SEMILINEAR VOLTERRA INTEGRODIFFERENTIAL EQUATIONS WITH NONLOCAL INITIAL CONDITIONS

where h ∈ L1(0,T ;X) and f : [0,T ]×X → X. This is obtained if one takes F(u)(t) = h(t)−∫ t 0 a(t−s)f (s,u(s))ds in (1.1). Such problems are important from the viewpoint of applications since they cover nonlocal generalizations of integrodifferential equations arising in the mathematical modeling of heat conduction in materials with memory. Byszewski [6, 7] initiated the work concerning abstract nonlocal semilinear initial-value problems. He used fixedpoint methods to prove the existence and uniqueness of mild solutions to the Cauchy

The paper most closely related to the present one is by Lin and Liu [15].They developed an existence theory for the nonlocal integrodifferential equation u (t) + A u(t) + t 0 a(t − s)u(s) ds = f t, u(t) , 0 < t < T , u(0) + g t 1 , . . ., t p , u t 1 , . . ., u t p = u 0 (1.4) in X.Here A, g, f , and u 0 are as in (1.3), and a(t), 0 < t < T , is a bounded linear operator on X.
Our results concerning (1.1) and (1.2) can be viewed as a counterpart to the results in [15], in the sense that the operator A appears now only behind the integral sign.We first establish the existence and uniqueness of mild solutions to (1.1) under Lipschitz conditions on F and g.We then replace the Lipschitz conditions by weaker sublinear growth conditions, at the expense of some compactness restrictions.
The outline of the paper is as follows.In Section 2, we review some basic facts concerning resolvent operators and mild solutions for abstract linear integrodifferential equations.Section 3 is concerned with the existence and uniqueness of solutions to (1.1) under Lipschitz conditions on F and g, while Section 4 is devoted to the existence theory for (1.1) and (1.2) under compactness assumptions.Finally, two examples involving integro-partial differential equations are discussed in Section 5.

Preliminaries
For further background and details of this section, we refer the reader to [10,11,12,19,20] and the references therein.Let X be a Banach space with norm • , and A : D(A) ⊂ X → X be a linear, closed operator with dense domain D(A).Consider the Volterra integrodifferential equation
where Y stands for D(A) equipped with the graph norm.Definition 2.2.A family {S(t)} 0≤t≤T of bounded linear operators in X is said to be a resolvent for (2.1) (equivalently (2.2)) if the following conditions are satisfied: (i) S(t) is strongly continuous on [0, T ] and S(0 x ds, for all x ∈ D(A), 0 ≤ t ≤ T .Note that S(t) Ꮾ(X) ≤ M S on [0, T ] due to (i) and the uniform boundedness principle.(Here Ꮾ(X) designates the space of all bounded linear operators on X.) General conditions on a and A that guarantee the existence and uniqueness of a resolvent family for (2.1) can be found in [12].
It is convenient to represent a mild solution of (2.1) using a variation of parameters type formula involving the resolvent S(t).Specifically, we have the following proposition.
Proposition 2.3.Suppose that (2.1) admits a resolvent S(t).Then, for any x 0 ∈ X and h ∈ L 1 (0, T ; X), equation (2.1) has a unique mild solution u, given by Next we consider (2.1) under the following stronger restrictions on a and A: (C1) a ∈ L 1 (0, T ) is positive, nonincreasing, and convex; (C2) A is a linear, densely-defined, closed, invertible operator such that A −1 is compact on X; (C3) either (i) −a is convex, X is a separable Hilbert space, and A is self-adjoint and strictly positive definite, or (ii) a is log-convex (with a locally absolutely continuous on (0, T ] in the case when a(0 + ) < ∞), and −A generates a strongly continuous cosine family on X.These conditions ensure (cf.[10,11,14,19]) that (2.1) has a resolvent S(t), in the sense of Definition 2.2.Now let : L 1 (0, T ; X) → C([0, T ]; X) be defined by (2.4) The following two compactness results have been established in [2].

The case of a Lipschitz continuous nonlinearity
Consider (1.1) in a Banach space X under the following assumptions: is allowed to depend only on the restriction of u on [0, t].In addition, there exists M F > 0 such that where q is the conjugate of p (i.e., p −1 + q −1 = 1).Remark 3.1.Naturally, by a mild solution u ∈ C([0, T ]; X) to (1.1), we mean a mild solution to (2.1) with u 0 = g(u) and h = F (u).According to Proposition 2.3, u is a mild solution of (1.1) if and only if Proof.Let v ∈ C([0, T ]; X) and consider the problem , where u v is the mild solution to (3.3).We show that Ᏺ is a strict contraction.To this end, let v, w ∈ C([0, T ]; X).By using (2.3), (H2), (H3), (H4), and (H5), together with Hölder's inequality, we obtain Since M S (M g +M F T 1/q ) < 1 (cf.(H6)), we conclude that Ᏺ is indeed a strict contraction.So, by the contraction mapping principle, Ᏺ has a unique fixed point u, which is clearly the mild solution to (1.1) that we seek.This completes the proof.
We conclude this section with some comments on (1.2) where g is given by Obviously g, as given by (3.5), satisfies (H5) with M g = 1.Since always M S ≥ 1, it is clear that condition (H8) does not hold.To incorporate (3.5) in our theory, we consider that the kernel a, and the functions f and h are defined on [0, ∞) (rather than on a fixed interval [0, T ]).Specifically, we assume (H9) a ∈ L 1 (0, ∞); (H10) f : [0, ∞) × X → X is continuous and satisfies (for some M > 0) We also suppose that the pair (a, A) generates a resolvent S(t), in the sense of Definition 2.2, on [0, ∞) such that (H12) S(t) Ꮾ(X) ≤ Le −ωt , for all t ≥ 0, for some constants L ≥ 1 and ω > 0. For conditions on a and A that guarantee (H12), see [12] and [19, pages 42-44].(Note, in particular, that (H9) and (H12) are compatible.)We can now prove that the problem has a unique mild solution, provided that T is large enough.Proof.By standard arguments, it follows that for each (fixed) T > 0 and x ∈ X, the initial-value problem has a unique mild solution u x on [0, T ], with u x given by (3.9) On account of (H9), (H10), and (H12), (3.9) yields Define Q T : X → X by Q T x = u x (T ), for all x ∈ X, and observe that (3.11) and (H13) imply that Q T is a contraction on X for a sufficiently large T .Therefore, if T is chosen such that (H13) is satisfied, Q T has a unique fixed point x 0 .The corresponding u x 0 = u is obviously the (unique) mild solution of (3.7) and the proof is complete.Remark 3.5.It is easy to generalize Theorem 3.4 to the case when (1.2a) is coupled with a nonlocal condition of the form (1.3b).The details are left to the reader.

The case when A −1 is compact
We now investigate (1.1) in the case where the Lipschitz conditions on F and g are dropped.The following assumptions will be used instead. (H14) ) is a continuous hereditary map satisfying where where d 1 , d 2 > 0. Proof.Consider (3.3) and define the solution map Ᏺ as in the proof of Theorem 3.2.We appeal to Schaefer's theorem to prove that Ᏺ has a fixed point.The continuity of Ᏺ is easily verified.We show that Ᏺ is a compact map.Let K r = {v ∈ C([0, T ]; X) : v C([0,T ];X) ≤ r} and observe that (cf.(2.3) and (2.4) . (Here, B(x i , /M S ) denotes the ball in X with radius /M S centered at Bi and therefore, it is totally bounded.Thus, K is precompact in C([0, T ]; X), as desired.
To apply Schaefer's fixed point theorem, we also need to show that the set ξ(Ᏺ), as defined in Theorem 2.6 (with C([0, T ]; X) in place of X) is bounded.If v ∈ ξ(Ᏺ) and 134 Semilinear Volterra integrodifferential equations … 0 ≤ t ≤ T , we have by (3.2), (H14), and (H15) where M S has the same meaning as in (H3).Taking into account that λ ≥ 1 and (We hereafter employ C to denote a generic positive constant.)Hence, ξ(Ᏺ) is bounded and consequently Ᏺ has a fixed point u ∈ C([0, T ]; X).Obviously, u satisfies (3.2) and the proof is complete.
Next, we consider (1.2) under conditions similar to those in Theorem 4.1.Precisely, we replace assumption (H14) by (H16) f : [0, T ] × X → X is a map satisfying the Carathéodory conditions (i.e., f is measurable in t and continuous in x) and f (t,x) ≤ c 1 (t) x + c 2 (t), for almost all t ∈ (0, T ) and all x ∈ X, where c 1 , c 2 ∈ L 1 (0, T ).
Proof.Let v ∈ C([0, T ]; X) and consider the problem , where u v is the mild solution of (4.4).Invoking (2.3) and (2.4), we can write, for v ∈ C([0, T ]; X), Again, we use Schaefer's theorem to establish that Ᏺ has at least one fixed point.The continuity of Ᏺ follows easily.Let r > 0 and define the set K r as in the proof of Theorem 4.1.We show that Ᏺ(K r ) is precompact in C([0, T ]; X).Since {S(•)g(v) : v ∈ K r } was earlier shown to be precompact in C([0, T ]; X) and S * h is independent of v, we only focus on the last term on the right-hand side of (4.5).By (H16), we have Thus, Ᏺ is a compact map.Finally, let ξ(Ᏺ) = {v ∈ C([0, T ]; X) : λv = Ᏺv, for some λ ≥ 1} and let v ∈ ξ(Ᏺ).We seek a constant C, independent of λ and v, such that v C([0,T ];X) ≤ C, for all S. Aizicovici and M. McKibben 135 v ∈ ξ(Ᏺ).Owing to (H15) and (H16), we have for t ∈ [0, T ], Recalling that λ ≥ 1 and M S (d 1 + a * c 1 L 1 (0,T ) ) < 1, (4.6) yields the desired bound for v C([0,T ];X) .Therefore, we conclude by Theorem 2.6 that Ᏺ has a fixed point, which is a mild solution to (1.2).This completes the proof.
Proof.Let X = L 2 ( ) and define A as in (5.4).Define f : [0, T ] × X → X and g : C([0, T ]; X) → X by f (t, u)(x) = f t, x, u(x) , ∀u ∈ X, a.e. on , (5.6) g(u)(x) = T 0 g t, x, z, u(t, z) dt dz, a.e. on , (5.7) respectively, and remark that our problem can be rewritten in the abstract form (1.2) in X.We show that Theorem 4.3 is applicable.First, note that from (5.6) and (H21), it follows that f is a well-defined mapping from [0, T ] × X into X which satisfies the Carathéodory conditions.Let k ∈ N be fixed and u ∈ X be such that u ≤ k.Then, (H21) implies that f (t,u) ≤ m 1 (t)k + m 2 (t, •) = g k (t), with g k ∈ L 1 (0, T ; R + ).(5.9) Since σ ∈ L 2 ((0, T )× × ), we conclude that g satisfies all conditions of (H15).(In particular, d 1 = δ.)Finally, note that (C2) and (C3)(i) are fulfilled because of (5.4) and our assumptions on a, and that M S = 1 in this case.Hence, all conditions of Theorem 4.3 are satisfied.As a result, the conclusion of Theorem 5.2 follows readily.
.4) It is well known that A is a positive definite, self-adjoint operator in X.Moreover, by (C1) and [20, page 38], condition (H3) is satisfied with M S = 1.Next, f generates a function