Nonlinear functional integrodifferential equations in Hilbert space

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NONLINEAR FUNCTIONAL INTEGRODIFFERENTIAL

EQUATIONS IN HILBERT SPACE

J. Y. PARK, S. Y. LEE, and M. J. LEE

(Received 7 November 1996 and in revised form 3 November 1997)

Abstract.LetXbe a Hilbert space and letΩ⊂Rn_{be a bounded domain with smooth} boundary∂Ω. We establish the existence and norm estimation of solutions for the parabolic partial functional integro-differential equation inXby using the fundamental solution.

Keywords and phrases. Functional integro-differential equation, elliptic differential opera-tors, fundamental solution, Gårding’s inequality, successive approximation, norm estima-tion.

1991 Mathematics Subject Classification. 35A05, 35J60, 45K05.

1. Introduction. LetXbe a Hilbert space and letΩ⊂Rn_{be a bounded domain with} smooth boundary∂Ω. We consider the following parabolic partial functional integro-differential equation.

∂u

∂t =Ꮽ0u(t,x)+Ꮽ1u(t−h,x)+

0

−ha(s)Ꮽ2u(t+s,x)ds

+

_t

0

k(t,s)Gs,u(s−h),x+Ht,s,u(s−h,x)ds

+Ft,u(t−h,x)+f (t,x), 0< t≤T , x∈Ω,

(1.1)

whereᏭi(i=0,1,2)are elliptic differential operators,f is a forcing function,h >0 is a delay time,a(s) is a real scalar function on[−h,0],G,H, and F are nonlinear functions, andkis a kernel. The boundary condition attached to (1.1) is, e.g., given by the Dirichlet boundary condition

u|∂Ω=0, 0< t≤T , (1.2) and the initial condition is given by

u(θ,x)=g(θ,x), θ∈[−h,0], x∈Ω. (1.3) From [4], the above mixed problems (1.1), (1.2), and (1.3) can be formulated abstractly as

du(t)

dt =A0u(t)+A1u(t−h)+

₀

−ha(s)A2u(t+s)ds

+

t

0

k(t,s)G(s,us)+H(t,s,us)ds

+F(t,ut)+f (t), 0< t≤T ,

(1.4)

where the state u(x) of the system (1.5) lies in an appropriate Hilbert space and

Ai(i=0,1,2)are unbounded operators associated withᏭi(i=0,1,2), respectively. Next, we explain the notationutin (1.5). LetI=[−h,0]. If a functionu(t)is continuous fromI∪[0,T ]into a Hilbert spaceX, thenutis an element inC=C([−h,0];X), which has the point-wise definition

ut(θ)=u(t+θ) forθ∈I. (1.6) Let∆T= {(s,t);0≤s≤t≤T}. We assume in (1.5) thatG:[0,T ]×C→X,H:∆T×C→

X,F :[0,T ]×C→Xand the kernelk:∆T →R (R denotes the set of real numbers) are continuous,f:[0,T ]→V∗_{with some enlarged spaceV}∗_{⊃Handg:[−h,0]→V} with some dense subspaceV⊂H. It is assumed that the inclusionsV⊂H⊂V∗ _are continuous andV∗_{is the dual space ofV.}

Many authors [2, 8] studied the following delay differential equation:

du(t)

dt =A0u(t)+A1u(t−h)+

0

−ha(s)A2u(t+s)ds+f (t), a.e.t≥0,

u(θ)=g(θ), θ∈[−h,0].

(1.7)

The fundamental solution is constructed in Tanabe [8]. In this paper, we establish the existence and norm estimation of solutions for the equation (1.5) by using the fundamental solution.

2. Preliminaries. LetHbe a pivot complex Hilbert space andVbe a complex Hilbert space such thatV is dense inHand the inclusion mapi:V→H is continuous. The norms ofH,V, and the inner product of H are denoted by| · |, · , and·,·, re-spectively. Identifying the antidual ofH withH, we may consider thatV⊂H⊂V∗_. The norm of the dual spaceV∗_{is denoted by·∗. We consider the following linear} functional differential equation on the Hilbert spaceH.

du(t)

dt =A0u(t)+A1u(t−h)+

0

−ha(s)A2u(t+s)ds+f (t), a.e.t≥0,

u(0)=g0_{, u(s)=g}1_{(s), a.e.s∈[−h,0].}

(2.1)

Let a(u,v) be a bounded sesquilinear form defined in V×V satisfying Gårding’s inequality

Rea(u,u)≥c0u2−c1|u|2, (2.2) wherec0>0 andc1≥0 are real constants. LetA0be the operator associated with this sesquilinear form

v,A0u = −a(u,v), u,v∈V, (2.3) where·,·denotes the duality pairing betweenVandV∗_{. The operatorA0is bounded} linear fromVintoV∗_{. The realization ofA}

0inH, which is the restriction ofA0to the domainD(A0)= {u∈V:A0u∈H}, is also denoted byA0. It is proved in Tanabe [6] thatA0generates an analytic semigroupetA0=T (t)both inHandV∗and thatT (t):

V∗_{→V for eacht >0. Throughout this paper, it is assumed that eachA}

i(i=1,2) is bounded and linear fromV toV∗ _(i.e.,A

endowed with the graph norm ofA0toHcontinuously. The real valued scalar function

a(s)is assumed to be Hölder continuous on[−h,0]. We introduce a Stieltjes measure

ηgiven by

η(s)= −χ(−∞,−h](s)A1− 0

s a

ξdξ A2 :V →V∗, s∈[−h,0], (2.4) whereχ(−∞,−h]denotes the characteristic function of(−∞,−h]. Then the delay term in (2.1) is written simply as₋0_hdη(s)u(t+s). The fundamental solutionW (t)of (2.1) is defined as a unique solution of

W (t)=

      

T (t)+

_t

0T (t−s) ₀

−hdη

ξWξ+sds, t≥0,

0, t <0, (2.5)

andW (t)is constructed by Tanabe [7] under the Hölder continuity ofa(s).

Theorem2.1[2]. The fundamental solutionW (t) is strongly continuous inV,H, andV∗_{, and for eacht >0,W (t):V}∗_{→V. Furthermore,W (t)satisfies}

d

dtW (t)=A0W (t)+

0

−hdη(s)W (t+s), a.e.t >0. (2.6) For eacht >0, we define the operator valued functionUt(·)by

Ut(s)= s

−hW

t−s+ξdηξ :V →V, a.e.s∈[−h,0]. (2.7)

LetT >0 be fixed. Associated withUt(·), we consider the operatorᐁ:L2(−h,0;V)→

L2_{(0,T;V)defined by}

ᐁg1_(t)=0 −hUt(s)g

1_{(s)ds, t∈[0,T ] (2.8)} forg1_∈L2_{(−h,0;V ).}

Theorem2.2[8]. LetT >0be fixed. Assume thatf∈L2_(0,T;V∗_)andg=(g0_,g1_)∈

H×L2_{(−h,0;V ). Then there exists a unique solutionu(t)=u(t;f ,g)of (2.1) on[0,T ]} satisfying

u∈L2_{(0,T;V )∩W}1,2_(0,T;V∗_{)⊂C([0,T ];H). (2.9)} Further, for eachT >0, there is a constantKT such that

T

0

_u(t)2_dt+T 0

du(t)_dt 2_∗dt≤KT

_g02₊0 −h

_g1_(s)2_ds+T 0

_{f (t)}2 ∗dt

.

(2.10) This solutionu(t)is represented by

ut;f ,g=W (t)g0_+ᐁg1_(t)+t

0W (t−s)f (s)ds. (2.11) In what follows, in order to consider the solutions in the state spaceC=C([−h,0];H), we assume thatg=(g0_,g1_{)is continuous inH, i.e.,}

Let

ˆ

ut;f ,g=

  u

t;f ,g, t∈[0,T ],

g(t), t∈[−h,0]. (2.13)

Then, by Theorem 2.2, we get ˆ

u·;f ,g∈C[−h,T ];H (2.14) if (2.12) is satisfied.

3. Existence and uniqueness of functional integro-differential equations. Using the fundamental solutionW (t)in Section 2, we consider the following abstract func-tional integral equation.

v(t)=ut;f ,g

+

t

0W (t−s) s

0

k(s,τ)Gτ,vτ

+H(s,τ,vτ)dτ+F(s,vs)

ds, 0< t≤T , v(θ)=g(θ), θ∈[−h,0],

(3.1)

whereu(t;f ,g)is given by (2.11). We list the following hypotheses.

(A1) The nonlinear functionsG:[0,T ]×C→H,H:∆T×C→H,F:[0,T ]×C→H, and the kernelk:∆T→R(Rdenotes the set of real numbers) are continuous.

(A2) Letb1,b3:[0,T ]→R,b2:∆T→R+be continuous functions such that _G(t,φ)−G

t,φX≤b1(t)φ−φC; _{H(t,s,φ)−H}

t,s,φ_X≤b2(t,s)φ−φC; _F(t,φ)−F

t,φX≤b3(t)φ−φC

(3.2)

fort,s∈[0,T ],φ,φ∈C.

(A3) The functionk(t,s)is Hölder continuous with exponentα, i.e., there exists a positive constantasuch that

_{k(t1,s1)−k(t2,s2)≤a}

|t1−t2|α+|s1−s2|α (3.3) fort1,t2,s1,s2∈[0,T ], 0< α≤1.

(A4) For all 0≤s≤t≤T,

G(t,0)=0, H(t,s,0)=0, F(t,0)=0. (3.4)

Theorem3.1. Letf∈L2_(0,T;V∗_{)andg=(g(0),g(·))∈H×L}2_{(−h,0;V)satisfy} (2.12). Assume that the hypotheses (A1)–(A4) hold. Then there exists a timet1>0such that the functional integral equation (3.1) admits a unique solutionv(t)on[0,t1].

Proof. We prove this theorem by using the method of successive approximations. Setv0_{(t)=u(t;f ,g),t≥0. Let ˆv}0_{(t)be the extension ofv}0_{(t)on[−h,T ]by (2.13).} Then, by the assumptions onf andg, we have ˆv0_{(t)∈C([−h,T ];H). By hypotheses} (A1)–(A4), we define{vˆn_}∞

ˆ

vn_{(t)=u(t;f ,g)}

+

_t

0W (t−s) s

0

k(s,τ)Gτ,vˆn−1

τ

+Hs,τ,vˆn−1

τ dτ+Fs,vsn−1ds, 0< t≤T ,

(3.5)

ˆ

vn_{(θ)=g(θ), θ∈[−h,0]. (3.6)}

It is obvious thatM=supt∈[0,T ]W (t)L(H)is finite and that ˆ

vn+1_(θ)−vˆn_{(θ)=0, θ∈[−h,0]. (3.7)} For 0≤t≤T, we have, by (A1)–(A4) and the strong continuity ofW (t)on[0,T ], _vˆn+1_(t)−vˆn_(t)

=

_t

0W (t−s) s

0

k(s,τ)Gτ,vˆn

τ+Hs,τ,vˆτndτ+Fs,vˆsnds

−

t

0W (t−s) s

0

k(s,τ)Gτ,vˆn−1 τ

+Hs,τ,vˆn−1 τ

dτ+Fs,vˆn−1 s ds ≤M t 0 s 0

_k(s,τ)G

τ,vˆn

τ−Gτ,vˆτn−1dτ ds

+

_s

0 _H

s,τ,vˆn

τ−Hs,τ,vˆτn−1dτ

+M

_t

0 _F

s,vˆn

s−Fs,vˆsn−1ds

≤M _t 0 s 0

k(s,τ)b1(τ)vˆτn−vˆτn−1+b2(s,τ)vˆτn−vˆτn−1dτ

ds

+M

t

0

_b3(s)vˆn

s −vˆsn−1ds

≤M

t

0

KL1+L2vˆτn−vˆτn−1s ds+M t

0L3 _vˆn

s −vˆsn−1ds

≤M(KL1+L2)1₂t2+ML3tvˆn−vˆn−1C([−h,T ];H)

=c1t+c2tvˆn−vˆn−1C([−h,T ];H),

(3.8)

wherec1=(1/2)M(KL1+L2)andc2=ML3. We now choose a sufficiently small con-stantt1>0 such that

L=c1t1+c2t1<1. (3.9) Then by (3.6), (3.8), and (3.9), we get

_vˆn+1_−vˆn

C([−h,T ];H)≤Lvˆn−vˆn−1C([−h,T ];H)

··· ≤Ln_vˆ1_−vˆ0

C([−h,T ];H).

(3.10)

This implies that{vˆn_}∞

n=0converges uniformly to some ˆv∈C([−h,0];H). Therefore, lim

n→∞_t∈sup_[0,t 1]

_vˆn

Hence, by lettingn→ ∞in (3.5), in view of (A1)–(A4) and (3.11), we get

ˆ

v(t)=ut;f ,g

+

t

0

W (t−s)s

0

k(s,τ)G(τ,vˆτ)

+H(s,τ,vˆτ)dτ

+F(s,vˆs)

ds, 0< t≤t1, ˆ

v(θ)=g(θ), θ∈[−h,0].

(3.12)

This shows the local existence of a solutionv(t)=v(t)ˆ |[0,t1]of (3.1) on[0,t1]. Letv1 andv2be the solution of (3.1) on[0,t1]. Then it is easy to see, similarly to the above, that

_vˆ1_−vˆ2

C([−h,t1];H)≤Lvˆ 1_−vˆ2

C([−h,t1];H), (3.13) so that byL <1,v1(t)=v2(t)on[0,t1]. This proves the uniqueness.

Sincek(s,τ)G(τ,vˆτ),H(s,τ,vˆn),F(s,vs)∈L2(0,t1;H)⊂L2(0,t1;V∗), by Theorem 2.1, we see that the solutionv(t)of (3.1) satisfies

dv(t)

dt =A0v(t)+A1v(t−h)+

0

−ha(s)A2v(t+s)ds

+

t

0

k(t,s)G(s,vs)+H(t,s,vs)ds

+F(t,vt)+f (t), a.e.t∈[0,t1],

v(θ)=g(θ), θ∈[−h,0],

(3.14)

and v∈L2_{(0,t1;V )∩W}1,2_(0,t1;V∗_{). In this sense, we call thisv a mild solution of} (1.5) on[0,t1]. We give a norm estimation of the mild solution of (1.5) and estab-lish the global existence of solutions with the aid of norm estimations. It is well known (cf. Lions and Magenes [3, Prop. 2.1, Thm. 3.1]) that the inclusionL2_{(0,T;V )∩}

W1,2_(0,T;V∗_{)⊂C([0,T ];H)is continuous, that is, there exists a constantc}

0such that

uC([0,T ];H)≤c0

uL2_{(0,T;V )}+du

dt

_L2(0,T;V∗) (3.15)

for allu∈L2_{(0,T;V )∩W}1,2_(0,T;V∗_).

Lemma3.1[5]. Leta(t),b(t), andc(t)be real valued nonnegative continuous func-tions defined onR+_{, for which the inequality}

c(t)≤c0+ t

0a(s)c(s)ds+ t

0a(s) s

0b(τ)c(τ)dτ

ds (3.16)

holds for allt∈R+_{, wherec0is a nonnegative constant. Then}

c(t)≤c0

1+

_t

0a(s)exp s

0

a(τ)+b(τ)dτ

ds

Theorem3.2. Assume that the conditions in Theorem 3.1 hold. Then for any solu-tionv(t)=v(t;f ,g)of (3.1) on[−h,T ], we have the estimate

_vt

·;f ,gC([0,T ];H)≤c

g(0)+gL2_{(−h,0;V )}+f_L2_(0,T;V∗₎

eKt_{, (3.18)} wherecis a positive constant which does not depend onv.

Proof. From hypotheses (A1)–(A4), we have _v

t+θ;f ,g

≤ut+θ;f ,g+

_t+θ

0 W (t+θ−s)

×

s

0

k(s,τ)G(τ,vτ)+H(s,τ,vτ)dτ+F(s,vs)

ds

≤u·;f ,gC([0,T ];H)

+M

t+θ 0

s

0

K|b1(τ)|vτ+b2(s,τ)vτdτ+b3(s)vs

ds.

(3.19) Hence, by (2.10) and (3.15),

_vt

·;f ,g= sup θ∈[−h,0]

_v

t+θ;f ,g

≤KTc0g(0)+gL2_{(−h,0;V )}+f_L2_(0,T;V∗₎

+

t

0c1 _vs

·;f ,gds+

t

0 s

0c2 _vτ

·;f ,gdτ ds

≤c_g(0)+g

L2_{(−h,0;V )}+f_L2_(0,T;V∗₎

+M

t

0 _vs

·;f ,gds+

t

0 s

0 _vτ

·;f ,gdτ ds

.

(3.20)

By using Lemma 3.1, we get _vt

·;f ,g_{C[0,T ];H}≤cg(0)+gL2_{(−h,0;V )}+f_L2_(0,T;V∗₎

×1+M

t

0exp s

0(M+1)dτ

ds

≤c_g(0)+g

L2_{(−h,0;V )}+f_L2_(0,T;V∗₎

×1+Mexp(M+1)Tt

≤cg(0)+gL2_{(−h,0;V )}+f_L2_(0,T;V∗₎

eKt

(3.21)

for some constantscandK. This completes the proof. By using Theorems 3.1, 3.2, we get the following theorem:

Theorem3.3. Assume that the conditions in Theorem 3.1 hold. Then there exists a unique solutionv(t)on[0,T ]of (3.1) which satisfies the estimate

_v

·;f ,gC([0,T ];H)≤c

g(0)+gL2_{(−h,0;V )}+f_L2_(0,T;V∗₎

Acknowledgement. This work was supported by KOSEF, 1996.

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