Impulsive Periodic Type Boundary Value Problems for Multi-Term Singular Fractional Differential Equations

MALAYSIANMATHEMATICAL

SCIENCESSOCIETY http://math.usm.my/bulletin

Impulsive Periodic Type Boundary Value Problems for Multi-Term Singular Fractional Differential Equations

YUJILIU

Department of Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, P. R.China liuyuji888@sohu.com

Abstract. A class of impulsive periodic type boundary value problems for multi-term sin- gular fractional differential equations is presented. Results on the existence of solutions to these problems are established. The analysis relies on the well known fixed point theorems.

An example is given to illustrate the efficiency of the main theorems.

2010 Mathematics Subject Classification: 92D25, 34A37, 34K15

Keywords and phrases: Solution, impulsive singular fractional differential equation, peri- odic type boundary value problem, fixed point theorem.

1. Introduction

In recent papers [1–6, 8, 11–14, 16–20], the authors studied the existence or uniqueness of positive solutions or solutions of boundary value problems for fractional differential equa- tions. While the existence of solutions of impulsive boundary value problems for Riemann- Liouville fractional differential equations has not been given up to now, the research pro- ceeds slowly and appears some new difficulties.

In [15], the authors studied the existence and uniqueness of solutions of the following periodic boundary value problem of the impulsive fractional differential equation

(1.1)







D^α₀₊u(t)−λu(t) =f(t,u(t)), t∈(0,1],t6=t₁∈(0,1),0<α≤1, limt→0t^1−αu(t) =u(1),

lim_t→t+

1[t−t₁]^1−α[u(t)−u(t₁)] =I(u(t₁)),

whereD^α₀+is the Riemann-Liouville fractional derivative of orderα,I:R→Ris continuous, f :[0,1]×R→R is continuous, λ ∈Ris a constant. The existence and uniqueness of solutions of BVP(1.1) are established under some assumptions by using Banach contraction principle. One of the main assumptions in [15] is as follows: there exist positive numbers Mandmsuch that

(1.2) |f(t,x)| ≤M, |I(x)| ≤m, t∈[0,1], x∈R.

Communicated bySyakila Ahmad.

Received:February 14, 2012;Revised:May 30, 2012.

In this paper, we discuss the periodic type boundary value problem of the nonlinear singular fractional differential equation with the multi-terms

(1.3)







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













D^β₀+[Φ(ρ(t)D^α₀+u(t))] =q(t)f(t,u(t),D^α₀+u(t)), t∈(0,1), lim_t→1t^1−αu(t)−lim_t→0t^1−αu(t) =^R₀¹G(s,u(s),D^α₀+u(s))ds,

limt→1t^1−βΦ(ρ(t)D^α₀+u(t))−limt→0t^1−βΦ(ρ(t)D^α₀+u(t))

=^R₀¹H(s,u(s),D^α₀+u(s))ds, lim_t→t+

1 u(t) =I(t₁,u(t₁),D^α₀+u(t₁)), lim_t→t+

1 Φ(ρ(t)D^α

0⁺u(t)) =J(t₁,u(t₁),D^α₀+u(t₁)).

where

• 0<α,β≤1,D^α₀+( orD^β₀+) is the Riemann-Liouville fractional derivative of order α ( orβ),

• Φ:R→Ris a sup-multiplicative-like function with supporting functionω, its in- verse function is denoted byΦ⁻¹:R→Rwith supporting functionν,

• 0<t₁<1,I,J:(0,1)×R²→Rare continuous functions,(•) φ,ψ :(0,1)→R withφ|_(0,t1],ρ|_(0,t1]∈L¹(0,t₁)andφ|_(t1,1],ρ|_(t1,1]∈L¹(t₁,1),

• ρ:(0,1)→[0,+∞)withρ|_(0,t

1]∈C⁰(0,t₁]andρ|_(t

1,1]∈C⁰(t₁,1)and satisfies that there exist numbers L>0 andk>−α such thatρ(t)≥(t^−kν(t^β−1))/Lfor all t∈(0,1),t6=t₁,

• q:(0,1)→Rwithq|_(0,t

1]∈C⁰(0,t₁]andq|_(t

1,1]∈C⁰(t₁,1)and there exist numbers L₁>0 andk₁>−β such that|q(t)| ≤L₁t^k1 for allt∈(0,1),

• f,G,Hdefined on(0,t₁)^S(t₁,1)×R×Rareimpulsive Caratheodory functions that may be singular att=0,t₁and 1.

A functionxdefined on(0,1)is called a solution of BVP(1.3), ifx|_(0,t1]∈C⁰(0,t₁]and x|_(t1,1]∈C⁰(t₁,1], D^α₀+x|_(0,t1] ∈C⁰(0,t₁] and D^α₀+x|_(t1,1] ∈C⁰(t₁,1], D^β₀+[Φ(ρ(t)D^α₀+x(t))]

∈L¹(0,1)andxsatisfies all equations in (1.3).

We obtain the results on the existence of at least one solution of BVP(1.3). An example is given to illustrate the efficiency of the main theorem. The results in this paper general- ize those ones in [18], i.e., assumption (1.2) is replaced by a weaker one. The impulsive functions are different those ones in [18].

Remark 1.1. In the special case:α=β=1,g(t,x,y) =h(t,x,y)≡0 and the impulse dis- appears, BVP(1.3) becomes the periodic boundary value problem for ordinary differential equation

([Φ(ρ(t)u⁰(t))]⁰=q(t)f(t,u(t),u⁰(t)), t∈(0,1), u(0) =u(1),u⁰(0) =u⁰(1).

So we call BVP(1.3) the impulsive periodic type boundary value problem of the nonlinear fractional differential equation.

The remainder of this paper is as follows: in Section 2, we present preliminary results.

In Section 3, the main theorems and their proof are given. In Section 4, an example is given to illustrate the main results.

2. 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 literatures [7,10]. Let the Gamma and beta functionsΓ(α)andB(p,q)be defined by

Γ(α) = Z +∞

0

x^α−1e^−xdx, B(p,q) = Z 1

0

x^p−1(1−x)^q−1dx.

Definition 2.1. The Riemann-Liouville fractional integral of order α >0 of a function g:(0,∞)→R is given by

I₀₊^α g(t) = 1 Γ(α)

Z _t 0

(t−s)^α−1g(s)ds, provided that the right-hand side exists.

Definition 2.2. The Riemann-Liouville fractional derivative of orderα>0of a continuous function g:(0,∞)→R is given by

D^α₀+g(t) = 1 Γ(n−α)

dⁿ dtⁿ

Z t 0

g(s) (t−s)^α−n+1ds,

where n−1≤α<n, provided that the right-hand side is point-wise defined on(0,∞).

Definition 2.3. Let X and Y be Banach spaces. L: D(L)⊂X→Y is called a Fredholm operator of index zero if Im L is closed in X anddimKerL=codimImL<+∞.

It is easy to see that ifLis a Fredholm operator of index zero, then there exist the projec- torsP: X→X, andQ: Y →Y such that

ImP=KerL, KerQ=ImL, X=KerL⊕KerP, Y=ImL⊕ImQ.

IfL: D(L)⊂X→Y is called a Fredholm operator of index zero, the inverse of L|_D(L)∩KerP: D(L)∩KerP→ImL

is denoted byK_p.

Definition 2.4. Suppose that L: D(L)⊂X→Y is called a Fredholm operator of index zero.

The continuous map N:X→Y is called L-compact if both QN(Ω)and K_p(I−Q)N:Ω→X are compact for each nonempty open subsetΩof X satisfying D(L)∩Ω6=/0.

To obtain the main results, we need the abstract existence theorem [9].

Lemma 2.1. Leray-Schauder Nonlinear Alternative.Let X,Y be Banach spaces and L: D(L)^TX →Y a Fredholm operator of index zero with KerL={0∈X}, N : X →Y L- compact. SupposeΩis a nonempty open subset of X satisfying D(L)∩Ω6=/0. Then either there exists x∈∂Ω and θ ∈(0,1)such that Lx=θNx or there exists x∈Ω such that Lx=Nx.

Definition 2.5. An odd homeomorphism Φ of the real line R onto itself is called a sup- multiplicative-like function if there exists a homeomorphismω of[0,+∞)onto itself which supportsΦin the sense that for all v1,v2≥0it holds

(2.1) Φ(v₁v₂)≥ω(v₁)Φ(v₂).

ωis called the supporting function ofΦ.

Remark 2.1. Note that any sup-multiplicative function is sup-multiplicative-like function.

Also any function of the form

Φ(u):=

k

∑

j=0

c_j|u|^ju, u∈R

is sup-multiplicative-like, provided thatc_j≥0. Here a supporting function is defined by ω(u):=min{u^k+1,u},u≥0.

Remark 2.2. It is clear that a sup-multiplicative-like functionΦ and any corresponding supporting functionωare increasing functions vanishing at zero and moreover their inverses Φ⁻¹andνrespectively are increasing and such that

(2.2) Φ⁻¹(w₁w₂)≤ν(w₁)Φ⁻¹(w₂), for allw₁,w₂≥0 andνis called the supporting function ofΦ⁻¹.

Remark 2.3. IfΦ_p(x) =|x|^p−2xforp>1, we callΦ_pa one-dimensionalp-Laplacian. By Remark 2.1,Φ_pis a sup-multiplicative-like function with its supporting functionω(x) =

|x|^p−2xand its inverse functionΦ⁻¹_p (x) =|x|^q−2x. The supporting function ofΦ⁻¹_p isν(x) =

|x|^q−2x. Hereqsatisfies 1/p+1/q=1. It is easy to see that limt→0

1

ν(t^1−β)ν(t^β−1)=lim

t→0

ν(t^β−1)

ν(t^β−1)=1<+∞.

In this paper we always suppose that Φ is a sup-multiplicative-like function with its supporting functionω, the inverse functionΦ⁻¹has its supporting functionν.

Definition 2.6. We call F:(0,t1)∪(t₁,1)×R²→R animpulsive Caratheodory function if it satisfies the following items:

(i) t→F t,t^α−1u,(Φ⁻¹(t^β−1v))/(ρ(t))

is continuous both on(0,t₁]and(t₁,1)and the limits exist

lim_t→0+F

t,t^α−1u,^{Φ−1(tβ−1v)}

ρ(t)

, lim_t→t+

1 F

t,t^α−1u,^{Φ−1(tβ−1v)}

ρ(t)

,

lim_t→1−F

t,t^α−1u,^Φ−1_ρ(t)^(tβ−1v) for any(u,v)∈R²,

(ii) (u,v)→F t,t^α−1u,(Φ⁻¹(t^β−1v))/(ρ(t))

is continuous on R²for all t∈(0,t₁)∪ (t₁,1).

Define

x(t) =u(t), y(t) =Φ(ρ(t)D^α₀+x(t)).

Then BVP (1.3) is transformed to

(2.3)































D^α₀+x(t) =^Φ−1_ρ(t)^(y(t)), t∈(0,1),t6=t₁, D^β₀+y(t) =q(t)f

t,x(t),^Φ−1(y(t))

ρ(t)

, t∈(0,1),t6=t₁, limt→1t^1−αx(t)−limt→0t^1−αx(t) =^R₀¹φ(t)G

t,x(t),^Φ−1_ρ(t)^(y(t)) dt, limt→1t^1−βy(t)−limt→0t^1−βy(t) =^R₀¹ψ(t)H

t,x(t),^Φ−1_ρ(t)^(y(t)) dt, lim_t→t+

1 x(t) =I

t₁,x(t₁),^Φ−1_ρ(t^(y(t1))

1)

, lim_t→t+

1 y(t) =J

t₁,x(t₁),^Φ−1_ρ(t^(y(t1))

1)

.

It is easy to see that if(x,y)is a solution of BVP (2.3), thenxis a solution of BVP (1.3).

We use the Banach spaces

X=







x:(0,1]→R:

x|_(0,t1]∈C⁰(0,t₁],x|_(t1,1]∈C⁰(t₁,1]

there exist the limits lim_t→0+t^1−αx(t),lim_t→t+

1 x(t)





 with the norm

||x||=||x||_∞= sup

t∈(0,1]

t^1−α|x(t)|,

Y=







y:(0,1]→R: y|_(0,t

1]∈C⁰(0,t₁],y|_(t

1,1]∈C⁰(t₁,1]

there exist the limits lim_t→0+t^1−βy(t),lim_t→t+

1 y(t)





 with the norm

||y||=||y||_∞= sup

t∈(0,1)

t^1−β|y(t)|,

L¹[0,1]with the norm

||u||₁= Z 1

0

|u(s)|ds.

ChooseE=X×Ywith the norm

||(x,y)||=max{||x||_∞,||y||_∞} for (x,y)∈X×Y.

ChooseZ=L¹(0,1)×L¹(0,1)×R⁴with the norm















 u v c d c d

















=max{||u||₁,||v||₁,|a|,|b|,|c|,|d|} for















 u v c d c d

















∈Z.

DefineLto be the linear operator fromD(L)^TEtoZwith D(L) =n

(x,y)∈E:D^α₀+x,D^β₀+y∈L¹(0,1)o and

L(x,y)(t) =



















D^α₀+x(t) D^β₀+y(t)

limt→1t^1−αx(t)−limt→0t^1−αx(t) limt→1t^1−βy(t)−limt→0t^1−βy(t)

lim_t→t+ 1 x(t) lim_t→t+

1 y(t)



















T

for(x,y)∈D(L)^TE.DefineN:E→Zby

N(x,y)(t) =





























Φ⁻¹(y(t)) ρ(t)

q(t)f

t,x(t),^Φ−1_ρ(t)^(y(t)) R1

0φ(t)G

t,x(t),^Φ−1_ρ(t)^(y(t)) dt R1

0ψ(t)H

t,x(t),^Φ−1_ρ(t)^(y(t)) dt I

t1,x(t₁),^Φ−1_ρ(t^(y(t1))

1)

J

t1,x(t₁),^{Φ−1(y(t1))}

ρ(t1)





























T

for (x,y)∈E.

Then BVP(2.3) can be written as

L(x,y) =N(x,y), (x,y)∈E.

Lemma 2.2. Suppose that f,G,H areimpulsive Caratheodory functions, I,J are contin- uous functions andΦis a sup-multiplicative-like function withνsatisfying

(2.4) lim

t→0

1

ν(t^1−β)ν(t^β−1)=lim

t→0

ν(t^β−1) ν(t^β−1)<+∞.

Then L is a Fredhold operator with index zero and N:E→Z is L-compact.

Proof. To prove thatLis a Fredhold operator with index zero, we should do the following three steps.

Step (i)Prove that KerL={(0,0)∈E}.

We know that(x,y)∈KerLif and only if

(2.5)



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

















D^α₀+x(t) =0, D^β₀+y(t) =0,

limt→1t^1−αx(t)−limt→0t^1−αx(t) =0, lim_t→1t^1−βy(t)−lim_t→0t^1−βy(t) =0 lim_t→t+

1 x(t) =0 lim_t→t+

1 y(t) =0.

Hence(x,y)∈KerLif and only ifx(t) =0 andy(t) =0. Thus KerL={(0,0)∈E}.

Step (ii)Prove that ImL={(u,v,a,b,c,d)∈Z}=Z.

For(u,v,a,b,c,d)∈Z, we know that (u,v,a,b,c,d)∈ImL if and only if there exist (x,y)∈Esuch that

(2.6)























D^α₀+x(t) =u(t), D^β₀+y(t) =v(t),

lim_t→1t^1−αx(t)−lim_t→0t^1−αx(t) =a, limt→1t^1−βy(t)−limt→0t^1−βy(t) =b, lim_t→t⁺

1 x(t) =c, lim_t→t+

1 y(t) =d.

So we get

x(t) =













 R_t

0 (t−s)^α−1

Γ(α) u(s)ds+

R1 0

(1−s)^α−1 Γ(α) u(s)ds +^cΓ(α)−

Rt1

0 (t₁−s)^α−1u(s)ds Γ(α)t₁^α−1 −a

t^α−1, t∈(0,t₁],

1 Γ(α)

R_t

0(t−s)^α−1u(s)ds+^cΓ(α)−

Rt1

0 (t₁−s)^α−1u(s)ds

Γ(α)t₁^α−1 t^α−1, t∈(t₁,1]

y(t) =













 Rt

0 (t−s)^β−1

Γ(β) v(s)ds+

R1 0

(1−s)^β−1 Γ(β) v(s)ds +^dΓ(β)−

Rt1

0 (t₁−s)^β−1v(s)ds Γ(β)t₁^β−1 −b

t^β−1, t∈(0,t₁],

1 Γ(β)

R_t

0(t−s)^β−1v(s)ds+^dΓ(β)−

Rt1

0 (t₁−s)^β−1v(s)ds

Γ(β)t₁^β−1 t^β−1, t∈(t₁,1].

(2.7)

It is easy to show that(x,y)∈D(L)^TE. Then ImL=Z. Furthermore,(x,y)satisfies (2.7) if and only if(x,y)satisfies (2.6).

Step (iii)Prove that ImLis closed inX and dim KerL= co dim ImL<+∞.

From Step (ii) ImL=Z is closed in Z. It follows from KerL={(0,0)∈E} that dim KerL=0. Define the projectorP: E→Eby

(2.8) P(x,y)(t) = (0, 0) for (x,y)∈E.

It is easy to prove that

(2.9) ImP=KerL, X=KerL⊕KerP.

Define the projectorQ: Z→Zby

(2.10) Q(u,v,a,b,c,d)(t) = (0,0,0,0,0,0) for(u,v,a,b,c,d)∈Z.

It is easy to show that

(2.11) ImL=KerQ, Y=ImQ⊕ImL.

From above discussion, we see that dim KerL= co dim ImL=0<+∞. SoL is a Fredholm operator of index zero.

Now, we prove thatNisL-compact. This is divided into three steps.

Step (i) We prove thatN is continuous. Let(x_n,y_n)∈E with(x_n,y_n)→(x₀,y₀)asn→∞.

We will show thatN(x_n,y_n)→N(x₀,y₀)asn→∞.

In fact, we have

||(x_n,y_n)||= sup

n=0,1,2,...

( sup

t∈(0,1]

t^1−α|x_n(t)|, sup

t∈(0,1]

t^1−β|y_n(t)|

)

=r<+∞

and

(2.12) sup

t∈(0,1]

t^1−α|x_n(t)−x₀(t)| →0, sup

t∈(0,1]

t^1−β|y_n(t)−y₀(t)| →0,n→∞.

By

N(x_n,y_n)(t) =





























Φ⁻¹(yn(t)) ρ(t)

q(t)f

t,x_n(t),^Φ−1_ρ(t)^(yn(t)) R1

0φ(t)G

t,x_n(t),^{Φ−1(yn(t))}

ρ(t)

dt R1

0ψ(t)H

t,x_n(t),^{Φ−1(yn(t))}

ρ(t)

dt I

t₁,x_n(t₁),^Φ−1_ρ(t^(yn(t1))

1)

J

t₁,x_n(t₁),^Φ−1_ρ(t^(yn(t1))

1)





























T

for(x,y)∈E.

SinceΦis a sup-multiplicative-like function, we get from (2.2) that 1

ν(t^1−β)ν(t^β−1)Φ⁻¹(x)≤Φ⁻¹(t^β−1x)

ν(t^β−1) ≤ν(t^β−1)

ν(t^β−1)Φ⁻¹(x),x≥0 and

1

ν(t^1−β)ν(t^β−1)Φ⁻¹(x)≥Φ⁻¹(t^β−1x)

ν(t^β−1) ≥ν(t^β−1)

ν(t^β−1)Φ⁻¹(x),x≤0 Then (2.4) implies that ^{Φ−1(tβ−1x)}

ν(t^β−1) is continuous on(0,t₁]×R² and there exists the limit lim_t→0+Φ⁻¹(t^β−1x)

ν(t^β−1) . Hence^{Φ−1(tβ−1x)}

ν(t^β−1) is continuous on[0,t₁]×R². It follows that ^{Φ−1(tβ−1x)}

ν(t^β−1)

is uniformly continuous on[0,t₁]×[−r,r]×[−r,r].

Similarly, we can see that^{Φ−1(tβ−1x)}

ν(t^β−1) is uniformly continuous on[t₁,1]×[−r,r]×[−r,r].

For anyε>0 there existsδ>0 such that (2.13)

Φ⁻¹(t^β−1u₁)

ν(t^β−1) −Φ⁻¹(t^β−1u₂) ν(t^β−1)

< ε

R1 0

ν(t^β−1) ρ(t) dt

,t∈(0,1],|u₁−u₂|<δ. From (2.12), there existsNsuch that

(2.14) t^1−α|x_n(t)−x₀(t)|<δ,t^1−β|y_n(t)−y₀(t)|<δ,t∈(0,1],n>N.

Hence

Z 1 0

Φ⁻¹(y_n(t))

ρ(t) −Φ⁻¹(y₀(t)) ρ(t)

dt

= Z 1

0

ν(t^β−1) ρ(t)

Φ⁻¹ t^β−1t^1−βy_n(t)

ν(t^β−1) −Φ⁻¹ t^β−1t^1−βy₀(t) ν(t^β−1)

dt

<

Z ₁ 0

ν(t^β−1) ρ(t)

ε R1

0 ν(t^β−1)

ρ(t) dt

dt=ε,n>N.

Since f is a Caratheodory function, we know that f

t,t^α−1u,^Φ−1_ρ(t)^(tβ−1v)

is continu- ous on[0,t₁]×R². So f

t,t^α−1u,^Φ−1_ρ(t)^(tβ−1v)

is uniformly continuous on[0,t₁]×[−r,r]× [−r,r]. Similarly we can see thatf

t,t^α−1u,^Φ−1_ρ(t)^(tβ−1v)

is uniformly continuous on[t₁,1]× [−r,r]×[−r,r].

So for anyε>0, there existsδ>0 such that

f t,t^α−1u1,Φ⁻¹(t^β−1v₁) ρ(t)

!

−f t,t^α−1u2,Φ⁻¹(t^β−1v₂) ρ(t)

!

< ε

R₁ 0q(t)dt for allt∈(0,1]and|u₁−u2|<δ and|v₁−v2|<δ. Hence we get

Z ₁ 0

q(t)f

t,x_n(t),Φ⁻¹(y_n(t)) ρ(t)

−q(t)f

t,x0(t),Φ⁻¹(y₀(t)) ρ(t)

dt

= Z 1

0

q(t)

f t,t^α−1t^1−αx_n(t),Φ⁻¹ t^β−1t^1−βy_n(t) ρ(t)

!

−f t,t^α−1t^1−αx₀(t),Φ⁻¹ t^β−1t^1−βy0(t) ρ(t)

!

dt

<

Z 1 0

q(t) ε R1

0q(t)dtdt=ε,n>N.

Similarly we get forn>Nthat

Z 1 0

φ(t)G

t,x_n(t),Φ⁻¹(y_n(t)) ρ(t)

dt−

Z 1 0

φ(t)G

t,x₀(t),Φ⁻¹(y₀(t)) ρ(t)

dt

<ε,

and

Z 1 0

ψ(t)H

t,x_n(t),Φ⁻¹(y_n(t)) ρ(t)

dt−

Z 1 0

ψ(t)H

t,x₀(t),Φ⁻¹(y₀(t)) ρ(t)

dt

<ε,

and

I

t₁,x_n(t₁),Φ⁻¹(y_n(t₁)) ρ(t₁)

−I

t₁,x₀(t₁),Φ⁻¹(y₀(t₁)) ρ(t₁)

<ε,

J

t₁,x_n(t₁),Φ⁻¹(y_n(t₁)) ρ(t₁)

−J

t₁,x₀(t₁),Φ⁻¹(y₀(t₁)) ρ(t₁)

<ε.

Then

||N(x_n,y_n)−N(x₀,y₀)|| →0,n→∞.

It follows thatNis continuous.

LetP:X→XandQ:Y→Ybe defined by (2.8) and (2.10). For(u,v,a,b,c,d)∈ImL= Z, let

(2.15) K_P(u,v,a,b,c,d)(t) = (x₁(t),y₁(t))

where (2.16)

x₁(t) =





















 Rt

0 (t−s)^α−1

Γ(α) u(s)ds

+

R₁ 0

(1−s)^α−1

Γ(α) u(s)ds+^cΓ(α)−

Rt1

0 (t₁−s)^α−1u(s)ds Γ(α)t₁^α−1 −a

t^α−1,t∈(0,t₁],

1 Γ(α)

Rt

0(t−s)^α−1u(s)ds+^cΓ(α)−

Rt1

0 (t₁−s)^α−1u(s)ds

Γ(α)t₁^α−1 t^α−1,t∈(t₁,1],

y₁(t) =





















 Rt

0 (t−s)^β−1

Γ(β) v(s)ds

+

R₁ 0

(1−s)^β−1

Γ(β) v(s)ds+^dΓ(β)−

Rt1

0 (t1−s)^β−1v(s)ds Γ(β)t₁^β−1 −b

t^β−1,t∈(0,t₁],

1 Γ(β)

Rt

0(t−s)^β−1v(s)ds+^dΓ(β)−

Rt1

0 (t₁−s)^β−1v(s)ds

Γ(β)t^β−1₁ t^β−1,t∈(t₁,1].

One seesK_P(u,v,a,b,c,d)∈D(L)^TEandK_Pis the inverse ofL:D(L)^TKerP→ImL.

The isomorphism∧: KerL→Y/ImLis given by

∧(0,0) = (0,0,0,0,0,0).

Furthermore, one has

(2.17) QN(x,y)(t) =Q(0,0,0,0,0,0), and

K_p(I−Q)N(x,y)(t) = (x₂(t),y₂(t)), where

(2.18) x₂(t) =













































 R_t

0 (t−s)^α−1

Γ(α)

Φ⁻¹(y(s)) ρ(s) ds+

R₁ 0

(1−s)^α−1 Γ(α)

Φ⁻¹(y(s)) ρ(s) ds

+

I

t₁,x(t₁),^Φ

−1(y(t1))

ρ(t1)

Γ(α)−^R₀^t1(t₁−s)^α−1Φ−1_ρ(s)^(y(s))ds Γ(α)t^α₁⁻¹

−^R₀¹φ(t)G

t,x(t),^Φ−1_ρ(t)^(y(t)) dt

t^α−1, t∈(0,t₁],

1 Γ(α)

R_t

0(t−s)^α−1Φ−1_ρ(s)^(y(s))ds

+

I

t1,x(t1),^Φ

−1(y(t1))

ρ(t1)

Γ(α)−^R₀^t1(t1−s)^{α−1Φ−1(y(s))}

ρ(s) ds

Γ(α)t^α−1₁ t^α−1,t∈(t₁,1],

(2.19)

y₂(t) =





























































 R_t

0 (t−s)^β−1

Γ(β) q(s)f

s,x(s),^Φ−1_ρ(s)^(y(s)) ds

+ R1

0

(1−s)^β−1 Γ(β) q(s)f

s,x(s),^Φ−1(y(s))

ρ(s)

ds

+

J

t₁,x(t₁),^Φ

−1(y(t1))

ρ(t1)

Γ(β)−^R₀^t1(t₁−s)^β−1q(s)f

s,x(s),^Φ−1_ρ(s)^(y(s))

ds Γ(β)t^β−1₁

−^R₀¹ψ(t)H

t,x(t),^Φ−1_ρ(t)^(y(t)) dt

t^β−1, t∈(0,t1],

1 Γ(β)

Rt

0(t−s)^β−1q(s)f

s,x(s),^Φ−1_ρ(s)^(y(s)) ds+

J

t1,x(t1),^Φ

−1(y(t1))

ρ(t1)

t₁^β−1 t^β−1

−

Rt1

0 (t₁−s)^β−1q(s)f

s,x(s),^Φ−1_ρ(s)^(y(s))

ds

Γ(β)t^β−1₁ t^β−1,t∈(t₁,1].

LetΩbe a bounded open subset ofEwithΩ^TD(L)6=/0. We have (2.20) ||(x,y)||= sup

n=0,1,2,...

( sup

t∈(0,1]

t^1−α|x(t)|, sup

t∈(0,1]

t^1−β|y(t)|

)

=r<+∞,(x,y)∈Ω.

Since f,G,H are impulsive Caratheodory functions, I,J are continuous functions, to- gether with (2.20), there existsM>0 such that

(2.21)

f

t,x(t),^Φ−1_ρ(t)^(y(t)) =

f

t,t^α−1t^1−αx(t),^Φ

−1(t^β−1t^1−βy(t))

ρ(t)

≤M,

G

t,x(t),^Φ−1_ρ(t)^(y(t)) ≤M,

H

t,x(t),^Φ−1_ρ(t)^(y(t)) ≤M,

I

t₁,x(t₁),^Φ−1_ρ(t^(y(t1))

1)

≤M,

J

t₁,x(t₁),^Φ−1_ρ(t^(y(t1))

1)

≤Mhold for allt∈[0,1].

Step (ii) Prove thatQN(Ω)is bounded.

It is easy to see from (2.17) thatQN(Ω)is bounded.

Step (iii) Prove thatK_P(I−Q)N:Ω→E is compact, i.e., prove that K_P(I−Q)N(Ω)is relatively compact.

We must prove thatK_P(I−Q)N(Ω)is uniformly bounded and equi-continuous both on each subinterval[e,f]⊆(0,t₁]and(t₁,1]respectively, and equi-convergent both att=0 and t=t₁respectively.

By (2.18) and (2.2), we have t^1−αx2(t)

≤t^1−α Z _t

0

(t−s)^α−1 Γ(α)

Φ⁻¹(|y(s)|) ρ(s) ds+

Z ₁ 0

(1−s)^α−1 Γ(α)

Φ⁻¹(|y(s)|) ρ(s) ds +

I

t₁,x(t₁),^Φ−1_ρ(t^(|y(t1)|)

1)

Γ(α) +^R₀^t1(t₁−s)^α−1Φ−1_ρ(s)^(|y(s)|)ds Γ(α)t₁^α−1

+ Z 1

0

φ(t)G

t,x(t),Φ⁻¹(y(t)) ρ(t)

dt

≤Lt^1−α Z _t

0

(t−s)^α−1

Γ(α) s^kds+L Z ₁

0

(1−s)^α−1 Γ(α) s^kds +MΓ(α) +L^R₀^t1(t₁−s)^α−1s^kds

Γ(α)t₁^α−1 +M Z ₁

0

|φ(t)|dt

=Lt^k+1 Z 1

0

(1−w)^α−1

Γ(α) w^kdw+L Z 1

0

(1−s)^α−1 Γ(α) s^kds +MΓ(α) +L^R₀^t1(t₁−s)^α−1s^kds

Γ(α)t₁^α−1 +M Z 1

0

|φ(t)|dt

≤L Z 1

0

(1−w)^α−1

Γ(α) w^kdw+L Z 1

0

(1−s)^α−1 Γ(α) s^kds +MΓ(α) +L^R₀^t1(t₁−s)^α−1s^kds

Γ(α)t₁^α−1 +M Z 1

0

|φ(t)|dt<+∞

and

t^1−βy₂(t)

≤t^1−β Z t

0

(t−s)^β−1 Γ(β)

q(s)f

s,x(s),Φ⁻¹(y(s)) ρ(s)

ds

+ Z 1

0

(1−s)^β−1 Γ(β)

q(s)f

s,x(s),Φ⁻¹(y(s)) ρ(s)

ds

+ J

t1,x(t₁),^Φ−1_ρ(t^(y(t1))

1)

Γ(β) +^R₀^t1(t₁−s)^β−1 q(s)f

s,x(s),^Φ−1_ρ(s)^(y(s)) ds Γ(β)t₁^β−1

+ Z 1

0

ψ(t)H

t,x(t),Φ⁻¹(y(t)) ρ(t)

dt

≤L₁Mt^k1+1 Z 1

0

(1−w)^β−1

Γ(β) w^k1ds+L₁M Z 1

0

(1−s)^β−1 Γ(β) s^k1ds +MΓ(β) +L1M^R₀^t1(t₁−s)^β−1s^k1ds

Γ(β)t₁^β−1

+M Z 1

0

|ψ(t)|dt

≤L1M Z ₁

0

(1−w)^β−1

Γ(β) w^k1ds+L1M Z ₁

0

(1−s)^β−1 Γ(β) s^k1ds +MΓ(β) +L₁M^R₀^t1(t₁−s)^β−1s^k1ds

Γ(β)t₁^β−1

+M Z 1

0

|ψ(t)|dt<+∞.

It is easy to see thatK_P(I−Q)N(Ω)is uniformly bounded.

For each[e,f]⊆(0,t₁], ands₁,s₂∈[e,f]withs₂≥s₁, use (2.18), we have

|s^1−α₁ x₂(s₁)−s^1−α₂ x₂(s₂)|

= 1

Γ(α)

s^1−α₁ Z s₁

0

(s₁−s)^α−1Φ⁻¹(y(s))

ρ(s) ds−s^1−α₂ Z s₂

0

(s₂−s)^α−1Φ⁻¹(y(s)) ρ(s) ds

≤ 1 Γ(α)

s^1−α₁ −s^1−α₂

Z s₁ 0

(s₁−s)^α−1Φ⁻¹ s^β−1s^1−β|y(s)|

ρ(s) ds + 1

Γ(α)s^1−α₂ Z _s2

s1

(s₂−s)^α−1Φ⁻¹ s^β−1s^1−β|y(s)|

ρ(s) ds

+ 1 Γ(α)s^1−α₂

Z s₂ 0

|(s₁−s)^α−1−(s₂−s)^α−1|Φ⁻¹ s^β−1s^1−β|y(s)|

ρ(s) ds

≤ 1 Γ(α)

s^1−α₁ −s^1−α₂

Z s₁ 0

(s₁−s)^α−1ν(s^β−1)Φ⁻¹(||y||) ρ(s) ds + 1

Γ(α)s^1−α₂ Z _s2

s₁

(s₂−s)^α−1ν(s^β−1)Φ⁻¹(||y||) ρ(s) ds + 1

Γ(α)s^1−α₂ Z _s2

0

|(s₁−s)^α−1−(s₂−s)^α−1|ν(s^β−1)Φ⁻¹(||y||) ρ(s) ds

≤Φ⁻¹(r) Γ(α)

s^1−α₁ −s^1−α₂

Z s₁ 0

(s₁−s)^α−1ν(s^β−1) ρ(s) ds +Φ⁻¹(r)

Γ(α) s^1−α₂ Z s₂

s₁

(s₂−s)^α−1ν(s^β−1) ρ(s) ds +Φ⁻¹(r)

Γ(α) s^1−α₂ Z _s2

0

|(s₁−s)^α−1−(s₂−s)^α−1|ν(s^β−1) ρ(s) ds

≤Φ⁻¹(r) LΓ(α)

s^1−α₁ −s^1−α₂

Z _s1 0

(s₁−s)^α−1s^kds +Φ⁻¹(r)

LΓ(α)s^1−α₂ Z _s2

s1

(s₂−s)^α−1s^kds+Φ⁻¹(r) LΓ(α)s^1−α₂

Z _s2 0

[(s₁−s)^α−1−(s₂−s)^α−1]s^kds

≤Φ⁻¹(r) LΓ(α)

"

s^1−α₁ −s^1−α₂

s^k+α₁ B(α,k+1) +s^k+1₂ Z ₁

k1 k2

(1−w)^α−1w^kdw

+ s₂

s1

1−α

s^k+1₁ Z ^s2

s1

0

(1−w)^α−1w^kdw−s^k+1₂ B(α,k+1)

#

→0 uniformly ass₁→s₂.Similarly we get

|s^1−β₁ y₂(s₁)−s^1−β₂ y₂(s₂)| →0 uniformly ass₁→s₂.

SoK_P(I−Q)N(Ω)is equi-continuous on each subinterval[e,f]⊆(0,t₁]. Similarly we can show thatK_P(I−Q)N(Ω)is equi-continuous on each subinterval[e,f]⊆(t₁,1].

Since

t^1−αx₂(t)− R1

0

(1−s)^α−1 Γ(α)

Φ⁻¹(y(s)) ρ(s) ds +

I

t1,x(t1),^Φ

−1(y(t1))

ρ(t1)

Γ(α)−^R₀^t1(t1−s)^{α−1Φ−1(y(s))}

ρ(s) ds Γ(α)t₁^α−1

−^R₀¹φ(t)G

t,x(t),^Φ−1_ρ(t)^(y(t)) dt

≤^t

1−αR_t

0(t−s)^α−1Φ

−1(^{sβ−1s1−β|y(s)|})

ρ(s) ds

Γ(α)

≤^t

1−αRt

0(t−s)^{α−1ν(sβ−1)Φ−1(r)}

ρ(s) ds

Γ(α) ≤LΦ⁻¹(r)^t1−α

R_t

0(t−s)^α−1s^kds Γ(α)

=LΦ⁻¹(r)t^k+1B(α,k+1)→0 uniformly ast→0.

Similarly we can show that

t^1−βy₂(t)− R1

0 (1−s)^β−1

Γ(β) q(s)f

s,x(s),^Φ−1_ρ(s)^(y(s)) ds

+

J

t1,x(t₁),^Φ

−1(y(t1))

ρ(t1)

Γ(β)−^R₀^t1(t₁−s)^β−1q(s)f

s,x(s),^Φ−1_ρ(s)^(y(s))

ds Γ(β)t₁^β−1

−^R₀¹ψ(t)H

t,x(t),^Φ−1_ρ(t)^(y(t)) dt

→0

uniformly ast→0.HenceK_P(I−Q)N(Ω)is equi-convergent att=0. Similarly we can show thatK_P(I−Q)N(Ω)is equi-convergent att=t₁.

SoK_P(I−Q)N(Ω)is relatively compact. ThenN isL-compact. The proofs are com- pleted.

3. Main results

Now, we prove the main theorem in this paper.

Theorem 3.1. Suppose that

(B) there exist nonnegative numbers A,B,C,ai,bi,ci(i=1,2)and Ai,Bi,C_i(i=1,2)such

that

f

t,t^α−1x,^Φ−1_ρ(t)^(tβ−1y)

≤C+BΦ(|x|) +A|y|,

G

t,t^α−1x,^Φ−1_ρ(t)^(tβ−1y)

≤c₁+b₁|x|+a₁Φ⁻¹(|y|),

H

t,t^α−1x,^Φ−1_ρ(t)^(tβ−1y)

≤c2+b₂Φ(|x|) +a2|y|,

I

t₁,t₁^α−1x,^Φ

−1(t₁^β−1y) ρ(t₁)

≤C₁+B₁|x|+A₁Φ⁻¹(|y|),

J

t₁,t₁^α−1x,^Φ

−1(t₁^β−1y) ρ(t1)

≤C₂+B₂Φ(|x|) +A₂|y|.