Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 4, 1-12; http://www.math.u-szeged.hu/ejqtde/

EXISTENCE AND UNIQUENESS OF A SOLUTION TO A PARTIAL INTEGRO-DIFFERENTIAL EQUATION BY THE METHOD OF LINES

D. BAHUGUNA & J. DABAS DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF TECHNOLOGY, KANPUR - 208 016, INDIA.

Abstract. In this work we consider a partial integro-differential equation. We reformulate it a functional integro-differential equation in a suitable Hilbert space. We apply the method of lines to establish the existence and uniqueness of a strong solution.

1. Introduction

In the present analysis we are concerned with the following partial integrodifferential equation,

\{ \begin{matrix}\frac{∂^{2}w(x,t)}{∂t^{2}} = \frac{∂^{2}w(x,t)}{∂x^{2}} + q(x)\frac{∂w(x,t)}{∂t} + f_{1}(x, t, w(x, t), \frac{∂w(x,t)}{∂t}) \\ + ∫_{0}^{t} k(t, s) [\frac{∂^{2}w(x,s)}{∂x^{2}} + f_{2}(x, s, w(x, s), \frac{∂w(x,s)}{∂s})] ds, \\ (x, t) ∈ (0, 1) × (0, T], 0 < T < ∞, \\ w(x, 0) = g_{1}(x), \frac{∂w}{∂t} (x, 0) = g_{2}(x), x ∈ (0, 1), \\ w(0, t) = 0 = w(1, t), t ∈ [0, T], \end{matrix}  (1.1)

where the unknown function w : [0, 1] × [0, T] → ℂ, the coefficient function of the damping term \frac{∂w}{∂t}, q : [0, 1] → ℂ satisfies certain integrability conditions, stated later, f_{i} : (0, 1) × [0, T] × ℂ^{2} → ℂ, g_{i} : (0, 1) → ℂ, i = 1, 2; and k : [0, T]^{2} → ℂ are given functions satisfying certain required conditions.

In the next section we transform (1.1) as a Cauchy problem for the following functional integro-differential equation in the product Hilbert space H := H_{0}^{1}(0, 1)× L^{2}(0, 1),

\{ \begin{matrix}\frac{du}{dt} − Au = ∫_{0}^{t} k(t, s)Au(s)ds + F (t, u_{t}), \\ u_{0} = ϕ, \end{matrix}  (1.2)

where A : D(A) ⊂ H → H is shown to be the infinitesimal generator of a contraction semigroup in H and the nonlinear function F : [0, T] × C_{0} → H. Here the space C_{t} := C([−T, t]; H), t ∈ [0, T], is the Banach space of all continuous functions from [−T, t] into H endowed with the norm

∥χ∥_{C_{t}} := sup_{−T ≤s≤t} ∥χ(s)∥_{H}, χ ∈ C_{t},

EJQTDE, 2008 No. 4, p. 1 ∥.∥_{H} being the norm in H given by

2000 Mathematics Subject Classification. 34K30, 34G20, 47H06.

Key words and phrases. method of lines, integro-differential equation, semigroups, contractions, strong solution.

∥u∥_{H}^{2} := ∥u_{1}∥_{H_{0}^{1}}^{2} + ∥u_{2}∥_{L^{2}}^{2}, u = (u_{1}, u_{2}) ∈ H_{0}^{1}(0, 1) × L^{2}(0, 1),
∥u_{1}∥_{H_{0}^{1}}^{2} = ∫_{0}^{1} [|u′(x)|^{2} + |u(x)|^{2}] dx,

∥u_{2}∥_{L^{2}}^{2} = ∫_{0}^{1} |u(x)|^{2}dx.

The space C_{0} is called the “history space” or the phase space. We also show that F verifies a Lipschitz condition under certain assumptions on the functions f_{1} and f_{2}.

Our aim is to apply Rothe’s method to (1.2) involving delays to establish the existence and uniqueness of a strong solution which in turn will guarantee the wellposedness of (1.1). The delay differential equations arise in the study of various population dynamical models [9]. The method of lines is a powerful tool for proving the existence and uniqueness of solutions to evolution equations. This method is oriented towards the numerical approximations. For instance, we refer to Rektorys [11] for a rich illustration of the method applied to various interesting physical problems. Until today, the application of method of lines includes only those nonlinear differential and Volterra integro-differential equations (VIDEs) in which bounded, though nonlinear, operators appear inside the integrals, see Kacur [6, 7], Rektorys [11], Bahuguna and Raghavendra [2] and Bahuguna, Pani and Raghavendra [5]. In the present study we extend the application of the method of lines to a class of nonlinear VIDEs in which differential operators occur inside the integrals and hence are unbounded. Motivation for considering such problems arises from the theory of wave propagation under the influence of damping, see Bahuguna [1], and Bahuguna and Shukla [4] and references cited therein.

2. Reformulation and Main Result

In order to reformulate (1.1) as (1.2) we choose the following settings. We consider the complex Hilbert space L^{2}(0, 1) of all square integrable complex-valued functions on (0, 1) with the norm ∥.∥_{L^{2}} introduced earlier. We shall also be dealing with the Sobolev spaces H^{d}(0, 1) and H_{0}^{d}(0, 1) for d = 1, 2, ⋯ (cf. Pazy [10] or Engel and Nagel [8] for definitions and details).

We assume that q : [0, 1] → ℂ is measurable and satisfies the following conditions. (C1) There exist constants γ ≥ 0 and δ > 0 such that

|Im q(x)| ≤ γ Re q(x), Re q(x) ≤ −δ, a.e. x ∈ [0, 1],

where Re q(x) and Im q(x) denote the real and imaginary parts of q(x).

(C2) For every 0 < ϵ < 1/2, q ∈ L^{2}[ϵ, 1 − ϵ] and the map x ↦ x(1 − x)q^{2}(x) is in L^{1}[0, 1].

We define

D(P) = {u ∈ L^{2}(0, 1) : q(.)u(.) ∈ L^{2}(0, 1)}, (P u)(x) = q(x)u(x),  (2.1)
D(A_{0}) = H^{2}(0, 1) × (D(P) ∩ H_{0}^{1}(0, 1)), A_{0} = ( \begin{matrix}0 & I \\ \frac{∂^{2}}{∂x^{2}} & P \end{matrix} ) .  (2.2)

Under the conditions (C1) and (C2) on q, it follows that the closure A, of A_{0} is given by

D(A) = H_{0}^{2}(0, 1) × H_{0}^{1}(0, 1),  (2.3)
A = ( \begin{matrix}0 & I \\ −C^{∗}C & Q \end{matrix} ) ,  (2.4)

where Q = \overline{C^{∗}^{−1}P C^{−1}}, is a bounded linear operator on L^{2}(0, 1), is the infinitesimal generator of a C_{0}-semigroup of contractions in H (cf. Engel and Nagel [8], page 381). Here, for any operator L, \overline{L} represents the closure of L.

With these operators introduced and for g_{1} ∈ H_{0}^{1}(0, 1) and g_{2} ∈ L^{2}(0, 1), we rewrite (1.1) as

\{ \begin{matrix}\frac{du}{dt} − Au = F_{1}(t, u(t)) + ∫_{0}^{t} k(t, s)[Bu(s) + F_{2}(s, u(s))]ds, \\ u(0) = (g_{1}, g_{2}), \end{matrix}  (2.5)

where

D(B) = D(A),  (2.6)
B = ( \begin{matrix}0 & 0 \\ −C^{∗}C & 0 \end{matrix} ) ,  (2.7)

and, for i = 1, 2, F_{i} : [0, T] × H → H are given by

F_{i}(t, (u_{1}, u_{2}))(x) = (0, f_{i}(x, t, u_{1}(x, t), u_{2}(x, t))), u = (u_{1}, u_{2}) ∈ H.

We put t − s = −η in the integral term in (2.5) to obtain

∫_{0}^{t} k(t, s)F_{2}(s, u(s))ds = ∫_{−t}^{0} k(t, t + η)F_{2}(t + η, u(t + η))dη  (2.8)
= ∫_{−t}^{0} k(t, t + η)F_{2}(t + η, u_{t}(η))dη.  (2.9)

Thus, if for any u ∈ C_{T} := C([0, T]; H) and t ∈ [0, T], we let u_{t} ∈ C_{0} := C([−T, 0]; H) given by u_{t}(η) = u(t + η), η ∈ [−t, 0] and u_{t}(η) = u(0) for η ∈ [−T, −t], we may rewrite (2.5) as (1.2) where F : [0, T] × C_{0} → H, given by

F (t, χ) = F_{1}(t, χ(0)(t)) + ∫_{−t}^{0} k(t, t + η)[(B − A)χ(η)(s) + F_{2}(s, χ(η)(s))]ds, χ ∈ C_{0},

and ϕ ∈ C_{0} is given by

ϕ(η) ≡ u(0) = (g_{1}, g_{2}), η ∈ [−T, 0].

We list here the properties of the linear operator A, the nonlinear map F and the kernel k.

(P1) The operator A : D(A) ⊂ H → H is the infinitesimal generator of a C_{0} semigroup S(t) of contractions in H.

(P2) The function F : [0, T]×C_{0} → H satisfies the Lipschitz condition, i.e., there exists a positive constant L_{F} such that

∥F (t_{1}, χ_{1}) − F (t_{2}, χ_{2})∥_{H} ≤ L_{F} [|t_{1} − t_{2}| + ∥χ_{1} − χ_{2}∥_{C_{0}}],

for (t_{i}, χ_{i}) ∈ [0, T] × C_{0}, i = 1, 2.

EJQTDE, 2008 No. 4, p. 3

(P3) The function k : [0, T]^{2} → ℝ is continuous and there exists a positive constant L_{k} such that

|k(t_{1}, s) − k(t_{2}, s)| ≤ L_{k}|t_{1} − t_{2}|, t_{1}, t_{2} ∈ [0, T].

We have the following main result.

Theorem 2.1. Suppose that (P1)-(P3) are satisfied and ϕ ∈ C_{0} is Lipschitz continuous. Then there exists a unique u ∈ C_{T}, with u_{0} = ϕ, u(t) ∈ D(A), a.e. t ∈ [0, T], u(t) is Lipschitz continuous on [0, T] and satisfies (1.2) a.e. on [0, T].

3. Discretization Scheme and A Priori Estimates

To apply Rothe’s method, We use the following procedure. For any positive integer n we consider a partition t_{j}^{n} defined by t_{j}^{n} = jh; h = T/n, j = 0, 1, 2, …, n. Set u_{0}^{n} = ϕ(0) for all n ∈ ℕ. For j = 1, 2, …, n, we define u_{j}^{n} ∈ D(A) the unique solutions of each of the equations

\frac{u_{j}^{n} − u_{j−1}^{n}}{h} − Au_{j}^{n} = F_{j}^{n} + h ∑_{i=0}^{j−1} k_{ji}^{n} Au_{i}^{n},  (3.1)

Where F_{j}^{n} = F (t_{j}^{n}, ũ_{j−1}^{n}) and k_{ji}^{n} = k(t_{j}^{n}, t_{i}^{n}) 1 ≤ i ≤ j ≤ n. and ũ_{0}^{n}(t) = ϕ(t) for t ∈ [−T, 0], ũ_{0}^{n}(t) = ϕ(0) for t ∈ [0, T] and for 2 ≤ j ≤ n,

ũ_{j}^{n}(θ) = \begin{cases}ϕ(t_{j}^{n} + θ), θ ≤ −t_{j}^{n}, \\ u_{i−1}^{n} + (θ − t_{j+1−i}^{n})δu_{i}^{n}, θ ≥ −t_{j}^{n}, θ ∈ [−t_{j+1−i}^{n}, −t_{j−i}^{n}], 1 ≤ i ≤ j. \end{cases}

Now, the existence of a unique u_{j}^{n} ∈ D(A) satisfying (3.1) follows from the mdissipativity of A and by Theorems 1.4.2 and 1.4.3 in Pazy [10]. In order to ensure the existence of a unique solution u_{j}^{n} ∈ D(A) of (3.1) we rewrite it as

u_{j}^{n} = (I − hA)^{−1} [u_{j−1}^{n} + hF_{j}^{n} + h^{2} ∑_{i=0}^{j−1} k_{ji}^{n}Au_{i}^{n}] ,

as (I − hA)^{−1} exists for all h > 0. The existence of unique u_{j}^{n} ∈ D(A) satisfying (3.1) is ensured.

Definition 3.1. We define the Rothe sequence {U^{n}} ⊂ C([−T, T]; H) given by

U^{n}(t) = \begin{cases}ϕ(t), t ∈ [−T, 0] \\ u_{j−1}^{n} + (t − t_{j−1}^{n})δu_{j}^{n}, t ∈ [t_{j−1}^{n}, t_{j}^{n}], j = 1, 2, …, n. \end{cases}

and a sequence {X^{n}} of step functions from [−T, T] into H given by

X^{n}(t) = ϕ(t), t ∈ [−T, 0] X^{n}(t) = u_{j}^{n}, t ∈ (t_{j−1}^{n}, t_{j}^{n}], j = 1, 2, …, n.

We prove the convergence of the sequence {U^{n}} to the unique solution of the problem as n → ∞ using some a priori estimates on {U^{n}}. For convenience, we shall denote by C a generic constant, i.e., KC, e^{KC}, etc., will be replace by C where K is a positive constant independent of j, h and n.

We shall use later the following lemma due to Sloan and Thomee [12]. EJQTDE, 2008 No. 4, p. 4 Lemma 3.2. Let {w_{n}} be a sequence of nonnegative real numbers satisfying

w_{n} ≤ α_{n} + ∑_{i=0}^{n−1} β_{i}w_{i}, n > 0,

where {α_{n}} is a nondecreasing sequence of nonnegative real numbers and β_{n} ≥ 0. Then

w_{n} ≤ α_{n} exp{∑_{i=0}^{n−1} β_{i}}, n > 0.

Furthermore, we also require the following lemma for later use.

Lemma 3.3. Let C > 0, h > 0 and let {α_{j}}_{j=1}^{n} be a sequence of nonnegative real numbers satisfying

α_{j} ≤ (1 + Ch)α_{j−1} + Ch^{2} ∑_{i=1}^{j−1} α_{i} + Ch, 2 ≤ j ≤ n.  (3.2)

Then

α_{j} ≤ (1 + Ch)^{j}[α_{1} + jCh^{2} ∑_{i=1}^{j−1} α_{i} + jCh], 2 ≤ j ≤ n.

Proof. From (3.2)

α_{j−1} ≤ (1 + Ch)α_{j−2} + Ch^{2} ∑_{p=1}^{j−2} α_{p} + Ch

≤ (1 + Ch)α_{j−2} + Ch^{2} ∑_{p=1}^{j−1} α_{p} + Ch.  (3.3)

Putting in (3.2)

α_{j} ≤ (1 + Ch)^{2}α_{j−2} + Ch^{2}[1 + (1 + Ch)] ∑_{p=1}^{j−1} α_{p} + Ch[1 + (1 + Ch)].  (3.4)

By repeating the above process

α_{j} ≤ (1 + Ch)^{(j−1)}α_{1} + Ch^{2}[1 + (1 + Ch) + ⋯ + (1 + Ch)^{(j−1)}] ∑_{p=1}^{j−1} α_{p}
+ Ch[1 + (1 + Ch) + ⋯ + (1 + Ch)^{(j−1)}]
≤ (1 + Ch)^{j}[α_{1} + jCh^{2} ∑_{p=1}^{j−1} α_{p} + jCh].  (3.5)

This completes the proof of the lemma.

□

Lemma 3.4. There exists a constant C independent of j, h and n such that

∥δu_{j}^{n}∥_{H} ≤ C.

EJQTDE, 2008 No. 4, p. 5 Proof. From (3.1) for j = 1 we get

δu_{1}^{n} − hAδu_{1}^{n} = Au_{0} + F_{1}^{n} + hk_{10}^{n}Au_{0}.

Theorem 1.4.2 [10] implies that

∥δu_{1}^{n}∥_{H} ≤ ∥Au_{0} + F_{1}^{n} + hk_{10}^{n}Au_{0}∥_{H} ≤ C.

Hence ∥Au_{1}^{n}∥_{H} ≤ C. Let 2 ≤ j ≤ n. Subtracting (3.1) for j − 1 from (3.1) for j, we get

δu_{j}^{n} − hAδu_{j}^{n} = δu_{j−1}^{n} + F_{j}^{n} − F_{j−1}^{n} + hk_{jj−1}^{n}Au_{j−1}^{n} + ∑_{i=0}^{j−2}[k_{ji}^{n} − k_{j−1i}^{n}]Au_{i}^{n}.

Applying Theorem 1.4.2 of [10] again, we get

∥δu_{j}^{n}∥_{H} ≤ ∥δu_{j−1}^{n}∥_{H} + ∥F_{j}^{n} − F_{j−1}^{n}∥_{H} + h|k_{jj−1}^{n}|∥Au_{j−1}^{n}∥_{H}
+ h ∑_{i=0}^{j−2} |k_{ji}^{n} − k_{j−1i}^{n}|∥Au_{i}^{n}∥_{H}.  (3.6)

Now using Lipschitz continuity of the function F

∥F_{j}^{n} − F_{j−1}^{n}∥_{H} = ∥F (t_{j}^{n}, ũ_{j−1}^{n}) − F (t_{j−1}^{n}, ũ_{j−2}^{n})∥_{H}
≤ L_{F} (|t_{j}^{n} − t_{j−1}^{n}| + ∥ũ_{j−1}^{n} − ũ_{j−2}^{n}∥_{C_{0}})
≤ Ch(1 + max_{1≤p≤j−1} ∥δu_{p}^{n}∥_{H}).  (3.7)

Then (3.6) becomes

∥δu_{j}^{n}∥_{H} ≤ ∥δu_{j−1}^{n}∥_{H} + Ch(1 + max_{1≤p≤j−1} ∥δu_{p}^{n}∥_{H}) + Ch^{2} ∑_{i=0}^{j−1} ∥Au_{i}^{n}∥_{H}

≤ (1 + Ch) max_{1≤p≤j−1} ∥δu_{p}^{n}∥_{H} + Ch^{2} ∑_{i=0}^{j−1} ∥Au_{i}^{n}∥_{H} + Ch.  (3.8)

From (3.1), for 2 ≤ i ≤ j, we have

∥Au_{i}^{n}∥_{H} ≤ ∥δu_{i}^{n}∥_{H} + ∥F_{i}^{n}∥_{H} + Ch ∑_{p=0}^{i−1} ∥Au_{p}^{n}∥_{H}.  (3.9)

Again using Lipchitz continuity of F

∥F_{i}^{n}∥_{H} = ∥F (t_{i}^{n}, ũ_{i−1}^{n}) − F (t_{i−1}^{n}, 0)∥_{H} + ∥F (t_{i−1}^{n}, 0)∥_{H}
≤ L_{F} (|t_{i}^{n} − t_{i−1}^{n}| + ∥ũ_{i−1}^{n}∥_{C_{0}}) + max_{0≤t≤T} ∥F (t, 0)∥_{H}

≤ C(h + h ∑_{p=1}^{i−1} ∥δũ_{p}^{n}∥_{C_{0}} + ∥ũ_{0}∥_{C_{0}}) + max_{0≤t≤T} ∥F (t, 0)∥_{H}. (3.10)

Then (3.9) becomes

∥Au_{i}^{n}∥_{H} ≤ C(1 + max_{1≤p≤i} ∥δu_{p}^{n}∥_{H}) + Ch + Ch ∑_{p=1}^{i−1} ∥Au_{p}^{n}∥_{H}.  (3.11)

Applying Lemma 3.2 in (3.11), we get

∥Au_{i}^{n}∥_{H} ≤ Ce^{CT} (1 + max_{1≤p≤i} ∥δu_{p}^{n}∥_{H}).  (3.12)

Using (3.12) in (3.8), we have

max_{1≤p≤j} ∥δu_{p}^{n}∥_{H} ≤ (1 + Ch) max_{1≤p≤j−1} ∥δu_{p}^{n}∥_{H} + Ch^{2} ∑_{p=1}^{j−1} max_{1≤p≤i} ∥δu_{p}^{n}∥_{H} + Ch. (3.13)

To use Lemma 3.3 in (3.13), we take α_{j} = max_{1≤p≤j} ∥δu_{p}^{n}∥_{H} and the fact that (1 + Ch)^{j} ≤ e^{CT}, 2 ≤ j ≤ l and α_{1} ≤ C to get the estimate

max_{1≤p≤j} ∥δu_{p}^{n}∥_{H} ≤ C + Ch ∑_{p=1}^{j−1} max_{1≤p≤j−1} ∥δu_{p}^{n}∥_{H}.  (3.14)

Again we apply Lemma 3.2 to get the required estimate. This completes the proof of the lemma. □

Remark 3.5. Each of the functions {U^{n}} is Lipschitz continuous with uniform Lipschitz constant, i.e.,

∥U^{n}(t) − U^{n}(s)∥_{H} ≤ C|t − s|, t, s ∈ [0, T].

Furthermore,

∥U^{n}(t) − X^{n}(t)∥_{H} ≤ \frac{C}{n} .

Definition 3.6. We define the sequence {F^{n}} and {K^{n}} of step functions [0, T] into H by

K^{n}(0) = 0, K^{n}(t) = h ∑_{i=0}^{j−1} k_{ji}^{n}Au_{i}^{n}, t ∈ (t_{j−1}^{n}, t_{j}^{n}],  (3.15)
F^{n}(0) = F (0, ϕ), F^{n}(t) = F (t_{j}^{n}, ũ_{j−1}^{n}), t ∈ (t_{j−1}^{n}, t_{j}^{n}].  (3.16)

Lemma 3.7. Under the given assumptions we have (a) {K^{n}(t)} is uniformly bounded;

(b) ∫_{0}^{t} AX^{n}(s)ds = u_{0} − U^{n}(t) − ∫_{0}^{t} K^{n}(s)ds − ∫_{0}^{t} F^{n}(s);

(c) \frac{d^{−}}{dt} U^{n}(t) − AX^{n}(t) = K^{n}(t) + F^{n}(t), t ∈ (0, T], where \frac{d^{−}}{dt} is the left-derivative.

Proof. (a) This is a direct consequence of the estimates in (3.12) and (3.14). EJQTDE, 2008 No. 4, p. 7 (b) For 2 ≤ j ≤ n and t ∈ (t_{j−1}^{n}, t_{j}^{n}], by Definition 3.1, we have

∫_{0}^{t} AX^{n}(s)ds = ∑_{i=1}^{j−1} ∫_{t_{i−1}^{n}}^{t_{i}^{n}} AX^{n}(s)ds + ∫_{t_{j−1}^{n}}^{t} AX^{n}(s)ds

= ∑_{i=1}^{j−1}(u_{i}^{n} − u_{i−1}^{n}) + \frac{1}{h}(t − t_{j−1}^{n})(u_{j}^{n} − u_{j−1}^{n})

− h ∑_{i=1}^{j−1} [h ∑_{p=0}^{i−1} k_{ip}^{n}Au_{p}^{n}] − (t − t_{j−1}^{n})[h ∑_{p=0}^{j−1} k_{ip}^{n}Au_{p}]

− h ∑_{i=0}^{j−1} F_{i}^{n} − (t − t_{j−1}^{n})F_{j}^{n}

= u_{0} − U^{n}(t) − ∫_{0}^{t} K^{n}(s)ds − ∫_{0}^{t} F^{n}(s)ds

When j = 1, t ∈ (0, t_{1}^{n}]. we have

∫_{0}^{t} AX^{n}(s)ds = tAu_{1}^{n}
= \frac{t}{h}(u_{1}^{n} − u_{0}^{n}) − thk_{10}^{n}Au_{0} − tF_{1}^{n}
= u_{0} − U^{n}(t) − ∫_{0}^{t} K^{n}(s)ds − ∫_{0}^{t} F^{n}(s)ds.

(c) for t ∈ (t_{j−1}^{n}, t_{j}^{n}],

AX^{n}(t) = Au_{j}^{n} and \frac{d^{−}u^{n}}{dt} (t) = \frac{1}{h}(u_{j}^{n} − u_{j−1}^{n}).

Therefore,

\frac{d^{−}u^{n}}{dt} (t) − AX^{n}(t) = \frac{1}{h}(u_{j}^{n} − u_{j−1}^{n}) − Au_{j}^{n}
= h ∑_{i=0}^{j−1} k_{ji}^{n}Au_{i}^{n} + F_{j}^{n}
= K^{n}(t) + F^{n}(t).

This completes the proof of the lemma. □

In the next lemma we prove the local uniform convergence of the Rothe sequence.

Lemma 3.8. There exist a subsequence {U^{n_{p}}} of {U^{n}} and a function u : [0, T] → D(A) such that U^{n_{p}} → u in C([0, T]; H), and AU^{n_{p}}(t) ⇀ Au(t) uniformly in H as p → ∞, where ⇀ denotes the weak convergence in H. Furthermore, Au(t) is weakly continuous on [0, T].

Proof. Since {U^{n}(t)} and {AX^{n}(t)} are uniformly bounded in the Hilbert space H, there exist weakly convergent subsequences {U^{n_{p}}(t)} and {AX^{n_{p}}(t)} (we take the same indices without loss of generality otherwise we first take the subsequence

EJQTDE, 2008 No. 4, p. 8 {U^{n_{p}}(t)} of {U^{n}(t)} and then take the subsequence {U^{n_{pn}} (t)} and {AX^{n_{pn}} (t)} of {U^{n_{p}}(t)} and {AX^{n_{p}}(t)}, respectively). Thus, there exist functions u, w : [0, T] → H such that U^{n_{p}}(t) ⇀ u(t) and AX^{n_{p}}(t) ⇀ w(t) as p → ∞. Also, we have X^{n_{p}}(t) ⇀ u(t) as p → ∞. Clearly, {X^{n_{p}}(t)} and {AX^{n_{p}}(t)} are uniformly bounded and X^{n_{p}}(t) ⇀ u(t) and AX^{n_{p}}(t) ⇀ w(t) as p → ∞. Since D(A) = H_{0}^{2}(0, 1) × H_{0}^{1}(0, 1) is compactly embedded in H = H_{0}^{1}(0, 1) × L^{2}(0, 1), it follows that (I + A)^{−1} : H → H is compact. The boundedness of V^{n} = (I + A)U^{n} and the compactness of (I + A)^{−1} imply that U^{n} = (I + A)^{−1}V^{n} has a convergent subsequence. For convenience, we again denote this convergent subsequence by {U^{n_{p}}}. Thus, U^{n_{p}}(t) → u(t) as p → ∞. Also, X^{n_{p}}(t) → u(t) as p → ∞. By the maximal dissipativity of A, it follows that u(t) ∈ D(A) and AX^{n_{p}}(t) ⇀ Au(t). Since U^{n_{p}} is Lipschitz continuous with uniform Lipschitz constant, it follows that {U^{n_{p}}} is equi-continuous in C([0, T]; H) and {U^{n_{p}}(t)} is relatively compact in H. Hence by Ascoli-Arzela theorem, U^{n_{p}} → u as p → ∞ in C([0, T]; H).

To show the weak continuity of Au(t) in t, let {t_{p}} ⊂ [0, T] such that t_{p} → t as p → ∞, t ∈ [0, T]. Then u(t_{p}) → u(t) and since ∥Au(t_{p})∥_{H} ≤ C, there exists a subsequence {Au(t_{k_{p}})} ⊂ {Au(t_{p})} such that Au(t_{p_{m}}) ⇀ z(t) as m → ∞. Since u(t_{p_{m}}) → u(t) and Au(t_{p_{m}}) ⇀ z(t) as m → ∞, it follows as above that u(t) ∈ D(A) and Au(t) = z(t). Hence Au(t) is weakly continuous. This completes the proof of the lemma. □

Lemma 3.9. Au(t) is Bochner integrable on [0, T].

For a proof of this lemma we refer to Bahuguna and Raghavendra [2].

Lemma 3.10. Let {K^{n}(t)} be the sequence of functions defined by (3.15) and

K(ϕ)(t) = ∫_{0}^{t} k(t, s)ϕ(s)ds,

where ϕ : [0, T] → H is Bochner integrable. We have

K^{n_{p}}(t) ⇀ K(Au)(t),

uniformly on [0, T] as p → ∞.

Proof. We first show that K^{n_{p}}(t)−K(AX^{n_{p}})(t) → 0 uniformly on [0, T] as p → ∞. For t ∈ (t_{j−1}^{n_{p}}, t_{j}^{n_{p}}], we have

K^{n_{p}}(t) − K(AX^{n_{p}})(t) = h ∑_{i=0}^{j−1} k_{ji}^{n_{p}}Au_{i}^{n_{p}} − ∫_{0}^{t} k(t, s)AX^{n_{p}}(s) ds

= ∑_{i=1}^{j−1} [∫_{t_{i−1}^{np}}^{t_{i}^{np}} [k_{ji}^{n_{p}} − k(t, s)] ds] Au_{i}^{n_{p}}

+hk(t_{j}^{n_{p}}, t_{0}^{n_{p}})Au_{0}^{n_{p}} − [ ∫_{t_{j−1}^{np}}^{t} k(t, s) ds]Au_{j}^{n_{p}}.

Since ∥Au_{j}^{n_{p}}∥_{H} ≤ C, and k : [0, T] → ℝ Lipschitz continuous imply that the last two terms on the right hand side tend to zero strongly and uniformly on [0, T] as

EJQTDE, 2008 No. 4, p. 9 p → ∞. we have

∥K^{n_{p}}(t) − K(AX^{n_{p}})(t)∥_{H} ≤ C[∑_{i=0}^{j−2} ∫_{t_{i}^{np}}^{t_{i+1}^{np}} |k_{ji}^{n_{p}} − k(t, s)|ds.

Now, since k satisfies (K3) hence k(t, s) is uniformly continuous in t as well as in s on [0, T]. Hence for each ϵ > 0 we can choose n sufficiently large such that for |t_{1} − t_{2}| + |s_{1} − s_{2}| < h = \frac{T}{n} , t_{i}, s_{i} ∈ [0, T], i = 1, 2, we have

|k(t_{1}, s_{1}) − k(t_{2}, s_{2})| < \frac{ϵ}{CT} .

Then for sufficiently large n, we have

∥K^{n_{p}}(t) − K(AX^{n_{p}})(t)∥_{H} ≤ \frac{ϵ}{CT} Cjh < ϵ,

Which show that K^{n_{p}}(t) − K(AX^{n_{p}})(t) → 0 as p → ∞, uniformly on [0, T]. Now we show that K(X^{n_{p}})(t) ⇀ ∫_{0}^{t} k(t, s)Au(s) ds uniformly as p → ∞. For any v ∈ H, We note that ⟨Au(t), v⟩ is continuous hence we may write

⟨∫_{0}^{t} k(t, s)Au(s) ds, v⟩ = ∫_{0}^{t} k(t, s)⟨Au(s), v⟩ ds.

Now, for any v ∈ H,

⟨K(X^{n_{p}})(t), v⟩ = ⟨∫_{0}^{t} k(t, s)AX^{n_{p}}(s)ds, v⟩

= ∑_{i=0}^{j−2} ∫_{t_{i}^{np}}^{t_{i+1}^{np}} k(t, s)⟨Au_{i+1}^{n_{p}}, v⟩ds

+ ∫_{t_{j−1}^{np}}^{t} k(t, s)⟨Au_{j}^{n_{p}}, v⟩ds → ∫_{0}^{t} k(t, s)⟨Au(s), v⟩ ds,

as p → ∞. This completes the proof of the lemma. □

Proof of the Theorem 2.1.

Proof. From Lemma 3.7, for each v ∈ H, and t ∈ (0, T] we have

⟨U^{n_{p}}(t), v⟩ = ⟨u_{0}, v⟩ + ∫_{0}^{t}⟨AX^{n_{p}}(s), v⟩ds + ∫_{0}^{t}⟨K^{n_{p}}(s) + F^{n_{p}}(s), v⟩ ds.

Passing to the limit as p → ∞ using bounded convergence theorem and Lemmas 3.8 and 3.10, we obtain

⟨u(t), v⟩ = ⟨u_{0}, v⟩ + ∫_{0}^{t}⟨Au(s), v⟩ds + ∫_{0}^{t}⟨K(u)(s) + F (s, u_{s}), v⟩ ds. (3.17)

The integrands in (3.17) are continuous on [0, T] for each fixed v ∈ H and hence ⟨u(t), v⟩ is continuously differentiable. The boundedness of Au(t) on [0, T] implies that K(u)(t) is Lipschitz continuous. Making use of Lemma 3.10 and continuity of

EJQTDE, 2008 No. 4, p. 10 K(u)(t) and F (t, u_{t}) on [0, T] in (3.17) we obtain the existence of a strong derivative on u(t) almost everywhere on [0, T] and

\frac{du(t)}{dt} − Au(t) = F (t, u_{t}) + ∫_{0}^{t} k(t, s)Au(s) ds, a.e. t ∈ [0, T], u_{0} = ϕ. (3.18)

This show that u(t) ia a strong solution to (1.2) since u(0) = ϕ and u(t) is absolutely (in fact Lipschitz) continuous on [0, T] satisfying (1.2) a.e on [0, T]. Now we prove the uniqueness. Since k is a real valued Lipschitz continuous function on [0, T], it is differentiable a.e. on [0, T] and its derivative is essentially bounded on [0, T]. Let u_{1} and u_{2} be two solutions of (1.2) and let u = u_{1} − u_{2}. Then

u(t) = ∫_{0}^{t} T (t − s)[F (s, (u_{1})_{s}) − F (s, (u_{2})_{s})
+ ∫_{0}^{s} k(s, τ)Au(τ) dτ] ds
= ∫_{0}^{t} T (t − s)[F (s, (u_{1})_{s}) − F (s, (u_{2})_{s})]ds
+ ∫_{0}^{t} (∫_{0}^{s} k(s, τ)T (t − s)Au(τ)) dτ) ds
= ∫_{0}^{t} T (t − s)[F (s, (u_{1})_{s}) − F (s, (u_{2})_{s})]ds
+ ∫_{0}^{t} (∫_{τ}^{t} k(s, τ)T (t − s)Au(τ)) ds) dτ
= ∫_{0}^{t} T (t − s)[F (s, (u_{1})_{s}) − F (s, (u_{2})_{s})]ds
+ ∫_{0}^{t} (∫_{0}^{t−τ} k(t − η, τ)T (η)Au(τ)) dη) dτ.  (3.19)

Since u(τ) ∈ D(A) for τ ∈ [0, T], we have T (η)Au(τ) = \frac{∂}{∂η} (T (η)u(τ)) (cf. Theorem 1.2.4 in Pazy). Thus, we have

u(t) = ∫_{0}^{t} T (t − s)[f(s, (u_{1})_{s}) − f(s, (u_{2})_{s})]ds

+ ∫_{0}^{t} (∫_{0}^{t−τ} k(t − η, τ)\frac{∂}{∂η} (T (η)u(τ)) dη) dτ

= ∫_{0}^{t} T (t − s)[f(s, (u_{1})_{s}) − f(s, (u_{2})_{s})]ds

+ ∫_{0}^{t} k(τ, τ)T (t − τ)u(τ)dτ − ∫_{0}^{t} k(t, τ)u(τ)dτ

+ ∫_{0}^{t} (∫_{0}^{t−τ} k′(t − η, τ)T (η)u(τ)dη) dτ.  (3.20)

EJQTDE, 2008 No. 4, p. 11 Now taking the norm and using the fact that ∥T (t)∥_{H} ≤ 1, we have

max_{0≤r≤t} ∥u(r)∥_{H} ≤ C ∫_{0}^{t} max_{0≤r≤s} ∥u(r)∥_{H} ds.

Gronwall’s inequality implies that u(t) ≡ 0. This completes the proof of the theorem. □

Acknowledgements: The authors would like to thank the referee for his valuable suggestions which has helped us to improve the original manuscript.

References

[1] Bahuguna, D., Rothe’s method to strongly damped wave equations, Acta. Applicandae Mathematicae. 38 (1995), 185–196.

[2] Bahuguna, D. and Raghavendra, V., Rothe’s method to parabolic integro-differntial equations via abstract integro-differential equations, Appl. Anal. 33 (1989), 3-4, 153–167.

[3] Bahuguna, D. and Raghavendra, V., Application of Rothe’s method to nonlinear integrodifferential equations in Hilbert spaces, Nonlinear Analysis, TMA. 23 (1994), no. 1, 75-81.

[4] Bahuguna, D. and Shukla, R., Approximations of solutions to second order semilinear integrodifferential equations.Numer. Funct. Anal. Optim. 24 (2003), no. 3-4, 365–390.

[5] Bahuguna, D. Pani, A.K. and Raghavendra, V., Rothe’s method to semilinear hyperbolic integro-differential equations, Journal of Applied Mathematics and Stochatic Analysis. 3 (1990), no. 4, 245-252.

[6] Kacur, J., Application of Rothe’s method to perturbed linear hyperbolic equations and variantional inequalities, Czechoslovak Mathematical Journal. 34 (1984),(109), 92-106.

[7] Kacur, J., Method of Rothe in evolution equations, Lecture Notes in Mathematics Springer, Berlin. 1192 (1985), 23-34.

[8] Klaus, J.E and Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Springer Verlag (2000).

[9] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics, Academic Press, Inc. v-191 (1993).

[10] Pazy, A., Semi-groups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag (1983).

[11] Rektorys, K., The Method of Discretization in Time and Partial Differential Equations, Dordrecht-Boston-London, Reidel (1982).

[12] Sloan, I.H. and Thomee, V., Time discretization of an integro-differential equation of parabolic type, SIAM J. Numer. Anal. 23 (1986), 1052-1061.

(Received July 5, 2007)

Dhirendra Bahuguna: Department of Mathematics Indian Institute of Technology, Kanpur - 208 016, India.

E-mail address: dhiren@iitk.ac.in

Jaydev Dabas: Department of Mathematics Indian Institute of Technology, Kanpur - 208 016, India.

E-mail address: jaydev@iitk.ac.in