# Existence Results for Impulsive Fractional q-Difference Equation with Antiperiodic Boundary Conditions.

1. Introduction

In this paper, we are concerned with the existence and uniqueness of solutions for the following impulsive fractional q-difference equation with antiperiodic boundary conditions

[mathematical expression not reproducible] (1)

where q [member of] (0,1), 1 < [alpha] [less than or equal to] 2, 0 < [beta] < 1, [alpha] - [beta] -1 > 0, J = [0, 1], [D.sub.q] is q-derivative, [sup.c][D.sup.[alpha].sub.q], and [sup.c][D.sup.[beta].sub.q] denote the Caputo q-derivative of orders [alpha] and [beta], respectively. f [member of] C(J x R x R x R, R), [I.sub.k], [I.sup.*.sub.k] [member of] C(R, R) (k = 1, 2, ..., m), R is the set of all real numbers, and 0 = [t.sub.0] < [t.sub.1] < ... < [t.sub.m] < [t.sub.m+1] = 1. T and S are linear operators defined by

[mathematical expression not reproducible], (2)

where [mathematical expression not reproducible] represent the right and left limits of [mathematical expression not reproducible] has a similar meaning.

Fractional q-difference calculus plays a very important role in modern applied mathematics due to their deep physical background and has been studied extensively [1-4]. Impulsive differential equations are important in both theory and applications. Considerable effort has been devoted to differential equations with or without impulse, for example, [5-21]. In recent years, impulsive fractional difference and differential equations with antiperiodic conditions have received much attention; see [22-27] and the references therein. Zhang and Wang [24] have applied cone contraction fixed point theorem to establish the existence of solutions to nonlinear fractional differential equation with impulses and antiperiodic boundary conditions

[mathematical expression not reproducible], (3)

where [sup.c][D.sup.[alpha]] is the Caputo fractional derivative, f [member of] C(J x R, R), [I.sub.k], [I.sup.*.sub.k] [member of] C(R, R). By using Banach fixed point theorem, Schaefer fixed point theorem, and nonlinear alternative of Leray-Schauder type theorem, some existence results of solutions for problem (3) are obtained in [25]. Ahmad et al. [28] studied existence of solutions for the following antiperiodic boundary value problem (BVP for short) of impulsive fractional q-difference equation

[mathematical expression not reproducible], (4)

where [mathematical expression not reproducible] denotes the Caputo [q.sub.k]-fractional derivative of order [mathematical expression not reproducible] denote the Riemann-Liouville [q.sub.k]-integral of orders [[beta].sub.k] and [[gamma].sub.k], respectively.

In this paper we are concerned with the existence and uniqueness of solutions for impulsive fractional q-difference equation antiperiodic BVP. By applying the theorem of nonlinear alternative of Leray-Schauder type and Banach contraction mapping principle, we show the existence and uniqueness of solutions for the BVP (1). Some ideas of this paper are from [29, 30].

2. Preliminaries and Lemmas

For q [member of] (0, 1), let

[mathematical expression not reproducible]. (5)

We define the q-analogue of the power function [(a - b).sup.n] with n [member of] [N.sup.0] is

[mathematical expression not reproducible], (6)

and, for [alpha] [member of] R,

[mathematical expression not reproducible]. (7)

The q-derivative of f is defined by

[mathematical expression not reproducible] (8)

and q-derivative of higher order by

[mathematical expression not reproducible]. (9)

The q-integral of f is defined by

[mathematical expression not reproducible] (10)

Lemma 1 (see [31]). (1) If [absolute value of f] is q-integral on the interval [0, x], then [absolute value of [[integral].sup.x.sub.0] f(t)[d.sub.q]t] [less than or equal to] [[integral].sup.x.sub.0] [absolute value of f(t)][d.sub.q]t.

(2) If f and g are q-integral on the interval [0, x], f(t) [less than or equal to] g(t) for all t [member of] [0, x], then [[integral].sup.x.sub.0] f(t)[d.sub.q]t [less than or equal to] [[integral].sup.x.sub.0] g(t)[d.sub.q]t.

Definition 2 (see [2]). Let [alpha] [greater than or equal to] 0 and f be a function defined on [0, b]. The fractional q-integral of the Riemann-Liouville type is defined by ([I.sup.0.sub.q] f)(x) = f(x) and

[mathematical expression not reproducible]. (11)

Definition 3 (see [3]). The fractional q-derivative of the Caputo type of order [alpha] [greater than or equal to] 0 is defined by

[mathematical expression not reproducible], (12)

where [[alpha]] is the smallest integer greater than or equal to [alpha]. If [mathematical expression not reproducible].

Lemma 4 (see [2,3]). Let [alpha], [beta] [greater than or equal to] 0 and f be a function defined on [0,b]. The following formulas hold:

(1) ([I.sup.[beta].sub.q] [I.sup.[alpha].sub.q])(x) = ([I.sup.[alpha]+[beta].sub.q] f)(x);

(2) ([D.sup.[beta].sub.q] [I.sup.[alpha].sub.q] f)(x) = (f)(x);

Lemma 5 (see [3]). Let a [member of] [R.sup.+] \ N and a < x. Then

[mathematical expression not reproducible]. (13)

Lemma 6 (see [3]). For [beta] [member of] [R.sup.+], [lambda] [member of] (-1, +[infinity]) and 0 [less than or equal to] a < t [less than or equal to] b,

[mathematical expression not reproducible]. (14)

In particular, when [lambda] = 0 and a = 0 using q-integration by part,

[mathematical expression not reproducible]. (15)

Lemma 7 (see [32] (nonlinear alternative of Leray-Schauder type)). Let X be a Banach space, U be a bounded open subset of X with 0 [member of] U, and P : [bar.U] [right arrow] X be a completely continuous operator. Then, either there exists x [member of] [partial derivative]U such that x = [lambda]Px for [lambda] [member of] (0, 1) or there exists a fixed point [x.sup.*] [member of] [bar.U].

Let PC(J, R) = {u : u ia a map from J into R such that u(t) is continuous at t [not equal to] [t.sub.k], left continuous at t = [t.sub.k] and its right limit at t [not equal to] [t.sub.k] exists for k = 1, ..., m}; then PC(J, R) is a Banach space with the norm [[parallel]u[parallel].sub.PC] = sup{[absolute value of u(t)] : t [member of] J}.

Lemma 8. For h [member of] PC(J, R), the solution of impulsive BVP,

[mathematical expression not reproducible], (16)

is given by

[mathematical expression not reproducible]. (17)

Proof. In view of Definitions 2 and 3 and Lemma 5, for t [member of] [J.sub.k] = [[t.sub.k], [t.sub.k+1]], k = 0, 1, 2, ..., m, we have

[mathematical expression not reproducible], (18)

and

[mathematical expression not reproducible]. (19)

It follows from Definition 3, Lemma 5, and (18) that

[mathematical expression not reproducible], (20)

Applying [sup.c][D.sup.[beta].sub.q]u(0) = - [sup.c][D.sup.[beta].sub.q]u(1) in (20), we obtain

[mathematical expression not reproducible]. (21)

Note boundary conditions [mathematical expression not reproducible]; we get

[mathematical expression not reproducible]. (22)

Applying (21) and (22), we have

[mathematical expression not reproducible]. (23)

Thanks to u(0) = -u(1), it is derived that

[mathematical expression not reproducible]. (24)

and

[mathematical expression not reproducible]. (25)

By (22), we get

[mathematical expression not reproducible]. (26)

Combining (25) and (26), we have

[mathematical expression not reproducible]. (27)

Therefore, for t [member of] [J.sub.k], k = 0, 1, 2, ..., m - 1,

[mathematical expression not reproducible]. (28)

and, for t [member of] [J.sub.m],

[mathematical expression not reproducible]. (29)

3. Main Results

Define an operator A : PC(J, R) [right arrow] PC(J, R) by

[mathematical expression not reproducible]. (30)

Theorem 9. Assume that

([H.sub.1]) There exist nonnegative functions [L.sub.j](t) [member of] C(J) (j = 1, 2, 3) such that

[mathematical expression not reproducible]. (31)

for t [member of] J, [u.sub.i], [v.sub.i], [[omega].sub.j] [member of] R, i = 1, 2.

([H.sub.2]) There exist positive numbers N and [N.sup.*] such that

[mathematical expression not reproducible]. (32)

(H.sub.3)

[mathematical expression not reproducible]. (33)

where [[bar.L].sub.i], = max{[L.sub.i],(t) : t [member of] J}, i = 1, 2, 3, [k.sub.0] = max{[absolute value of k(f, s)] : (t, s) [member of] D}, [h.sub.0] = max{[absolute value of h(t, s)] : (t, s) [member of] [D.sub.0].

Then BVP (1) has a unique solution.

Proof. For u, v [member of] PC(J, R) and t [member of] J, we have

[mathematical expression not reproducible], (34)

and then [[parallel]Au - Av[parallel].sub.PC] [less than or equal to] [chi][parallel]u - v[parallel].sub.PC], and hence A is a contraction operator. It follows from Banach contraction mapping principle that BVP (1) has a unique solution.

Theorem 10. Assume the following:

([H.sub.4]) There exist continuous and nondecreasing function g : [0, +[infinity]) [right arrow] (0, +[infinity]) and a(t) [member of] C[0,1] such that

[absolute value of f (t, u, v, w)] [less than or equal to] a (t) g (max{[absolute value of u], [absolute value of v], [absolute value of w]}),

T [member of] [0, 1] u, v, w, [member of] R. (35)

([H.sub.5]) There exist continuous and nondecreasing functions [phi], [psi] : [0, +[infinity]) [right arrow] (0, +[infinity]) such that

[absolute value of [I.sub.k](u)] [less than or equal to] [psi]([absolute value of u]),

[absolute value of [I.sup.*.sub.k](u)] [less than or equal to] [psi]([absolute value of u]),

u [member of] R, k = 1, ..., m. (36)

([H.sub.6]) There exists constant M > 0 such that

[mathematical expression not reproducible]. (37)

Then BVP (1) has at least one solution.

Proof. The continuity of f, [I.sub.k], [I.sup.*.sub.k] implies that operator A : PC(J, R) [right arrow] PC(J, R) is continuous. Let B [subset] PC(J, R) be bounded; then there exist positive constants [P.sub.1], [P.sub.2], and [P.sub.3] such that [absolute value of f(t, u(t), Tu(t), Su(t))] [less than or equal to] [P.sub.1], [absolute value of [I.sub.i](u([t.sup.-.sub.i]))] [less than or equal to] [P.sub.2], and [I.sup.*.sub.i](u([t.sup.-.sub.i]))] [less than or equal to] [P.sub.3] for all t [member of] J, u [member of] B, i = 1, 2, ..., m. Thus, we have

[mathematical expression not reproducible]. (38)

Consequently, operator A is uniformly bounded on B.

On the other hand, for [t.sub.k] [less than or equal to] [[tau].sub.1] < [[tau].sub.2] [less than or equal to] [t.sub.k+1], u [member of] B, we have

[mathematical expression not reproducible], (39)

which tends to zero as [[tau].sub.2] [right arrow] [[tau].sub.1]; then A is equicontinuous on [J.sub.k]. Henceby PC-type Arzela-Ascoli Theorem ([33]), operator A : PC(J, R) [right arrow] PC(J, R) is completely continuous.

Let u [member of] PC(J, R) be such that u(t) = [lambda](Au)(t) for some [lambda] [member of] (0, 1); then

[mathematical expression not reproducible]. (40)

and hence

[mathematical expression not reproducible]. (41)

Let U = {u [member of] PC(J, R) : [parallel]u[parallel] < M}; then operator A : [bar.U] [right arrow] PC(J, R) is completely continuous. By ([H.sub.6]), one has u [not equal to] [lambda]Au for any [lambda] [member of] (0, 1) and u [member of] [partial derivative]U. By Lemma 7, BVP (1) has at least one solution.

4. Examples

Example 1. Consider the BVP

[mathematical expression not reproducible]. (42)

Let

[mathematical expression not reproducible]. (43)

By direct computation, [k.sub.0] = max{[e.sup.-(t+s)] : 0 [less than or equal to] s [less than or equal to] t [less than or equal to] 1} = 1, [h.sub.0] = max{1/(2 + t + s) : 0 [less than or equal to] s, t [less than or equal to] 1} = 1/2. For any [u.sub.1], [u.sub.2], [v.sub.1], [v.sub.2], [[omega].sub.1], [[omega].sub.2] [member of] R and t [member of] J, we have

[mathematical expression not reproducible]. (44)

Let [L.sub.1](t) = 1/100, [L.sub.2](t) = 1/50, [L.sub.3](t) = 1/80, N = 1/10, and [N.sup.*] = 1/20; then

[mathematical expression not reproducible]. (45)

Then, ([H.sub.1])-([H.sub.3]) hold. It follows from Theorem 9 that BVP (42) has a unique solution.

Example 2. Consider the BVP

[mathematical expression not reproducible]. (46)

Let

[mathematical expression not reproducible]. (47)

then

[absolute value of f(t, u(t), Tu(t), Su(t))] [less than or equal to] 1/60 (1 + t + 1)

[less than or equal to] 6(r + 1). (48)

Let g(r) = 6, a(t) = t + 1; then a' = max{t + 1 : t [member of] [0,1]} = 2. Choose [phi](w) = 1 and [psi](u) = 1; we have

[mathematical expression not reproducible]. (49)

Let M = 20; then condition ([H.sub.6]) holds. Therefore, by Theorem 10, BVP (46) has at least one solution.

https://doi.org/10.1155/2018/3798342

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors' Contributions

The authors contributed equally to this paper. All authors read and approved the final manuscript.

Acknowledgments

Supported financially by the National Natural Science Foundation of China (11501318,11871302), the China Postdoctoral Science Foundation (2017M612230), the Natural Science Foundation of Shandong Province of China (ZR2017MA036), and the International Cooperation Program of Key Professors by Qufu Normal University.

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Mingyue Zuo (iD) and Xinan Hao (iD)

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

Correspondence should be addressed to Xinan Hao; haoxinan2004@163.com

Received 17 July 2018; Accepted 17 September 2018; Published 9 October 2018