R E S E A R C H
Open Access
Existence of solutions for nonlinear mixed
type integro-differential functional evolution
equations with nonlocal conditions
Shengli Xie
**Correspondence: [email protected]
Department of Mathematics and Physics, Anhui University of Architecture, Jinzhai Road, Hefei, Anhui 230022, P.R. China
Abstract
Using the Mönch fixed point theorem, this article proves the existence of mild solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions in Banach spaces. Some restricted conditions ona priori estimation and measure of noncompactness estimation have been deleted, and compactness conditions of evolution operators or compactness conditions on a nonlinear termf(t,Xr,Xr,Xr) have been weakened. Our results extend and improve
many known results. MSC: 34G20; 34K30
Keywords: integro-differential functional evolution equation; mild solution; nonlocal conditions; fixed point; Banach spaces
1 Introduction
Let (X, · ) be a Banach space,C[J,X] ={x:J= [,a]→X,x(t) is continuous inJ}with the normxC=supt∈Jx(t). It is easy to verify thatC[J,X] is a Banach space. The space ofX-valued Bochner integrable functions onJ with the normx=
a
x(s)dsis de-noted byL[J,X]. Consider the following nonlinear mixed type integro-differential func-tional evolution equations with nonlocal conditions in a Banach spaceX(IVP),
x(t) =A
x(t) +
t
F(t–s)x(s)ds
+ft,xσ(t)
, (Kxσ)(t), (Hxσ)(t)
, t∈J, (.)
x() =g(x) +x, (.)
where
(Kxσ)(t) =
t
kt,s,xσ(s)ds, (Hxσ)(t) =
a
ht,s,xσ(s)ds, (.)
Ais the generator of a strongly continuous semigroup in the Banach spaceX, andF(t) is a bounded operator fort∈J,x∈X,f ∈C[J×X,X],g:C[J,X]→X,k∈C[ ×X,X],
={(t,s)∈J×J:s≤t},h∈C[J×J×X,X],σi∈C[J,J] andσi(t)≤t(i= , , ).
For the existence of mild solutions of integro-differential functional evolution equations in abstract spaces, there are many research results, see [–], and references therein. In order to obtain the existence and controllability of mild solutions in these study papers,
usually, some restricted conditions on a priori estimation and compactness conditions of an evolution operator or compactness conditions onf(t,Xr,Xr,Xr) are used.
Recently, using a fixed point theorem, Haribhau Laxman Tidkey and Machindra Babu-rao Dhakne [] have studied the existence of mild solutions ofIVP(.)-(.) whenσi(t) =t (i= , , ), the compactness of the resolvent operator and the restricted condition
M
x+G+Lrb+LKrb+LKb+LHrb+LHb+Lb ≤r
withM[Lb+LKb+LHb] < is used. Malar [] and Shi [] studied the existence of mild solutions of semilinear mixed type integrodifferential evolution equations with the equicontinuous semigroup
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x(t) =Ax(t) +f(t,x(t),ta(t,s)k(s,x(s))ds,
a
b(t,s)h(s,x(s))ds), t∈[,a], x() =x+g(x).
(.)
Solvability of the scalar equation
m(t) =K+K
t
hs,m(s),n(s),q(s)ds, t∈J
and the restricted condition on measure of noncompactness estimation
t
η(s) +kη(s) +kη(s) ds≤K
are used in []. But estimations (.) and (.) in [] seem to be incorrect, as they have no meaning.
In this paper, using the Mönch fixed point theorem, we investigate the existence of mild solutions ofIVP(.)-(.). Some restricted conditions ona prioriestimation and measure of noncompactness estimation have been deleted, and compactness conditions of a resol-vent operator or compactness conditions on a nonlinear termf(t,Xr,Xr,Xr) have been weakened. Our results extend and improve some corresponding results in papers [–, –].
2 Preliminaries
We will make the following assumptions:
(H) Agenerates a strongly continuous semigroup in the Banach spaceX.
(H) F(t)∈B(X),≤t≤a.F(t) :Y→Yand forx(·)continuous inY,AF(·)x(·)∈L[J,X]. For x∈X, F(t)x is continuous int∈J, whereB(X)is the space of all linear and bounded operators onX, andYis the Banach space formed fromD(A), the domain ofA, endowed with the graph norm.
Definition .[] R(t) is a resolvent operator of (.) withf ≡ ifR(t)∈B(X) for ≤t≤a and satisfies the following conditions:
() for allx∈X,R(t)xis continuous for≤t≤a,
() R(t)∈B(Y),≤t≤a; fory∈Y,R(·)y∈C[J,X]∩C[J,Y]and
d
dtR(t)y=A
R(t)y+
t
F(t–s)R(s)y ds
=R(t)Ay+
t
R(t–s)AF(s)y ds, ≤t≤a. (.)
The resolvent operatorR(t) is said to be equicontinuous if{t→R(t)x:x∈B}is equicon-tinuous for the entire bounded setB⊂Xandt> . Ifx∈C[J,X] satisfies the following integral equation:
x(t) =R(t)x+g(x)
+
t
R(t–s)fs,xσ(s)
, (Kxσ)(s), (Hxσ)(s)
ds, t∈J,
thenxis said to be a mild solutionIVP(.)-(.).
Lemma . [] Let the conditions (H), (H) be satisfied. Then (.) with f ≡ has a unique resolvent operator.
The following lemma is obvious.
Lemma . Let the resolvent operator R(t)be equicontinuous. If there isρ∈L[J,R+]such
thatx(t) ≤ρ(t)for a.e. t∈J, then the set{tR(t–s)x(s)ds}is equicontinuous.
Lemma .[] Let V∈C[J,E]be an equicontinuous bounded subset. Thenα(V(t))∈ C[J,R+](R+= [,∞)),α(V) =max
t∈Jα(V(t)).
Lemma .[] Let V={xn} ⊂L[J,E]and there existsσ∈L[J,R+]such thatxn(t) ≤
σ(t)for any x∈V and a.e. t∈J. Thenα(V(t))∈L[J,R+]and
α
t
xn(s)ds:n∈N
≤
t
αV(s)ds, t∈J.
Lemma .[] (Mönch) Let E be a Banach space,a closed convex subset in E and
y∈. Suppose that the continuous operator F:→has the following property:
V⊂countable,V⊂co{y} ∪F(V)
⇒V is relatively compact.
Then F has a fixed point in.
ForV⊂C[J,X], letV(t) ={x(t) :x∈V},Vσi(t) ={x(σi(t)) :x∈V}(i= , , ), (KV)(t) = {(Kx)(t) :x∈V}, (HV)(t) ={(Hx)(t) :x∈V}(t∈J),Xr={x∈X:x ≤r}andSr={x∈ C[J,X] :xC≤r} for anyr> .α(·) andαC(·) denote the Kuratowski measure of
3 Existence of a mild solution
We make the following assumptions for convenience.
(H) There exist constantslg> ,M> andlgM< such that
g(x) –g(y)≤lgx–yC, x,y∈C[J,X],
andg() = .
(H) g:C[J,X]→Eis continuous, compact and there exists a constantN≥such that
g(x) ≤N.
(H) There existsq∈C[J,R+]such that
f(t,x,y,z)≤q(t)x+y+z, t∈J,x,y,z∈X.
(H) There existk∈C[,R+],h∈C[J×J,R+]such that
k(t,s,x)≤k(t,s)x, (t,s)∈ ,x∈X,
h(t,s,x)≤h(t,s)x, t,s∈J,x∈X.
(H) For anyr> and a bounded setVi⊂Xr, there exist constantsli> (i= , , ) such that
αf(t,V,V,V)≤lα(V) +lα(V) +lα(V), t∈J.
(H) For anyr> and a bounded setV⊂Xr,
αk(t,s,V)≤k(t,s)α(V), (t,s)∈ ,
αh(t,s,V)≤h(t,s)α(V), t,s∈J.
(H) The resolvent operatorR(t)is equicontinuous andR(t) ≤Me–wtfort∈Jand some positive number
w=maxMq( +Ka+Ha), M(l+ laK+ laH)
,
whereK=max(t,s)∈k(t,s),H=maxt,s∈Jh(t,s),q=maxt∈Jq(t).
Without loss of generality, we always suppose thatx= .
Theorem . Let conditions (H), (H), (H)-(H) be satisfied. Then IVP (.)-(.) has at least one mild solution.
Proof Let
(Fx)(t) =R(t)g(x)
+
t
R(t–s)fs,xσ(s), (Kxσ)(s), (Hxσ)(s)
We have by (H), (H) and (H),
(Fx)(t)
≤R(t)g(x)+
t
R(t–s)fs,xσ(s)
, (Kxσ)(s), (Hxσ)(s)ds
≤Mg(x)+M
t
e–w(t–s)q(s)xσ(s)+(Kx)σ(s)+(Hx)σ(s)ds
≤lgMxC
+Mq
t
ew(s–t)x(s)+
s
k(s,r)x(r)dr+
a
h(s,r)x(r)dr
ds
≤lgMxC+Mq
t
ew(s–t)( +Ka+Ha)xCds
≤lgMxC+Mq( +Ka+Ha)w–xC≤ xC. (.)
Let
BR=
x∈C[J,X] :xC≤R
.
ThenBRis a closed convex subset inC[J,X], ∈BRandF:BR→BR. Similar to the proof of [] and [], it is easy to verify thatFis a continuous operator fromBRintoBR. Forx∈BR, s∈J, (H) and (H) imply
fs,xσ(s), (Kxσ)(s), (Hxσ)(s)≤q(s)
+
s
k(s,r)dr+
a
h(s,r)dr
R. (.)
We can show from (.), (H) and Lemma . thatF(BR) is an equicontinuous subset in C[J,X].
LetV⊂BRbe a countable set andV⊂co({} ∪F(V)), then
V(t)⊂co{} ∪(FV)(t). (.)
From equicontinuity ofF(BR) and (.), we know thatV is an equicontinuous subset in C[J,X]. By the properties of noncompact measure, the conditions (H), (H), (H), (.) and Lemma ., we have
αV(t)≤α(FV)(t)
≤R(t)αg(V)+
t
R(t–s)αfs,Vσ(s), (KVσ)(s), (HVσ)(s)
ds
≤lgMαC(V) + M
t
ew(s–t)
lα
Vσ(s)+ l
s
k(s,r)αVσ(r)dr
+ l
a
h(s,r)αVσ(r)dr
ds
≤lgMαC(V)
+ M
t
ew(s–t)
lα
V(s)ds+
l
s
K+l
a
H
αV(r)dr
≤lgMαC(V) + M(l+ laK+ laH)αC(V)
t
ew(s–t)ds
≤lgMαC(V) + M(l+ laK+ laH)w
–α
C(V)≤
αC(V), t∈J. (.)
(.) together with Lemma . imply thatαC(V)≤αC(V), and soαC(V) = . HenceVis relatively compact inC[J,X]. Lemma . implies thatFhas a fixed point inC[J,X]. Then IVP(.)-(.) has at least one mild solution. The proof is completed.
Theorem . Let the conditions (H), (H) and (H)-(H) be satisfied. Then IVP (.)-(.) has at least one mild solution.
Proof Similar to (.) and (.), it is easy to verify
(Fx)(t)≤MN+Mq( +Ka+Ha)w–xC=MN+ηxC,
whereη=Mq(+Ka+Ha)w–< . TakingR>MN(–η)–, letBR={x∈C[J,X] :xC≤ R}. We haveF:BR→BRand the inequality (.) is transformed intoα(V(t))≤αC(V),
t∈J.
The other proof is similar to the proof of Theorem ., we omit it.
4 An example
LetX=L[,π]. Consider the following partial functional integro-differential equation with a nonlocal condition,
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ut(t,y) =uy(t,y) +
t
F(t–s)uy(s,y)ds+γsinu(t–r,y) +tγu(s–r,y)ds
(+t) +
a
γu(s–r,y)ds
(+t)(+s) , ≤t≤a, u(,y) =u(y) +γu(y),
(.)
wherer,γi∈R(i= , , , ),σ(t) =σ(t) =σ(t) =t–r, ≤r≤t≤a,F(t) satisfies the condition (H),
ft,uσ(t), (Kuσ)(t), (Suσ)(t)
(y)
=γsinu(t–r,y) +
t
γu(s–r,y)ds ( +t) +
a
γu(s–r,y)ds
( +t)( +s) , (.)
kt,s,uσ(s)(y) =u(s–r,y) +t , h
t,s,uσ(s)(y) = u(s–r,y)
( +t)( +s), (.)
g(u)(y) =γu(y). (.)
Let the operatorAbe defined byAw=w,w∈D(A) with the domain
D(A) =w∈E:w∈E,wis almost everywhere bounded.
ThenAgenerates a translation semigroupR(t) andR(t) is equicontinuous. The problem (.) can be regarded as a form ofIVP(.)-(.). We have by (.), (.) and (.),
f(t,u,v,z)≤ |γ|u+v+z, |γ|=max|γ|,|γ|,|γ|
k(t,s,u)≤ u, h(t,s,u)≤ u, u∈X,
and
g(u) –g(v)≤ |γ|u–vC, g() = .
γandMcan be chosen such that M|γ|< . In addition, for anyr> and a bounded set Vi⊂Xr(i= , , ), we can show that by the diagonal method,
αf(t,V,V,V)≤ |γ|α(V) +α(V) +α(V), t∈J,
αk(t,s,V)
≤α(V), t,s∈ ,
αh(t,s,V)≤α(V), t,s∈[,a].
Hence all the conditions of Theorem . are satisfied, the problem (.) has at least one mild solution inC[J,X].
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The work was supported by Natural Science Foundation of Anhui Province (11040606M01) and Education Department of Anhui (KJ2011A061, KJ2011Z057), China.
Received: 5 May 2012 Accepted: 22 August 2012 Published: 7 September 2012 References
1. Tidke, HL, Dhakne, MB: Existence of solutions and controllability of nonlinear mixed integrodifferential equation with nonlocal conditions. Appl. Math. E-Notes11, 12-22 (2011)
2. Yan, Z, Wei, P: Existence of solutions for nonlinear functional integrodifferential evolution equations with nonlocal conditions. Aequ. Math.79, 213-228 (2010)
3. Tidke, HL, Dhakne, MB: Nonlocal Cauchy problems for nonlinear mixed integro-differential equations. Tamkang J. Math.41, 361-373 (2010)
4. Balachandran, K, Kumar, RR: Existence of solutions of integrodifferential evolution equations with time varying delays. Appl. Math. Lett.7, 1-8 (2007)
5. Kumar, RR: Nonlocal Cauchy problem for analytic resolvent integrodifferential equations in Banach spaces. Appl. Math. Comput.204, 352-362 (2008)
6. Yang, YL, Wang, JR: On some existence results of mild solutions for nonlocal integro-differential Cauchy problems in Banach spaces. Opusc. Math.31, 443-455 (2011)
7. Liu, Q, Yuan, R: Existence of mild solutions for semilinear evolution equations with non-local initial conditions. Nonlinear Anal.71, 4177-4184 (2009)
8. Boucherif, A, Precup, R: Semilinear evolution equations with nonlocal initial conditions. Dyn. Syst. Appl.16, 507-516 (2007)
9. Chalishajar, DN: Controllability of mixed type Volterra-Fredholm integro-differential systems in Banach space. J. Franklin Inst.344, 12-21 (2007)
10. Xue, X: Existence of solutions of a semilinear nonlocal Cauchy problem in a Banach space. Elect. J. Diff. Eqs.2005(64), 1-7 (2005)
11. Liu, JH, Ezzinbi, K: Non-autonomous integrodifferential equations with nonlocal conditions. J. Integral Equ. Appl.15, 79-93 (2003)
12. Ntouyas, S, Tsamotas, P: Global existence for semilinear evolution equations with nonlocal conditions. J. Math. Anal. Appl.210, 679-687 (1997)
13. Byszewski, L, Akca, H: Existence of solutions of a semilinear functional-differential evolution nonlocal problem. Nonlinear Anal.34, 65-72 (1998)
14. Lin, Y, Liu, J: Semilinear integrodifferential equations with nonlocal Cauchy problem. Nonlinear Anal.26, 1023-1033 (1996)
15. Byszewski, L: Existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem. Zesz. Nauk. Politech. Rzesz., Mat. Fiz.18, 109-112 (1993)
16. Byszewski, L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl.162, 494-505 (1991)
17. Malar, K: Existence of mild solutions for nonlocal integro-differential equations with measure of noncompactness. Int. J. Math. Sci. Comput.1, 86-91 (2011)
19. Zhu, T, Song, C, Li, G: Existence of mild solutions for abstract semilinear evolution equations in Banach spaces. Nonlinear Anal.75, 177-181 (2012)
20. Fan, Z, Dong, Q, Li, G: Semilinear differential equations with nonlocal conditions in Banach spaces. Int. J. Nonlinear Sci. 2(3), 131-139 (2006)
21. Dong, Q, Li, G: Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces. Electron. J. Qual. Theory Differ. Equ.47, 1-13 (2009)
22. Dajun, G, Lakshmikantham, V, Liu, X: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic, Dordrecht (1996)
23. Heine, HP: On the behavior of measure of noncompactness with respect to differentiation and integration of vector valued functions. Nonlinear Anal.7, 1351-1371 (1983)
24. Mönch, H: Boundary value problems for nonlinear ordinary equations of second order in Banach spaces. Nonlinear Anal.4, 985-999 (1980)
doi:10.1186/1687-2770-2012-100