion in Remark 2.

An example
To illustrate the usefulness of our main results, we present an example that Theorem 1 can be easily applied.