# Existence Results for Impulsive Fractionalq-Difference Equation with Antiperiodic Boundary Conditions

Authors: Mingyue Zuo; Xinan Hao
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this paper, we investigate the impulsive fractionalq-difference equation with antiperiodic conditions. The existence and uniqueness results of solutions are established via the theorem of nonlinear alternative of Leray-Schauder type and the Banach contraction mapping principle. Two examples are given to illustrate our results.


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## Body

## 1. Introduction

In this paper, we are concerned with the existence and uniqueness of solutions for the following impulsive fractionalq-difference equation with antiperiodic boundary conditions(1)Dcqαut=ft,ut,Tut,Sut,t∈J′=J∖t1,t2,…,tm,Δut=tk=Ikutk-,ΔDqut=tk=Ik∗utk-,k=1,2,…,m,u0=-u1,Dcqβu0=-Dcqβu1,where q∈(0,1), 1<α≤2, 0<β<1, α-β-1>0, J=[0,1], Dq is q-derivative, Dcqα, and Dcqβ denote the Caputo q-derivative of orders α and β, respectively. f∈C(J×R×R×R,R), Ik,Ik∗∈C(R,R)(k=1,2,…,m), R is the set of all real numbers, and 0=t0<t1<⋯<tm<tm+1=1. T and S are linear operators defined by (2)Tut=∫0tkt,susdqs,Sut=∫01ht,susdqs,t∈J,where k∈C(D,R), h∈C(J×J,R), D={(t,s)∈J×J:t≥s}. Δu|t=tk=u(tk+)-u(tk-), where u(tk+) and u(tk-) represent the right and left limits of u(t) at t=tk,ΔDqu|t=tk=Dqu(tk+)-Dqu(tk-) has a similar meaning.Fractionalq-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 (3)Dcαut=ft,ut,1<α≤2,t∈J∖t1,t2,…,tm,J=0,T,Δut=tk=Ikutk,Δu′t=tk=Ik∗utk,k=1,2,…,p,u0=-uT,u′0=-u′T,where Dcα is the Caputo fractional derivative, f∈C(J×R,R), Ik,Ik∗∈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 (4)​tkcDqkαkxt=ft,xt,t∈Jk⊆0,T,t≠tk,Δxt=tk=xtk+-xtk=φktk-1Iqk-1βk-1xtk,k=1,2,…,m,​tkDqkxtk+-​tk-1Dqk-1xtk=φk∗​tk-1Iqk-1γk-1xtk,k=1,2,…,m,x0=-xT,​0Dq0x0=-​tmDqmxT,where ​tkcDqkαk denotes the Caputo qk-fractional derivative of order αk on Jk, 1<αk≤2, 0<qk<1, f∈C(J×R,R), φk,φk∗∈C(R,R), k=1,2,…,m. ​tkIqkβk and ​tkIqkγk denote the Riemann-Liouville qk-integral of orders βk and γk, respectively.In this paper we are concerned with the existence and uniqueness of solutions for impulsive fractionalq-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

Forq∈(0,1), let(5)aq=1-qa1-q,a;q∞=∏i=0∞1-aqi,a;qα=a;q∞aqα;q∞,a,α∈R.We define the q-analogue of the power function (a-b)n with n∈N0 is (6)a-b0=1,a-bn=∏k=0n-1a-bqk,n∈N,a,b∈R,and, for α∈R, (7)a-bα=aα∏n=0∞a-bqna-bqα+n.The q-derivative of f is defined by(8)Dqfx=fx-fqx1-qx,Dqf0=limx→0Dqfxand q-derivative of higher order by (9)Dq0fx=fx,Dqnfx=DqDqn-1fx,n∈N.The q-integral of f is defined by (10)Iqf=∫0xftdqt=x1-q∑n=0∞fxqnqn,x∈0,b.Lemma 1 (see [31]).
(1) If|f| is q-integral on the interval [0,x], then |∫0xf(t)dqt|≤∫0x|f(t)|dqt.
(2) Iff and g are q-integral on the interval [0,x], f(t)≤g(t) for all t∈[0,x], then ∫0xf(t)dqt≤∫0xg(t)dqt.Definition 2 (see [2]).
Letα≥0 and f be a function defined on [0,b]. The fractional q-integral of the Riemann-Liouville type is defined by (Iq0f)(x)=f(x) and (11)Iqαfx=1Γqα∫0xx-qtα-1ftdqt,α>0,x∈0,b.Definition 3 (see [3]).
The fractionalq-derivative of the Caputo type of order α≥0 is defined by (12)​cDqαfx=Iqα-αDqαfx,α≥0,where [α] is the smallest integer greater than or equal to α. If f(x)=xβ-1, β>0, then ​cDqαf(x)=Γq(β)/Γq(β-α)xβ-α-1.Lemma 4 (see [2, 3]).
Letα,β≥0 and f be a function defined on [0,b]. The following formulas hold:
(1)(IqβIqαf)(x)=(Iqα+βf)(x);
(2)(DqαIqαf)(x)=f(x).Lemma 5 (see [3]).
Letα∈R+∖N and a<x. Then (13)Iqα​cDqαfx=fx-∑k=0α-1DqkfaΓqk+1xkax:qk.If α≥m≥β, then ​cDqβIqαf(x)=Iqm-βIqα-mf(x)=Iqα-βf(x).Lemma 6 (see [3]).
Forβ∈R+, λ∈(-1,+∞) and 0≤a<t≤b, (14)Iqβt-aλ=Γqλ+1Γqβ+λ+1t-aβ+λ.In particular, when λ=0 and a=0, using q-integration by part,(15)Iqβ1t=1Γqβ∫0tt-qsβ-1dqs=1Γqβ+1tβ.Lemma 7 (see [32] (nonlinear alternative of Leray-Schauder type)).
LetX be a Banach space, U be a bounded open subset of X with 0∈U, and P:U¯→X be a completely continuous operator. Then, either there exists x∈∂U such that x=λPx for λ∈(0,1) or there exists a fixed point x∗∈U¯.
LetPC(J,R)={u:u ia a map from J into R such that u(t) is continuous at t≠tk, left continuous at t=tk and its right limit at t=tk exists for k=1,…,m}; then PC(J,R) is a Banach space with the norm uPC=sup{|u(t)|:t∈J}.Lemma 8.
Forh∈PC(J,R), the solution of impulsive BVP, (16)​cDqαut=ht,t∈J′,Δut=tk=Ikutk-,ΔDqut=tk=Ik∗utk-,k=1,2,…,m,u0=-u1,​cDqβu0=-​cDqβu1,is given by(17)
u
t
=
1
Γ
q
α
∫
0
t
t
-
q
s
α
-
1
h
s
d
q
s
-
1
2
Γ
q
α
∫
0
1
1
-
q
s
α
-
1
h
s
d
q
s
-
1
2
∑
i
=
1
m
I
i
u
t
i
-
+
Γ
q
2
-
β
Γ
q
α
-
β
1
2
-
t
∫
0
1
1
-
q
s
α
-
β
-
1
h
s
d
q
s
+
1
2
∑
i
=
1
m
t
i
I
i
∗
u
t
i
-
-
∑
i
=
1
m
t
I
i
∗
u
t
i
-
,
t
∈
0
,
t
1
,
1
Γ
q
α
∫
0
t
t
-
q
s
α
-
1
h
s
d
q
s
-
1
2
Γ
q
α
∫
0
1
1
-
q
s
α
-
1
h
s
d
q
s
+
Γ
q
2
-
β
Γ
q
α
-
β
1
2
-
t
∫
0
1
1
-
q
s
α
-
β
-
1
h
s
d
q
s
+
1
2
∑
i
=
1
m
I
i
u
t
i
-
-
1
2
∑
i
=
1
m
t
i
I
i
∗
u
t
i
-
-
∑
i
=
k
+
1
m
I
i
u
t
i
-
+
∑
i
=
k
+
1
m
t
i
-
t
I
i
∗
u
t
i
-
,
t
∈
t
k
,
t
k
+
1
,
k
=
1
,
…
,
m
-
1
,
1
Γ
q
α
∫
0
t
t
-
q
s
α
-
1
h
s
d
q
s
-
1
2
Γ
q
α
∫
0
1
1
-
q
s
α
-
1
h
s
d
q
s
+
Γ
q
2
-
β
Γ
q
α
-
β
1
2
-
t
∫
0
1
1
-
q
s
α
-
β
-
1
h
s
d
q
s
+
1
2
∑
i
=
1
m
I
i
u
t
i
-
-
1
2
∑
i
=
1
m
t
i
I
i
∗
u
t
i
-
,
t
∈
t
m
,
1
.Proof.
In view of Definitions2 and 3 and Lemma 5, for t∈Jk=[tk,tk+1], k=0,1,2,…,m, we have (18)
u
t
=
I
q
α
h
t
+
d
k
+
e
k
t
=
1
Γ
q
α
∫
0
t
t
-
q
s
α
-
1
h
s
d
q
s
+
d
k
+
e
k
t
,and (19)Dqut=1Γqα-1∫0tt-qsα-2hsdqs+ek.It follows from Definition 3, Lemma 5, and (18) that (20)​cDqβut=1Γqα-β∫0tt-qsα-β-1hsdqs+ekt1-βΓq2-β,t∈Jk.Applying ​cDqβu(0)=-​cDqβu(1) in (20), we obtain (21)em=-Γq2-βΓqα-β∫011-qsα-β-1hsdqs.Note boundary conditions Δu|t=tk=Ik(u(tk-)) and ΔDqu|t=tk=Ik∗(u(tk-)); we get (22)dk-dk-1+ek-ek-1tk=Ikutk-,ek-ek-1=Ik∗utk-,k=1,2,…,m.Applying (21) and (22), we have (23)ek=em-∑i=k+1mIi∗uti-=-Γq2-βΓqα-β∫011-qsα-β-1hsdqs-∑i=k+1mIi∗uti-.Thanks to u(0)=-u(1), it is derived that (24)d0=-1Γqα∫011-qsα-1hsdqs-dm-em,and (25)dm+d0=-1Γqα∫011-qsα-1hsdqs-em=-1Γqα∫011-qsα-1hsdqs+Γq2-βΓqα-β∫011-qsα-β-1hsdqs.By (22), we get (26)dm-d0=∑i=1mIiuti--∑i=1mtiIi∗uti-.Combining (25) and (26), we have (27)dm=12-1Γqα∫011-qsα-1hsdqs+Γq2-βΓqα-β∫011-qsα-β-1hsdqs+∑i=1mIiuti--∑i=1mtiIi∗uti-,dk=dm-∑i=k+1mIiuti-+∑i=k+1mtiIi∗uti-=-12Γqα∫011-qsα-1hsdqs+Γq2-β2Γqα-β∫011-qsα-β-1hsdqs+12∑i=1mIiuti--12∑i=1mtiIi∗uti--∑i=k+1mIiuti-+∑i=k+1mtiIi∗uti-.Therefore, for t∈Jk, k=0,1,2,…,m-1, (28)
u
t
=
1
Γ
q
α
∫
0
t
t
-
q
s
α
-
1
h
s
d
q
s
-
1
2
Γ
q
α
∫
0
1
1
-
q
s
α
-
1
h
s
d
q
s
+
Γ
q
2
-
β
2
Γ
q
α
-
β
∫
0
1
1
-
q
s
α
-
β
-
1
h
s
d
q
s
+
1
2
∑
i
=
1
m
I
i
u
t
i
-
-
1
2
∑
i
=
1
m
t
i
I
i
∗
u
t
i
-
-
∑
i
=
k
+
1
m
I
i
u
t
i
-
+
∑
i
=
k
+
1
m
t
i
I
i
∗
u
t
i
-
-
Γ
q
2
-
β
Γ
q
α
-
β
t
∫
0
1
1
-
q
s
α
-
β
-
1
h
s
d
q
s
-
t
∑
i
=
k
+
1
m
I
i
∗
u
t
i
-
=
1
Γ
q
α
∫
0
t
t
-
q
s
α
-
1
h
s
d
q
s
-
1
2
Γ
q
α
∫
0
1
1
-
q
s
α
-
1
h
s
d
q
s
+
Γ
q
2
-
β
Γ
q
α
-
β
1
2
-
t
∫
0
1
1
-
q
s
α
-
β
-
1
h
s
d
q
s
+
1
2
∑
i
=
1
m
I
i
u
t
i
-
-
1
2
∑
i
=
1
m
t
i
I
i
∗
u
t
i
-
-
∑
i
=
k
+
1
m
I
i
u
t
i
-
+
∑
i
=
k
+
1
m
t
i
-
t
I
i
∗
u
t
i
-
,and, for t∈Jm, (29)ut=1Γqα∫0tt-qsα-1hsdqs-12Γqα∫011-qsα-1hsdqs+Γq2-βΓqα-β12-t∫011-qsα-β-1hsdqs+12∑i=1mIiuti--12∑i=1mtiIi∗uti-.


## 3. Main Results

Define an operatorA:PC(J,R)→PC(J,R) by (30)
A
u
t
=
1
Γ
q
α
∫
0
t
t
-
q
s
α
-
1
f
s
,
u
s
,
T
u
s
,
S
u
s
d
q
s
-
1
2
Γ
q
α
∫
0
1
1
-
q
s
α
-
1
f
s
,
u
s
,
T
u
s
,
S
u
s
d
q
s
+
Γ
q
2
-
β
Γ
q
α
-
β
1
2
-
t
∫
0
1
1
-
q
s
α
-
β
-
1
f
s
,
u
s
,
T
u
s
,
S
u
s
d
q
s
+
1
2
∑
i
=
1
m
I
i
u
t
i
-
-
1
2
∑
i
=
1
m
t
i
I
i
∗
u
t
i
-
-
∑
i
=
k
+
1
m
I
i
u
t
i
-
+
∑
i
=
k
+
1
m
t
i
-
t
I
i
∗
u
t
i
-
t
∈
J
k
,
k
=
0,1
,
…
,
m
.Theorem 9.
Assume that
( H 1 ) There exist nonnegative functions Lj(t)∈C(J)(j=1,2,3) such that (31)ft,u1,v1,w1-ft,u2,v2,w2≤L1tu1-u2+L2tv1-v2+L3tw1-w2for t∈J, ui,vi,ωi∈R, i=1,2.
( H 2 ) There exist positive numbers N and N∗ such that (32)Iku-Ikv≤Nu-v,Ik∗u-Ik∗v≤N∗u-v,u,v∈R,k=1,2,…,m.
( H 3 ) (33) χ = 3 L 1 ¯ + L 2 ¯ k 0 + L 3 ¯ h 0 2 Γ q α + 1 + Γ q 2 - β L 1 ¯ + L 2 ¯ k 0 + L 3 ¯ h 0 2 Γ q α - β + 1 + 3 2 m N + 5 2 m N ∗ < 1 ,where Li¯=max{Li(t):t∈J}, i=1,2,3, k0=max{|k(t,s)|:(t,s)∈D}, h0=max{|h(t,s)|:(t,s)∈D0}.Then BVP (1) has a unique solution.Proof.
Foru,v∈PC(J,R) and t∈J, we have (34)
A
u
t
-
A
v
t
≤
∫
0
t
t
-
q
s
α
-
1
Γ
q
α
f
s
,
u
s
,
T
u
s
,
S
u
s
-
f
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,
v
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,
T
v
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+
1
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1
-
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1
Γ
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,
u
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T
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q
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+
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q
2
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β
2
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q
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β
∫
0
1
1
-
q
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1
f
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,
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T
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T
v
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v
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+
1
2
∑
i
=
1
m
I
u
t
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-
-
I
i
v
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i
+
1
2
∑
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=
1
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∗
u
t
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-
I
i
∗
v
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+
∑
i
=
1
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i
u
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-
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+
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∑
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i
∗
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t
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-
≤
1
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q
α
∫
0
t
t
-
q
s
α
-
1
L
1
s
u
s
-
v
s
+
L
2
s
∫
0
s
k
s
,
τ
u
τ
-
v
τ
d
q
τ
+
L
3
s
∫
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1
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u
τ
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d
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+
1
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1
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1
L
1
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+
L
2
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∫
0
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k
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u
τ
-
v
τ
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q
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+
L
3
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∫
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1
h
s
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τ
u
τ
-
v
τ
d
q
τ
d
q
s
+
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q
2
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β
2
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q
α
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β
∫
0
1
1
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1
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+
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∫
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v
τ
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+
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3
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∫
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1
h
s
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u
τ
-
v
τ
d
q
τ
d
q
s
+
3
2
∑
i
=
1
m
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i
u
t
i
+
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i
v
t
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-
+
∑
i
=
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3
2
t
i
+
t
I
i
∗
u
t
i
+
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I
i
∗
v
t
i
-
≤
1
Γ
q
α
∫
0
t
t
-
q
s
α
-
1
L
1
¯
u
-
v
P
C
+
L
2
¯
k
0
u
-
v
P
C
+
L
3
¯
h
0
u
-
v
P
C
d
q
s
+
1
2
Γ
q
α
∫
0
1
1
-
q
s
α
-
1
L
1
¯
u
-
v
P
C
+
L
2
¯
k
0
u
-
v
P
C
+
L
3
¯
h
0
u
-
v
P
C
d
q
s
+
Γ
q
2
-
β
2
Γ
q
α
-
β
∫
0
1
1
-
q
s
α
-
β
-
1
L
1
¯
u
-
v
P
C
+
L
2
¯
k
0
u
-
v
P
C
+
L
3
¯
h
0
u
-
v
P
C
d
q
s
+
3
2
m
N
u
-
v
P
C
+
5
2
m
N
∗
u
-
v
P
C
≤
L
1
¯
+
L
2
¯
k
0
+
L
3
¯
h
0
1
Γ
q
α
+
1
+
1
2
Γ
q
α
+
1
+
Γ
q
2
-
β
2
Γ
q
α
-
β
+
1
u
-
v
P
C
+
3
2
m
N
+
5
2
m
N
∗
u
-
v
P
C
=
3
L
1
¯
+
L
2
¯
k
0
+
L
3
¯
h
0
2
Γ
q
α
+
1
+
Γ
q
2
-
β
L
1
¯
+
L
2
¯
k
0
+
L
3
¯
h
0
2
Γ
q
α
-
β
+
1
+
3
2
m
N
+
5
2
m
N
∗
u
-
v
P
C
=
χ
u
-
v
P
C
,and then Au-AvPC≤χu-vPC, 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 4 ) There exist continuous and nondecreasing function g:[0,+∞)→(0,+∞) and a(t)∈C[0,1] such that (35)ft,u,v,w≤atgmax⁡u,v,w,t∈0,1,u,v,w∈R.
( H 5 ) There exist continuous and nondecreasing functions φ,ψ:[0,+∞)→(0,+∞) such that (36)Iku≤φu,Ik∗u≤ψu,u∈R,k=1,…,m.
( H 6 ) There exists constant M>0 such that (37)M>a′gmaxM,k0M,h0M32Γqα+1+Γq2-β2Γqα-β+1+3m2φM+52mψM,where a′=max{a(t):t∈[0,1]}.Then BVP (1) has at least one solution.Proof.
The continuity off,Ik,Ik∗ implies that operator A:PC(J,R)→PC(J,R) is continuous. Let B⊂PC(J,R) be bounded; then there exist positive constants P1, P2, and P3 such that |f(t,u(t),Tu(t),Su(t))|≤P1, |Ii(u(ti-))|≤P2, and |Ii∗(u(ti-))|≤P3 for all t∈J, u∈B, i=1,2,…,m. Thus, we have (38)Aut≤∫0tt-qsα-1ΓqαP1dqs+12∫011-qsα-1ΓqαP1dqs+12Γq2-β∫011-qsα-β-1Γqα-βP1dqs+m2P2+P32∑i=1mti+mP2+P3∑i=1mti+mP3≤3P12Γqα+1+Γq2-β2Γqα-β+1P1+3m2P2+52mP3.Consequently, operator A is uniformly bounded on B.
On the other hand, fortk≤τ1<τ2≤tk+1,u∈B, we have (39)Auτ2-Auτ1=1Γqα∫0τ2τ2-qsα-1fs,us,Tus,Susdqs-1Γqα∫0τ1τ1-qsα-1fs,us,Tus,Susdqs+Γq2-βΓqα-βτ1-τ2∫011-qsα-β-1fs,us,Tus,Susdqs+∑i=k+1mIi∗uti-τ1-τ2≤1Γqα∫0τ1τ2-qsα-1-τ1-qsα-1fs,us,Tus,Susdqs+1Γqα∫τ1τ2τ2-qsα-1fs,us,Tus,Susdqs+Γq2-βΓqα-βP1τ2-τ1∫011-qsα-β-1dqs+mP3τ2-τ1≤P1τ2α-τ1αΓqα+1+P1Γq2-βΓqα-β+1τ2-τ1+mP3τ2-τ1,which tends to zero as τ2→τ1; then A is equicontinuous on Jk. Hence by PC-type Arzela-Ascoli Theorem ([33]), operator A:PC(J,R)→PC(J,R) is completely continuous.
Letu∈PC(J,R) be such that u(t)=λ(Au)(t) for some λ∈(0,1); then (40)
u
t
=
λ
A
u
t
≤
A
u
t
≤
∫
0
t
t
-
q
s
α
-
1
Γ
q
α
a
′
g
m
a
x
u
P
C
,
k
0
u
P
C
,
h
0
u
P
C
d
q
s
+
1
2
∫
0
1
1
-
q
s
α
-
1
Γ
q
α
a
′
g
m
a
x
u
P
C
,
k
0
u
P
C
,
h
0
u
P
C
d
q
s
+
1
2
Γ
q
2
-
β
∫
0
1
1
-
q
s
α
-
β
-
1
Γ
q
α
-
β
a
′
g
m
a
x
u
P
C
,
k
0
u
P
C
,
h
0
u
P
C
d
q
s
+
1
2
m
φ
u
P
C
+
1
2
ψ
u
P
C
∑
i
=
1
m
t
i
+
m
φ
u
P
C
+
ψ
u
P
C
∑
i
=
1
m
t
i
+
m
ψ
u
P
C
≤
a
′
g
m
a
x
u
P
C
,
k
0
u
P
C
,
h
0
u
P
C
3
2
Γ
q
α
+
1
+
Γ
q
2
-
β
2
Γ
q
α
-
β
+
1
+
3
m
2
φ
u
P
C
+
5
2
m
ψ
u
P
C
,and hence (41)uPC≤a′gmaxuPC,k0uPC,h0uPC32Γqα+1+Γq2-β2Γqα-β+1+3m2φuPC+52mψuPC.Let U={u∈PC(J,R):u<M}; then operator A:U¯→PC(J,R) is completely continuous. By (H6), one has u≠λAu for any λ∈(0,1) and u∈∂U. By Lemma 7, BVP (1) has at least one solution.


## 4. Examples

Example 1.
Consider the BVP(42)​cD1/23/2ut=ut+1100+150+t2∫0tuset+sdqs+e-t80∫01us2+t+sdqs,t∈0,1∖12,Δut=1/2=u1/210+u1/2,ΔD1/2ut=1/2=u1/220+u1/2,u0=-u1,​cD1/21/4u0=-​cD1/21/4u1.Let(43)ft,u,v,w=u+1100+150+t2v+e-t80w,Tut=∫0te-t+susdqs,Sut=∫01us2+t+sdqs.By direct computation, k0=max{e-(t+s):0≤s≤t≤1}=1,h0=max{1/(2+t+s):0≤s,t≤1}=1/2. For any u1,u2,v1,v2,ω1,ω2∈R and t∈J, we have (44)ft,u1,v1,w1-ft,u2,v2,w2≤1100u1-u2+150v1-v2+180w1-w2,Iku-Ikv≤110u-v,Ik∗u-Ik∗v≤120u-v.Let L1(t)=1/100, L2(t)=1/50, L3(t)=1/80, N=1/10, and N∗=1/20; then (45)χ=3L1¯+L2¯k0+L3¯h02Γ1/25/2+Γ1/27/4L1¯+L2¯k0+L3¯h02Γ1/29/4+32N+52N∗≈0.307<1.Then, (H1)-(H3) hold. It follows from Theorem 9 that BVP (42) has a unique solution.Example 2.
Consider the BVP(46)​cDq3/2ut=t260sinut+∫0tcosusdqs+∫011us2+t2+1dqs,t∈0,1∖12,Δut=1/2=u1/220+u1/2,ΔDqut=1/2=u1/230+u1/2,u0=-u1,​cDq1/4u0=-​cDq1/4u1.Let (47)ft,u,v,w=t260sinu+v+w,Tut=∫0tcosusdqs,Sut=∫01dqsus2+t2+1;then (48)ft,ut,Tut,Sut≤1601+t+1≤6t+1.Let g(r)=6, a(t)=t+1; then a′=max{t+1:t∈[0,1]}=2. Choose φ(u)=1 and ψ(u)=1; we have (49)2×6×32Γ1/25/2+Γ1/27/42Γ1/29/4+32+52≈19.87.Let M=20; then condition (H6) holds. Therefore, by Theorem 10, BVP (46) has at least one solution.



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*Source: 3798342-2018-10-09.xml*