© Hindawi Publishing Corp. METHOD OF SEMIDISCRETIZATION IN TIME FOR QUASILINEAR INTEGRODIFFERENTIAL EQUATIONS

We consider a class of quasilinear integrodifferential equations 
in a reflexive Banach space. We apply the method of 
semidiscretization in time to establish the existence, uniqueness, 
and continuous dependence on the initial data of strong solutions.


Introduction.
Let X and Y be two real reflexive Banach spaces such that the embedding Y X is dense and continuous. We consider the following quasilinear evolution equation: We use the ideas and techniques of Zeidler [10] and the method of semidiscretization in time to establish existence, uniqueness, and continuous dependence on initial data of strong solutions to (1.1) on [0,T ] for some 0 < T ≤ T . For the study of particular cases of (1.1) in which f (t,u,v) ≡ 0 and f (t,u,v) ≡ f (t,u), we refer to Crandall and Souganidis [2], Kato [6], and references cited therein. The crucial assumption in these works is that there exists an open subset W of Y such that for each w ∈ W , A(w) generates a C 0 -semigroup in X, A(·) is locally Lipschitz continuous on W from X into itself, f , defined from W into Y , is bounded and globally Lipschitz continuous from Y into itself, and there exists an isometric isomorphism S : Y → X such that where B(w) is in the space B(X) of all bounded linear operators from X into itself. A generalization to the quasilinear evolution equation, 3)

CORE
Metadata, citation and similar papers at core.ac.uk has been considered by Katō [4] under similar conditions on A(t, w) and f (t,w) for (t, w) ∈ [0,T ]× W . The method of semidiscretization in time is developed and applied to linear as well as nonlinear evolution equations by Rektorys [9], Kartsatos and Zigler [3], Nečas [7], Bahuguna and Raghavendra [1], and others. This method consists in replacing the time derivatives in an evolution equation by the corresponding difference quotients giving rise to a system of time-independent operator equations. With the help of the theory of semigroups and the theory of monotone operators, these systems are guaranteed to have unique solutions. An approximate solution to the evolution equation is defined in terms of the solutions of these time-independent systems. After proving a priori estimates for the approximate solution, the convergence of the approximate solution to the unique solution of the evolution equation is established. In these works, either global Lipschitz conditions or local Lipschitz conditions with some growth conditions on nonlinear forcing terms have been assumed.
In this paper, we assume only local Lipschitz conditions on the nonlinear maps f and G. We first prove that the discrete points lie in a ball in X of fixed radius R, where R is independent of the discretization parameters. Then using the local Lipschitz continuity, we establish a priori estimates on the difference quotients. With the help of these a priori estimates, we prove the convergence of a sequence of approximate solutions defined in terms of the discrete points to a unique solution of the problem.

Preliminaries.
Let X and Y be as in Section 1. Let x Z denote the norm of an element x belonging to a Banach space Z. For a real β, N(Z, β) represents the set of all densely defined linear operators L in Z such that if λ > 0 and λβ < 1, then (I + λL) is one-to-one with a bounded inverse defined everywhere on Z and where I is the identity operator on Z. The Hille-Yosida (cf. Pazy [8]) theorem states that L ∈ N(Z, β) if and only if −L is the infinitesimal generator of a strongly continuous where z, z * is the value of z * ∈ Z * at z ∈ Z and F : Z → 2 Z * is the duality map given by Here, 2 Z * denotes the power set of Z * . If L ∈ N(Z, β), then (L+βI) is m-accretive in Z, that is, (L+βI) is accretive and the range R(L+λI) = Z for some λ > β. If Z * is uniformly convex, then F is single-valued and uniformly continuous on bounded subsets of Z.
(H) We assume in addition that the embedding Y X is compact and the dual X * is uniformly convex. Furthermore, we state the following hypotheses: (H1) there exist an open subset W of Y and β ≥ 0 such that u 0 ∈ W and , and positive constants µ P and γ P such that for all w, w 1 ,w 2 ∈ W , where λ G (r ) is a nonnegative nondecreasing function. In particular, we may take operator G as a Volterra operator defined by in which a is a real-valued continuous function defined on [0,T ] and k is defined on is a nonnegative nondecreasing function. Also, this map satisfies the Lipschitz condition where θ = β + P B(X) and V (φ) is the total variation of φ on [0,T ]. Let z 0 ∈ Y and let T 0 , 0 < T 0 ≤ T , be such that (2.14) We note that (2.14) implies that We have the following main result for the existence, uniqueness, and continuous dependence on the initial data of the strong solutions to (1.1).

16)
where C is a positive constant depending only on T 0 .

Basic lemmas.
We will state and prove several lemmas required to prove Theorem 2.1. A proof of Theorem 2.1 will be given in the next section.
Let w 0 = Su 0 . Let h = T 0 /n for all positive integers n ≥ N where N is a positive integer such that θ(T 0 /N) < 1/2. For n ≥ N, we set u n 0 = u 0 ,ũ n 0 =ũ 0 , and t n j = jh for j = 1, 2,...,n. We consider the scheme δu n j + A u n j−1 u n j = f t n j ,u n j−1 ,G ũ n j−1 t n j , j = 1, 2,...,n, where, for j = 1, 2,...,n, n ≥ N, The following result establishes the fact that u n j ∈ W R , j = 1, 2,...,n, n ≥ N. Applying S on both sides in (3.3) using (H3) and putting w n 1 = Su n 1 , we have The estimates in [2, Lemma 2] imply that Since hθ < 1/2, we have Therefore, in view of the estimate (2.15). Hence, u n 1 ∈ W R . Now suppose that u n i ∈ W R for i = 1, 2,...,j − 1. Again, [2, Lemma 2] implies that for 2 ≤ j ≤ n there exist unique u n j ∈ Y such that u n j + hA u n j−1 u n j = u n j−1 + hf t n j ,u n j−1 ,G ũ n j−1 t n j . (3.8) Proceeding as before and putting w n j = Su n j , we get the estimate Reiterating the above inequality, we get Using the fact that jh ≤ T 0 , we arrive at The above estimate and (3.3) and (3.8) give the required result. This completes the proof.  for some positive constant C 1 independent of j, h, and n. We note that (3.17) Using (3.17) in (3.16), we obtain where C 2 is another positive constant independent of j, h, and n. Reiterating inequality (3.18) and using (3.14), we get where V (φ) is the total variation of φ. Hence, Furthermore, we define a sequence of step functions {X n } from (−h, T 0 ] into Y given by u j , t ∈ t n j−1 ,t n j , j = 1, 2,...,n. (3.22) Remark 3.3. We observe that X n (t) ∈ W R for all t ∈ (−h, T 0 ] and n ≥ N. Also, X n (t) − U n (t) → 0 in X uniformly on J 0 as n → ∞ and {U n } are in Lip(J 0 ,X) with uniform Lipschitz constant C.
For notational convenience, we will denote f n (t) = f t n j ,u n j−1 ,G ũ n j−1 t n j , t ∈ t n j−1 ,t n j , j = 1, 2,...,n. (3.23) We note that Proof. Since {X n } is uniformly bounded in Y , the compact embedding of Y implies that there exist a subsequence {X m } of {X n } and a function u : The uniform continuity of {U n } on J 0 implies that {X m } is an equicontinuous family in C(J 0 ,X) and the strong convergence of U m (t) to u(t) in X implies that {U m } is relatively compact in X. We use the Ascoli-Arzelá theorem to conclude that U m → u in C(J 0 ,X) as m → ∞. Since U m are in Lip(J 0 ,X) with uniform Lipschitz constant, u ∈ Lip(J 0 ,X). This completes the proof of the lemma.

Proof of Theorem 2.1. First we show that A(X m (t − h))X m (t) A(u(t))u(t) in
X as m → ∞, where " " denotes the weak convergence in X, (4.1) Now, as m → ∞, since X m (t) → u(t) in X uniformly on J 0 . Since A(u(t)) ∈ N(X, β), βI +A(u) is m-accretive in X. We use [5, Lemma 2.5] and the fact that to assert that A(u(t))X m (t) A(u(t))u(t) in X, and hence A(X m (t − h))X m (t) A(u(t))u(t) in X as m → ∞. To show that A(u(t))u(t) is weakly continuous on J 0 , let {t k } ⊂ J 0 be a sequence such that t k → t as k → ∞. Then u(t k ) → u(t) in X as k → ∞ and we may follow the same arguments as above to prove that A(u(t k ))u(t k ) A(u(t))u(t) in X as k → ∞. The Bochner integrability of A(u(t))u(t) can be established in the similar way as in Kato [5,Lemma 4.6]. Now from (3.24), for each x * ∈ X * , we have Letting m → ∞ using bounded convergence theorem and Lemma 3.4, we get The continuity of the integrand implies that u(t), x * is continuously differentiable on J 0 . The Bochner integrability of A(u(t))u(t) implies that the strong derivative of u(t) exists a.e. on J 0 and Since u(0) = u 0 , u is a strong solution to (1.1). Now, we establish the uniqueness and the continuous dependence on the initial data of a strong solution to (1.1).

Uniqueness.
Let v be another strong solution to (
Integrating the above inequality on (0,t) and taking the supremum, we get 1 2 U 2 C(J t ,X) ≤ C R t 0 U 2 C(Js ,X) ds. (4.9) Applying Gronwall's inequality, we get U ≡ 0 on J 0 .

Continuous dependence.
Let v 0 ∈ B Y (u 0 ,R 0 ). Then Hence, 1 + e 2θT Sv 0 − z 0 + T 0 γ A z 0 Y + γ P z 0 X + M ≤ 3 1 + e 2θT R 0 = R. We may proceed as before to prove the existence of v n j ∈ W R satisfying scheme (1.1) with u n j and u 0 replaced by v n j and v 0 , respectively. Convergence of v n j to v(t) can be proved in a similar manner. Let U = u − v. Then following the steps used to prove the uniqueness, we have for a.e. t ∈ J 0 , 1 2 d dt U(t) 2 X ≤ C R U 2 C(J t ,X) . (4.12) Integrating the above inequality on (0,t) and taking the supremum, we get Applying Gronwall's inequality, we get U C(J t ,X) ≤ C U(0) X , (4.14) where C is a positive constant. This proves the required result. This completes the proof of Theorem 2.1.