Malaya Journal of Matematik, Vol. 7, No. 1, 27-33, 2019

https://doi.org/10.26637/MJM0701/0006

Existence of mild solutions to partial neutraldifferential equations with non-instantaneousimpulsesR. Poongodi 1, V. T. Suveetha 2 and S. Dhanalakshmi 3

AbstractIn this article, we study the existence of PC -mild solutions for the initial value problems for a class of semilinearneutral equations. These equations have non-instantaneous impulses in Banach space and the correspondingsolution semigroup is noncompact. We assume that the nonlinear terms satisfies certain local growth conditionand a noncompactness measure condition. Also we assume the non-instantaneous impulsive functions satisfysome Lipschitz conditions.

KeywordsMild solutions, Non-instantaneous impulse, Noncompactness, Neutral systems.

AMS Subject Classification54B05.

1Research Scholar, Department of Mathematics, Kongunadu Arts and Science College, Coimbatore, Tamil Nadu, India.2, 3Department of Mathematics, Kongunadu Arts and Science College, Coimbatore, Tamil Nadu, India.*Corresponding author: 1 pookasc@gmail.com 2 suveetha94@gmail.comArticle History: Received 21 July 2018; Accepted 26 December 2018 c©2019 MJM.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1. IntroductionIn this article, we study the existence of PC -mild so-

lutions for a class of partial neutral functional differentialequations with non-instantaneous impulses described by theform

ddt

[x(t)−g(t,x(t))

]= −Ax(t)+ f (t,x(t)),

t ∈ ∪nk=0 (sk, tk+1],

x(t) = γk(t,x(t)),

t ∈ ∪nk=1(tk,sk],

x(0) = x0, (1.1)

where A : D(A) ⊂ E → E is a closed linear operator, −A isthe infinitesimal generator of a strong continuous semigroup

(C0-semigroup) U(t)(t ≥ 0) on a Banach space E, 0 < t1 <t2 < · · ·< tn < tn+1 := a, a > 0 is a constant, s0 := 0 and sk ∈(tk, tk+1) for each k = 1,2, · · · ,n, f ,g : [0,a]×E→ E are ap-propriate functions, γk : (tk,sk]×E→ E is non-instantaneousimpulsive neutral function for all k = 1,2, · · · ,n, x0 ∈ E.

In mathematical models for both physical sciences andsocial sciences impulsive differential equations have becamemore important in recent years. There are several process andphenomena in the real world, which are subjected during theirdevelopment to the short-term external influences. At definitepoints in time, many dynamic phenomena experience unfore-seen instantaneous, quick healing exhibited by a jump in theirstates. Their duration is negligible compared with the totalduration of the studies phenomena and process. Therefore, itcan be assumed that these external effects are “instantaneous”.For more facts on the results and applications of impulsivedifferential systems, one can refer to the books of Laksmikan-tham [15] the papers [1, 2, 4, 5-7, 10, 12, 17, 18, 22, 24, 25]and the references cited therein.

Neutral Differential Equations arise in many areas of sci-ence and engineering have received much attention in the lastten years. These models turned out to be very serviceable inthe situation where the system depends not only on the presentstates but also on the past states. See the monograph of [11,

Existence of mild solutions to partial neutral differential equations with non-instantaneous impulses — 28/33

19, 21, 22] and the references therein.In 2013, Pierri et al [20] studied the existence of mild

solution for a class of semi-linear abstract differential equa-tion with non-instantaneous impulses by using the theory ofanalytic semigroup. By a compactness criterion a certainclass of functions, Colao et al [8] investigated the existence ofsolutions for a second-order differential equations with non-instantaneous impulses and delay on an unbounded interval.Using the theory of semigroup and fixed point methods, Yuand Wang [25] discussed the existence of solution to periodicboundary value problems for nonlinear evolution equationwith non-instantaneous impulses on Banach space. The moti-vation of this article is as follows: to the best of the authorsknowledge, (see, for example [5, 8, 11, 20, 23]) used variousfixed point theorems to study the existence results of evo-lution equations with non-instantaneous impulses when thecorresponding semigroup U(t)(t ≥ 0) is compact, this is con-venient to the equations with compact resolvent. But for thisoccurrence that the corresponding semigroup U(t)(t ≥ 0) isnoncompact. Therefore, we study the existence of PC -mildsolution for (1.1) under the assumption that the correspondingsolution semigroup is noncompact by using the properties ofKuratowski measure of noncompactness, k-set-contractionmapping fixed point theorem and the measure of noncompact-ness.

Section 2 provides the definitions and preliminary resultsto be used in this article. In section 3, the existence of PC -mild solution of partial neutral differential equations withnon-instantaneous impulses is established.

2. PreliminariesLet E be a Banach space with the norm ‖ · ‖. We use θ to

present the zero element in E. For any costant a > 0, denoteD = [0,a]. Let C(D,E) be the Banach space of all continuousfunctions from D into E endowed with the supremum-norm‖x‖C = supt∈D ‖x(t)‖ for every x ∈C(D,E). From the asso-ciate literature, we consider the following space of piecewisecontinuous functions,

PC (D,E) = x : D→ E : x is continuous for t 6= tk,left continuous at t = tk and x(t+k ) exists for k = 1,2, · · · ,n.

It is easy to see that PC (D,E) is Banach space endowedwith PC -norms

‖x‖PC =max

supt∈D‖x(t+)‖, sup

t∈D‖x(t−)‖

, x∈PC (D,E),

where x(t+) and x(t−) represent respectively the right and leftlimits of x(t) at t ∈ D. For each finite constant r > 0, let

Ωr = x ∈PC (D,E) : ‖x(t)‖ ≤ r, t ∈ D,

then Ωr is a bounded closed and convex set in PC (D,E).Let L (E) be the Banach space of all linear and bounded

operators on E. Since the semigroup U(t)(t ≥ 0) generated

by −A is a C0-semigroup in E, denote

M := supt∈D‖U(t)‖L (E) (2.1)

then M ≥ 1 is a finite number.

Definition 2.1. ([3, 9]). The Kuratowski measure of noncom-pactness α(·) defined on bounded set S of Banach space E is

α(S) := inf

δ > 0 : S = ∪ni=1Siwith; diam(Si)≤ δ

for i = 1,2, · · · ,n.

The following properties about the Kuratowski measureof noncompactness are well known.

Lemma 2.2. ([3, 9]) Let E be a Banach space and S,T ⊂ Ebe bounded. The following properties are satisfied:

(i) α(T ) = 0 if and only if T is compact, where T meansthe closure hull of T;

(ii) α(T ) = α(T ) = α(convT ), where convT means theconvex hull of T;

(iii) α(λT ) = |λ |α(T ) for any λ ∈ R;

(iv) T ⊂ S implies α(T )≤ α(S);

(v) α(T ∪S) = maxα(T ),α(S);

(vi) α(T +S)≤ α(T )+α(S),whereT +S = u|u = v+w,v ∈ T,w ∈ S;

(vii) If the map F : D(Q) ⊂ E → X is Lipschitz continu-ous with constant k, then α(F(Y )) ≤ kα(Y ) for anybounded subset Y ⊂ D(Q), where X is another Banachspace.

In this article, we denote by α(·), αc(·) and αPC (·) theKuratowski measure of noncompactness on the bounded setof E, C(D,E) and PC (D,E), respectively. For any D ⊂C(D,E) and t ∈ D, set D(t) = x(t) |x ∈ D then D(t) ⊂ E.If D ⊂ C(D,E) is bounded, then D(t) is bounded in E andα(D(t)) ≤ αC(D). For more details about the properties ofthe Kuratowski measure of noncompactness, we refer to [3, 9].

Definition 2.3. ([9]). Let E be a Banach space and T bea nonempty subset of E. A continuous mapping F : T → Eis called to be k-set-contraction if there exists a constantk ∈ [0,1) such that, for every bounded set Ω⊂ T ,

α(F(Ω))≤ kα(Ω).

Lemma 2.4. ([7, 16]). Let E be a Banach space, and letD⊂ E be bounded. Then there exists a countable set D0 ⊂ D,such that

α(D)≤ 2α(D0).

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Existence of mild solutions to partial neutral differential equations with non-instantaneous impulses — 29/33

Lemma 2.5. ([9]). Let E be a Banach space. Assume thatΩ⊂ E is a bounded closed and convex set on E, the operatorF : Ω→Ω is k-set-contractive. Then F has at least one fixedpoint in Ω.

Lemma 2.6. ([13]). Let E be a Banach space, and let D=xn ⊂PC ([b1,b2],E) be a bounded and countable set forconstants −∞ < b1 < b2 < +∞. Then α(D(t)) is Lebesgueintegral on [b1,b2], and

α

(∫ b2

b1

xn(t)dt : n ∈ N)≤ 2

∫ b2

b1

α(D(t))dt.

Lemma 2.7. ([3]). Let E Banach space, and letD ⊂ C([b1,b2],E) be bounded and equicontinuous. Thenα(D(t)) is continuous on [b1,b2], and

αC(D) = maxt∈[b1,b2]

α(D(t)).

we give the definition of mild solution for (1.1) accordingto the developments of Hernandez and O’Regan [14].

Definition 2.8. A function x ∈PC (D,E) is called a mildsolution of (1.1) if x satisfies

x(t)=

g(t,x(t))+U(t)[x0−g(0,x0)]+∫ t

0 U(t− s)f (s,x(s))ds−

∫ t0 AU(t− s)g(s,x(s))ds,

t ∈ [0, t1],γk(t,x(t)), t ∈ (tk,sk], k = 1,2, · · · ,n.g(t,x(t))+U(t− sk)[γk(sk,x(sk))

−g(sk,γk(sk,x(sk)))]+∫ t

skU(t− s) f (s,x(s))ds

−∫ t

skAU(t− s)g(s,x(s))ds, t ∈ (sk, tk+1],

k = 1,2, · · · ,n.

.

3. Main ResultsTo obtain the existence of PC -mild solution for (1.1), we

introduce the following hypotheses:

(H1) The nonlinear function f : D×E → E is continuous,for some r > 0 there exists a constant ρ > 0, Lebesgueintegral function ϕ : D → [0,+∞) and a nondecreasingcontinuous function Ψ : [0,+∞)→ (0,+∞) such thatfor all t ∈ D and x ∈ E satisfying ‖x‖ ≤ r,

‖ f (t,x)‖ ≤ ϕ(t) Ψ(‖x‖)

and

lim infr→ +∞

Ψ(r)r

= ρ < +∞.

(H2) The function g : D×E→ E satisfies:

(i) If x : [·, a] → E be such that x0 = φ and x ∈PCthen the function t→ g(t,x) belongs to PC andt→ g(t,x) is strong measurable function.

(ii) The function g : D×E → E is continuous andthere exists a constant c1 > 0 such that,

‖g(t,x)‖E ≤ c1‖x‖E , ∀ (t,x) ∈ D×E.

(iii) The function g : D×E → E is continuous andthere exists a positive constant L > 0 such that,

‖g(t,x)−g(t,y)‖E ≤ L‖x− y‖E ,

∀ (t,x),(t,y) ∈ D×E.

(iv) There exists positive constant Pk (k = 0,1, · · · ,n)such that for any countable set D⊂ E,

α(g(t,D))≤ Pk α(D), t ∈ (sk, tk+1],

k = 0,1, · · · ,n.

(H3) The impulsive function γk : [tk,sk]×E→ E is continu-ous and there exists a constant Kγk > 0, k = 1,2 · · · ,n,such that for all x,y ∈ E

‖γk(t,x)− γk(t,y)‖ ≤ Kγk‖x− y‖, ∀ t ∈ (tk,sk].

(H4) For every x ∈ E, the function t→U(t)x is continuousfrom [0,∞) into E. Moreover, U(t)(E)⊂ D(A) for ev-ery t > 0 and there exists a positive continuous functionγ ∈ L1([0,a]) such that,

‖AU(t)‖L (E) ≤ γ(t), for every t ∈ D.

(H5) There exists positive constant Qk (k = 0,1, · · · ,n) suchthat for any countable set D⊂ E,

α( f (t,D))≤Qkα(D), t ∈ (sk, tk+1], k= 0,1, · · · ,n.

For concision of notation, we denote

K := maxk=1,2,··· ,n

Kγk , Λ := maxk=1,2,··· ,n

‖ϕ‖L [sk, tk+1],

Γ := maxk=1,2,··· ,n

‖m‖L [sk, tk+1] Q := maxk=1,2,··· ,n

Qk

(3.1)

b :=∫ a

0γ(s)ds P := max

k=1,2,··· ,nPk.

Remark 3.1. Let us report that the hypothesis (H2), (H4) arelinked to the integrability of the function s→ AU(t− s)g(s,x).In general, expect for the inconsiderable case in which A isa bounded linear operator, the operator function t→ AU(t)is not integrable over [0,a]. However, if condition (H4) holdsand g satisfies either assumption (H2) and Bochner’s criterionthen the estimate,

‖AU(t− s)g(s,x)‖ ≤ ‖AU(t− s)‖L (E)‖g(s,x)‖E ,

≤ γ(t− s)sups∈a‖g(s,x)‖E ,

that s→ AU(t− s)g(s,x) is integrable over [0, t), for everyt ∈ [0,a].

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Existence of mild solutions to partial neutral differential equations with non-instantaneous impulses — 30/33

Theorem 3.2. Assume that the semigroup U(t)(t ≥ 0) gener-ated by−A is equicontinuous, the function γk(·,θ) is boundedfor k = 1,2, · · · ,n. If the conditions (H1)− (H5) are satisfied,then (1.1) has at least one PC -mild solution x ∈PC (D,E)provided that

max(

2c1(1+b)+K +M(Kc1 +K +ρΛ),

2P+4MaQ−4bP+MK(1+ c1))< 1. (3.2)

Proof. Define the operator G on PC (D,E) by

Gx(t) = G1x(t)+G2x(t), (3.3)

where

G1x(t) =

U(t)[x0−g(0,x0)], t ∈ [0, t1],γk(t,x(t)), t ∈ (tk,sk] ,

k = 1,2, · · · ,n,U(t− sk)[γk(sk,x(sk))−g(sk, γk(sk,x(sk))], t ∈ (sk, tk+1],

k = 1,2, · · · ,n,(3.4)

and

G2x(t) =

g(t,x(t))+

∫ tsk

U(t− s) f (s,x(s))ds−∫ t

skAU(t− s)g(s,x(s))ds, t ∈ (sk, tk+1],

k = 1,2, · · · ,n,0 otherwise.

(3.5)

The operator G is well defined on PC (D,E) by directcalculation. The PC -mild solution of (1.1) is equivalent tothe fixed point of operator G defined by (3.3). Next, we willprove that the operator G has atleast one fixed point.

Firstly, we exhibit that Gx ∈ PC (D,E)for all x ∈ PC (D,E) for 0 ≤ τ < t ≤ t1 by the strong conti-nuity of the semigroup U(t)(t ≥ 0) and g(t,x) is continuous,we know that

‖Gx(t)−Gx(τ)‖ ≤∥∥g(t,x(t))−g(τ,x(τ))

∥∥+∥∥U(t)−U(τ)∥∥[x0−g(0,x0)]

+∫ t

τ

∥∥AU(t− s)g(s,x(s))∥∥ds+∫ t

τ

∥∥U(t− s) f (s,x(s))∥∥ds

+∫

τ

0

∥∥AU(t− s)−AU(τ− s)∥∥∥∥g(s,x(s))

∥∥ds

+∫

τ

0

∥∥U(t− s)−U(τ− s)∥∥∥∥ f (s,x(s))

∥∥ds

(3.6)

→ 0 as t→ τ.

From the above inequality it follows that Gx ∈ C([0, t1], E)and from (3.4) and the continuity of the non-instantaneous

impulsive functions γk(t,x(t)), k = 1,2, · · · ,n, it is easy toknow that Gx ∈ C((tk, sk], E) for every k = 1,2, · · · ,n. Assimilar with the proof for the continuity of Gx(t) with re-spect to t on [0, t1], we can prove that Gx ∈ C((sk, tk+1], E)for k = 1,2, · · · ,n. Therefore, we have proved that Gx ∈PC (D,E) for x ∈PC (D,E), namely, G maps PC (D,E)to PC (D,E), that is

G : PC (D,E)→PC (D,E).

Next, we prove that there exists a constant r > 0, suchthat G(Ωr) ⊂ Ωr. If it is false, then for each r > 0, therewould exist xr ∈Ωr and tr ∈ D such that ‖(Gxr)(tr)‖> r. Iftr ∈ [0, t1], then by (2.1), (3.3), remark 3.1 and the assumption(H1), (H2) and (H4), we have

‖(Gxr)(tr)‖ ≤ ‖g(tr,x(tr))‖+‖U(tr)[x0−g(0,x0)]‖+∫ tr

0‖U(tr− s) f (s,xr(s))‖ds

+∫ tr

0‖AU(tr− s)g(s,xr(s))‖ds

≤ c1‖xr‖+M[φ −g(0,φ)]+MΨ(r)∫ t

0ϕ(s)ds+∫ tr

0γ(tr− s)‖g(s,xr(s))ds‖

≤ c1r(1+b)+M[φ −g(0,φ)]+MΨ(r)‖ϕ‖L[0,t1].

(3.7)

If tr ∈ (tk, sk], k = 1,2, · · · ,n, then by (2.1), (3.3) and assump-tions (H3), we obtain

‖(Gxr)(tr)‖= ‖γk(tr,xr(tr))‖≤ Kγk‖xr‖+‖γk(tr,0)‖≤ Kγk‖xr‖+N

≤ Kγk r+N, (3.8)

whereN = max

k=1,2,··· ,nsupt∈D‖γk(tr,0)‖.

If tr ∈ (sk, tk+1] , k = 1,2, · · · ,n, then by (2.1), (3.3), remark3.1 and assumptions (H1), (H2), (H4), we obtain

‖(Gxr)(tr)‖ ≤ ‖U(tr− sk)‖‖[γk(sk,xr(sk))−g(sk, γk(sk,xr(sk)))]‖+‖g(tr,xr(tr))‖

+∫ t

sk

‖U(tr− s)‖‖ f (s,xr(s))‖ds+∫ tr

sk

‖AU(tr− s)g(s,xr(s))‖ds

≤M[[Kγkr+N]+ c1‖γk(sk,xr(sk))‖

]+

(c1r)+MΨ(r)‖ϕ‖L[sk,tk+1]

+(c1r)∫ tr

sk

γ(tr− s)ds

≤M[Kγkr+N](1+ c1)+(c1r)(1+b)+

MΨ(r)‖ϕ‖L[sk,tk+1]. (3.9)

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Existence of mild solutions to partial neutral differential equations with non-instantaneous impulses — 31/33

Combining (3.7)-(3.9), (2.1), (3.1), (3,3) and with the factr < ‖(Gxr)(tr)‖, we obtain

r < M[φ −g(0,φ)]+ [M(1+ c1)+1](Kr+N)

+2(c1r)(1+b)+MΨ(r)Λ. (3.10)

Dividing (3.10) by r and taking the limit as r→+∞ on bothside, we have

1 < 2c1(1+b)+K +M(Kc1 +K +ρΛ),

which is contradicts to (3.2).Next, we prove that the operator G1 : Ωr→Ωr is Lipschitz

continuous. For t ∈ (tk, sk], k = 1,2, · · · ,n and x,y ∈ ΩR, by(3.4) and assumption (H4), we have

‖(G1x)(t)− (G1y)(t)‖= ‖γk(t,x(t))− γk(t,y(t))‖,≤ Kγk‖x− y‖PC. (3.11)

For t ∈ (sk, tk+1], k = 1,2, · · · ,n and x,y ∈ΩR, by (2.1), (3.4)and assumption (H4), we have

‖(G1x)(t)− (G1y)(t)‖ ≤

M

∥∥∥γk(sk,x(sk))− γk(sk,y(sk))∥∥∥

+∥∥∥g(sk,γk(sk,x(sk))−g(sk,γk(sk,y(sk))

∥∥∥≤M

[Kγk‖x− y‖PC +LKγk‖x− y‖PC

]≤MKγk(1+L)‖x− y‖PC. (3.12)

From (2.1), (3.1), (3.11) and (3.12), we have

‖(G1x)(t)− (G1y)(t)‖ ≤ (1+M+ML)K‖x− y‖PC.(3.13)

Next we prove that G2 is continuous in Ωr. Let xn ∈Ωr bea sequence such that limn→+∞ xn = x in Ωr. By the continuityof nonlinear term f and g with respect to the second variablefor each s ∈ D, we have

limn→+∞

f (s,xn(s)) = f (s,x(s)) and

limn→+∞

g(s,xn(s)) = g(s,x(s)) (3.14)

From the hypotheses (H1) we have,

‖ f (s,xn(s))− f (s,x(s))‖= 2ϕ(s)Ψ(r). (3.15)

Then by the function s→ 2ϕ(s)Ψ(r) is Lebesgue inte-grable for s ∈ [sk, t] and t ∈ (sk, tk+1], k = 1,2, · · · ,n. Thenby (2.1), (3.1), (3.5), (3.14), (3.15), remark 3.1, assumption(H2), (H5) and the Lebesgue dominated converges theorem,

we have,

‖G2xn(t)−G2x(t)‖ ≤ ‖g(t,xn(t))−g(t,x(t))‖+∫ t

sk

‖U(t− s)‖[‖ f (s,xn(s))− f (s,x(s))‖

]ds

+∫ t

sk

‖AU(t− s)‖[‖g(s,xn(s))−

g(s,x(s))‖]ds

≤ ‖g(t,xn(t))−g(t,x(t))‖+

M∫ t

sk

‖ f (s,xn(s))− f (s,x(s))‖ds

+∫ t

sk

γ(t− s)‖g(s,xn(s))−

g(s,x(s))‖ds

→ 0 as n→+∞.(3.16)

Then we infer,

‖G2xn−G2x‖PC→ 0 as n→+∞.

Thus G2 is continuous in Ωr.Now, the operator G2 : Ωr→Ωr is equicontinuous. For anyx ∈Ωr and sk < t

′< t

′′ ≤ tk+1 for k = 0,1,2, · · · ,n, we obtainthat,

‖G2x(t′′)−G2x(t

′)‖ ≤

∥∥∥∥g(t′′,x(t

′′))−g(t

′,x(t

′))

∥∥∥∥+

∥∥∥∥∫ t′

sk[U(t

′′ − s)−U(t′ − s)] f (s,x(s))ds

∥∥∥∥+

∥∥∥∥∫ t′

sk[AU(t

′′ − s)−AU(t′ − s)]g(s,x(s))ds

∥∥∥∥+

∥∥∥∥∫ t′′

t ′ U(t′′−s) f (s,x(s))ds

∥∥∥∥+∥∥∥∥∫ t′′

t ′ AU(t′′−s)g(s,x(s))ds

∥∥∥∥= I1 + I2 + I3,where

I1 =

∥∥∥∥g(t′′,x(t

′′))−g(t

′,x(t

′))

∥∥∥∥I2 =

∥∥∥∥∫ t′

sk

[U(t′′ − s)−U(t

′ − s)] f (s,x(s))ds∥∥∥∥

+

∥∥∥∥∫ t′

sk

[AU(t′′ − s)−AU(t

′ − s)]g(s,x(s))ds∥∥∥∥

I3 =

∥∥∥∥∫ t′′

t ′U(t

′′ − s) f (s,x(s))ds∥∥∥∥+∥∥∥∥∫ t

′′

t ′AU(t

′′ − s) f (s,x(s))ds∥∥∥∥.

Therefore, we need to check I1, I2 and I3 tends to 0 indepen-dently of x ∈ Ωr when t

′′ → t′. For I1, g(t,x) is continuous,

we have,

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Existence of mild solutions to partial neutral differential equations with non-instantaneous impulses — 32/33

I1 =

∥∥∥∥g(t′′,x(t

′′))−g(t

′,x(t

′))

∥∥∥∥→ 0 as t

′′ − t ′→ 0.

Now for I2, by assumption (H1), (H2) and (H4), we have,

I2 ≤Ψ(r)∫ t′

sk

‖ [U(t′′ − s)−U(t

′ − s)]‖ϕ(s)ds+

(c1r)∫ t′

sk

‖[AU(t′′ − s)−AU(t

′ − s)]‖ds

→ 0 as t′′ → t ′.

For I3, by (2.1), assumption (H1), (H2) and (H4), we have,

I3 ≤MΨ(r)∫ t′′

t ′ϕ(s)ds+(c1r)

∫ t′′

t ′γ(t− s)ds

→ 0 as t′′ → t ′.

As a result ‖G2x(t′′)−G2x(t

′)‖ → 0 independently of

x ∈ Ωr as (t′′ − t ′)→ 0. Which means that G2 : Ωr → Ωr is

equicontinuous.For any bounded D⊂Ωr then we know that there exists a

countable set D0 = xn ⊂ D such that,

α(G2(D))PC ≤ 2α(G2(D0))PC . (3.17)

Since G2(D0)⊂ G2(Ωr) is bounded and equicontinuous,

α(G2(D0))PC = maxt∈[sk,tk+1],k=0,1,··· ,n

α(G2(D0)(t)).

(3.18)

For every t ∈ [sk, tk+1],k = 0,1, · · · ,n. From the Lemma 2.2(vi)(vii), (2.1), (3.2) and hypotheses (H2), (H5), we have,

α(G2(D0)(t))≤ Pkα(D0)+2MaQkα(D0)PC+

2bPkα(D0)PC

≤(Pk +2MaQk +2bPk

)α(D0)PC .

Therefore, from (3.2), (3.17)-(3.18) we know that,

α(G2(D))PC ≤ (2P+4MaQ−4bP)α(D)PC .(3.19)

For bounded D⊂Ωr and from Lemma 2.2 (vii) we have

α(G1(D))PC ≤M

Kα(D)PC + c1Kα(D)PC

≤MK(1+ c1)α(D)PC (3.20)

From (3.19), (3.20)

(G(D))PC ≤ (2P+4MaQ−4bP)α(D)PC+

MK(1+ c1)α(D)PC

‘≤(

2P+4MaQ−4bP+MK(1+ c1))

α(D)PC .

(3.21)

Combaining (3.21) and (3.2) and Definition 2.3 we knowthat, the operatorG : Ωr→Ωr is k-set-contractive.

Thus G has atleast one fixed point x ∈Ωr which is just aPC -mild solution of (1.1).

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