doi:10.1155/2011/806729
Research Article
Solvability of Nonautonomous Fractional
Integrodifferential Equations with Infinite Delay
Fang Li
School of Mathematics, Yunnan Normal University, Kunming 650092, China
Correspondence should be addressed to Fang Li,[email protected]
Received 4 September 2010; Revised 19 October 2010; Accepted 29 October 2010
Academic Editor: Toka Diagana
Copyrightq2011 Fang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the existence and uniqueness of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with infinite delay in a Banach spaceX. The existence of mild solution is obtained by using the theory of the measure of noncompactness and Sadovskii’s fixed point theorem. An application of the abstract results is also given.
1. Introduction
The Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decadescf., e.g.,1–15. This paper is concerned with existence results for nonautonomous fractional integrodifferential equations with infinite delay in a Banach spaceX
dqut
dtq −Atut f
t,
t
0
Kt, susds, ut
, t∈0, T,
ut φt, t∈−∞,0,
1.1
whereT > 0, 0 < q < 1,{At}t∈0,T is a family of linear operators inX, K ∈CD,Rwith
D{t, s∈R2: 0≤s≤t≤T}and
sup
t∈0,T
t
0
f : 0, T×X× P → X,ut : −∞,0 → X defined byutθ utθforθ ∈ −∞,0,φ
belongs to the phase spaceP, andφ0 0. The fractional derivative is understood here in the Riemann-Liouville sense.
In recent years, the fractional differential equations have been proved to be good tools in the investigation of many phenomena in engineering, physics, economy, chemistry, aerodynamics, electrodynamics of complex medium and they have been studied by many researchers cf., e.g., 13, 14, 16, 17 and references therein. Moreover, many phenomena cannot be described through classical differential equations but the integral and integrodifferential equations in abstract spaces in fields like electronic, fluid dynamics, biological models, and chemical kinetics. So many significant works on this topic have been appearedcf., e.g.,10,15,18–25and references therein.
In this paper, we study the existence of mild solution of 1.1 and obtain the existence theorem based on the measures of noncompactness without the assumptions that the nonlinearity f satisfies a Lipschitz type condition and the semigroup {exp−tAs}
generated by−As s∈0, Tis compactseeTheorem 3.1. An example is given to show an application of the abstract results.
2. Preliminaries
Throughout this paper, we setJ 0, T, a compact interval inR. We denote byXa Banach space,LXthe Banach space of all linear and bounded operators onX, andCJ, Xthe space of allX-valued continuous functions onJ. We set
Gut: t
0
Kt, susds, G∗:sup
t∈J
t
0
Kt, sds <∞. 2.1
Next, we recall the definition of the Riemann-Liouville integral.
Definition 2.1see26. The fractionalarbitraryorder integral of the functiong ∈L1R,R of orderν >0 is defined by
Iνgt 1
Γν
t
0
t−sν−1gsds, 2.2
whereΓis the Gamma function. Moreover,Iν1Iν2Iν1ν2, for allν
1, ν2>0.
Remark 2.2. 1Iν:L10, T → L10, T see26,
2obviously, forg∈L1J,R, it follows fromDefinition 2.1that
t
0
η
0
t−ηq−1η−sγ−1gsds dηBq, γ
t
0
t−sqγ−1gsds, 2.3
whereBq, γis a beta function.
Definition 2.3. A linear spacePconsisting of functions fromR−intoX, with seminorm · P, is called an admissible phase space ifPhas the following properties.
1Ifx:−∞, T → Xis continuous onJandx0∈ P, thenxt∈ Pandxtis continuous
int∈J, and
xt ≤LxtP, 2.4
whereL≥0 is a constant.
2There exist a continuous functionC1t>0 and a locally bounded functionC2t≥0
int≥0, such that
xtP≤C1tsup
s∈0,txsC2tx0P, 2.5
fort∈0, Tandxas in1.
3The spacePis complete.
Remark 2.4. Equation2.4in1is equivalent toφ0 ≤LφP, for allφ∈ P.
Next, we consider the properties of Kuratowski’s measure of noncompactness.
Definition 2.5. LetBbe a bounded subset of a seminormed linear spaceY. The Kuratowski’s measure of noncompactnessfor brevity,α-measureofBis defined as
αB infd >0 :Bhas a finite cover by sets of diameter≤d. 2.6
From the definition we can get some properties ofα-measure immediately, see28.
Lemma 2.6see28. LetAandBbe bounded subsets ofX, then 1αA≤αB, ifA⊆B;
2αA αA, whereAdenotes the closure ofA; 3αA 0if and only ifAis precompact; 4αλA |λ|αA,λ∈R;
5αA∪B max{αA, αB};
6αAB≤αA αB, whereAB{xy:x∈A, y∈B};
7αAx αA, for anyx∈X.
ForH⊂CJ, X, we define
t
0
Hsds
t
0
usds:u∈H
, fort∈J, 2.7
The following lemma will be needed.
Lemma 2.7. IfH⊂CJ, Xis a bounded, equicontinuous set, then
iαH supt∈JαHt,
iiα0tHsds≤0tαHsds, fort∈J.
For a proof refer to28.
Lemma 2.8see29. If{un}∞n1⊂L1J, Xand there exists anm∈L1J,Rsuch thatunt ≤
mt, a.e.t∈J, thenα{unt}∞n1is integrable and
α
t
0
unsds
∞
n1
≤2
t
0
α{uns}∞n1ds. 2.8
We need to use the following Sadovskii’s fixed point theorem here, see30.
Definition 2.9. LetPbe an operator in Banach spaceX. IfP is continuous and takes bounded sets into bounded sets, andαPB< αBfor every bounded setBofXwithαB>0, then
Pis said to be a condensing operator onX.
Lemma 2.10Sadovskii’s fixed point theorem30. LetP be a condensing operator on Banach spaceX. IfPH⊆Hfor a convex, closed, and bounded setHofX, thenPhas a fixed point inH.
In this paper, we denote thatC is a positive constant, and assume that a family of closed linear operators{At}t∈Jsatisfying the following.
A1The domain DAof {At}t∈J is dense in the Banach spaceX and independent oft.
A2The operatorAt λ−1exists inLXfor anyλwith Reλ≤0 and
At λ−1≤ C
|λ|1, t∈J. 2.9
A3There exist constantsγ∈0,1andCsuch that
At1−At2A−1s≤C|t1−t2|γ, t1, t2, s∈J. 2.10
Under conditionA2, each operator−As,s ∈ J, generates an analytic semigroup exp−tAs,t >0, and there exists a constantCsuch that
Ansexp−tAs≤ C
tn, 2.11
LetΩbe set defined by
Ω u:−∞, T−→X such that u|−∞,0∈ Pand u|J ∈CJ, X. 2.12
According to16, a mild solution of1.1can be defined as follows.
Definition 2.11. A functionu∈Ωsatisfying the equation
ut
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
φt, t∈−∞,0,
t
0
ψt−η, ηfη, Guη, uη
dη
t
0
η
0
ψt−η, ηϕη, sfs, Gus, usds dη, t∈J
2.13
is called a mild solution of1.1, where
ψt, s q
∞
0
θtq−1ξqθexp−tqθAsdθ, 2.14
andξq is a probability density function defined on0,∞such that its Laplace transform is
given by
∞
0
e−σxξqσdσ ∞
j0
−xj
Γ1qj, 0< q≤1, x >0,
ϕt, τ
∞
k1
ϕkt, τ,
2.15
where
ϕ1t, τ At−Aτψt−τ, τ,
ϕk1t, τ
t
τ
ϕkt, sϕ1s, τds, k1,2, . . . .
2.16
To our purpose the following conclusions will be needed. For the proofs refer to16.
Lemma 2.12see16. The operator-valued functionsψt−η, ηandAtψt−η, ηare continuous in uniform topology in the variablest,η, where0≤η≤t−ε,0≤t≤T, for anyε >0. Clearly,
ψt−η, η≤Ct−ηq−1. 2.17
Moreover, we have
ϕ
3. Main Results
We need the hypotheses as follows:
H1f :J×X× P → Xsatisfiesf·, v, w:J → Xis measurable for allv, w∈X× P
and ft,·,· : X× P → X is continuous for a.e.t ∈ J, and there exist a positive function μ· ∈ LpJ,Rp > 1/q > 1and a continuous nondecreasing function
W:0,∞ → 0,∞, such that
ft, v, w≤μtWvwP, t, v, w∈J×X× P, 3.1
and setTp,q:Tq−1/p,
H2for any bounded setsD1⊂X,D2 ⊂ P, and 0≤τ≤s≤t≤T,
αψt−s, sfs, D1, D2
≤β1t, sαD1 β2t, s sup
−∞<θ≤0
αD2θ,
αψt−s, sϕs, τfτ, D1, D2
≤β3t, s, ταD1 β4t, s, τ sup
−∞<θ≤0
αD2θ,
3.2
where supt∈J0tβit, sds:βi<∞i1,2,supt∈J
t
0
s
0βit, s, τdτ ds:βi<∞i
3,4andD2θ {wθ:w∈D2},
H3there existsM, with 0< M <1 such that
C1CBq, γTp,q,γMp,q
G∗C∗1μLpJ,Rlim inf τ→ ∞
Wτ
τ < M, 3.3
whereMp,q : p−1/pq−1p−1/p,C∗1 sup0≤η≤TC1ηandTp,q,γ max{Tp,q,
Tp,qγ}.
Theorem 3.1. Suppose thatH1–H3are satisfied, and if4G∗β12β3 β22β4<1, then
for1.1there exists a mild solution on−∞, T.
Proof. Consider the operatorΦ:Ω → Ωdefined by
Φut
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
φt, t∈−∞,0,
t
0
ψt−η, ηfη, Guη, uη
dη
t
0
η
0
ψt−η, ηϕη, sfs, Gus, usds dη, t∈J.
3.4
It is easy to see thatΦis well-defined.
Letx·:−∞, T → Xbe the function defined by
xt
⎧ ⎨ ⎩
φt, t∈−∞,0,
0, t∈J.
Letut xt yt,t∈−∞, T. It is easy to see thatysatisfiesy00 and
yt
t
0
ψt−η, ηfη, Gxηyη, xηyη
dη
t
0
η
0
ψt−η, ηϕη, sfs, Gxs ys, xsys
ds dη, t∈J
3.6
if and only ifusatisfies
ut
t
0
ψt−η, ηfη, Guη, uη
dη
t
0
η
0
ψt−η, ηϕη, sfs, Gus, usds dη, t∈J
3.7
andut φt,t∈−∞,0.
LetY0{y∈Ω:y00}. For anyy∈Y0,
yY
0sup
t∈J
yty0Psup
t∈J
yt. 3.8
ThusY0, · Y0is a Banach space.
We define the operatorΦ :Y0 → Y0byΦyt 0,t∈−∞,0and
Φyt
t
0
ψt−η, ηfη, Gxηyη, xηyη
dη
t
0
η
0
ψt−η, ηϕη, sfs, Gxs ys, xsys
ds dη, t∈J.
3.9
Obviously, the operatorΦhas a fixed point if and only ifΦ has a fixed point. So it turns out to prove thatΦ has a fixed point.
Let{yk}
k∈Nbe a sequence such thatyk → yinY0ask → ∞. Sincef satisfiesH1,
for almost everyt∈J, we get
ft, Gxt ykt, x tykt
−→ft, Gxt yt, xtyt
Fort∈−∞, T, we can prove thatΦ is continuous. In fact,
Φykt−Φyt
≤ t
0
ψt−η, ηfη, Gxηykη, xηykη
−fη, Gxηyη, xηyηdη
t
0
η
0
ψt−η, ηϕη, sfs, Gxs yks, xsyks
−fs, Gxs ys, xsysds dη.
3.11
LetC∗2sup0≤η≤TC2η, and noting2.4,2.5, we have
xtyt
P≤ xtPytP
≤C1tsup
0≤τ≤txτC2tx0PC1t0sup≤τ≤t
yτC2ty0P
C2tφPC1tsup 0≤τ≤t
yτ
≤C∗2φPC∗1sup
0≤τ≤t
yτ.
3.12
Moreover,
Gxt yt≤
t
0
Kt, τxτ yτdτ
t
0
Kt, τ·yτdτ.
3.13
Noting thatyk → yinY0, we can see that there existsε >0 such thatyk−y ≤ε, for
ksufficiently large. Therefore, we have
ft, Gxt ykt, xtykt
−ft, Gxt yt, xtyt
≤μt
WGxt yktxtytkP
WGxt ytxtytP
≤μtω1kt ω2kt,
where
ω1kt WG∗εG∗yY
0C
∗
2φPC∗1εC1∗yY0
,
ω2kt WG∗yY
0C
∗
2φPC∗1yY0
.
3.15
In view of2.17and the Lebesgue Dominated Convergence Theorem ensure that
t
0
ψt−η, ηfη, Gxηykη, x ηykη
−fη, Gxηyη, xηyηdη
≤C
t
0
t−ηq−1fη, Gxηykη, x ηykη
−fη, Gxηyη, xηyηdη
−→0, ask−→ ∞.
3.16
Similarly,by2.17and2.18, we have
t
0
η
0
ψt−η, ηϕη, sfs, Gxs yks, xsyks
−fs, Gxs ys, xsysds dη
≤C2
t
0
η
0
t−ηq−1η−sγ−1fs, Gxs yks, xsyks
−fs, Gxs ys, xsysds dη
−→0, ask−→ ∞.
3.17
Therefore, we deduce that
lim
k→ ∞
Φyk−Φy Y0
0. 3.18
This means thatΦ is continuous.
We show thatΦ maps bounded sets ofY0into bounded sets inY0. For anyr >0, we
setBr {y∈Y0 :yY0 ≤r}. Now, fory∈Br, by3.12,3.13, andH1, we can see
ft, Gxt yt, xtyt≤μtWM1, 3.19
Then for anyy∈Br, by2.17,2.18,3.19, andRemark 2.2, we have
Noting that the H ¨older inequality, we have
t
For every bounded setH ⊂ Bk and anyε > 0, we can take a sequence{hn}∞n1 ⊂ H
sinceεis arbitrary, we can obtain
αΦH≤4 G∗β12β3
β22β4
!
αH< αH. 3.31
In view of the Sadovskii’s fixed point theorem, we conclude thatΦ has at least one fixed pointyinBk. Letut xt yt,t∈−∞, T, thenutis a fixed point of the operator
Now we assume that
H1’there exists a positive functionl·∈L1J,R, such that
ft, v1, w1−ft, v2, w2
≤ltv1−v2w1−w2P, v1, v2∈X2, w1, w2∈ P2,
3.32
H2’there exists a constant, with 0< <1, such that the functionΛ:J → Rdefined by
Λt CG∗C∗1Γq Iqlt CΓγIγqlt!≤, t∈J. 3.33
Theorem 3.2. Assume that H1’andH2’are satisfied, then1.1has a unique mild solution.
Proof. LetΦ be defined as inTheorem 3.1. For anyy,y∗∈Y0, we have
f
t, Gxt yt, xtyt
−ft, Gxt y∗t, xty∗t
≤ltGyt−y∗tyt−y∗tP
≤lt
G∗y−y∗Y
0C1tsup
0≤τ≤t
yτ−y∗τ
≤G∗C∗1lty−y∗Y
0.
3.34
Thus, from2.17,2.18,Definition 2.1andRemark 2.2, we have
Φyt−Φy∗t
≤ t
0
ψ
t−η, ηfη, Gxηyη, xηyη
−fη, Gxηy∗η, xηy∗ηdη
t
0
η
0
ψt−η, ηϕη, sfs, Gxs ys, xsys
− fs, Gxs y∗s, xsy∗sds dη
≤CG∗C∗1y−y∗Y
0· t
0
t−ηq−1lηdηC
t
0
η
0
t−ηq−1η−sγ−1lsds dη
CG∗C∗1· ΓqIqlt CΓqΓγIγqlt!·y−y∗Y
0
Λty−y∗Y
0.
So, we get
Φyt−Φy∗t
Y0
<y−y∗Y
0, 3.36
and the result follows from the contraction mapping principle.
Example 3.3. We consider the following model:
∂q
∂tqvt, ξ at, ξ
∂2v ∂ξ2t, ξ
tn n2sin
"" "" "
t
0
t−svs, ξds
"" "" "
· t
0
e−|0ss−τvτ,ξdτ|ds t
n
n2
0
−∞ζθsin
""
"t2vtθ, ξ"""dθ,
vt,0 vt,1 0,
vθ, ξ v0θ, ξ, −∞< θ≤0,
3.37
where 0 ≤ t ≤ 1,ξ ∈ 0,1,n ∈ N,at, ξis a continuous function and is uniformly H ¨older continuous int, that is, there existC >0 andγ∈0,1such that
at1, ξ−at2, ξ ≤C|t1−t2|γ, ξ∈0,1, 0≤t1≤t2≤1, 3.38
ζ:−∞,0 → R,v0:−∞,0×0,1 → Rare continuous functions, and
0
−∞|ζθ|dθ <∞.
SetX L20,1,Rand defineAtby
DAt H20,1∩H010,1,
Atu−at, ξu. 3.39
Then−Asgenerates an analytic semigroup exp−tAssatisfying assumptionsA1–A3
see33.
Let the phase spaceP be BUCR−, X, the space of bounded uniformly continuous functions endowed with the following norm:
ϕP sup
−∞<θ≤0
""ϕθ"", ∀ϕ∈ P, 3.40
Moreover,
The author is grateful to the referees for their valuable suggestions. This work is supported by the NSF of Yunnan Provinceno. 2009ZC054M.
References
1 Z. Fan, “Existence and continuous dependence results for nonlinear differential inclusions with infinite delay,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2379–2392, 2008.
3 Faming Guo, B. Tang, and F. Huang, “Robustness with respect to small delays for exponential stability of abstract differential equations in Banach spaces,”The ANZIAM Journal, vol. 47, no. 4, pp. 555–568, 2006.
4 J. H. Liu, “Periodic solutions of infinite delay evolution equations,”Journal of Mathematical Analysis and Applications, vol. 247, no. 2, pp. 627–644, 2000.
5 J. Liu, T. Naito, and N. Van Minh, “Bounded and periodic solutions of infinite delay evolution equations,”Journal of Mathematical Analysis and Applications, vol. 286, no. 2, pp. 705–712, 2003.
6 J. Liang and T. J. Xiao, “Functional-differential equations with infinite delay in Fr´echet space,”Sichuan Daxue Xuebao, vol. 26, no. 4, pp. 382–390, 1989.
7 J. Liang and T. J. Xiao, “Functional-differential equations with infinite delay in Banach spaces,”
International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 497–508, 1991.
8 J. Liang and T. J. Xiao, “Solutions to abstract functional-differential equations with infinite delay,”
Acta Mathematica Sinica, vol. 34, no. 5, pp. 631–644, 1991.
9 J. Liang and T.-J. Xiao, “The Cauchy problem for nonlinear abstract functional differential equations with infinite delay,”Computers & Mathematics with Applications, vol. 40, no. 6-7, pp. 693–703, 2000.
10 J. Liang, T.-J. Xiao, and J. van Casteren, “A note on semilinear abstract functional differential and integrodifferential equations with infinite delay,”Applied Mathematics Letters, vol. 17, no. 4, pp. 473– 477, 2004.
11 J. Liang and T.-J. Xiao, “Solvability of the Cauchy problem for infinite delay equations,”Nonlinear Analysis: Theory, Methods & Applications, vol. 58, no. 3-4, pp. 271–297, 2004.
12 J. Liang and T.-J. Xiao, “Solutions to nonautonomous abstract functional equations with infinite delay,”Taiwanese Journal of Mathematics, vol. 10, no. 1, pp. 163–172, 2006.
13 G. M. Mophou and G. M. N’Gu´er´ekata, “A note on a semilinear fractional differential equation of neutral type with infinite delay,”Advances in Difference Equations, vol. 2010, Article ID 674630, 8 pages, 2010.
14 G. M. Mophou and G. M. N’Gu´er´ekata, “Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay,”Applied Mathematics and Computation, vol. 216, no. 1, pp. 61–69, 2010.
15 T.-J. Xiao and J. Liang, “Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e1442– e1447, 2009.
16 M. M. El-Borai, “The fundamental solutions for fractional evolution equations of parabolic type,”
Journal of Applied Mathematics and Stochastic Analysis, no. 3, pp. 197–211, 2004.
17 G. M. Mophou and G. M. N’Gu´er´ekata, “Existence of the mild solution for some fractional differential equations with nonlocal conditions,”Semigroup Forum, vol. 79, no. 2, pp. 315–322, 2009.
18 J.-H. Chen, T.-J. Xiao, and J. Liang, “Uniform exponential stability of solutions to abstract Volterra equations,”Journal of Evolution Equations, vol. 9, no. 4, pp. 661–674, 2009.
19 J. Liang and T.-J. Xiao, “Semilinear integrodifferential equations with nonlocal initial conditions,”
Computers & Mathematics with Applications, vol. 47, no. 6-7, pp. 863–875, 2004.
20 J. Liang, J. H. Liu, and T.-J. Xiao, “Nonlocal problems for integrodifferential equations,”Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 15, no. 6, pp. 815–824, 2008.
21 T. Naito, N. Van Minh, and J. H. Liu, “On the bounded solutions of Volterra equations,”Applicable Analysis, vol. 83, no. 5, pp. 433–446, 2004.
22 P. H. A. Ngoc, S. Murakami, T. Naito, J. S. Shin, and Y. Nagabuchi, “On positive linear Volterra-Stieltjes differential systems,”Integral Equations and Operator Theory, vol. 64, no. 3, pp. 325–335, 2009.
23 T.-J. Xiao and J Liang,The Cauchy Problem for Higher-Order Abstract Differential Equations, vol. 1701 of
Lecture Notes in Mathematics, Springer, Berlin, Germany, 1998.
24 T.-J. Xiao and J. Liang, “Approximations of Laplace transforms and integrated semigroups,”Journal of Functional Analysis, vol. 172, no. 1, pp. 202–220, 2000.
25 T.-J. Xiao, J Liang, and J. van Casteren, “Time dependent Desch-Schappacher type perturbations of Volterra integral equations,”Integral Equations and Operator Theory, vol. 44, no. 4, pp. 494–506, 2002.
26 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
27 J. K. Hale and J. Kato, “Phase space for retarded equations with infinite delay,”Funkcialaj Ekvacioj, vol. 21, no. 1, pp. 11–41, 1978.
29 H.-P. Heinz, “On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions,”Nonlinear Analysis: Theory, Methods & Applications, vol. 7, no. 12, pp. 1351–1371, 1983.
30 B. Sadovskii, “On a fixed point principle,”Functional Analysis and Its Applications, vol. 1, no. 2, pp. 151–153, 1967.
31 E. Hille and R. S. Phillips,Functional Analysis and Semi-Groups, vol. 31 ofAmerican Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1957.
32 D. Bothe, “Multivalued perturbations of m-accretive differential inclusions,” Israel Journal of Mathematics, vol. 108, pp. 109–138, 1998.