Existence results for a class of abstract nonlocal Cauchy problems

Nonlinear Analysis 39 (2000) 649 – 668

www.elsevier.nl/locate/na

Existence results for a class of abstract nonlocal Cauchy problems Sergiu Aizicovici∗ , Mark McKibben Department of Mathematics, Ohio University, Athens, OH 45701, USA Received 4 October 1997; accepted 29 January 1998

Keywords: Nonlocal initial condition; m-accretive operator; Compact semigroup

1. Introduction In this paper we study the global existence of solutions to nonlinear evolution equations with nonlocal initial conditions of the general form u0 (t) + Au(t) 3 F(u)(t);

0¡t¡T;

u(0) = g(u);

(1.1)

in a real Banach space X . Here A is a nonlinear m-accretive (possibly multivalued) operator on X , F : L1 (0; T ; X ) → L1 (0; T ; X ); and g : L1 (0; T ; X ) → D(A). Our work is a continuation of the study begun in [2]. There, A is assumed to be m-accretive, while both F : C([0; T ]; X ) → L1 (0; T ; X ) and g : C([0; T ]; X ) → X satisfy Lipschitz conditions. In the theory developed in this paper we will relax the Lipschitz conditions on F and g at the expense of stronger restrictions on A. Speci cally, we will assume that −A generates a compact contraction semigroup (for all t¿0). The case when A in Eq. (1.1) is linear has been studied extensively. The work concerning abstract nonlocal semilinear initial-value problems was initiated by Byszewski [7–9]. Among his several results, he proves the existence and uniqueness of solutions to the Cauchy problem u0 (t) + Au(t) = f(t; u(t));

0¡t¡T;

u(0) + g(t1 ; : : : ; tp ; u(t1 ); : : : ; u(tp )) = u0 ; ∗

Corresponding author. E-mail: [email protected]

0362-546X/99/$ - see front matter ? 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 2 2 7 - 2

(1.2)

650

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

where 0¡t1 ¡ · · · ¡tp ≤ T are xed, u0 ∈ X; −A generates a linear C0 -semigroup on X , while f : [0; T ] × X → X and g : [0; T ]p × X p → X satisfy Lipschitz-type conditions. Ntouyas and Tsamatos [21, 22] study semilinear problems of the form (1.1), (1.2) by compactness arguments. However, the proofs in [21, 22] are incomplete in the sense that when checking the equicontinuity and compactness conditions of the Arzela–Ascoli theorem, the authors omit to discuss the case when t = 0 (see [21, pp. 683, 684] or [22, pp. 103, 104]). In addition, assumption H (g) in [21, 22] should be replaced by the boundedness of g on C([0; T ]; X ), rather than on X , which is a strong restriction. This paper not only extends the above results to the fully nonlinear case, but it also completes them. The present work may be viewed as an attempt to develop a general existence theory for abstract nonlocal problems of the general form (1.1) under various compactness assumptions. Our basic tools are methods and results for dierential equations governed by m-accretive operators in Banach spaces, the theory of compact nonlinear semigroups, and xed-point techniques. Also, this study is important from the viewpoint of applications since it covers nonlinear generalizations of nonlocal parabolic equations arising in physics (e.g., heat conduction and diusion processes). Such problems in the semilinear case have been analyzed in [7, 9, 12, 18]. The outline of this paper is as follows. In Section 2, we recall some facts about m-accretive operators, nonlinear evolution equations, and multivalued mappings. The main existence results are stated in Section 3, and the corresponding proofs are given in Section 4. Finally, Section 5 is devoted to the discussion of some examples. 2. Preliminaries For further background and details of this section, we refer the reader to [1, 4, 5, 20, 24, 29, 30]. Let X be a real Banach space of norm k·k. A set-valued operator A in X with domain D(A) and range R(A) is said to be accretive if kx1 − x2 k ≤ kx1 − x2 + (y1 − y2 )k, for all ¿0 and yi ∈ Axi ; i = 1; 2. A is called m-accretive if it is accretive and R(I + A) = X , for all ¿0. (Here I stands for the identity on X ). In the case when X is a Hilbert space, m-accretivity is equivalent to maximal monotonicity. An important subclass of maximal monotone operators is that of subdierentials. If ’ is a proper, convex, and lower-semicontinuous function from the Hilbert space X into (−∞; ∞], then its subdierential @’ is given by @’(x) = {y ∈ X : ’(z) − ’(x) ≥ (y; z − x); ∀z ∈ D(’)}; where D(’) = {x ∈ X : ’(x)¡∞}, and (,) denotes the inner product on X . Next, consider the Cauchy problem u0 (t) + Au(t) 3 f(t);

0¡t¡T;

u(0) = u0 ; where A is m-accretive in X , f ∈ L1 (0; T ; X ), and u0 ∈ D(A).

(2.1)

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

651

De nition 2.1. A function u : [0; T ] → X is called an integral solution of the initialvalue problem (2.1) if u is continuous on [0; T ]; u(0) = u0 , and the inequality 2

2

ku(t) − xk ≤ ku(t) − xk + 2

Z

t

t

hf() − y; u() − xis d

(2.2)

holds for all [x; y] ∈ A and 0 ≤ t ≤ t ≤ T . Here the function h; is : X × X → R is de ned by hy; xis = sup{x∗ (y): x∗ ∈ J (x)}, where J : X → X ∗ is the duality mapping of X . It is well-known that Eq. (2.1) has a unique integral solution u ∈ C([0; T ]; D(A)): Moreover, we have the following theorem due to Benilan: Theorem 2.2. Let u and v be the integral solutions of u0 + Au 3 f; u(0) = u0 ; and v0 + Av 3 g; v(0) = v0 ; respectively; where A is m-accretive; f; g ∈ L1 (0; T ; X ) and u0 ; v0 ∈ D(A). Then Z ku(t) − v(t)k ≤ ku(s) − v(s)k +

t s

kf() − g()k d;

(2.3)

for all 0 ≤ s ≤ t ≤ T . If f = 0 in (2.1), the map u0 → u(t), where u is the corresponding integral solution, de nes a contraction semigroup on D(A), denoted by S(t). We say that S(t) is the nonlinear semigroup generated by −A. S(t) is said to be a compact semigroup if for each t¿0, the mapping S(t) : X → X maps bounded subsets of X into precompact subsets of X . Various characterizations of nonlinear compact semigroups in Banach or Hilbert spaces can be found in [6, 16, 26]. If A depends on t, then Eq. (2.1) becomes u0 (t) + A(t)u(t) 3 f(t);

0¡t¡T;

(2.4)

where {A(t): 0 ≤ t ≤ T } is a family of m-accretive operators on X . In this case, the semigroup S(t) is replaced by an evolution operator as de ned below. Let {A(t): 0 ≤ t ≤ T } be a family of (possibly multivalued) operators on X , of domains D(A(t)) such that D(A(t)) = D (independent of t). Also, assume (HA ) (i) {A(t): 0 ≤ t ≤ T } is a family of m-accretive operators on X . (ii) There exist a continuous function q : [0; T ] → X and a nondecreasing function L : [0; ∞) → [0; ∞) such that hy1 − y2 ; x1 − x2 is ≥ − kq(t) − q(s)kkx1 − x2 k · L(max{kx1 k; kx2 k}); for all x1 ∈ D(A(t)); y1 ∈ A(t)x1 ; x2 ∈ D(A(s)); and y2 ∈ A(s)x2 , with 0 ≤ s ≤ t ≤ T.

652

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

If assumption (HA ) is satis ed, the family {A(t): 0 ≤ t ≤ T } generates a so-called evolution operator U (t; s) on D given by U (t; s)x = lim

n→∞

n  Y

I+

i=1

 −1 t−s t−s A s+i x; n n

for all x ∈ D and all

(t; s) ∈
; where
= {(t; s): 0 ≤ s ≤ t ≤ T }. We say that the evolution operator U is compact if U (t; s) maps bounded subsets of D into precompact subsets of D, for all (t; s) ∈
, t 6= s. See [13] for an example. Note that De nition 2.1 can be extended to Eq. (2.4). More precisely, we have De nition 2.3. A function u ∈ C([0; T ]; D) is an integral solution of Eq. (2.4) if ku(t) − xk2 − ku(t) − xk2 Z t ≤ 2 (hf() − y; u() − xis + Cku() − xkkq() − q()k) d t

(2.5)

for all 0 ≤ t ≤ t ≤ T;  ∈ [0; T ]; x ∈ D(A()); y ∈ A()x; q as in (HA )(ii), and C = L(max{kxk; kukC([0;T ]; X ) }): It can be shown [24] that if (HA ) is ful lled, then for each u0 ∈ D and f ∈ L1 (0; T ; X ), Eq. (2.4) has a unique integral solution satisfying u(0) = u0 . Moreover, (2.3) holds. The following xed-point theorem, together with the well-known Schauder’s xedpoint theorem, plays a key role in the proofs of certain results stated in Section 3. Theorem 2.4 (Schaefer’s xed-point theorem [27]). Let F : X → X be continuous and compact and let (F) = {x ∈ X : x = Fx; for some  ≥ 1}. Then either (F) is unbounded or F has a xed point. We remark that Theorem 2.4 is, in fact, an immediate consequence of the Leray– Schauder Principle [17]. For more general results in this direction, see, e.g. [25]. Next, a subset B of RL1 (0; T ; X ) is called uniformly integrable if for each ¿0, there is a ()¿0 such that kf()k d¡, for every Lebesgue measurable ⊂ [0; T ] with Lebesgue measure m( )¡() and every f ∈ B. The next theorem, which is a direct consequence of the results in [3], will be used in the sequel. Theorem 2.5. Consider the initial-value problem (2.1). If −A generates a compact semigroup on D(A); then the following holds: R (1) The mapping R : D(A) × L1 (0; T ; X ) → Lq (0; T ; X ) (q¡∞) de ned by (u0 ; f) −→ u is compact; where u is the integral solution of Eq. (2.1) corresponding to u0 and f.

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

653

(2) Let B ⊆ L1 (0; T ; X ) be uniformly integrable and K be a precompact subset of D(A). Then the set {u: u is an integral solution to Eq. (2.1) for some f ∈ B; u0 ∈ K} is precompact in C ([0; T ]; X ). We conclude with some preliminaries on multivalued mappings. In what follows, we assume that X is separable. Let ( ; ; ) be a measure space and H : → 2X be closed-valued. We say that H is measurable if the function ! → d(x; H (!)) is measurable, ∀x ∈ X , where d(a; B) = inf {ka − bk: b ∈ B}. (In the case when H : × X → 2X , the measurability is understood in the sense of  × B(X ), where B(X ) is the -algebra of Borel subsets of X ). We de ne SHp (1 ≤ p ≤ ∞) to be the set of all selections of H that belong to p L ( ; X ); that is, SHp = {h ∈ Lp ( ; X ): h(!) ∈ H (!); -a:e:}. This set is nonempty if H is measurable and |H (·)| ∈ Lp ( ; R), where |H (!)| = sup{kxk: x ∈ H (!)}. Next, let U be a topological space and H : U → 2X a nonempty and closed-valued mapping. We say that H is lower semicontinuous (l.s.c.) if the set {H −1 (D)={v ∈ U : H (v) ∩ D 6= ∅} is open whenever D is open. 1 Let K be a nonempty subset in C([0; T ]; X ) and let H : K → 2L (0; T ; X ) be a nonempty and closed-valued mapping. We say that H is decomposable if for all u ∈ K, f; g ∈ H (u) and each measurable subset E in [0; T ], we have fXE + gX[0; T ]−E ∈ H (u), where XE is the characteristic function of E. We now state a speci c form of a selection theorem due to Fryszkowski [11] which will be used in the proofs when F in Eq. (1.1) is generated by a multivalued mapping. Theorem 2.6. Let X be a separable real Banach space; K a compact subset in 1 C([0; T ]; X ); and H : K → 2L (0; T ; X ) a nonempty; closed-valued mapping which is lowersemicontinuous and decomposable. Then there exists at least one continuous function f : K → L1 (0; T ; X ) such that f(u) ∈ H (u); for each u ∈ K. 3. Statement of results Consider the initial-value problem (1.1) in a Banach space X under the following assumptions: (H1) A is an m-accretive operator on X such that −A generates a nonlinear compact semigroup {S(t): t¿0}. (H2) F : L1 (0; T ; X ) → L1 (0; T ; X ) is a continuous map such that kF(u)kL1 (0; T ; X ) ≤ c1 kukL1 (0; T ; X ) + c2 ; ∀u ∈ L1 (0; T ; X ); for some positive constants c1 and c2 . (H3) g : L1 (0; T ; X ) → D(A) is a continuous map such that kg(u)k ≤ d1 kukL1 (0; T ; X ) +d2 , ∀u ∈ L1 (0; T ; X ); for some positive constants d1 and d2 . (H4) 1 − T (d1 + c1 )¿0: An integral solution of Eq. (1.1) is a function u : [0; T ] → X satisfying De nition 2.1 with F(u) and g(u) in place of f and u0 , respectively. Our rst result is: Theorem 3.1. Let (H1)–(H4) hold. Then Eq. (1.1) has at least one integral solution.

654

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

In some cases, the mapping F in Eq. (1.1) can be generated by a function f : [0; T ] × X → X which satis es the following Caratheodory conditions: (C1) For almost all t ∈ (0; T ), the function f(t; ·) : X → X is continuous, (C2) For all x ∈ X , the function f(·; x) : [0; T ] → X is measurable. If in place of (H2) and (H4), we respectively assume (H20 ) f : [0; T ] × X → X satis es conditions (C1) and (C2) and there are c1 ¿0 and k ∈ L1 (0; T ; R+ ) such that kf(t; x)k ≤ c1 kxk + k(t), for almost all t ∈ [0; T ] and all x ∈ X , (H40 ) 1 − d1 T ec1 T ¿0, then the conclusion of Theorem 3.1 applies to the nonlocal initial-value problem u0 (t) + Au(t) 3 f(t; u(t)); u(0) = g(u):

0¡t¡T:

(3.1)

Speci cally, one has Theorem 3.2. Suppose that (H1), (H20 ), (H3), and (H40 ) are satis ed. Then Eq. (3.1) has at least one integral solution. Next, we examine a situation in which the integral solutions of Eq. (1.1) have greater regularity than mere continuity (in particular, they are strong solutions in the sense of the following de nition). De nition 3.3. A strong solution to Eq. (1.1) on [0; T ] is a function u ∈ W 1;1 (0; T ; X ) ∩ C([0; T ]; D(A)) satisfying u(0) = g(u) and u0 (t) + Au(t) 3 F(u)(t) a.e. on [0; T ]. The next theorem is an immediate consequence of Theorem 3.2 and [5, Theorem 3.6]. Theorem 3.4. Let X be a Hilbert space and A = @’; where ’ : X → (−∞; ∞] is a proper; convex; and lower-semicontinuous function. Assume that (H1) – (H4) hold and let u be an integral solution of Eq. (1.1). If F(u) ∈ L2 (0; T ; X ) and g(u) ∈ D(’); then u ∈ W 1; 2 (0; T ; X ) is a strong solution of Eq. (1.1) on [0; T ]. Next, we would like to consider an initial-value problem of the form (3.1) in which the mapping g is de ned on C([0; T ]; X ) rather than L1 (0; T ; X ). We do indeed get a result if we replace (H20 ), (H3), and (H40 ) by (H200 ), (H30 ), and (H400 ), respectively, as follows: (H200 ) (i) f : [0; T ] × X → X satis es (C1) and (C2), and for each k ∈ N, there is gk ∈ L1 (0; T ; R+ ) such that for almost all t ∈ (0; T ); supkxk≤k kf(t; x)k ≤ gk (t). RT (ii) limk→∞ k −1 0 gk (s) ds = ¡∞. (H30 ) g : C([0; T ]; X ) → D(A) is a continuous, compact map such that kg(x)k ≤ kxkC([0; T ]: X ) + ; ∀x ∈ C([0; T ]; X ), where and are positive constants. (H400 )  + ¡1.

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

655

Note that (H200 ) and (H30 ) are more general than (H20 ) and (H3). Comparable conditions appear in [23, 28] Theorem 3.5. Let (H1), (H200 ), (H30 ), and (H400 ) be satis ed. Then Eq. (3.1) has at least one integral solution. In all of the previous theorems the operator A was independent of t. Now, we consider the nonautonomous problem u0 (t) + A(t)u(t) 3 f(t; u(t));

0¡t¡T;

u(0) = g(u):

(3.2)

Replace hypothesis (H1) by (H10 ) The family {A(t): 0 ≤ t ≤ T } satis es (HA ) and generates a compact evolution operator. Then, we have: Theorem 3.6. Let (H10 ), (H200 ), (H30 ), (where g takes values in D); and (H400 ) be satis ed. Then Eq. (3.2) has at least one integral solution in the sense of De nition 2.3. Next, by choosing {A(t): 0 ≤ t ≤ T } to be a family of subdierentials, we obtain a stronger result. More precisely, we let X be a Hilbert space and ’t : X → (−∞; ∞) (0 ≤ t ≤ T ) be proper, convex, and l.s.c. functions satisfying (cf. [15]): (H’ ): For each r¿0; there are r ∈ W 1; 2 (0; T ) and r ∈ W 1; 1 (0; T ) with the property that for each s; t ∈ [0; T ] with s ≤ t and z ∈ D(’s ) with |z| ≤ r, there exists z1 ∈ D(’t ) such that |z1 − z| ≤ |r (t) − r (s)|(1 + |’ s (z)|1=2 ) and ’t (z1 ) − ’ s (z) ≤ | r (t) − r (s)|(1 + |’ s (z)|). We then obtain Theorem 3.7. Consider the initial-value problem u0 (t) + @’t (u(t)) 3 f(t; u(t)); u(0) = g(u)

0¡t¡T

(3.3)

where {’t : 0 ≤ t ≤ T } is a family of proper; convex; l.s.c. functions from a real Hilbert space X into (−∞; ∞] satisfying (H’ ). In addition; assume that for each r¿0 and t ∈ [0; T ]; the level set Ltr = {x ∈ D(’t ): kxk ≤ r and k’t (x)k ≤ r} is precompact in X . Let f and g be as in Theorem 3.5, with the exception that g maps C([0; T ]; X ) into D(’0 ) and gk ∈ L2 (0; T ; X ) in (H200 ). If (H400 ) also holds; then Eq. (3.3) has an 1; 2 (0; T ; X ). integral solution u such that u ∈ Wloc Finally, note that the forcing term F has been single-valued in all of the previous theorems. For the nal result, we consider the initial-value problem (3.1), where f(t; u(t)) is now a multivalued perturbation. In this case u : [0; T ] → X will be called

656

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

an integral solution to Eq. (3.1) if it satis es De nition 2.1, for some f ∈ L1 (0; T ; X )  ∈ f(t; u(t)) a.e. on (0; T ), and g(u) in place of u0 . with f(t) Theorem 3.8. Let X be a real separable Banach space. Assume that (H1) holds and that g : L1 (0; T ; X ) → D(A) is such that kg(u) − g(v)k ≤ mku − vkL1 (0; T ; X )

∀u; v ∈ L1 (0; T ; X );

(3.4)

for some m¿0. Let f : [0; T ] × X → 2X be a measurable; closed-valued multifunction such that x → f(t; x) is l.s.c. for almost all t ∈ (0; T ) and |f(t; x)| ≤ a1 kxk + a2 (t);

a.e. on (0; T );

for all x ∈ X;

(3.5)

where a1 ¿0 and a2 ∈ L1 (0; T ; R+ ). If also 1 − (m + a1 )T ¿0; then the problem (3.1) has at least one integral solution. 4. Proofs Proof of Theorem 3.1. problem

Let v ∈ L1 (0; T ; X ) be xed and consider the initial-value

u0 (t) + Au(t) = F(v)(t);

0¡t¡T;

u(0) = g(v):

(4.1)

Since F(v) ∈ L1 (0; T ; X ) and g(v) ∈ D(A), there is a unique integral solution uv to Eq. (4.1) on [0; T ]. De ne the map G : L1 (0; T ; X ) → L1 (0; T ; X ) by Gv = uv . We want to show that G has a xed-point. To do this, we shall use Schaefer’s xed-point theorem. First, we show that G is a compact map. Let r¿0 be xed and de ne K = {v ∈ L1 (0; T ; X ): kvkL1 (0; T ; X ) ≤ r}. We need to show that G(K) is precompact in L1 (0; T ; X ). Observe that by (H2) and (H3), kF(v)kL1 (0; T ; X ) ≤ c1 kvkL1 (0; T ; X ) + c2 ≤ c1 r + c2

(4.2)

kg(v)k ≤ d1 kvkL1 (0; T ; X ) + d2 ≤ d1 r + d2

(4.3)

and

hold for all v ∈ K. Let K˜ = {(g(v); F(v) ∈ D(A) × L1 (0; T ; X ) : v ∈ K}. De ne the map T : K →K˜ by T(v) = (g(v); F(v)); ∀v ∈ K. From (4.2) and (4.3), we conclude that K˜ is a bounded subset of D(A) × L1 (0; T ; X ). Let M = {uv : uv is an integral solution of Eq. (4.1), for some v ∈ K}. Clearly, M ⊆ L1 (0; T ; X ). De ne the map R : K˜ → M by R(g(v); F(v)) = uv . From Theorem 2.5(1), we know that M is precompact in L1 (0; T ; X ). Observe that G = R ◦ T on K with G(K) = M . Since r¿0 was arbitrarily chosen, we conclude that G does indeed take bounded sets in L1 (0; T ; X ) into precompact sets in L1 (0; T ; X ); that is, G is a compact map.

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

657

Next, the continuity of G follows easily from that of F and g and the inequality Z T kF(v)(s) − F(w)(s)kds; (4.4) kG(v)(t) − G(w)(t)k ≤ kg(v) − g(w)k + 0

1

for all v; w ∈ L (0; T ; X ) and t ∈ [0; T ]. Finally, let (G) = {v ∈ L1 (0; T ; X ) : v = Gv; for some  ≥ 1}. Let v ∈ (G) be xed but arbitrary. We want to nd a constant C, independent of v and , such that  ApkvkL1 (0; T ; X ) ≤ C, for all v ∈ (G). To this end, let x ∈ D(A) be such that y ∈ Ax. plying Theorem 2.2 yields Z t kv(t) − xk  ≤ kg(v) − xk  + kF(v)(s) − yk  ds; Z ≤ kg(v) − xk  +

0

T 0

kF(v)(s)k ds + T kyk; 

∀t ∈ [0; T ]:

(4.5)

Applying the triangle inequality in (4.5) and using the growth conditions in (H2) and (H3), we obtain kvkL1 (0; T ; X ) ≤ 2kxk  + d1 kvkL1 (0; T ; X ) + d2 + c1 kvkL1 (0; T ; X ) + c2 + T kyk; 

(4.6)

for all t ∈ [0; T ]. Since  ≥ 1, we have upon integrating (4.6) over [0; T ]: kvkL1 (0; T ; X ) ≤ [2kxk  + d2 + c2 + T kyk]T  + (d1 + c1 )T kvkL1 (0; T ; X ) Hence, [1−T (d1 +c1 )]kvkL1 (0; T ; X ) ≤ [2kxk+d   . Since [1−T (d1 +c1 )]¿0 2 +c2 +T kyk]T by (H4), we conclude that kvkL1 (0; T ; X ) ≤

[2kxk  + d2 + c2 T kyk]T  = C: 1 − T (d1 + c1 )

Clearly, C is a constant that is independent of v and . Hence, kvkL1 (0; T ; X ) ≤ C for all v ∈ (G), so that (G) is bounded. Therefore, by Theorem 2.4, we know that there is a xed point u of G. It is clear that u is an integral solution to Eq. (1.1) on [0; T ]. The proof is complete. G

Proof of Theorem 3.2. Observe that (H20 ) guarantees that the map v −→ f(·; v(·)) is well-de ned and continuous from L1 (0; T ; X ) into itself (see, e.g., [30, Chapter 26, p. 561]). The proof is identical to that of Theorem 3.1 up to the application of Schaefer’s theorem. Here, a slight modi cation is required since (H40 ) is used in place of (H4). Speci cally, the following holds: Let x ∈ D(A) be such that y ∈ Ax and v ∈ (G). Applying Theorem 2.2 yields Z t kv(t) − xk  ≤ kg(v) − xk  + kf(s; v(s))k ds + T kyk;  ∀t ∈ [0; T ]: (4.7) 0

Using the triangle inequality, (H20 ), and (H3) in (4.7), we obtain the inequality Z t kv(t)k ≤ [K + d1 kvkL1 (0; T ; X ) ] + c1 kv(s)k ds; ∀t ∈ [0; T ]; (4.8) 0

658

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

where K is the positive constant 2kxk  + T kyk  + d2 + kkkL1 (0; T ; X ) . Applying Gronwall’s lemma to (4.8) yields Rt

kv(t)k ≤ [K + d1 kvkL1 (0; T ; X ) ]e

0 c1

ds

;

∀t ∈ [0; T ]

≤ [K + d1 kvkL1 (0; T ; X ) ]ec1 T :

(4.9)

Integrating (4.9) over [0; T ] and rearranging some terms yields kvkL1 (0; T ; X ) [1 − d1 ec1 T T ] ≤ Kec1 T T:

(4.10)

Since 1 − d1 ec1 T T ¿0 by (H40 ), we conclude from (4.10) that kvkL1 (0; T ; X ) ≤

Kec1 T T = C: 1 − d1 ec1 T T

(4.11)

Clearly, the constant C is independent of  and v, and (4.11) holds for all v ∈ (G). So, by virtue of Theorem 2.3, we infer that Eq. (3.1) has at least one integral solution, thereby completing the proof. Proof of Theorem 3.5. Let v ∈ C([0; T ]; X ) and consider the initial-value problem u0 (t) + A(u(t)) = f(t; v(t));

0¡t¡T;

(4.12)

u(0) = g(v):

Since v is continuous on [0; T ], it is bounded, say supt∈[0;T ] kv(t)k ≤ N ¡∞, for some N ∈ N. So, by (H200 ), there is gN ∈ L1 (0; T ; R+ ) such that for almost all t ∈ (0; T ), kf(t; v(t))k ≤ gN (t), which implies that f ∈ L1 (0; T ; X ). Thus, we conclude that there is a unique integral solution uv to Eq. (4.12) on [0; T ] . De ne the map G : C([0; T ]; X ) → C([0; T ]; X ) by Gv = uv . We want to show that G has a xed point. To do this, we will use Schauder’s xed-point theorem this time (see, e.g., [10, p. 456]). First, we show that G is continuous. Let {vn }∞ n=1 be a sequence in C([0; T ]; X ) such that vn → v in C([0; T ]; X ) as n → ∞. Since Gvn and Gv are integral solutions to Eq. (4.12), we apply Theorem 2.2 to obtain kGvn (t) − Gv(t)k≤kg(vn ) − g(v)k Z T + kf(s; vn (s)) − f(s; v(s))kds; 0

∀t ∈[0; T ]:

(4.13)

By (H200 ), we know that there is a gN ∈ L1 (0; T ; X ) such that kf(t; vn (t))k + kf(t; v(t))k ≤ gN (t)

for almost all t ∈ (0; T ):

This, along with the continuity of f with respect to the second variable, enables us to use Lebesgue’s dominated convergence theorem to infer that f(· ; vn (·)) → f(· ; v(·)) in L1 (0; T ; X ) as n → ∞. This, in conjunction with (4.13), yields the continuity of G. Next, we show that G is compact. Let r ∈ N be xed and let K = {v ∈ C([0; T ]; X ): kvkC([0;T ];X ) ≤ r} and K˜ = {f(· ; v(·)) ∈ L1 (0; T ; X ) : v ∈ K}. By (H200 ),

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

659

there exists gr ∈ L1 (0; T ; X ) such that {f(t; v(t))k ≤ gr (t) a.e. on (0; T ), for all v ∈ K. Thus, K˜ ⊆ {f ∈ L1 (0; T ; X ) : kf(t)k ≤ gr (t) a:e: on (0; T )} and K˜ is uniformly integrable. De ne the map T : K →K˜ by T(v)(t) = f(t; v(t)) a.e. on (0; T ), for all v ∈ K. Next, let M = {u(u0 ; f) : u(u0 ; f) is an integral solution of Eq. (2.1), ˜ for some u0 ∈ g(K) and f ∈K}: ˜ De ne the map R : g(K) × K˜ → M by R(u0 ; f) = u(u0 ; f) , for all u0 ∈ g(K) and f ∈K. In view of Theorem 2.5(2) (note that g(K) is precompact in X by (H30 )), M is precompact in C([0; T ]; X ). Observe that G = R ◦ T on K and G(K) = M . Since r¿0 was arbitrarily chosen, we conclude that G maps bounded sets in C([0; T ]; X ) into precompact sets in C([0; T ]; X ); that is, G is a compact map. Let Bn = {x ∈ C([0; T ]; X ): kxkC([0; T ]; X ) ≤ n}. We will show that there exists an n ∈ N such that GBn ⊆ Bn . Suppose, by way of contradiction, that for each k ∈ N, there is a uk ∈ Bk such that Guk 6∈Bk ; that is, kGuk kC([0; T ]; X ) ¿k. Hence, 1 ≤ lim k −1 kGuk kC([0; T ]; X ) :

(4.14)

k→∞

By the de nition of G and Theorem 2.2, we have for any y ∈ Ax Z T  ≤ kg(uk ) − xk  + kf(s; uk (s))k ds + T kyk;  ∀t ∈ [0; T ]: kGuk (t) − xk

(4.15)

Applying the triangle inequality in (4.15) yields Z T kGuk (t)k ≤ 2k xk  + kg(uk )k + kf(s; uk (s))k ds + T kyk; 

(4.16)

0

0

∀t ∈ [0; T ]:

Note that for each xed k ∈ N, uk ∈ Bk and hence, kuk (s)k ≤ k; ∀s ∈ [0; T ]. So, by (H200 ), there is a gk ∈ L1 (0; T ; X ) such that kf(s; uk (s))k ≤ gk (s), for almost all s ∈ (0; T ). Using this in (4.16) yields Z T kGuk (t)kC([0; T ]; X ) ≤ [2k xk  + T kyk]  + kg(uk )k + gk (s) ds: (4.17) 0

0

00

Invoking (H3 ) and (H4 ) in (4.17), we successively obtain limk→∞ k −1 kGuk kC([0; T ]; X )  ≤ lim

k→∞

(2k xk  + T kyk)k  −1 + k −1 kuk kC([0; T ]; X ) + k −1 + k −1

Z 0

T

 gk (s) ds

≤+ ¡1:

(4.18)

660

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

But, (4.18) contradicts Eq. (4.14). Consequently, there is an n0 ∈ N such that GBn0 ⊆ Bn0 . Hence, we may now apply Schauder’s xed-point theorem to conclude that there is a u ∈ Bn0 such that Gu = u. Clearly, this u is the integral solution we seek, and the proof is complete. Remark. An inspection of the proof shows that hypothesis (H30 ) can be weakened slightly in that rather than assuming the sublinear growth condition on g; we can only assume that kg(x)k = ¡∞: kxk C([0; T ]; X ) kxkC([0; T ]; X ) →∞ lim

Proof of Theorem 3.6. We follow the reasoning used in the proof of Theorem 3.5. The continuity of G and the fact that there is an n ∈ N such that GBn ⊆ Bn are proved in exactly the same way since the analog of Benilan’s inequality (2.3) still holds true in the time-dependent case, under our hypotheses (see, e.g., [14, 24]). The dierence comes in showing that G is a compact map. For this, we use the compactness of g and U and slightly modify the arguments given in [14, Theorems 2, 3, Lemma 4]. (In the proofs used in [14], the initial condition is xed, while in our case it varies in a compact set.) Proof of Theorem 3.7. Let v ∈ C([0; T ]; X ) be xed and consider the initial-value problem u0 (t) + @’t (u(t)) = f(t; v(t)); u(0) = g(v):

0¡t¡T;

(4.19)

From Kenmochi [15], it follows that there is a unique integral solution uv to Eq. (4.19) 1; 2 (0; T ; X ). De ne the map G : C([0; T ]; X ) → C([0; T ]; X ) on [0; T ] satisfying uv ∈ Wloc by Gv = uv . We will show that G has a xed point by applying Schauder’s xed-point theorem. The continuity of G and the fact that there is an n ∈ N such that GBn ⊆ Bn are proved as in Theorem 3.5. It remains to show that G is a compact map. Let K be a bounded subset of C([0; T ]; X ) and let K˜ = {(g(v); f(· ; v(·))): v ∈ K}. Since g is compact, there is an N such that kg(v)k ≤ N;

∀v ∈ K:

(4.20)

Also, {f(· ; v(·)): v ∈ K} is contained in a uniformly integrable set in L2 (0; T ; X ): (4.21) De ne the map T : K →K˜ by Tv = (g(v); f(· ; v(·))). Clearly, TK =K˜ is a bounded subset of D(’0 ) × L2 (0; T ; X ) because of (4.20) and (4.21). Let M = {uv : uv is a strong solution of Eq. (4.19), for some v ∈ K}. De ne R : K˜ → M by

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

661

R(g(v); f(· ; v(·))) = uv and observe that G = R ◦ T on K with G(K) = M . We will use the Arzela–Ascoli theorem to show that M is a precompact set in C([0; T ]; X ). First, let t = 0 and note that the set {uv (0): uv ∈ M } is clearly precompact because uv (0) = g(v) and g is compact. Now, to prove the equicontinuity, let wv be the integral solution of the initial-value problem wv0 + @’t (wv ) = 0;

0¡t¡T;

(4.22)

wv (0) = g(v): Let ¿0 be xed. Observe that by the triangle inequality, we have kuv (t) − uv (0)k ≤ kuv (t) − wv (t)k + kwv (t) − g(v)k:

(4.23)

Aslo by Theorem 2.2, as extended to the time-dependent case, we get Z t kf(s; v(s))k ds kuv (t) − wv (t)k ≤ 0

≤t

1=2

Z 0

t

2

1=2

kf(s; v(s))k ds

:

(4.24)

As t → 0+ , the right-hand side of (4.24) converges to 0 uniformly with respect to v since f varies in a uniformly integrable set in L2 (0; T ; X ) (cf. Eq. (4.21).) Thus, there is a 1 ¿0 such that 0¡t¡1 ⇒ kuv (t) − wv (t)k¡=2;

∀v ∈ K:

(4.25)

Next, we must control kwv (t) − g(v)k. Since K is a bounded subset of C([0; T ]; X ) and g is compact, we know that g(K) is contained in a compact set in X . So, g(K) ⊆

p [

B(g(vi ); =6)

for some p ∈ N;

(4.26)

i=1

where B(g(vi ); =6) denotes the ball centered at g(vi ) of radius =6 (in X ). Employing the triangle inequality and Theorem 2.2, along with Eq. (4.26), we see that for any v ∈ K, kwv (t) − g(v)k ≤ kwv (t) − wvi (t)k + kwvi (t) − wvi (0)k + kg(vi ) − g(v)k ≤ 2kg(vi ) − g(v)k + kwvi (t) − wvi (0)k ≤ =2;

(4.27)

provided that 0¡t¡2 , where 2 depends on  only. Thus, upon using (4.25) and (4.27) in (4.23), we conclude that kuv (t) − uv (0)k¡, for all v ∈ K, as soon as 0¡t¡ = min{1 ; 2 }. Hence, the family {uv (0) : v ∈ K} is equicontinuous.

662

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

Now, let t ∈ (0; T ] be xed. From [15, Lemmas 1.3.1, 1.3.2], it follows that there are positive constants N1 and N2 , both depending on kf(· ; v(·))kL2 (0; T ; X ) and kg(v)k, such that kuv (t)k ≤ N1 ; √ k tuv0 kL2 (0; T ; X ) ≤ N2 ;

(4.28) (4.29)

|t’t (uv (t))| ≤ N2 :

(4.30)

In view of (4.20) and (4.21), and [15, Remark 1.3.1], N1 and N2 can be chosen independent of v ∈ K, as genuine constants. Observe that (4.28) and (4.30) yield kuv (t)k + |’t (uv (t))| ≤ r

∀v ∈ K;

(4.31)

for some r¿0. Since, by assumption, the level subsets Ltr are precompact, (4.31) implies that {uv (t): v ∈ K} is precompact in X . It remains to show the equicontinuity. Choose 0¡¡t and consider the interval [; T ]. By (4.29), one has Z T Z T 1 √ 0 [ skuv (s)k]2 ds kuv0 (s)k2 ds = s   ≤

1 

Z

T



√ [ skuv0 (s)k2 ] ds

1 ≤ N2 ; 

(4.32)

for all v ∈ K. Thus, from (4.32), it follows that the family {uv (t): v ∈ K} is equicontinuous. We, therefore, conclude with the help of the Arzela–Ascoli theorem that the set M is precompact in C([0; T ]; X ), so that G is compact. By Schauder’s xedpoint theorem, G has a xed point u, which is the solution we seek. The proof is complete. Proof of Theorem 3.8. Throughout this proof C denotes a generic positive constant. Suppose that u is an integral solution to Eq. (3.1). We obtain a priori estimates on u in L1 (0; T ; X ) as follows: Let x ∈ D(A) be such that y ∈ Ax. Applying Theorem 2.2 along with the growth conditions imposed on g and f (see (3.4) and (3.5)), we derive the inequality:  + (m + a1 )kukL1 (0; T ; X ) : ku(t)k ≤ [2kxk + kg(0)k + ka2 kL1 (0; T ; X ) + T kyk]

(4.33)

Integrating (4.33) over the interval (0; T ) yields  (1 − (m + a1 )T )kukL1 (0; T ; X ) ≤ [2kxk + kg(0)k + ka2 kL1 (0; T ; X ) + T kyk]T: Since 1 − (m + a1 )T ¿0; we conclude that kukL1 (0; T ; X ) ≤

 [2kxk + kg(0)k + ka2 kL1 (0; T ; X ) + T kyk]T : 1 − (m + a1 )T

(4.34)

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

663

Using (4.34) in (4.33), we obtain that ku(t)k ≤ C; for all t ∈ [0; T ]. Therefore, by (3.5), |f(t; u(t))| ≤ a1 ku(t)k + a2 (t); ≤ a1 C + a2 (t);

a:e: on (0; T ):

(4.35)

Let (t) = a1 C + a2 (t). In view of (4.35), we will assume, without loss of generality, that |f(t; x)| ≤ (t), a.e on (0; T ); ∀x ∈ X . (This can be accomplished by replacing f, if necessary, by f(· ; rC (·)); where rC denotes the C-radial retraction on X .) Let K = {p ∈ L1 (0; T ; X ): kp(t)k ≤ (t); a:e on (0; T )}. Consider the initial-value problem u0 (t) + Au(t) 3 p(t);

0¡t¡T;

u(0) = g(u);

(4.36)

where p ∈ K. By minor adjustments to the proof of Theorem 3.3 in [2], we conclude that there is a unique integral solution up to Eq. (4.36) on [0; T ]. In addition, by the work leading to (4.34), {up : p ∈ K} is bounded in L1 (0; T ; X ). Hence, {g(up ): p ∈ K} is bounded in X . Clearly, K is a uniformly integrable subset of L1 (0; T ; X ). Since −A generates a compact semigroup, it follows from Theorem 2.5(1) that the set D = clL1 (0; T ; X ) {up : p ∈ K} is compact in L1 (0; T ; X ). (Here cl stands for closure.) De ne the map G : D → L1 (0; T ; X ) as follows: 1 1   G(u) = Sf(· ; u(·)) = {f ∈ L (0; T ; X ) :f(t) ∈ f(t; u(t)) a:e:}:

It is easy to show that G is l.s.c and has decomposable values. Thus, by Theorem 2.6, there is a continuous function  : D → L1 (0; T ; X ) such that (u) ∈ G(u), for all u ∈ D. Remark that (D) ⊆ K and that Q = co((D)) is a compact, convex subset of L1 (0; T ; X ) (cf, e.g., [10, pp. 414–416]) such that Q ⊆ K. (Here, co refers to the closed convex hull.) De ne the map : Q → L1 (0; T ; X ) by (p) = (up ); and observe that (Q) ⊆ Q since for any p ∈ Q; we have (p) = (up ) ∈ (D) ⊆ co((D)) = Q. Furthermore, note that is the composition of two continuous mappings (in particular, the fact that p → up is continuous from L1 (0; T ; X ) into itself follows from Theorem 2.2). Hence, we can apply Schauder’s xed-point theorem to on Q to conclude that there is a p0 ∈ Q such that (p0 ) = p0 . The corresponding u = up0 is the integral solution we seek. This completes the proof. 5. Examples Example 5.1. Let be a bounded domain in Rn (n¿2) with smooth boundary let p¿(n − 2)=n be xed. We consider the initial-boundary value problem Z t ut (t; x) −
x u|u|p−1 (t; x) 3 a(t − s)f(s; u(s; x)) ds; a:e: on (0; T ) × ; 0

and

664

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

Z u(0; x) =

T

0

u(t; x) = 0;

h(s; u(s; x)) ds;

a:e: on ;

(5.1)

a:e: on (0; T ) × :

The following conditions will be imposed on the data of Eq. (5.1): (i) f : [0; T ] × R → R satis es the Caratheodory conditions (see Section 3, (C1) and (C2)) and there are positive constants 1 and 1 such that |f(t; u)| ≤ 1 |u| + 1 for all u ∈ R; and almost all t ∈ (0; T );

(5.2)

(ii) a ∈ L1 (0; T ); (iii) h : [0; T ] × R → R satis es the Caratheodory conditions and there are positive constants 2 and 2 such that |h(t; u)| ≤ 2 |u| + 2 for all u ∈ R and almost all t ∈ (0; T ):

(5.3)

Theorem 5.1. Assume (i)–(iii). If we also have 1 − T (1 kakL1 (0;T ) + 2 )¿0; then Eq. (5.1) has at least one integral solution u ∈ C([0; T ]; L1 ( )). Proof. Let X = L1 ( ) and de ne Au =−
u|u|p−1 ; D(A) = {u ∈ L1 ( ): u|u|p−1 ∈ W01;1 ( ) and
u|u|p−1 ∈ L1 ( )}: Denote ut by u0 (t); and set Z t F(u)(t; x) = a(t − s)f(s; u(s; x)) ds Z g(u)(x) =

0

0

T

h(s; u(s; x)) ds

(5.4)

∀u ∈ L1 (0; T ; X );

∀u ∈ L1 (0; T ; X ):

With these substitutions, Eq. (5.1) can be written in the abstract form (1.1). We show that A; F; and g satisfy the conditions of Theorem 3.1. It is known that A (as given by Eq. (5.4)) is m-accretive in X with D(A) = X and −A generates a compact semigroup on X (see, e.g. [3]). Clearly, F and g are well-de ned and continuous from L1 (0; T ; X ) into L1 (0; T ; X ) and X , respectively. Next, let u ∈ L1 (0; T ; X ) be xed. Observe that by (5.2) kF(u)kL1 (0; T ; X ) ≤ kakL1 (0; T ) kfkL1 (0; T ; X ) ≤ kakL1 (0; T ) [1 kukL1 (0; T ; X ) + 1 T ]: Let c1 = 1 kakL1 (0; T ) and c2 = 1 T kakL1 (0; T ) . Then, kF(u)kL1 (0; T ; X ) ≤ c1 kukL1 (0; T ; X ) +c2 , for all u ∈ L1 (0; T ; X ); so that (H2) holds. Next, using (5.3), we get the inequalities Z T kh(s; u(s; x))kds ≤ 2 kukL1 (0; T ; X ) + 2 T: kg(u)k ≤ 0

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

665

Observe that if we choose d1 = 2 and d2 = 2 T , then g satis es (H3). Thus, all conditions of Theorem 3.1 are satis ed. As a consequence, Eq. (5.1) has at least one integral solution u ∈ C([0; T ]; L1 ( )). This completes the proof. Example 5.2. Let be a bounded domain in Rn (n ≥ 1) with smooth boundary and let : D( ) ⊆ R → 2R be m-accretive with 0 ∈ (0). Let p ∈ [2; ∞) and ¿0 be given. For each u ∈ W 1;p ( ), we de ne the pseudo-Laplacian operator by ! p−2 n X @ @u @u  (5.5) − u|u|p−2 ;
p u = @xi @xi @xi i=1

where the partial derivatives are taken in the sense of distributions over . Let Lp : D(Lp ) ⊆ L2 ( ) → L2 ( ) be given by Lp u = −
p u

∀u ∈ D(Lp );

(5.6)

where D(Lp ) = {u ∈ W 1;p ( ):
p u ∈ L2 ( ) and − (@u=@p )(x) ∈ (u(x)) a:e: on Here n X @u p−2 @u @u = cos(n · ei ); @x @p @xi

}:

i=1

where n is the outward unit normal to and {e1 ; : : : ; en } is the canonical basis of Rn . Consider the initial-boundary value problem ut (t; x) −
p u(t; x) 3 f1 (t; x; u(t; x)); a:e: on (0; T ) × ; Z Z T h(t; x; z; u(t; z)) dt dz; a:e on ; u(0; x) =

−

(5.7)

0

@u (t; x) ∈ (u(t; x)); @p

a:e: on (0; T ) × :

We impose the following restrictions on the data in Eq. (5.7): h : [0; T ] × × ×R → R is a function satisfying the Caratheodory conditions; that is, • For almost every (t; x; z) ∈ (0; T ) × × ; h(t; x; z; r) is a continuous function of r. • For each xed r ∈ R; h(t; x; z; r) is a measurable function of (t; x; z). Moreover, (i) |h(t; x; z; r) − h(t; x ; z; r)| ≤ k (t; x; x ; z);

(5.8) 1

for (t; x; z; r) and (t; x ; z; r) in [0; T ] × × × R with |r| ≤ k; where k ∈ L ([0; T ] × R RT

× × R; R+ ) is such that limx→x 0 k (t; x; x ; z)d t dz = 0 uniformly in x ∈ . (ii)  |h(t; x; z; r)| ≤ |r| + (t; x; z) for all r ∈ R; (5.9) Tm( ) where  ∈ L2 ((0; T ) × × ; R+ ) and ¿0.

666

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

Next, let f1 : [0; T ] × × R → R be a function satisfying the Caratheodory conditions (that is, measurable in (t; x) and continuous in its third variable) and the growth condition |f1 (t; x; u)| ≤ m1 (t)|u| + m2 (x);

(5.10)

where m1 ∈ L1 (0; T ; R+ ) and m2 ∈ L2 ( ; R+ ). Theorem 5.2. Assume the above conditions. If we also have  + ¡1; where  = km1 kL1 (0; T ; R+ ) ; then Eq. (5.7) has at least one integral solution u ∈ C([0; T ]; L2 ( )). Proof. Let X = L2 ( ) and set A = Lp (cf. Eqs. (5.5) and (5.6)). De ne f : [0; T ] × X → X and g : C([0; T ]; X ) → X by f(t; u)(x) = f1 (t; x; u(x)) ∀u ∈ X; a:e: on (0; T ) ×

and

Z Z g(u)(x) =



0

T

h(t; x; z; u(t; z)) dt dz

(5.11)

∀u ∈ C([0; T ]; X );

a:e: on :

(5.12)

With these de nitions and by letting u0 stand for ut , we can rewrite Eq. (5.7) in the abstract form (3.1). We now show that A; f; and g satisfy the conditions of Theorem 3.5. It is known that A is m-accretive on X and −A generates a compact semigroup on X (see [29, pp. 22, 23]); also, D(A) = X . Next, on account of (5.8), (5.9), and (5.12), it follows directly from Theorem 4.2 of [19, p. 175] that g is both compact and continuous. We now show that g satis es the growth condition in (H30 ). To this end, let u ∈ C([0; T ]; X ) be xed. Observe that Z Z T |h(t; x; z; u(t; z))| dt dz |g(u)(x)| ≤

0

 ≤ Tm( ) ≤

Z Z

 p T m( )

T

0

Z 0

T

Z Z |u(t; z)| dt dz + Z

|u(t; z)|2 dz

 = p kukC([0;T ]; L2 ( )) + m( )



(t; x; z) dt dz

1=2

Z Z dt +

Z Z

0

T

0

T



0

T

(t; x; z) dt dz

(t; x; z) dt dz:

(5.13)

In as much as  ∈ L2 ((0; T ) × × ), Eq. (5.13) implies that g satis es all of (H30 ) (with  and = (m( )T )1=2 kkL2 ([0; T ]× × ) ). Next, we show that f satis es (H200 ). First, note that from (5.10) and (5.11), it follows that f is a well-de ned mapping from [0; T ] × X into X which satis es the Caratheodory conditions. Secondly, let k ∈ N be xed and let u ∈ X be such that kukL2 ( ) ≤ k. Then, (5.10) yields

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

667

kf(t; u)kL2 ( ) ≤ m1 (t)k + km2 kL2 ( ) : Let gk (t) = m1 (t)k + km2 kL2 ( ) and note that gk ∈ L1 (0; T ; R+ ). Hence, sup kf(t; u)k ≤ gk (t);

kuk≤k

a:e: on (0; T );

as needed. Moreover, lim k

−1

k→∞

Z 0

T

gk (t) dt = lim k

−1

Z

k→∞

0

Z = lim

k→∞

T

0

T

[m1 (t)k + km2 kL2 ( ) ]dt

km2 kL2 ( ) T m1 (t) dt + k



= km1 kL1 (0;T ;R+ ) = ¡∞: Thus, (H200 ) satis ed. Consequently, Theorem 3.5 is applicable and Eq. (5.7) has at least one integral solution u ∈ C([0; T ]; L2 ( )). This completes the proof. Remark. When de ning the pseudo-Laplacian operator in Example 5.2; we can choose in various ways in order to handle a variety of applications including the black body radiation heat emission; natural convection; and thermostat control processes (see [29; pp: 23;24]). References [1] S. Aizicovici, Y. Ding, N.S. Papageorgiou, Time-dependent Volterra integral inclusions in Banach spaces, Discrete Continuous Dyn. Systems 2 (1996) 53– 63. [2] S. Aizicovici, Y. Gao, Functional dierential equations with nonlocal initial conditions, J. Appl. Math. Stochastic Anal. 10 (1997) 145–156. [3] P. Baras, Compacite de l’operateur f → u solution d’une e quation non-lineaire, (du=dt) + Au 3 f; C. R. Acad. Sci. Paris 286 (1978) 1113–1116. [4] V. Barbu, Nonlinear Semigroups and Dierential Equations in Banach Spaces, Noordho, Leyden, 1976. [5] H. Brezis, Operateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973. [6] H. Brezis, New results concerning monotone operators and nonlinear semigroups, Functional analysis and numerical analysis for nonlinear problems, Proc. Symp., Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1974, vol. 258, 1975, pp. 2–27. [7] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991) 494 –505. [8] L. Byszewski, Existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problem, Zeszyty Nauk. Politech. Rzeszowskiej Mat. Fiz. 18 (1993) 109 –112. [9] L. Byszewski, Dierential and Functional–Dierential Problems together with Nonlocal Conditions, Cracow University of Technology, Cracow, 1995 (in Polish). [10] N. Dunford, J.T. Schwartz, Linear Operators, Part I, Wiley Interscience, New York, 1958. [11] A. Fryszkowski, Continuous selections for a class of nonconvex multivalued maps, Studia Math. 76 (1983) 163–174. [12] D. Jackson, Existence and uniqueness of solutions to semilinear nonlocal parabolic equations, J. Math. Anal. Appl. 172 (1993) 256 –265.

668

S. Aizicovici, M. McKibben / Nonlinear Analysis 39 (2000) 649 – 668

[13] A.G. Kartsatos, A compact evolution operator generated by a nonlinear time-dependent m-accretive operator in a Banach space, Math. Ann. 302 (1995) 473 – 487. [14] A.G. Kartsatos, K.Y. Shin, Solvability of functional evolutions via compactness methods in general Banach spaces, Nonlinear Anal. 21 (1993) 517–535. [15] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Ed. Chiba Univ. 30 (1981) 1– 87. [16] Y. Konishi, Sur la compacite des semigroupes nonlineaires dans les espaces de Hilbert, Proc. Japan Acad. 48 (1972) 278–280. [17] J. Leray, J. Schauder, Topologie et e quations fonctionnelles, Ann. Sci. Ecole Norm. Sup. 51 (1934) 45–78. [18] Y. Lin, Analytical and numerical solutions for a class of nonlocal nonlinear parabolic dierential equations, SIAM J. Math. Anal. 25 (1994) 1577–1596. [19] R.H. Martin, Nonlinear Operators and Dierential Equations in Banach Spaces, Wiley, New York, 1976. [20] E. Mitidieri, I.I. Vrabie, Dierential inclusions governed by nonconvex perturbations of m-accretive operators, Dierential Integral Equations 2 (1989) 525–531. [21] S.K. Ntouyas, P.Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl. 210 (1997) 679– 687. [22] S.K. Ntouyas, P.Ch. Tsamatos, Global existence for semilinear evolution integrodierential equations with delay and nonlocal conditions, Appl. Anal. 64 (1997) 99–105. [23] N.S. Papageorgiou, Boundary value problems for evolution inclusions, Comm. Math. Univ. Carolinae 29 (1988) 355–363. [24] N.H. Pavel, Nonlinear Evolution Operators and Semigroups, Lecture Notes in Math., vol. 1260, Springer, Berlin, 1987. [25] S. Reich, Fixed points of condensing functions, J. Math. Anal. Appl. 41 (1973) 460 – 467. [26] S. Reich, Product formulas, nonlinear semigroups, and accretive operators, J. Funct. Anal. 36 (1980) 147–168.  [27] H. Schaefer, Uber die methode der apriori-shranken, Math. Ann. 129 (1955) 415– 416. [28] J.R. Ward, Boundary value problems for dierential equations in Banach space, J. Math. Anal. Appl. 70 (1979) 589–598. [29] I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Monographs Surveys Pure Appl. Math, vol. 32, Longman Scienti c and Technical, Harlow, 1987. [30] E. Zeidler, Nonlinear Functional Analysis and its Applications II=B. Nonlinear Monotone Operators, Springer, New York, 1990.