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A class of nonlinear evolution equations subjected to nonlocal initial conditions
Nonlinear Analysis 72 (2010) 4091–4100
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
A class of nonlinear evolution equations subjected to nonlocal initial conditions Angela Paicu a , Ioan I. Vrabie b,c,∗ a
University of Suceava, 720225, Romania
b
‘‘Al. I. Cuza’’ University, Iaşi 700506, Romania
c
‘‘O. Mayer’’ Mathematics Institute of the Romanian Academy, Iaşi 700506, Romania
article
info
Article history: Received 10 December 2009 Accepted 20 January 2010 MSC: primary 34A60 34C25 secondary 34G20 47H06 Keywords: Differential inclusion C 0 -solution Accretive operator Compact semigroup
abstract We prove the existence of C 0 -solutions for a class of nonlinear evolution equations subjected to nonlocal initial conditions, of the form:
u0 (t ) + Au(t ) 3 f (t ) f (t ) ∈ F (t , u(t )) u(0) = g (u), where A : D(A) ⊆ X ; X is an m-accretive operator acting on the infinite-dimensional Banach space X , F : [0, 2π] × D(A) ; X is an almost strongly weakly u.s.c. multi-function which satisfies an appropriate ‘‘sign’’ condition, while g : C ([0, 2π ]; D(A)) → D(A) is a continuous function. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Let X be a Banach space with norm k · k, A : D(A) ⊆ X ; X an m-accretive operator, let F : [0, 2π ] × D(A) ; X be an almost strongly weakly u.s.c. multi-function, and let g : C ([0, 2π ]; D(A)) → D(A) be a given continuous function. Let us consider the nonlinear evolution equation subjected to a nonlocal initial condition u0 (t ) + Au(t ) 3 f (t ) f (t ) ∈ F (t , u(t )) u(0) = g (u),
(
(1.1)
which represents the abstract formulation of many nonlinear problems of practical interest. Concerning the function g, appearing in the nonlocal condition, we mention here four remarkable cases covered by our general framework, i.e.: (i) g (u) = u(2π ) (ii) g (u) = −u(2π )
R 2π
(iii) g (u) = 21π 0 u(s) ds Pn Pn (iv) g (u) = i=1 αi u(ti ), where 0 < t1 < t2 < · · · < tn = 2π are arbitrary, but fixed and i=1 |αi | ≤ 1.
∗
Corresponding author at: ‘‘O. Mayer’’ Mathematics Institute of the Romanian Academy, Iaşi 700506, Romania. E-mail addresses: [email protected] (A. Paicu), [email protected] (I.I. Vrabie).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.01.041
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We notice that (i)–(iv) correspond to particular instances of the general choice of g as g ( u) =
1
2π
Z
µ([0, 2π ])
N(u(s)) dµ(s),
(1.2)
0
where N : D(A) → D(A) is a (possible nonlinear) nonexpansive operator and µ is a σ -finite and complete measure on [0, 2π] which is continuous with respect to the Lebesgue measure at t = 0, i.e. limh↓0 µ([0, h]) = 0. We emphasize that g given by (1.2) falls also into our general framework. Furthermore, the case in which µ : Σ → L(X ), Σ being the σ -field of Lebesgue measurable subsets in [0, 2π ], is a continuous with respect to the Lebesgue measure at t = 0 operator valued measure, is also covered by our main result, Theorem 3.2. Various specific forms of (1.1) have been intensively studied over the past two decades by many authors, mainly by using an interplay of compactness arguments and monotonicity methods. The problem (1.1), subjected to the periodic condition (i), was considered by Caşcaval–Vrabie [1], Hirano [2], Hirano–Shioji [3], Paicu [4], Vrabie [5] – for F single-valued – and Aizicovici–Papageorgiou–Staicu [6], Castaing–Monteiro Marques [7], Lakshmikantham–Papageorgiou [8], Papageorgiou [9], Shushuan Hu–Papageorgiou [10] and Paicu [11] – for F multi-valued. For the case of anti-periodic condition (ii), see [12] and the references therein. As far as we know, the first paper dealing with abstract nonlocal initial conditions for semilinear evolution equations is due to [13]. The fully nonlinear case was considered by Aizicovici–Lee [14], Aizicovici–McKibben [15], Aizicovici–Staicu [16], García-Falset [17] and García-Falset–Reich [18]. All these studies are motivated by the practical interests of such nonlocal Cauchy problems. For example, the diffusion of a gas through a thin transparent tube is described by a parabolic equation subjected to a nonlocal initial condition very close to the one imposed in (1.1). See [19]. In the present paper we prove a multi-valued variant of a result established in [18]. Our main theorem also extends the main results in both [11,4]. Unlike the case considered in [16], where F is assumed to be measurable in t ∈ [0, 2π ] and strongly weakly u.s.c. in u ∈ D(A) and X is reflexive, here, we focus our attention on the case when F is almost strongly weakly u.s.c. – which is nothing but a Scorza Dragoni type property for multi-functions – and we allow X to be nonreflexive. The paper is divided into five sections, Section 2 being mainly concerned with basic prerequisites. In Section 3 we state our main result, i.e. Theorem 3.2, in Section 4 we prove Theorem 3.2, while Section 5 contains two illustrating examples. 2. Preliminaries We assume familiarity with the theory of m-accretive operators and nonlinear evolution equations in Banach spaces, and we refer the reader to [20,21,5] for details. However, we recall for easy references some basic concepts and results in the field which we will use in what follows. Let X be a real Banach space X with norm k · k and let r > 0. We denote by D(0, r ) the closed ball with center 0 and radius r. Let x, y ∈ X and h ∈ R \ {0}. We denote by
[x, y]h :=
1
(kx + hyk − kxk), h and we recall that there exist the limit [x, y]+ = lim[x, y]h . h ↓0
Remark 2.1. For each x, y ∈ X and α > 0, we have (i) [α x, y]+ = [x, y]+ (ii) |[x, y]+ | ≤ kyk. For further details see [21]. An operator A : D(A) ⊆ X X is called m-accretive if it is accretive, and, in addition, R(I + λA) = X , for each λ > 0. Let f ∈ L1 (a, b; X ) and let us consider the evolution equation u0 (t ) + Au(t ) 3 f (t ).
(2.1)
A function u : [a, b] → X is called a C 0 -solution, or integral solution of (2.1) on [a, b], if u ∈ C ([a, b]; X ), u(t ) ∈ D(A) for each t ∈ [a, b] and u satisfies:
ku(t ) − xk ≤ ku(s) − xk +
t
Z
[u(τ ) − x, f (τ ) − y]+ dτ
(2.2)
s
for each x ∈ D(A), y ∈ Ax and a ≤ s ≤ t ≤ b. Remark 2.2. If u : [a, b] → D(A) is a C 0 -solution of (2.1) on then, in view of (ii) in Remark 2.1, it follows that
ku(t ) − xk ≤ ku(s) − xk +
t
Z
kf (τ ) − yk dτ s
for each x ∈ D(A), y ∈ Ax and a ≤ s ≤ t ≤ b.
(2.3)
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Theorem 2.1. Let A : D(A) ⊆ X X be an m-accretive operator. Then, for each x ∈ D(A) and f ∈ L1 (a, b; X ) there exists a unique C 0 -solution of (2.1) on [a, b] which satisfies u(a) = x. If f , g ∈ L1 (a, b; X ) and u, v are two C 0 -solutions of (2.1) corresponding to f and g, respectively then:
ku(t ) − v(t )k ≤ ku(s) − v(s)k +
t
Z
kf (τ ) − g (τ )k dτ
(2.4)
s
for each a ≤ s ≤ t ≤ b. See [20, Theorem 2.1, p. 124]. Let ξ ∈ D(A), τ ∈ [a, b) and f ∈ L1 (a, b; X ). We denote by u(·, τ , ξ , f ) the unique C 0 -solution v : [τ , b] → D(A), of the problem (2.1) which satisfies v(τ ) = ξ . We denote by {S (t ) : D(A) → D(A), t ≥ 0} the semigroup generated by −A on D(A), i.e., S (t )ξ = u(t , 0, ξ , 0) for each ξ ∈ D(A) and t ≥ 0. We say that the semigroup generated by −A on D(A) is compact if, for each t > 0, S (t ) is a compact operator. A subset F in L1 (a, b; X ) is uniformly integrable if, for each ε > 0 there exists δ(ε) > 0 such that, for each measurable subset E in [a, b] whose Lebesgue measure λ(E ) < δ(ε), we have
Z
kf (s)k ds ≤ ε, E
uniformly for f ∈ F. Remark 2.3. Let F ⊆ L1 (a, b; X ). It is easy to see that: (i) if F is uniformly integrable then it is norm bounded in L1 (a, b; X ); (ii) if F is bounded in Lp (a, b; X ) for some p > 1, then it is uniformly integrable; (iii) if there exists k ∈ L1 (a, b; R+ ) such that
kf (t )k ≤ k(t ) for each f ∈ F and a.e. t ∈ (a, b), then F is uniformly integrable. The following compactness result will be useful in that follows. Theorem 2.2. Let A : D(A) ⊆ X ; X be m-accretive and such that −A generates a compact semigroup. Let B ⊆ D(A) be bounded and let F be uniformly integrable in L1 (a, b; X ). Then, for each c ∈ (a, b), the C 0 -solutions set
{u(·, a, ξ , f ); ξ ∈ B, f ∈ F} is relatively compact in C ([c , b]; X ). If, in addition B is relatively compact then the C 0 -solutions set is relatively compact even in C ([a, b]; X ). See [22] or Theorem 2.3.3, p. 47, in [23]. Theorem 2.3. Let (Ω , Σ , µ) be a finite measure space and let X be a Banach space. Let F ⊆ L1 (Ω , µ; X ) be bounded and uniformly integrable. If for each ε > 0 there exist a weakly compact subset Cε ⊆ X and a measurable subset Eε ∈ Σ with µ(Ω \ Eε ) ≤ ε and f (Eε ) ⊆ Cε for all f ∈ F, then F is weakly relatively compact in L1 (Ω , µ; X ). See [24], or [25, p. 117]. We conclude this section with a variant of a general result due to [26]. Theorem 2.4. Let K be a nonempty, convex and compact set in a separated locally convex space and let Q : K ; K be a nonempty, closed and convex valued multi-function with closed graph. Then Q has at least one fixed point, i.e. there exists f ∈ K such that f ∈ Q (f ). Since, in a Banach space, the weak closure of a weakly relatively compact set coincides with its weak sequential closure – see [27, Theorem 8.12.1, p. 549] –, from Theorem 2.4, we deduce: Theorem 2.5. Let K be a nonempty, convex and weakly compact set in Banach space and let Q : K ; K be a nonempty, closed and convex valued multi-function with sequentially closed graph. Then Q has at least one fixed point, i.e. there exists f ∈ K such that f ∈ Q (f ). 3. The main result Definition 3.1. The m-accretive operator A is called of compact type if for each sequences (fn )n in L1 (0, 2π ; X ) and (un )n in C ([0, 2π]; X ), with un a C 0 -solution on [0, 2π ] of the problem u0n (t ) + Aun (t ) 3 fn (t ),
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with limn fn = f weakly in L1 (0, 2π; X ) and limn un = u strongly in C ([0, 2π ]; X ), it follows that u is a C 0 solution of the limit problem u0 (t ) + Au(t ) 3 f (t ) on [0, 2π]. If the topological dual of X is uniformly convex and −A generates a compact semigroup, then A is of complete continuous type. See Corollary 2.3.1, p. 49, in [23]. An example of an m-accretive operator of complete continuous type in a nonreflexive Banach space (and, by consequence, whose dual is not uniformly convex) is the nonlinear diffusion operator 1ϕ in L1 (Ω ). Namely, let Ω be a bounded domain in Rn , n ≥ 1, with smooth boundary Γ , and let ∆ be the Laplace operator in the sense of distributions over Ω . If ϕ : D(ϕ) ⊆ R ; R, and u : Ω → D(ϕ), we denote by Sϕ (u) = {v ∈ L1 (Ω ); v(x) ∈ ϕ(u(x)), a.e. for x ∈ Ω }.
Theorem 3.1. Let Ω be a nonempty, bounded and open subset in Rn with C 1 boundary Γ and let ϕ : D(ϕ) ⊆ R ; R be maximal monotone with 0 ∈ ϕ(0). Then the operator 1ϕ : D(1ϕ) ⊆ L1 (Ω ) ; L1 (Ω ), defined by 1 ,1
D(1ϕ) = {u ∈ L1 (Ω ); ∃v ∈ Sϕ (u) ∩ W0 (Ω ), 1v ∈ L1 (Ω )} 1ϕ(u) = {1v; v ∈ Sϕ (u) ∩ W01,1 (Ω )} ∩ L1 (Ω ) for u ∈ D(1ϕ),
is m-dissipative on L1 (Ω ). If, in addition, ϕ : R → R is continuous on R and C 1 on R \ {0} and there exist two constants C > 0 and a > 0 if n ≤ 2 and a > (n − 2)/n if n ≥ 3 such that
ϕ 0 (r ) ≥ C |r |a−1 for each r ∈ R \ {0}, then 1ϕ generates a compact semigroup. Moreover, for each arbitrary but fixed ξ ∈ L1 (Ω ), the mapping f 7→ u(·, τ , ξ , f ) is weakly strongly sequentially continuous from L1 (τ , T ; L1 (Ω )) to C ([τ , T ]; L1 (Ω )) and thus 1ϕ is of complete continuous type. See [28, Theorems 1.7.7–1.7.9, p. 22], and [29, Lemma 2.7.2, p. 71]. Definition 3.2. A multi-function F : [0, 2π ] × D(A) ; X is said to be almost strongly weakly u.s.c. if for each ε > 0 there exists a Lebesgue measurable subset Eε ⊆ [0, 2π ] whose Lebesgue measure λ(Eε ) ≤ ε and such that F it is a u.s.c. from ([0, 2π ] \ Eε ) × D(A) – endowed with the strong topology – to X – endowed with the weak topology. Throughout, we denote by C ([0, 2π ]; D(A)) the subset of C ([0, 2π ]; X ) consisting in all functions u in with u(t ) ∈ D(A) for each t ∈ [0, 2π ]. The assumptions we need in that follows are listed below.
(H1 ) A : D(A) ⊆ X X is an operator with the properties: (a1 ) A is m-accretive operator and 0 ∈ A0; (a2 ) the semigroup generated by −A on D(A) is compact; (a3 ) A is of complete continuous type. (H2 ) F : [0, 2π ] × D(A) X is a nonempty, convex and weakly compact valued almost strongly weakly upper semicontinuous multi-function.
(H3 ) There exists r > 0 such that for each t ∈ R+ , each u ∈ D(A) with kuk = r and for each z ∈ F (t , u) we have [u, z ]+ ≤ 0. (H3 ) There exists r > 0 such that for each t ∈ R+ , each u ∈ D(A) with kuk ≥ r and for each z ∈ F (t , u) we have 0
[u, z ]+ ≤ 0. (H4 ) There exists k ∈ L1 (0, 2π ; R+ ) such that kyk ≤ k(t ) a.e. for t ∈ (0, 2π ), for each u ∈ D(0, r ) and each y ∈ F (t , u). (H4 0 ) There exists k ∈ L1 (0, 2π ; R+ ) such that
kyk ≤ k(t ) a.e. for t ∈ (0, 2π ), for each u ∈ X and y ∈ F (t , u). (H5 ) g : C ([0, 2π ]; D(A)) → D(A) is a function with the properties: (g1 ) for each U ⊆ C ([0, 2π ]; D(A)) which is bounded in C ([0, 2π ]; D(A)) and relatively compact in C ([δ, 2π ]; D(A)) for each δ ∈ (0, 2π ], the set g (U) is relatively compact; (g2 ) for each u ∈ C ([0, 2π ]; D(A)), we have kg (u)k ≤ kuk∞ ;
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(g3 ) for each u, v ∈ C ([0, 2π ]; D(A)), we have kg (u) − g (v)k ≤ ku − vk∞ . Remark 3.1. Condition (H3 ) ensures the invariance of D(0, r ) with respect to C 0 -solutions of the problem
u0 (t ) ∈ Au(t ) + f (t ) f (t ) ∈ F (t , u(t )).
Namely, it implies that each C 0 -solution issuing from an initial point in D(0, r ) does not escape D(0, r ). Condition (g1 ) is satisfied by all functions g of the general form (1.2) with µ continuous with respect to the Lebesgue measure at t = 0. Thus, it is satisfied by any g of the form (i)–(iv) in the Introduction. Condition (g2 ) substitutes the first part of the condition (C2) in [18] which assumes that for n ≥ 1 and each u ∈ C ([0, 2π ]; D(A)) with kuk∞ ≤ r, we have kS ( 1n )g (u)k ≤ r, where r > 0 is the constant appearing in (H3 ) and {S (t ) : D(A) → D(A); t ≥ 0} is the semigroup of nonexpansive mappings generated by −A. Finally, (g3 ) is slightly more general than condition (P5) in [18]. Remark 3.2. We notice that the condition
(g3 0 ) for each u, v ∈ C ([0, 2π ]; D(A)), we have 1 ku − vkL1 (0,2π ;X ) , 2π implies both (g1 ) and (g3 ). Indeed, (g1 ) is a simple consequence of Theorem A.1.1, p. 292 and Lemma A.1.3, p. 295 in [30], while (g3 ) is obvious.
kg (u) − g (v)k ≤
Remark 3.3. We emphasize that whenever the function g is defined as in (i)–(iv) in the Introduction, then g satisfies (H5 ). Now we may proceed to the statement of our main result. Theorem 3.2. If (H1 )–(H5 ) are satisfied, then the problem (1.1) has at least one C 0 -solution. We will prove Theorem 3.2 with the help of: Theorem 3.3. If (H1 ), (H2 ), (H3 0 ), (H4 0 ) and (H5 ) are satisfied, then the problem (1.1) has at least one C 0 -solution, u : [0, 2π ] → D(0, r ) ∩ D(A). The proof of Theorem 3.3 is divided into four steps. Let F = {f ∈ L1 (0, 2π; X ); kf (t )k ≤ k(t )},
where k is the function given by (H4 ). Firstly, we show that, for each ε ∈ (0, 1) and f ∈ F, the problem
u0 (t ) + Au(t ) = f (t ) u(0) = (1 − ε)g (u)
(3.1)
has a unique C 0 -solution ufε ∈ C ([0, 2π ]; X ). Secondly, we prove that for each fixed ε ∈ (0, 1), the operator f 7→ ufε , which associates to f the unique C 0 -solution ufε of the problem (3.1), is compact from F to C ([0, 2π ]; X ). Thirdly, as F is almost strongly weakly u.s.c., for the very same ε > 0, there exists Eε ⊆ [2, π] whose Lebesgue measure λ(Eε ) ≤ ε and such that F|([0,2π ]\Eε )×D(A) is strongly weakly u.s.c. Let us define the multi-function Fε : [0, 2π ] × D(A) ; X , by Fε (t , u) =
F (t , u) {0}
for (t , u) ∈ ([0, 2π ] \ Eε ) × D(A) for (t , u) ∈ Eε × D(A).
(3.2)
Further, we will show that the multi-function f 7→ Sel (Fε (·, ufε (·))), where Sel(Fε (·, ufε (·))) = {h ∈ L1 (0, 2π; X ); h(t ) ∈ Fε (t , ufε (t )) a.e. t ∈ [0, 2π ]}, maps a suitably chosen nonempty, convex and weakly compact set ⊆ L1 (0, 2π ; X ) into itself, and has weakly × weakly sequentially closed graph. Then, in view of Theorem 2.5, this mapping has at least one fixed point which, by means of f 7→ ufε , produces a C 0 -solution for the approximating problem u0 (t ) + Au(t ) = f (t ) f (t ) ∈ Fε (t , u(t )) u(0) = (1 − ε)g (u).
(
(3.3)
Fourthly and finally, for each ε > 0, we fix a C 0 -solution uε of the problem (3.3), and we show that there exists a sequence εn ↓ 0 such that (uεn )n converges in C ([0, 2π ]; X ) to a C 0 -solution of the problem (1.1).
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4. Proof of Theorem 3.2 As we already have noticed, for the sake of convenience and clarity, we divide the proof of Theorem 3.2 into four steps which are labeled as four lemmas. Lemma 4.1. Let us assume that (a1 ) in (H1 ) and (g3 ) in (H5 ) are satisfied. Then, for each ε > 0 and each f ∈ L1 (0, 2π ; X ), the problem (3.1) has a unique C 0 -solution ufε which satisfies
kufε k∞ ≤
1
2π
Z
ε
kf (s)k ds.
(4.1)
0
Proof. Let ε ∈ (0, 1) be arbitrary but fixed. In view of Theorem 2.1, for each v ∈ C ([0, 2π]; D(A)), the Cauchy problem
u0 (t ) + Au(t ) = f (t ) u(0) = (1 − ε)g (v)
(4.2)
has a unique C 0 -solution u ∈ C ([0, 2π ]; D(A)). Let Pε : C ([0, 2π ]; D(A)) → C ([0, 2π ]; D(A)) be defined by Pε (v) = u, where u is the unique C 0 -solution of the problem (3.1). According to (2.4) and (g3 ) in (H5 ), we have
kPε (v)(t ) − Pε (˜v )(t )k ≤ (1 − ε)kg (v) − g (˜v )k ≤ (1 − ε)kv − v˜ k∞ . In view of Banach fixed point theorem, the operator Pε has a unique fixed point ufε ∈ C ([0, 2π ]; D(A)) which, clearly, is a C 0 -solution of (3.1). Next, let tm ∈ [0, 2π ] be any maximum point of the mapping t 7→ ku(t )k, i.e. kufε (tm )k = kufε k∞ . Since 0 ∈ A0, from (2.2), we get
kufε k∞ = kufε (tm )k ≤ (1 − ε)kufε k∞ +
tm
Z
kf (s)k ds.
0
From this inequality we deduce that tm 6= 0 unless ufε ≡ 0. Thus
kufε k∞ ≤
1
ε
2π
Z
kf (s)k ds,
0
as claimed, and this completes the proof.
Lemma 4.2. Let us assume that (a1 ), (a2 ) in (H1 ) and (H5 ) are satisfied, let k ∈ L1 (0, 2π ; R+ ) and let ε > 0 be fixed. Then the operator f 7→ ufε , where ufε is the unique solution of the problem (3.1) corresponding to f , is compact from F = {f ∈ L1 (0.2π ; X ); kf (t )k ≤ k(t ) a.e. for t ∈ (0, 2π )}
to C ([0, 2π ]; X ). In particular, the image of F by the operator f 7→ ufε is compact. Proof. From (4.1), it follows that {ufε ; f ∈ F} is bounded in C ([0, 2π ]; D(A)). In view of (g2 ) in (H5 ), it follows that {ufε (0); f ∈ F} is bounded in X . Since F is uniformly integrable, from (a2 ) and Theorem 2.2, we conclude that, for each δ ∈ (0, 2π], {ufε ; f ∈ F} is relatively compact in C ([δ, 2π ]; D(A)). Thanks to (g1 ) in (H1 ), we deduce that the set {g (ufε ); f ∈ F}, which coincides with {ufε (0); f ∈ F}, is relatively compact in X . Again, from (a2 ) and the last part of Theorem 2.2, it follows that {ufε ; f ∈ F} is relatively compact in C ([0, 2π ]; D(A)). In order to complete the proof, we have to show that f 7→ ufε is continuous from F, endowed with the norm of 1 L (0, 2π; X ), to C ([0, 2π ]; D(A)) endowed with the uniform convergence topology, i.e. with the sup-norm topology. To this aim, let us observe that, in view of (2.4) and (g3 ) in (H5 ), we successively have
kufε (t ) − uhε (t )k ≤ (1 − ε)kufε − uhε k∞ +
2π
Z
kf (s) − h(s)k ds
0
and
kufε − uhε k∞ ≤
1
ε
2π
Z
kf (s) − h(s)k ds
0
for each f , h ∈ F. Thus f 7→ ufε is Lipschitz, with constant 1/ε , from F, endowed the L1 (0, 2π ; X )-norm, to C ([0, 2π ]; D(A)) endowed with the sup-norm. This completes the proof. Lemma 4.3. Let us assume that (H1 ), (H2 ), (H4 0 ) and (H5 ) are satisfied. Then, for each ε > 0 the problem (3.3) has at least a solution uε .
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Proof. Let k ∈ L1 (0, 2π ; R+ ) be given by (H4 0 ) and let F be defined as in Lemma 4.2. Since by (iii) in Remark 2.3, F is uniformly integrable in L1 (0, 2π ; X ), from (a2 ) in (H1 ) and Theorem 2.2, we deduce that the set
n
f
C = uε (t ); f ∈ F, t ∈ [0, 2π ]
o
is compact in D(A). Further, since the restriction of Fε to ([0, 2π ]\ Eε )× D(A) is strongly weakly u.s.c. and has weakly compact values, from Lemma 2.6.1, p. 47, in [28] and Theorem 4, p. 434 in [31], we deduce that the set Gε = conv Fε (([0, 2π ] \ Eε ) × C ) is weakly compact X . Hence Hε = conv Fε ([0, 2π ] × C ) = conv [Fε (([0, 2π ] \ Eε ) × C ) ∪ {0}] is weakly compact X too. Let ε
= {f ∈ F; f (t ) ∈ Hε , a.e. for t ∈ (0, 2π )}.
Let us now define the operator Qε :
ε
L1 (0, 2π ; X ) by
Qε f := Sel Fε (·, ufε (·)), where ufε is the unique solution of the problem (3.3) corresponding to f ∈ ε . We may easily see that Qε is well defined and maps the set ε into itself. In addition, thanks to (H2 ), (H4 0 ) and Theorem 2.3, it follows that Qε has nonempty, convex and weakly compact values in ε . More than this, its graph is weakly × weakly sequentially closed. Indeed, let ((fn , gn ))n be a sequence in the graph of Qε , weakly × weakly convergent to (f , g ) ∈ L1 (0, 2π ; X ) × L1 (0, 2π ; X ). Then, taking into account Lemma 4.2 and the fact that A is of complete continuous type – see (H1 ) –, we get lim ufεn (t ) = ufε (t )
n→∞
uniformly for t ∈ [0, 2π ]. Since gn (t ) ∈ Fε (t , ufεn (t )) for each n ∈ N and a.e. t ∈ [0, 2π ], by Theorem 3.1.2, p. 88, in [23], it follows that g (t ) ∈ Fε (t , ufε (t ))
(4.3)
a.e. t ∈ [0, 2π ]\ Eε . On the other hand, gn (t ) = g (t ) = 0 a.e. for t ∈ Eε , and consequently (4.3) holds true a.e. for t ∈ [0, 2π ]. So, the graph of Qε is weakly × weakly sequentially closed. By Theorem 2.5, Qε has at least a fixed point f ∈ . Since by means of f 7→ ufε , this fixed point f produces a C 0 -solution of the problem (3.3), this completes the proof of Lemma 4.3. Lemma 4.4. Let us assume that (H1 ), (H2 ), (H3 0 ), (H4 0 ) and (H5 ) are satisfied. Then, for each ε ∈ (0, 1), each C 0 -solution u of the problem (3.3) is uniformly bounded by r given by (H3 ), i.e., ku(t )k ≤ r for each t ∈ [0, 2π ]. Proof. Let u be an arbitrary integral solution of (3.3) and let us assume, by contradiction, kuk∞ > r, i.e., that there exists tm ∈ [0, 2π ] such that
kuk∞ = ku(tm )k > r . Clearly tm 6= 0 since, otherwise, from the nonlocal initial condition and (g2 ) in (H5 ), we get
kuk∞ = ku(0)k = (1 − ε)kg (u)k ≤ (1 − ε)kuk∞ which implies that 0 < r < kuk∞ = 0 — a contradiction. So, tm ∈ (0, 2π ]. Further, let us observe that u cannot be constant on [0, tm ]. Indeed, if u(t ) = ξ for each t ∈ [0, tm ], again from the nonlocal initial condition and (g2 ) in (H5 ), we deduce r < kuk∞ = kξ k ≤ (1 − ε)kuk∞ = (1 − ε)kξ k which shows that ξ = 0 which is impossible as long as 0 < r < kξ k. Consequently, we know that u is not constant on [0, tm ] and 0 < r < ku(tm )k, with tm ∈ (0, 2π ]. This shows that there exists t0 ∈ (0, tm ] such that r < ku(t0 )k < ku(s)k ≤ ku(tm )k = kuk∞ for each s ∈ (t0 , tm ]. Recalling that 0 ∈ A0 – see (a1 ) in (H1 ) – and using (2.2) with x = 0 and y = 0, we get r < ku(tm )k ≤ ku(t0 )k +
tm
Z 0
[u(s), f (s)]+ ds.
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Then, using (H3 0 ) – with z = f (s) – and (g2 ) if s ∈ [0, tm ] \ Eε and (ii) in Remark 2.1 with f (s) = y = 0 for s ∈ Eε , we conclude r < ku(tm )k ≤ ku(t0 )k < ku(tm )k, which is a contradiction. So, kuk∞ ≤ r, and this completes the proof.
We can now pass to the proof of Theorem 3.3. Proof. Let (εn )n be a sequence with εn ↓ 0, let (un )n be the sequence of the C 0 -solutions of the problem (3.3) corresponding to ε = εn and let (fn )n be such that u0n (t ) + Aun (t ) 3 fn (t ), t ∈ [0, 2π ] fn (t ) ∈ Fεn (t , un (t )), a.e. for t ∈ (0, 2π ) un (0) = (1 − εn )g (un ).
(
From (H4 0 ), we deduce that the set {fn ; n ∈ N} is uniformly integrable in L1 (0, 2π ; X ). From this observation, Lemma 4.4, (a2 ) in (H1 ) and Theorem 2.2, it follows that, for each δ ∈ (0, 2π ), the set {un ; n ∈ N} is relatively compact in C ([δ, 2π ]; D(A)). In view of (g1 ) in (H5 ), we deduce that the set
{un (0); n ∈ N} = {(1 − εn )g (un ); n ∈ N} is relatively compact in D(A). From the second part of Theorem 2.2, we conclude that {un ; n ∈ N} is relatively compact in C ([0, 2π ]; D(A)). So, C = {un (t ); n ∈ N, t ∈ [0, 2π ]} is compact in X . Since the restriction of Fεn to ([0, 2π ] \ Eεn ) × D(A) is strongly weakly u.s.c., again by Lemma 2.6.1, p. 47 in [28] combined with Krein–Šmulian Theorem 4, p. 434 in [31], we have that the set conv Fεn ([0, 2π] × C ) = conv[Fεn (([0, 2π ] \ Eεn ) × C ) ∪ {0}] is weakly compact in X . By Theorem 2.3, it follows that {fn ; n ∈ N} is sequentially relatively compact in L1 (0, 2π ; X ). So, on a subsequence at least, we have
lim f = f weakly in L1 (0, 2π ; X ) n n lim un (t ) = u(t ) uniformly for t ∈ [0, 2π] n lim(1 − ε )g (u ) = g (u). n n n
Hence, from [23, Theorem 3.1.2, p. 88], we get f (t ) ∈ F (t , u(t )) a.e. t ∈ [0, 2π ]. Since As of complete continuous type, it follows that u is a C 0 -solution of the problem (1.1) corresponding to the selection f of t 7→ F (t , u(t )). From Lemma 4.4 it follows that u : [0, 2π ] → D(0, r ) ∩ D(A), and this concludes the proof of Theorem 3.3. Finally, we can proceed to the proof of Theorem 3.2. Proof. Let ρ : X → D(0, r ) be defined by
ρ(u) =
u r kuk−1 u
if u ∈ D(0, r ) if u ∈ X \ D(0, r ).
Since ρ is continuous, it follows that the multi-function Fρ : [0, 2π ] × X ; X defined by Fρ (t , u) = F (t , ρ(u)), for each (t , u) ∈ [0, 2π ] × X , satisfies the conditions (H2 ), and (H4 0 ). Thanks to (i) in Remark 2.1, Fρ satisfies (H3 0 ) too. Hence, by virtue of Theorem 3.3, the problem u0 (t ) + Au(t ) = f (t ) f (t ) ∈ Fρ (t , u(t )) u(0) = g (u)
(
has at least one C 0 -solution u : [0, 2π ] → D(0, r ) ∩ D(A). Inasmuch as ku(t )k ≤ r for each t ∈ [0, 2π ], we conclude that Fρ (t , u(t )) = F (t , u(t )) for each t ∈ [0, 2π ], and thus u is a C 0 -solution of (1.1) as claimed. The proof is complete.
A. Paicu, I.I. Vrabie / Nonlinear Analysis 72 (2010) 4091–4100
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5. Examples Example 5.1. Let Ω be a nonempty bounded and open subset in Rn with C 2 boundary Γ , let p ∈ [2, ∞) and λ > 0 and let us consider the nonlinear problem
! p−2 n X ∂u ∂ ∂ u ∂ u − λ|u|p−2 u + F (t , u) ∈ ∂x ∂ t ∂ x ∂ x i i i i=1 ∂u ∈ β(u) − ∂νp Z T 1 u(0) = N(u(t )) dµ(t ). µ([0, T ]) 0
on (0, 2π ) × Ω on (0, 2π ) × Γ
(5.1)
Here
n X ∂ u p−2 ∂ u ∂u − → − → = cos( n , ei ), ∂x ∂νp ∂ xi i i=1 − →
− → − →
− →
where n is the outward normal of Γ and { e1 , e2 , . . . , en } is the canonical base in Rn , F (t , u) = [f1 (t , u) + h, f2 (t , u) + h] with fi : [0, 2π ] × R → R for i = 1, 2 and h ∈ L2 (Ω ). From Theorem 3.2, we deduce: Theorem 5.1. Let β : D(β) ⊆ R ; R be a maximal monotone operator with 0 ∈ D(β) and 0 ∈ β(0), let h ∈ L2 (Ω ) and let fi : [0, 2π] × R → R, i = 1, 2 be two given functions satisfying
(F1 ) (F2 ) (F3 ) (F4 )
f1 (t , u) ≤ f2 (t , u) for each (t , u) ∈ [0, 2π ] × R f1 is i.s.c. f2 is u.s.c. there exist a, b ∈ R+ such that |fi (t , u)| ≤ a|u| + b for i = 1, 2 and all (t , u) ∈ [0, 2π ] × R there exists c > 0 such that ufi (t , u) ≤ −cu2 for i = 1, 2 and all (t , u) ∈ [0, 2π ] × R.
Let N : L2 (Ω ) → L2 (Ω ) be a nonexpansive (possible nonlinear) operator and let µ be a σ -finite and complete measure on [0, 2π ] which is continuous with respect to the Lebesgue measure at t = 0. Then, the problem (5.1) has at least one C 0 -solution u ∈ C ([0, 2π ]; L2 (Ω )) which, for each δ ∈ (0, 2π ), satisfies u ∈ AC ([0, 2π ]; W 1,p (Ω )) ∩ W 1,2 ([δ, 2π ]; L2 (Ω )). Proof. Let A : D(A) ⊆ L2 (Ω ) → L2 (Ω ) be defined by
( ! ) n X ∂ ∂ u p−2 ∂ u 1 ,p p−2 2 D(A) = u ∈ W (ω); − λ|u| u ∈ L (Ω ) ∂ xi ∂ xi ∂ xi i =1 ! n X ∂ ∂ u p−2 ∂ u − λ|u|p−2 u, Au = ∂x ∂ x ∂ x i i i i=1 and g : C ([0, 2π ]; L2 (Ω )) → L2 (Ω ), g (u) =
1
µ([0, 2π])
2π
Z
N(u(s)) dµ(s)
0
for u ∈ C ([0, 2π ]; L2 (Ω )). With A, F and g as above, the problem (5.1) can be rewritten in the form (1.1). By Example 1.5.4, p. 18, in [23], we know that A is m-dissipative on L2 (Ω ) and 0 ∈ A0. Moreover, it generates a compact semigroup of nonexpansive mappings on L2 (Ω ) and is of complete continuous type. See Example 2.2.4, p. 43 and Corollary 2.3.2, p. 50 in [23]. Hence A satisfies (H1 ). A similar argument as the one in the proof of Problem 2.6.1, p. 46 in [28], shows that F is a nonempty, convex and weakly compact valued strongly weakly u.s.c. multi-function. So F satisfies (H2 ). From (F3 ) and (F4 ), we conclude that F satisfies (H3 ) and (H4 ) with r ≥ c −1 khkL2 (Ω ) . Since µ is continuous with respect to the Lebesgue measure at t = 0, it follows that limh↓0 µ([0, h]) = 0, and thus g satisfies (g1 ). Since (g2 ) and (g3 ) are obvious, we deduce that (H5 ) holds. An appeal to Theorem 3.2 completes the proof. Example 5.2. Let Ω be a nonempty, bounded and open subset in Rn with C 1 boundary Γ and let ϕ : R → R be continuous and nondecreasing function. Let us consider the nonlinear diffusion equation subjected to nonlocal initial conditions
ut (t , x) ∈ 1ϕ(u(t , x)) + f (t , x) in [0, 2π ] × Ω f (t , x) ∈ F (t , u(t , x)) in [0, 2π ] × Ω ϕ(u(t , x)) = 0 on [0, 2π ] × Γ Z 2π 1 u(0, x) = N(u(s, ·))(x) dµ(s) in [0, 2π ] × Ω . µ([0, 2π ]) 0
(5.2)
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Theorem 5.2. Let Ω be a nonempty, bounded and open subset in Rn with C 1 boundary Γ and let ϕ : R → R be continuous on R and C 1 on R \ {0} and for which there exist two constants C > 0 and a > 0 if n ≤ 2 and a > (n − 2)/n if n ≥ 3 such that
ϕ 0 (r ) ≥ C |r |a−1 for each r ∈ R \ {0}. Let F (t , u) = [f1 (t , u) + h, f2 (t , u) + h] with fi : [0, 2π ] × R → R, i = 1, 2, satisfying (F1 )–(F4 ) in Theorem 5.1 and let h ∈ L1 (Ω ). Let N : L1 (Ω ) → L1 (Ω ) a (possible nonlinear) nonexpansive operator. Then the problem (5.2) has at least one C 0 -solution u ∈ C ([τ , T ]; L1 (Ω )). Proof. Clearly (H1 ) is implied by Theorem 3.1. Since the proofs of (H2 )–(H5 ) follow the very same lines as before, this completes the proof. Acknowledgement The second author was partially supported by PN-II-ID-PCE-2007-1 Grant ID 397. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
R. Caşcaval, I.I. Vrabie, Existence of periodic solutions for a class of nonlinear evolution equations, Rev. Mat. Univ. Complut. Madrid 7 (1994) 325–338. N. Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. Amer. Math. Soc. 120 (1994) 185–192. N. Hirano, N. Shioji, Invariant sets for nonlinear evolution equations, Cauchy problems and periodic problems, Abstr. Appl. Anal. 3 (2004) 183–203. A. Paicu, Periodic solutions for a class of nonlinear evolution equations in Banach spaces, An. Ştiinţ. ‘‘Al. I. Cuza’’, Iaşi, Ser. Nouă Mat. LV (2009) 107–118. I.I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (3) (1990) 653–661. S. Aizicovici, N.S. Papageorgiou and, V. Staicu, Periodic solutions of nonlinear evolution inclusions in Banach spaces, J. Nonlinear & Convex. Anal. 7 (2006) 163–177. C. Castaing, D.P. Monteiro-Marques, Periodic solutions of evolution problems associated with a moving convex set, C.R. Acad. Sci. Paris, Série A 321 (1995) 531–536. V. Lakshmikantham, N.S. Papageorgiou, Periodic solutions of nonlinear evolution inclusions, Comput. Appl. Math. 52 (1994) 277–286. N.S. Papageorgiou, Periodic trajectories for evolution inclusions associated with time-dependent subdifferentials, Ann. Univ. Sci. Budapest. 37 (1994) 139–155. Hu Shuchuan, N.S. Papageorgiou, On the existence of periodic solutions for a class of nonlinear inclusions, Boll. Unione Mat. Ital. 71 (1993) 591–605. A. Paicu, Periodic solutions for a class of differential inclusions in general Banach spaces, J. Math. Anal. Appl. 337 (2008) 1238–1248. S. Aizicovici, N.H. Pavel, I.I. Vrabie, Anti-periodic solutions to strongly nonlinear evolution equations in Hilbert spaces, An. Ştiinţ. Univ. ‘‘Al. I. Cuza’’ Iaşi, Secţ. I a Mat. XLIV (1998) 227–234. L. Byszewski, Theorems about the existence and uniqueness of solutions of semilinear evolution nonlocal Cauchy problems, J. Math. Anal. Appl. 162 (1991) 494–505. S. Aizicovici, H. Lee, Nonlinear nonlocal Cauchy problems in Banach spaces, Appl. Math. Lett. 18 (2005) 401–407. S. Aizicovici, M. McKibben, Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear Anal. 39 (2000) 649–668. S. Aizicovici, V. Staicu, Multivalued evolution equations with nonlocal initial conditions in Banach spaces, NoDEA Nonlinear Differential Equations Appl. 14 (2007) 361–376. J. García-Falset, Existence results and asymptotic behaviour for nonlocal abstract Cauchy problems, J. Math. Anal. Appl. 338 (2008) 639–652. J. García-Falset, S. Reich, Integral solutions to a class of nonlocal evolution equations, Commun. Contemp. Math. (in press). K. Deng, Exponential decay of solutions of semilinear parabolic equations with initial boundary conditions, J. Math. Anal. Appl. 179 (1993) 630–637. V. Barbu, Nonlinear Semigroups and Differential Equation in Banach Spaces, Editura Academiei, Bucureşti, Noordhoff, 1976. V. Lakshmikantham, S. Leela, Nonlinear Differential Equations in Abstract Spaces, in: International Series in Nonlinear Mathematics, vol. 2, Pergamon Press, 1981. P. Baras, Compacité de l’opérateur definissant la solution d’une équation d’évolution non linéaire (du/dt ) + Au 3 f , C. R. Acad. Sci. Sér. I Math. 286 (1978) 1113–1116. I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, second ed., in: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 75, Longman and John Wiley & Sons, 1995. J. Diestel, Remarks on weak compactness in L1 (µ; X ), Glasgow. Math. J. 18 (1977) 87–91. J. Diestel, J.J. Uhl Jr., Vector measures, Math. Surveys 15 (1977). I.L. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952) 170–174. R.E. Edwards, Functional Analysis, Holt, Rinehart and Winston, 1965. O. Cârjă, M. Necula and, I.I. Vrabie, Viability, Invariance and Applications, in: Elsevier North-Holland Mathematics Studies, vol. 207, 2007. J.I. Diaz and, I.I. Vrabie, Existence for reaction diffusion systems: A compactness method approach, J. Math. Anal. Appl. 188 (1994) 521–540. I.I. Vrabie, C0 -Semigroups and Applications, in: North-Holland Mathematics Studies, vol. 191, Elsevier, 2003. N. Dunford, J.T. Schwartz, Linear Operators Part I: General Theory, Interscience Publishers, Inc., New York, 1958.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
A class of nonlinear evolution equations subjected to nonlocal initial conditions Angela Paicu a , Ioan I. Vrabie b,c,∗ a
University of Suceava, 720225, Romania
b
‘‘Al. I. Cuza’’ University, Iaşi 700506, Romania
c
‘‘O. Mayer’’ Mathematics Institute of the Romanian Academy, Iaşi 700506, Romania
article
info
Article history: Received 10 December 2009 Accepted 20 January 2010 MSC: primary 34A60 34C25 secondary 34G20 47H06 Keywords: Differential inclusion C 0 -solution Accretive operator Compact semigroup
abstract We prove the existence of C 0 -solutions for a class of nonlinear evolution equations subjected to nonlocal initial conditions, of the form:
u0 (t ) + Au(t ) 3 f (t ) f (t ) ∈ F (t , u(t )) u(0) = g (u), where A : D(A) ⊆ X ; X is an m-accretive operator acting on the infinite-dimensional Banach space X , F : [0, 2π] × D(A) ; X is an almost strongly weakly u.s.c. multi-function which satisfies an appropriate ‘‘sign’’ condition, while g : C ([0, 2π ]; D(A)) → D(A) is a continuous function. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Let X be a Banach space with norm k · k, A : D(A) ⊆ X ; X an m-accretive operator, let F : [0, 2π ] × D(A) ; X be an almost strongly weakly u.s.c. multi-function, and let g : C ([0, 2π ]; D(A)) → D(A) be a given continuous function. Let us consider the nonlinear evolution equation subjected to a nonlocal initial condition u0 (t ) + Au(t ) 3 f (t ) f (t ) ∈ F (t , u(t )) u(0) = g (u),
(
(1.1)
which represents the abstract formulation of many nonlinear problems of practical interest. Concerning the function g, appearing in the nonlocal condition, we mention here four remarkable cases covered by our general framework, i.e.: (i) g (u) = u(2π ) (ii) g (u) = −u(2π )
R 2π
(iii) g (u) = 21π 0 u(s) ds Pn Pn (iv) g (u) = i=1 αi u(ti ), where 0 < t1 < t2 < · · · < tn = 2π are arbitrary, but fixed and i=1 |αi | ≤ 1.
∗
Corresponding author at: ‘‘O. Mayer’’ Mathematics Institute of the Romanian Academy, Iaşi 700506, Romania. E-mail addresses: [email protected] (A. Paicu), [email protected] (I.I. Vrabie).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.01.041
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We notice that (i)–(iv) correspond to particular instances of the general choice of g as g ( u) =
1
2π
Z
µ([0, 2π ])
N(u(s)) dµ(s),
(1.2)
0
where N : D(A) → D(A) is a (possible nonlinear) nonexpansive operator and µ is a σ -finite and complete measure on [0, 2π] which is continuous with respect to the Lebesgue measure at t = 0, i.e. limh↓0 µ([0, h]) = 0. We emphasize that g given by (1.2) falls also into our general framework. Furthermore, the case in which µ : Σ → L(X ), Σ being the σ -field of Lebesgue measurable subsets in [0, 2π ], is a continuous with respect to the Lebesgue measure at t = 0 operator valued measure, is also covered by our main result, Theorem 3.2. Various specific forms of (1.1) have been intensively studied over the past two decades by many authors, mainly by using an interplay of compactness arguments and monotonicity methods. The problem (1.1), subjected to the periodic condition (i), was considered by Caşcaval–Vrabie [1], Hirano [2], Hirano–Shioji [3], Paicu [4], Vrabie [5] – for F single-valued – and Aizicovici–Papageorgiou–Staicu [6], Castaing–Monteiro Marques [7], Lakshmikantham–Papageorgiou [8], Papageorgiou [9], Shushuan Hu–Papageorgiou [10] and Paicu [11] – for F multi-valued. For the case of anti-periodic condition (ii), see [12] and the references therein. As far as we know, the first paper dealing with abstract nonlocal initial conditions for semilinear evolution equations is due to [13]. The fully nonlinear case was considered by Aizicovici–Lee [14], Aizicovici–McKibben [15], Aizicovici–Staicu [16], García-Falset [17] and García-Falset–Reich [18]. All these studies are motivated by the practical interests of such nonlocal Cauchy problems. For example, the diffusion of a gas through a thin transparent tube is described by a parabolic equation subjected to a nonlocal initial condition very close to the one imposed in (1.1). See [19]. In the present paper we prove a multi-valued variant of a result established in [18]. Our main theorem also extends the main results in both [11,4]. Unlike the case considered in [16], where F is assumed to be measurable in t ∈ [0, 2π ] and strongly weakly u.s.c. in u ∈ D(A) and X is reflexive, here, we focus our attention on the case when F is almost strongly weakly u.s.c. – which is nothing but a Scorza Dragoni type property for multi-functions – and we allow X to be nonreflexive. The paper is divided into five sections, Section 2 being mainly concerned with basic prerequisites. In Section 3 we state our main result, i.e. Theorem 3.2, in Section 4 we prove Theorem 3.2, while Section 5 contains two illustrating examples. 2. Preliminaries We assume familiarity with the theory of m-accretive operators and nonlinear evolution equations in Banach spaces, and we refer the reader to [20,21,5] for details. However, we recall for easy references some basic concepts and results in the field which we will use in what follows. Let X be a real Banach space X with norm k · k and let r > 0. We denote by D(0, r ) the closed ball with center 0 and radius r. Let x, y ∈ X and h ∈ R \ {0}. We denote by
[x, y]h :=
1
(kx + hyk − kxk), h and we recall that there exist the limit [x, y]+ = lim[x, y]h . h ↓0
Remark 2.1. For each x, y ∈ X and α > 0, we have (i) [α x, y]+ = [x, y]+ (ii) |[x, y]+ | ≤ kyk. For further details see [21]. An operator A : D(A) ⊆ X X is called m-accretive if it is accretive, and, in addition, R(I + λA) = X , for each λ > 0. Let f ∈ L1 (a, b; X ) and let us consider the evolution equation u0 (t ) + Au(t ) 3 f (t ).
(2.1)
A function u : [a, b] → X is called a C 0 -solution, or integral solution of (2.1) on [a, b], if u ∈ C ([a, b]; X ), u(t ) ∈ D(A) for each t ∈ [a, b] and u satisfies:
ku(t ) − xk ≤ ku(s) − xk +
t
Z
[u(τ ) − x, f (τ ) − y]+ dτ
(2.2)
s
for each x ∈ D(A), y ∈ Ax and a ≤ s ≤ t ≤ b. Remark 2.2. If u : [a, b] → D(A) is a C 0 -solution of (2.1) on then, in view of (ii) in Remark 2.1, it follows that
ku(t ) − xk ≤ ku(s) − xk +
t
Z
kf (τ ) − yk dτ s
for each x ∈ D(A), y ∈ Ax and a ≤ s ≤ t ≤ b.
(2.3)
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Theorem 2.1. Let A : D(A) ⊆ X X be an m-accretive operator. Then, for each x ∈ D(A) and f ∈ L1 (a, b; X ) there exists a unique C 0 -solution of (2.1) on [a, b] which satisfies u(a) = x. If f , g ∈ L1 (a, b; X ) and u, v are two C 0 -solutions of (2.1) corresponding to f and g, respectively then:
ku(t ) − v(t )k ≤ ku(s) − v(s)k +
t
Z
kf (τ ) − g (τ )k dτ
(2.4)
s
for each a ≤ s ≤ t ≤ b. See [20, Theorem 2.1, p. 124]. Let ξ ∈ D(A), τ ∈ [a, b) and f ∈ L1 (a, b; X ). We denote by u(·, τ , ξ , f ) the unique C 0 -solution v : [τ , b] → D(A), of the problem (2.1) which satisfies v(τ ) = ξ . We denote by {S (t ) : D(A) → D(A), t ≥ 0} the semigroup generated by −A on D(A), i.e., S (t )ξ = u(t , 0, ξ , 0) for each ξ ∈ D(A) and t ≥ 0. We say that the semigroup generated by −A on D(A) is compact if, for each t > 0, S (t ) is a compact operator. A subset F in L1 (a, b; X ) is uniformly integrable if, for each ε > 0 there exists δ(ε) > 0 such that, for each measurable subset E in [a, b] whose Lebesgue measure λ(E ) < δ(ε), we have
Z
kf (s)k ds ≤ ε, E
uniformly for f ∈ F. Remark 2.3. Let F ⊆ L1 (a, b; X ). It is easy to see that: (i) if F is uniformly integrable then it is norm bounded in L1 (a, b; X ); (ii) if F is bounded in Lp (a, b; X ) for some p > 1, then it is uniformly integrable; (iii) if there exists k ∈ L1 (a, b; R+ ) such that
kf (t )k ≤ k(t ) for each f ∈ F and a.e. t ∈ (a, b), then F is uniformly integrable. The following compactness result will be useful in that follows. Theorem 2.2. Let A : D(A) ⊆ X ; X be m-accretive and such that −A generates a compact semigroup. Let B ⊆ D(A) be bounded and let F be uniformly integrable in L1 (a, b; X ). Then, for each c ∈ (a, b), the C 0 -solutions set
{u(·, a, ξ , f ); ξ ∈ B, f ∈ F} is relatively compact in C ([c , b]; X ). If, in addition B is relatively compact then the C 0 -solutions set is relatively compact even in C ([a, b]; X ). See [22] or Theorem 2.3.3, p. 47, in [23]. Theorem 2.3. Let (Ω , Σ , µ) be a finite measure space and let X be a Banach space. Let F ⊆ L1 (Ω , µ; X ) be bounded and uniformly integrable. If for each ε > 0 there exist a weakly compact subset Cε ⊆ X and a measurable subset Eε ∈ Σ with µ(Ω \ Eε ) ≤ ε and f (Eε ) ⊆ Cε for all f ∈ F, then F is weakly relatively compact in L1 (Ω , µ; X ). See [24], or [25, p. 117]. We conclude this section with a variant of a general result due to [26]. Theorem 2.4. Let K be a nonempty, convex and compact set in a separated locally convex space and let Q : K ; K be a nonempty, closed and convex valued multi-function with closed graph. Then Q has at least one fixed point, i.e. there exists f ∈ K such that f ∈ Q (f ). Since, in a Banach space, the weak closure of a weakly relatively compact set coincides with its weak sequential closure – see [27, Theorem 8.12.1, p. 549] –, from Theorem 2.4, we deduce: Theorem 2.5. Let K be a nonempty, convex and weakly compact set in Banach space and let Q : K ; K be a nonempty, closed and convex valued multi-function with sequentially closed graph. Then Q has at least one fixed point, i.e. there exists f ∈ K such that f ∈ Q (f ). 3. The main result Definition 3.1. The m-accretive operator A is called of compact type if for each sequences (fn )n in L1 (0, 2π ; X ) and (un )n in C ([0, 2π]; X ), with un a C 0 -solution on [0, 2π ] of the problem u0n (t ) + Aun (t ) 3 fn (t ),
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with limn fn = f weakly in L1 (0, 2π; X ) and limn un = u strongly in C ([0, 2π ]; X ), it follows that u is a C 0 solution of the limit problem u0 (t ) + Au(t ) 3 f (t ) on [0, 2π]. If the topological dual of X is uniformly convex and −A generates a compact semigroup, then A is of complete continuous type. See Corollary 2.3.1, p. 49, in [23]. An example of an m-accretive operator of complete continuous type in a nonreflexive Banach space (and, by consequence, whose dual is not uniformly convex) is the nonlinear diffusion operator 1ϕ in L1 (Ω ). Namely, let Ω be a bounded domain in Rn , n ≥ 1, with smooth boundary Γ , and let ∆ be the Laplace operator in the sense of distributions over Ω . If ϕ : D(ϕ) ⊆ R ; R, and u : Ω → D(ϕ), we denote by Sϕ (u) = {v ∈ L1 (Ω ); v(x) ∈ ϕ(u(x)), a.e. for x ∈ Ω }.
Theorem 3.1. Let Ω be a nonempty, bounded and open subset in Rn with C 1 boundary Γ and let ϕ : D(ϕ) ⊆ R ; R be maximal monotone with 0 ∈ ϕ(0). Then the operator 1ϕ : D(1ϕ) ⊆ L1 (Ω ) ; L1 (Ω ), defined by 1 ,1
D(1ϕ) = {u ∈ L1 (Ω ); ∃v ∈ Sϕ (u) ∩ W0 (Ω ), 1v ∈ L1 (Ω )} 1ϕ(u) = {1v; v ∈ Sϕ (u) ∩ W01,1 (Ω )} ∩ L1 (Ω ) for u ∈ D(1ϕ),
is m-dissipative on L1 (Ω ). If, in addition, ϕ : R → R is continuous on R and C 1 on R \ {0} and there exist two constants C > 0 and a > 0 if n ≤ 2 and a > (n − 2)/n if n ≥ 3 such that
ϕ 0 (r ) ≥ C |r |a−1 for each r ∈ R \ {0}, then 1ϕ generates a compact semigroup. Moreover, for each arbitrary but fixed ξ ∈ L1 (Ω ), the mapping f 7→ u(·, τ , ξ , f ) is weakly strongly sequentially continuous from L1 (τ , T ; L1 (Ω )) to C ([τ , T ]; L1 (Ω )) and thus 1ϕ is of complete continuous type. See [28, Theorems 1.7.7–1.7.9, p. 22], and [29, Lemma 2.7.2, p. 71]. Definition 3.2. A multi-function F : [0, 2π ] × D(A) ; X is said to be almost strongly weakly u.s.c. if for each ε > 0 there exists a Lebesgue measurable subset Eε ⊆ [0, 2π ] whose Lebesgue measure λ(Eε ) ≤ ε and such that F it is a u.s.c. from ([0, 2π ] \ Eε ) × D(A) – endowed with the strong topology – to X – endowed with the weak topology. Throughout, we denote by C ([0, 2π ]; D(A)) the subset of C ([0, 2π ]; X ) consisting in all functions u in with u(t ) ∈ D(A) for each t ∈ [0, 2π ]. The assumptions we need in that follows are listed below.
(H1 ) A : D(A) ⊆ X X is an operator with the properties: (a1 ) A is m-accretive operator and 0 ∈ A0; (a2 ) the semigroup generated by −A on D(A) is compact; (a3 ) A is of complete continuous type. (H2 ) F : [0, 2π ] × D(A) X is a nonempty, convex and weakly compact valued almost strongly weakly upper semicontinuous multi-function.
(H3 ) There exists r > 0 such that for each t ∈ R+ , each u ∈ D(A) with kuk = r and for each z ∈ F (t , u) we have [u, z ]+ ≤ 0. (H3 ) There exists r > 0 such that for each t ∈ R+ , each u ∈ D(A) with kuk ≥ r and for each z ∈ F (t , u) we have 0
[u, z ]+ ≤ 0. (H4 ) There exists k ∈ L1 (0, 2π ; R+ ) such that kyk ≤ k(t ) a.e. for t ∈ (0, 2π ), for each u ∈ D(0, r ) and each y ∈ F (t , u). (H4 0 ) There exists k ∈ L1 (0, 2π ; R+ ) such that
kyk ≤ k(t ) a.e. for t ∈ (0, 2π ), for each u ∈ X and y ∈ F (t , u). (H5 ) g : C ([0, 2π ]; D(A)) → D(A) is a function with the properties: (g1 ) for each U ⊆ C ([0, 2π ]; D(A)) which is bounded in C ([0, 2π ]; D(A)) and relatively compact in C ([δ, 2π ]; D(A)) for each δ ∈ (0, 2π ], the set g (U) is relatively compact; (g2 ) for each u ∈ C ([0, 2π ]; D(A)), we have kg (u)k ≤ kuk∞ ;
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(g3 ) for each u, v ∈ C ([0, 2π ]; D(A)), we have kg (u) − g (v)k ≤ ku − vk∞ . Remark 3.1. Condition (H3 ) ensures the invariance of D(0, r ) with respect to C 0 -solutions of the problem
u0 (t ) ∈ Au(t ) + f (t ) f (t ) ∈ F (t , u(t )).
Namely, it implies that each C 0 -solution issuing from an initial point in D(0, r ) does not escape D(0, r ). Condition (g1 ) is satisfied by all functions g of the general form (1.2) with µ continuous with respect to the Lebesgue measure at t = 0. Thus, it is satisfied by any g of the form (i)–(iv) in the Introduction. Condition (g2 ) substitutes the first part of the condition (C2) in [18] which assumes that for n ≥ 1 and each u ∈ C ([0, 2π ]; D(A)) with kuk∞ ≤ r, we have kS ( 1n )g (u)k ≤ r, where r > 0 is the constant appearing in (H3 ) and {S (t ) : D(A) → D(A); t ≥ 0} is the semigroup of nonexpansive mappings generated by −A. Finally, (g3 ) is slightly more general than condition (P5) in [18]. Remark 3.2. We notice that the condition
(g3 0 ) for each u, v ∈ C ([0, 2π ]; D(A)), we have 1 ku − vkL1 (0,2π ;X ) , 2π implies both (g1 ) and (g3 ). Indeed, (g1 ) is a simple consequence of Theorem A.1.1, p. 292 and Lemma A.1.3, p. 295 in [30], while (g3 ) is obvious.
kg (u) − g (v)k ≤
Remark 3.3. We emphasize that whenever the function g is defined as in (i)–(iv) in the Introduction, then g satisfies (H5 ). Now we may proceed to the statement of our main result. Theorem 3.2. If (H1 )–(H5 ) are satisfied, then the problem (1.1) has at least one C 0 -solution. We will prove Theorem 3.2 with the help of: Theorem 3.3. If (H1 ), (H2 ), (H3 0 ), (H4 0 ) and (H5 ) are satisfied, then the problem (1.1) has at least one C 0 -solution, u : [0, 2π ] → D(0, r ) ∩ D(A). The proof of Theorem 3.3 is divided into four steps. Let F = {f ∈ L1 (0, 2π; X ); kf (t )k ≤ k(t )},
where k is the function given by (H4 ). Firstly, we show that, for each ε ∈ (0, 1) and f ∈ F, the problem
u0 (t ) + Au(t ) = f (t ) u(0) = (1 − ε)g (u)
(3.1)
has a unique C 0 -solution ufε ∈ C ([0, 2π ]; X ). Secondly, we prove that for each fixed ε ∈ (0, 1), the operator f 7→ ufε , which associates to f the unique C 0 -solution ufε of the problem (3.1), is compact from F to C ([0, 2π ]; X ). Thirdly, as F is almost strongly weakly u.s.c., for the very same ε > 0, there exists Eε ⊆ [2, π] whose Lebesgue measure λ(Eε ) ≤ ε and such that F|([0,2π ]\Eε )×D(A) is strongly weakly u.s.c. Let us define the multi-function Fε : [0, 2π ] × D(A) ; X , by Fε (t , u) =
F (t , u) {0}
for (t , u) ∈ ([0, 2π ] \ Eε ) × D(A) for (t , u) ∈ Eε × D(A).
(3.2)
Further, we will show that the multi-function f 7→ Sel (Fε (·, ufε (·))), where Sel(Fε (·, ufε (·))) = {h ∈ L1 (0, 2π; X ); h(t ) ∈ Fε (t , ufε (t )) a.e. t ∈ [0, 2π ]}, maps a suitably chosen nonempty, convex and weakly compact set ⊆ L1 (0, 2π ; X ) into itself, and has weakly × weakly sequentially closed graph. Then, in view of Theorem 2.5, this mapping has at least one fixed point which, by means of f 7→ ufε , produces a C 0 -solution for the approximating problem u0 (t ) + Au(t ) = f (t ) f (t ) ∈ Fε (t , u(t )) u(0) = (1 − ε)g (u).
(
(3.3)
Fourthly and finally, for each ε > 0, we fix a C 0 -solution uε of the problem (3.3), and we show that there exists a sequence εn ↓ 0 such that (uεn )n converges in C ([0, 2π ]; X ) to a C 0 -solution of the problem (1.1).
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4. Proof of Theorem 3.2 As we already have noticed, for the sake of convenience and clarity, we divide the proof of Theorem 3.2 into four steps which are labeled as four lemmas. Lemma 4.1. Let us assume that (a1 ) in (H1 ) and (g3 ) in (H5 ) are satisfied. Then, for each ε > 0 and each f ∈ L1 (0, 2π ; X ), the problem (3.1) has a unique C 0 -solution ufε which satisfies
kufε k∞ ≤
1
2π
Z
ε
kf (s)k ds.
(4.1)
0
Proof. Let ε ∈ (0, 1) be arbitrary but fixed. In view of Theorem 2.1, for each v ∈ C ([0, 2π]; D(A)), the Cauchy problem
u0 (t ) + Au(t ) = f (t ) u(0) = (1 − ε)g (v)
(4.2)
has a unique C 0 -solution u ∈ C ([0, 2π ]; D(A)). Let Pε : C ([0, 2π ]; D(A)) → C ([0, 2π ]; D(A)) be defined by Pε (v) = u, where u is the unique C 0 -solution of the problem (3.1). According to (2.4) and (g3 ) in (H5 ), we have
kPε (v)(t ) − Pε (˜v )(t )k ≤ (1 − ε)kg (v) − g (˜v )k ≤ (1 − ε)kv − v˜ k∞ . In view of Banach fixed point theorem, the operator Pε has a unique fixed point ufε ∈ C ([0, 2π ]; D(A)) which, clearly, is a C 0 -solution of (3.1). Next, let tm ∈ [0, 2π ] be any maximum point of the mapping t 7→ ku(t )k, i.e. kufε (tm )k = kufε k∞ . Since 0 ∈ A0, from (2.2), we get
kufε k∞ = kufε (tm )k ≤ (1 − ε)kufε k∞ +
tm
Z
kf (s)k ds.
0
From this inequality we deduce that tm 6= 0 unless ufε ≡ 0. Thus
kufε k∞ ≤
1
ε
2π
Z
kf (s)k ds,
0
as claimed, and this completes the proof.
Lemma 4.2. Let us assume that (a1 ), (a2 ) in (H1 ) and (H5 ) are satisfied, let k ∈ L1 (0, 2π ; R+ ) and let ε > 0 be fixed. Then the operator f 7→ ufε , where ufε is the unique solution of the problem (3.1) corresponding to f , is compact from F = {f ∈ L1 (0.2π ; X ); kf (t )k ≤ k(t ) a.e. for t ∈ (0, 2π )}
to C ([0, 2π ]; X ). In particular, the image of F by the operator f 7→ ufε is compact. Proof. From (4.1), it follows that {ufε ; f ∈ F} is bounded in C ([0, 2π ]; D(A)). In view of (g2 ) in (H5 ), it follows that {ufε (0); f ∈ F} is bounded in X . Since F is uniformly integrable, from (a2 ) and Theorem 2.2, we conclude that, for each δ ∈ (0, 2π], {ufε ; f ∈ F} is relatively compact in C ([δ, 2π ]; D(A)). Thanks to (g1 ) in (H1 ), we deduce that the set {g (ufε ); f ∈ F}, which coincides with {ufε (0); f ∈ F}, is relatively compact in X . Again, from (a2 ) and the last part of Theorem 2.2, it follows that {ufε ; f ∈ F} is relatively compact in C ([0, 2π ]; D(A)). In order to complete the proof, we have to show that f 7→ ufε is continuous from F, endowed with the norm of 1 L (0, 2π; X ), to C ([0, 2π ]; D(A)) endowed with the uniform convergence topology, i.e. with the sup-norm topology. To this aim, let us observe that, in view of (2.4) and (g3 ) in (H5 ), we successively have
kufε (t ) − uhε (t )k ≤ (1 − ε)kufε − uhε k∞ +
2π
Z
kf (s) − h(s)k ds
0
and
kufε − uhε k∞ ≤
1
ε
2π
Z
kf (s) − h(s)k ds
0
for each f , h ∈ F. Thus f 7→ ufε is Lipschitz, with constant 1/ε , from F, endowed the L1 (0, 2π ; X )-norm, to C ([0, 2π ]; D(A)) endowed with the sup-norm. This completes the proof. Lemma 4.3. Let us assume that (H1 ), (H2 ), (H4 0 ) and (H5 ) are satisfied. Then, for each ε > 0 the problem (3.3) has at least a solution uε .
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Proof. Let k ∈ L1 (0, 2π ; R+ ) be given by (H4 0 ) and let F be defined as in Lemma 4.2. Since by (iii) in Remark 2.3, F is uniformly integrable in L1 (0, 2π ; X ), from (a2 ) in (H1 ) and Theorem 2.2, we deduce that the set
n
f
C = uε (t ); f ∈ F, t ∈ [0, 2π ]
o
is compact in D(A). Further, since the restriction of Fε to ([0, 2π ]\ Eε )× D(A) is strongly weakly u.s.c. and has weakly compact values, from Lemma 2.6.1, p. 47, in [28] and Theorem 4, p. 434 in [31], we deduce that the set Gε = conv Fε (([0, 2π ] \ Eε ) × C ) is weakly compact X . Hence Hε = conv Fε ([0, 2π ] × C ) = conv [Fε (([0, 2π ] \ Eε ) × C ) ∪ {0}] is weakly compact X too. Let ε
= {f ∈ F; f (t ) ∈ Hε , a.e. for t ∈ (0, 2π )}.
Let us now define the operator Qε :
ε
L1 (0, 2π ; X ) by
Qε f := Sel Fε (·, ufε (·)), where ufε is the unique solution of the problem (3.3) corresponding to f ∈ ε . We may easily see that Qε is well defined and maps the set ε into itself. In addition, thanks to (H2 ), (H4 0 ) and Theorem 2.3, it follows that Qε has nonempty, convex and weakly compact values in ε . More than this, its graph is weakly × weakly sequentially closed. Indeed, let ((fn , gn ))n be a sequence in the graph of Qε , weakly × weakly convergent to (f , g ) ∈ L1 (0, 2π ; X ) × L1 (0, 2π ; X ). Then, taking into account Lemma 4.2 and the fact that A is of complete continuous type – see (H1 ) –, we get lim ufεn (t ) = ufε (t )
n→∞
uniformly for t ∈ [0, 2π ]. Since gn (t ) ∈ Fε (t , ufεn (t )) for each n ∈ N and a.e. t ∈ [0, 2π ], by Theorem 3.1.2, p. 88, in [23], it follows that g (t ) ∈ Fε (t , ufε (t ))
(4.3)
a.e. t ∈ [0, 2π ]\ Eε . On the other hand, gn (t ) = g (t ) = 0 a.e. for t ∈ Eε , and consequently (4.3) holds true a.e. for t ∈ [0, 2π ]. So, the graph of Qε is weakly × weakly sequentially closed. By Theorem 2.5, Qε has at least a fixed point f ∈ . Since by means of f 7→ ufε , this fixed point f produces a C 0 -solution of the problem (3.3), this completes the proof of Lemma 4.3. Lemma 4.4. Let us assume that (H1 ), (H2 ), (H3 0 ), (H4 0 ) and (H5 ) are satisfied. Then, for each ε ∈ (0, 1), each C 0 -solution u of the problem (3.3) is uniformly bounded by r given by (H3 ), i.e., ku(t )k ≤ r for each t ∈ [0, 2π ]. Proof. Let u be an arbitrary integral solution of (3.3) and let us assume, by contradiction, kuk∞ > r, i.e., that there exists tm ∈ [0, 2π ] such that
kuk∞ = ku(tm )k > r . Clearly tm 6= 0 since, otherwise, from the nonlocal initial condition and (g2 ) in (H5 ), we get
kuk∞ = ku(0)k = (1 − ε)kg (u)k ≤ (1 − ε)kuk∞ which implies that 0 < r < kuk∞ = 0 — a contradiction. So, tm ∈ (0, 2π ]. Further, let us observe that u cannot be constant on [0, tm ]. Indeed, if u(t ) = ξ for each t ∈ [0, tm ], again from the nonlocal initial condition and (g2 ) in (H5 ), we deduce r < kuk∞ = kξ k ≤ (1 − ε)kuk∞ = (1 − ε)kξ k which shows that ξ = 0 which is impossible as long as 0 < r < kξ k. Consequently, we know that u is not constant on [0, tm ] and 0 < r < ku(tm )k, with tm ∈ (0, 2π ]. This shows that there exists t0 ∈ (0, tm ] such that r < ku(t0 )k < ku(s)k ≤ ku(tm )k = kuk∞ for each s ∈ (t0 , tm ]. Recalling that 0 ∈ A0 – see (a1 ) in (H1 ) – and using (2.2) with x = 0 and y = 0, we get r < ku(tm )k ≤ ku(t0 )k +
tm
Z 0
[u(s), f (s)]+ ds.
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Then, using (H3 0 ) – with z = f (s) – and (g2 ) if s ∈ [0, tm ] \ Eε and (ii) in Remark 2.1 with f (s) = y = 0 for s ∈ Eε , we conclude r < ku(tm )k ≤ ku(t0 )k < ku(tm )k, which is a contradiction. So, kuk∞ ≤ r, and this completes the proof.
We can now pass to the proof of Theorem 3.3. Proof. Let (εn )n be a sequence with εn ↓ 0, let (un )n be the sequence of the C 0 -solutions of the problem (3.3) corresponding to ε = εn and let (fn )n be such that u0n (t ) + Aun (t ) 3 fn (t ), t ∈ [0, 2π ] fn (t ) ∈ Fεn (t , un (t )), a.e. for t ∈ (0, 2π ) un (0) = (1 − εn )g (un ).
(
From (H4 0 ), we deduce that the set {fn ; n ∈ N} is uniformly integrable in L1 (0, 2π ; X ). From this observation, Lemma 4.4, (a2 ) in (H1 ) and Theorem 2.2, it follows that, for each δ ∈ (0, 2π ), the set {un ; n ∈ N} is relatively compact in C ([δ, 2π ]; D(A)). In view of (g1 ) in (H5 ), we deduce that the set
{un (0); n ∈ N} = {(1 − εn )g (un ); n ∈ N} is relatively compact in D(A). From the second part of Theorem 2.2, we conclude that {un ; n ∈ N} is relatively compact in C ([0, 2π ]; D(A)). So, C = {un (t ); n ∈ N, t ∈ [0, 2π ]} is compact in X . Since the restriction of Fεn to ([0, 2π ] \ Eεn ) × D(A) is strongly weakly u.s.c., again by Lemma 2.6.1, p. 47 in [28] combined with Krein–Šmulian Theorem 4, p. 434 in [31], we have that the set conv Fεn ([0, 2π] × C ) = conv[Fεn (([0, 2π ] \ Eεn ) × C ) ∪ {0}] is weakly compact in X . By Theorem 2.3, it follows that {fn ; n ∈ N} is sequentially relatively compact in L1 (0, 2π ; X ). So, on a subsequence at least, we have
lim f = f weakly in L1 (0, 2π ; X ) n n lim un (t ) = u(t ) uniformly for t ∈ [0, 2π] n lim(1 − ε )g (u ) = g (u). n n n
Hence, from [23, Theorem 3.1.2, p. 88], we get f (t ) ∈ F (t , u(t )) a.e. t ∈ [0, 2π ]. Since As of complete continuous type, it follows that u is a C 0 -solution of the problem (1.1) corresponding to the selection f of t 7→ F (t , u(t )). From Lemma 4.4 it follows that u : [0, 2π ] → D(0, r ) ∩ D(A), and this concludes the proof of Theorem 3.3. Finally, we can proceed to the proof of Theorem 3.2. Proof. Let ρ : X → D(0, r ) be defined by
ρ(u) =
u r kuk−1 u
if u ∈ D(0, r ) if u ∈ X \ D(0, r ).
Since ρ is continuous, it follows that the multi-function Fρ : [0, 2π ] × X ; X defined by Fρ (t , u) = F (t , ρ(u)), for each (t , u) ∈ [0, 2π ] × X , satisfies the conditions (H2 ), and (H4 0 ). Thanks to (i) in Remark 2.1, Fρ satisfies (H3 0 ) too. Hence, by virtue of Theorem 3.3, the problem u0 (t ) + Au(t ) = f (t ) f (t ) ∈ Fρ (t , u(t )) u(0) = g (u)
(
has at least one C 0 -solution u : [0, 2π ] → D(0, r ) ∩ D(A). Inasmuch as ku(t )k ≤ r for each t ∈ [0, 2π ], we conclude that Fρ (t , u(t )) = F (t , u(t )) for each t ∈ [0, 2π ], and thus u is a C 0 -solution of (1.1) as claimed. The proof is complete.
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5. Examples Example 5.1. Let Ω be a nonempty bounded and open subset in Rn with C 2 boundary Γ , let p ∈ [2, ∞) and λ > 0 and let us consider the nonlinear problem
! p−2 n X ∂u ∂ ∂ u ∂ u − λ|u|p−2 u + F (t , u) ∈ ∂x ∂ t ∂ x ∂ x i i i i=1 ∂u ∈ β(u) − ∂νp Z T 1 u(0) = N(u(t )) dµ(t ). µ([0, T ]) 0
on (0, 2π ) × Ω on (0, 2π ) × Γ
(5.1)
Here
n X ∂ u p−2 ∂ u ∂u − → − → = cos( n , ei ), ∂x ∂νp ∂ xi i i=1 − →
− → − →
− →
where n is the outward normal of Γ and { e1 , e2 , . . . , en } is the canonical base in Rn , F (t , u) = [f1 (t , u) + h, f2 (t , u) + h] with fi : [0, 2π ] × R → R for i = 1, 2 and h ∈ L2 (Ω ). From Theorem 3.2, we deduce: Theorem 5.1. Let β : D(β) ⊆ R ; R be a maximal monotone operator with 0 ∈ D(β) and 0 ∈ β(0), let h ∈ L2 (Ω ) and let fi : [0, 2π] × R → R, i = 1, 2 be two given functions satisfying
(F1 ) (F2 ) (F3 ) (F4 )
f1 (t , u) ≤ f2 (t , u) for each (t , u) ∈ [0, 2π ] × R f1 is i.s.c. f2 is u.s.c. there exist a, b ∈ R+ such that |fi (t , u)| ≤ a|u| + b for i = 1, 2 and all (t , u) ∈ [0, 2π ] × R there exists c > 0 such that ufi (t , u) ≤ −cu2 for i = 1, 2 and all (t , u) ∈ [0, 2π ] × R.
Let N : L2 (Ω ) → L2 (Ω ) be a nonexpansive (possible nonlinear) operator and let µ be a σ -finite and complete measure on [0, 2π ] which is continuous with respect to the Lebesgue measure at t = 0. Then, the problem (5.1) has at least one C 0 -solution u ∈ C ([0, 2π ]; L2 (Ω )) which, for each δ ∈ (0, 2π ), satisfies u ∈ AC ([0, 2π ]; W 1,p (Ω )) ∩ W 1,2 ([δ, 2π ]; L2 (Ω )). Proof. Let A : D(A) ⊆ L2 (Ω ) → L2 (Ω ) be defined by
( ! ) n X ∂ ∂ u p−2 ∂ u 1 ,p p−2 2 D(A) = u ∈ W (ω); − λ|u| u ∈ L (Ω ) ∂ xi ∂ xi ∂ xi i =1 ! n X ∂ ∂ u p−2 ∂ u − λ|u|p−2 u, Au = ∂x ∂ x ∂ x i i i i=1 and g : C ([0, 2π ]; L2 (Ω )) → L2 (Ω ), g (u) =
1
µ([0, 2π])
2π
Z
N(u(s)) dµ(s)
0
for u ∈ C ([0, 2π ]; L2 (Ω )). With A, F and g as above, the problem (5.1) can be rewritten in the form (1.1). By Example 1.5.4, p. 18, in [23], we know that A is m-dissipative on L2 (Ω ) and 0 ∈ A0. Moreover, it generates a compact semigroup of nonexpansive mappings on L2 (Ω ) and is of complete continuous type. See Example 2.2.4, p. 43 and Corollary 2.3.2, p. 50 in [23]. Hence A satisfies (H1 ). A similar argument as the one in the proof of Problem 2.6.1, p. 46 in [28], shows that F is a nonempty, convex and weakly compact valued strongly weakly u.s.c. multi-function. So F satisfies (H2 ). From (F3 ) and (F4 ), we conclude that F satisfies (H3 ) and (H4 ) with r ≥ c −1 khkL2 (Ω ) . Since µ is continuous with respect to the Lebesgue measure at t = 0, it follows that limh↓0 µ([0, h]) = 0, and thus g satisfies (g1 ). Since (g2 ) and (g3 ) are obvious, we deduce that (H5 ) holds. An appeal to Theorem 3.2 completes the proof. Example 5.2. Let Ω be a nonempty, bounded and open subset in Rn with C 1 boundary Γ and let ϕ : R → R be continuous and nondecreasing function. Let us consider the nonlinear diffusion equation subjected to nonlocal initial conditions
ut (t , x) ∈ 1ϕ(u(t , x)) + f (t , x) in [0, 2π ] × Ω f (t , x) ∈ F (t , u(t , x)) in [0, 2π ] × Ω ϕ(u(t , x)) = 0 on [0, 2π ] × Γ Z 2π 1 u(0, x) = N(u(s, ·))(x) dµ(s) in [0, 2π ] × Ω . µ([0, 2π ]) 0
(5.2)
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Theorem 5.2. Let Ω be a nonempty, bounded and open subset in Rn with C 1 boundary Γ and let ϕ : R → R be continuous on R and C 1 on R \ {0} and for which there exist two constants C > 0 and a > 0 if n ≤ 2 and a > (n − 2)/n if n ≥ 3 such that
ϕ 0 (r ) ≥ C |r |a−1 for each r ∈ R \ {0}. Let F (t , u) = [f1 (t , u) + h, f2 (t , u) + h] with fi : [0, 2π ] × R → R, i = 1, 2, satisfying (F1 )–(F4 ) in Theorem 5.1 and let h ∈ L1 (Ω ). Let N : L1 (Ω ) → L1 (Ω ) a (possible nonlinear) nonexpansive operator. Then the problem (5.2) has at least one C 0 -solution u ∈ C ([τ , T ]; L1 (Ω )). Proof. Clearly (H1 ) is implied by Theorem 3.1. Since the proofs of (H2 )–(H5 ) follow the very same lines as before, this completes the proof. Acknowledgement The second author was partially supported by PN-II-ID-PCE-2007-1 Grant ID 397. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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