NONLINEAR FUNCTIONAL INTEGRODIFFERENTIAL EQUATIONS IN HILBERT SPACE

Let X be a Hilbert space and let Ω ⊂ 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 in X by using the fundamental solution.


Introduction.
Let X be a Hilbert space and let Ω ⊂ R n be a bounded domain with smooth boundary ∂Ω.We consider the following parabolic partial functional integrodifferential equation.
where Ꮽ i (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, and k is a kernel.The boundary condition attached to (1.1) is, e.g., given by the Dirichlet boundary condition and the initial condition is given by From [4], the above mixed problems (1.1), (1.2), and (1.3) can be formulated abstractly as where the state u(x) of the system (1.5) lies in an appropriate Hilbert space and A i (i = 0, 1, 2) are unbounded operators associated with Ꮽ i (i = 0, 1, 2), respectively.Next, we explain the notation u t in (1.5).
Many authors [2,8] studied the following delay differential equation: 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.

Preliminaries.
Let H be a pivot complex Hilbert space and V be a complex Hilbert space such that V is dense in H and the inclusion map i : V → H is continuous.The norms of H, V , and the inner product of H are denoted by | • |, • , and •, • , respectively.Identifying the antidual of H with H, we may consider that V ⊂ H ⊂ V * .The norm of the dual space V * is denoted by • * .We consider the following linear functional differential equation on the Hilbert space H.
Let a(u, v) be a bounded sesquilinear form defined in V × V satisfying Gårding's inequality where c 0 > 0 and c 1 ≥ 0 are real constants.Let A 0 be the operator associated with this sesquilinear form where •, • denotes the duality pairing between V and V * .The operator A 0 is bounded linear from V into V * .The realization of A 0 in H, which is the restriction of A 0 to the domain D(A 0 ) = {u ∈ V : A 0 u ∈ H}, is also denoted by A 0 .It is proved in Tanabe [6] that A 0 generates an analytic semigroup e tA 0 = T (t) both in H and V * and that T (t) : endowed with the graph norm of A 0 to H continuously.The real valued scalar function a(s) is assumed to be Hölder continuous on [−h, 0].We introduce a Stieltjes measure η given by where χ (−∞,−h] denotes the characteristic function of (−∞, −h].Then the delay term in (2.1) is written simply as 0 −h dη(s)u(t +s).The fundamental solution W (t) of (2.1) is defined as a unique solution of and W (t) is constructed by Tanabe [7] under the Hölder continuity of a(s).

Existence and uniqueness of functional integro-differential equations.
Using the fundamental solution W (t) in Section 2, we consider the following abstract functional integral equation.
We list the following hypotheses.
(A 1 ) The nonlinear functions G : The function k(t, s) is Hölder continuous with exponent α, i.e., there exists a positive constant a such that

12). Assume that the hypotheses (A 1 )-(A 4 ) hold. Then there exists a time t 1 > 0 such that the functional integral equation (3.1) admits a unique solution v(t) on [0,t 1 ].
Proof.We prove this theorem by using the method of successive approximations.Set v 0 (t) = u(t; f ,g), t ≥ 0. Let v0 (t) be the extension of v 0 (t) on [−h, T ] by (2.13).
Proof.From hypotheses (A 1 )-(A 4 ), we have for some constants c and K.
and the kernel k : ∆ T → R (R denotes the set of real numbers) are continuous.(A 2 ) Let b 1 ,b 3 : [0,T ] → R, b 2 : ∆ T → R + be continuous functions such that