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A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions multi-valued reaction-diffusion systems. The paper is divided into 5 sections. In Section 2 we recall some useful results throughout the paper. In Section 3 we formulate the main result, i.e. Theorem 3.1. In Section 4 we give some auxiliary lemmas and we prove the main result. In the last Section 5 we give an example referring to the some nonlinear reaction-diffusion system with delay, subjected to nonlocal initial conditions: one of the unknown function is subjected to a periodic condition and the other one to a mean condition. 2. Preliminaries We recall some basic concepts and results concerning m-dissipative operators and nonlinear evolution equations in Banach spaces. For other definitions, results and details regarding these topics, we refer the reader to Barbu [3] and Vrabie [21]. For more details concerning functional differential equations with delay we refer to Hale [16]. Let X be a Banach space with norm · . An operator A : D(A) X X is called dissipative if for each xi D(A) and yi Axi , i = 1, 2, we 1 have [x1 - x2 , y2 - y1 ]+ 0. Here, [x, y]h := h ( x + hy - x ) and [x, y]+ := lim[x, y]h = inf{[x, y]h ; h > 0}, h0 for each x, y X. We remark that |[x, y]+ | y for each x, y X. For other properties of the mapping (x, y) [x, y]+ , see Barbu [3, Proposition 3.7, p. 100]. The operator A : D(A) X X is called m-dissipative if it is dissipative and the range of I - A is R(I - A) = X, for each > 0. We consider the evolution equation (2.1) u (t) Au(t) + f (t), where f L1 (a, b ; X). A continuous function u C([ a, b ]; D(A)) is called a C 0 -solution, or integral solution of (2.1) on [ a, b ], if it satisfies: u(t) - x u(s) - x + [u( ) - x, f ( ) + y]+ d for each x D(A), y Ax and a s t b. Theorem 2.1. Let > 0 and let A : D(A) X X be an mdissipative operator such that A+I is dissipative. Then, for each 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) = . 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 : (2.2) u(t) - v(t) e-(t-s) u(s) - v(s) + s t e-(t-) f () - g() d MONICA-DANA BURLICA and DANIELA ROSU ¸ for each a s t b. In particular, if x D(A) and y Ax, we have (2.3) u(t) - x e-(t-s) u(s) - x + s t e-(t-) f () + y d for each a s t b. See Barbu [3, Theorem 4.1, p. 128]. For D(A), f L1 (a, b ; X) and [ a, b), we denote by u(·, , , f ) the unique C 0 -solution u : [ , b ] D(A), of the problem (2.1) which satisfies the initial condition u( ) = . The semigroup generated by A on D(A) is denoted by {S(t) : D(A) D(A), t 0} and is defined by S(t) = u(t, 0, , 0) for each D(A) and t 0. We say that the operator A generates a compact semigroup if, for each t > 0, the operator S(t) is a compact one. Definition 2.1. The m-dissipative operator A : D(A) X X is called of complete continuous type if for each (fn )n in L1 (a, b ; X) with limn fn = f weakly in L1 (a, b ; X) and (un )n in C([ a, b ]; D(A)), with un a C 0 -solution on [ a, b ] of u (t) Aun (t) + fn (t), t [ a, b ] and limn un = u n strongly in C([ a, b ]; X), it follows that u is a C 0 -solution on [ a, b ] of the limit problem u (t) Au(t) + f (t), t [ a, b ]. If A generates a compact semigroup and the topological dual of X is uniformly convex or A is linear, then A is of complete continuous type. See Vrabie [21, Corollary 2.3.1, p. 49]. The nonlinear diffusion operator in is an example of a m-dissipative operator which generates a compact semigroup and is of complete continuous type in a Banach space whose dual is not uniformly convex. See Theorem 5.1. Definition 2.2. A subset F L1 (a, b; X) is called uniformly integrable if, for each > 0 there exists () > 0 such that f (s) ds for each f F and each measurable subset E [ a, b ] whose Lebesgue measure satisfies (E) < (). Remark 2.1. One may easily check out that each uniformly integrable subset in L1 (a, b; X) is norm bounded in L1 (a, b; X). We also remark that if F L1 (a, b; X) is bounded in Lp (a, b; X) for some p (1, + ], then F is uniformly integrable. Theorem 2.2. Let A : D(A) X X be m-dissipative operator which 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, , f ) ; B, f F} is relatively compact in C([ c, b ]; X). If, in addition, B is relatively compact in X, then the C 0 -solutions set is relatively compact even in C([ a, b ]; X). See Baras [2] and Vrabie [21, Theorems 2.3.2 and 2.3.3, p. 46-47]. Now, we recall a version of a general fixed point theorem, for multifunctions, due to Glicksberg [15] in a locally convex space. Theorem 2.3. 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 ). 3. The main result Let a R. On the space Cb ([ a, +); Y ) we consider the family of seminorms { · k ; k N, k > a }, defined by v k = v C([ a,k ];Y ) which generates a locally convex and separated topology. This space will be denoted by Cb ([ a, +); Y ). We also denote by L1 (0, +; Y ) the space L1 (0, +; Y ) endowed with loc the family of semi-norms { · k,x ; x L1 (0, k; Y ) , k = 1, 2, . . . }, defined by g k,x = |x (g)|. This space is also locally convex and separated. Moreover, the convergence in L1 (0, +; Y ) is nothing but the weak convergence in L1 (0, k; Y ) for k = 1, 2, . . . . In the sequel we need the next hypotheses: (HA ) the operator A : D(A) X X is m-dissipative, 0 D(A), 0 A0 and there exists > 0 such that A + I is dissipative ; (HB ) the operator B : D(B) Y Y satisfies: (B1 ) B is m-dissipative, 0 D(B), 0 B0 and there exists > 0 such that B + I is dissipative ; MONICA-DANA BURLICA and DANIELA ROSU ¸ (B2 ) B generates a compact semigroup ; (B3 ) B is of complete continuous type ; (HF ) the function F : R+ × C([ -, 0 ]; D(A)) × C([ -, 0 ]; D(B)) X is continuous and satisfies: (F1 ) there exists > 0 such that F (t, u, v) - F (t, u, v) u-u C([-,0];X) + v-v C([-,0];Y ) for each t R+ , each u, u C([ -, 0 ]; D(A)) and each v, v C([ -, 0 ]; D(B)) ; (F2 ) there exists m > 0 such that F (t, 0, 0) [ 0, +) ; m for each t (HG ) the multi-function G : R+ ×C([ -, 0 ]; D(A))×C([ -, 0 ]; D(B)) Y has nonempty, convex and weakly compact values and it is stronglyweakly u.s.c. ; (G1 ) with > 0 and m > 0 given by (F1 ) and (F2 ), we have y u C([ -,0 ];X) C([ -,0 ];Y ) +m for each u C([ -, 0 ]; D(A)), each v C([ -, 0 ]; D(B)), each y G(t, u, v) and each t 0 ; (Hc ) the constants > 0, > 0 and > 0 satisfy the nonresonance condition < ; + (Hp ) p : Cb ([-, +); D(A))×Cb ([-, +); D(B)) C([-, 0]; D(A)) is continuous from the space Cb ([-, +); D(A))×Cb ([-, +); D(B)) to C([ -, 0 ]; D(A)) and satisfies: (p1 ) for each u Cb ([ -, +); D(A)) and each v Cb ([ -, +); D(B)), we have p(u, v) C([ -,0 ];X) u Cb ([ 0,+);X) ; (p2 ) there exists a > 0 such that for each u, u Cb ([ -, +); D(A)) and each v, v Cb ([ -, +); D(B)), we have p(u, v) - p(u, v) C([ -,0 ];X) Cb ([ a,+);X) , max{ u - u v-v Cb ([ 0,+);Y ) } ; (Hq ) q : Cb ([ -, +); X) × Cb ([ -, +); D(B)) C([ -, 0 ]; D(B)) is continuous from Cb ([ -, +); D(A)) × Cb ([ -, +); D(B)) to C([ -, 0 ]; D(B)) and satisfies: (q1 ) for each u Cb ([ -, +); X) and each v Cb ([ -, +); D(B)), we have q(u, v) C([ -,0 ];Y ) v Cb ([ 0,+);Y ) ; (q2 ) for each u, u Cb ([ -, +); X) and v, v Cb ([ -, +); D(B)), we have q(u, v) - q(u, v) C([ -,0 ];Y ) max{ u - u Cb ([ 0,+);X) , v - v Cb ([ 0,+);Y ) } ; (q3 ) for each bounded set U in Cb ([ -, +); X) and each bounded V in Cb ([ -, +); D(B)) which is relatively compact in Cb ([ , +); Y ) for each (0, +), the set q(U, V) is relatively compact in the space C([ -, 0 ]; Y ) . Our main result is: Theorem 3.1. If (HA ), (HB ), (HF ), (HG ), (Hp ), (Hq ) and (Hc ) are satisfied, then (1.1) has at least one C 0 -solution, (u, v) Cb ([ -, +); D(A)) × Cb ([ -, +); D(B)), satisfying u v Cb ([ -,+);X) Cb ([ -,+);Y ) (3.1) m - m . - 4. Proof of Theorem 3.1 Let > 0, let (f, g) Cb ([ 0, + ); X)×L (0, +; Y ) and let us consider the system u (t) Au(t) + f (t), t R+ , v (t) Bv(t) + g(t), t R+ , (4.1) t [ -, 0 ], u(t) = (1 - )p(u, v)(t), v(t) = (1 - )q(u, v)(t), t [ -, 0 ]. Lemma 4.1. If the hypotheses (HA ), (B1 ), (p1 ), (p2 ), (q1 ) and (q2 ) are satisfied, then the system (4.1) has a unique C 0 -solution (u, v) Cb ([ -, ); D(A)) × Cb ([ -, ); D(B)). MONICA-DANA BURLICA and DANIELA ROSU ¸ If, in addition, f (4.2) For the proof, see Burlica, Rosu and Vrabie [10, Lemma 5.1]. ¸ Idea of the proof of Theorem 3.1. Let us consider the approximate problem u (t) Au(t) + F (t, ut , vt ), t R+ , v (t) Bv(t) + g(t), t R+ , (4.3) g(t) [ 0,1/ ] (t)G(t, ut , vt ), t R+ , u(t) = (1 - )p(u, v)(t), t [ -, 0 ], v(t) = (1 - )q(u, v)(t), t [ -, 0 ], as well as the following two auxiliary systems (4.4) and (4.5) u (t) Au(t) + F (t, ut , vt ), u(t) = (1 - )p(u, v)(t), t [ 0, +), t [ -, 0 ], v (t) Bv(t) + g(t), v(t) = (1 - )q(u, v)(t), t [ 0, +), t [ -, 0 ] m and g m, then we have m u Cb ([ -,+);X) m v Cb ([ -,+);Y ) . where (0, 1) is arbitrary fixed. We will use a fixed point device described below. Fix an arbitrary (u, g) Cb ([ -, + ); X) × L (0, +; Y ). Using Vrabie [24, Lemma 4.5], the problem (4.4) has a unique C 0 -solution v Cb ([ -, + ); D(B)). Next, by Burlica and Rosu [8, Theorem 3.1], the ¸ problem (4.5) has a unique C 0 -solution u Cb ([ -, + ); D(A)). Now we define the multifunction : Cb ([ -, + ); X)×L (0, +; Y ) by (u, g) := (u, g); g L1 (0, +; Y ), g(t) G (t, ut , vt ) (4.6) a.e. for t [ 0, +) , Cb ([ -, + ); X)×L (0, +; Y ) for each (u, g) Cb ([ -, + ); X) × L (0, +; Y ), where G (t, u, v) = [ 0,1/ ] (t)G(t, u, v), for each (t, u, v) [ 0, +) × C([ -, 0 ]; D(A)) × C([ -, 0 ]; D(B)), and [ 0,1/ ] is the characteristic function of [ 0, 1/ ]. Obviously (4.3) has a C 0 solution if and only if the multi-function has a fixed point on a suitably defined set. Finally, we consider a family {(u , v ); (0, 1)} of C 0 -solutions for the problem (4.3) and we show that we can pick up a sequence ((un , vn ))n , such that limn n =0, limn (un , vn )=(u, v) in Cb ([ -, +); X)×Cb ([ -, +); Y ) and (u, v) is a C 0 -solution of (1.1). We begin by showing that we can suitably define a nonempty, convex and compact set K in the product space Cb ([ -, +); X)× L (0, +; Y ) such that maps K into itself and has sequentially closed graph with respect to the strong topology on Cb ([ -, +); X) and the locally convex topology on L1 (0, +; Y ). We will do this with the help of the next lemmas. Lemma 4.2. Let (HA ), (B1 ), (B2 ) in (HB ), (HF ), (Hp ), (Hq ) and (Hc ) be satisfied, and let (4.7) r= m . - ( + ) r Then, for each (u, g) Cb ([ -, +); X) × L (0, +; Y ) satisfying (4.8) u g Cb ([ -,+);X) L (0,+;Y ) r, the pair (u, v), where v is the unique C 0 -solution of (4.4) and u the unique C 0 -solution of (4.5), satisfies r u Cb ([ -,+);X) (4.9) r v Cb ([ -,+);Y ) , and (4.10) for each t [ 0, +). F (t, ut , vt ) r, MONICA-DANA BURLICA and DANIELA ROSU ¸ For the proof, see Burlica, Rosu and Vrabie [10, Lemma 5.2]. ¸ The lemma below is a continuity with respect to the data result and it was proved by Burlica, Rosu and Vrabie [9]. ¸ Lemma 4.3. Let {Fn : R+ × C([ -, 0 ]; D(A)) X; n N} be a family of continuous functions satisfying : (h1 ) there exists > 0 such that Fn (t, x) - Fn (t, y) x - y C([ -,0 ];X) for each n N, each t [ 0, +) and x, y C([ -, 0 ]; D(A)) ; (h2 ) there exists m > 0 such that Fn (t, 0) m for each n N and each t [ 0, +) ; (h3 ) limn Fn (t, x) = F (t, x) uniformly for t [ 0, +) (for t in bounded intervals in [ 0, +)) and x in bounded subsets in C([ -, 0 ]; D(A)). Let {pn : Cb ([-, +); D(A)) C([ -, 0 ]; D(A)); n N} be a family of functions satisfying : (h4 ) for each nN and uCb ([ -, +); D(A)), we have pn (u) u Cb ([ 0,+);X) ; C([ -,0 ];X) (h5 ) there exists a > 0 such that for each n N and u, u Cb ([ -, +); D(A)), we have pn (u) - pn (u) C([ -,0 ];X) u - u Cb ([ a,+);X) ; (h6 ) limn pn (u) = p(u) uniformly for u in bounded subsets in Cb ([ -, +); D(A)) (and p is continuous from Cb ([ -, +); D(A)) to C([ -, 0 ]; D(A))). Let us assume further that A satisfies (HA ) and < holds true. Let (un )n be the sequence of C 0 -solutions of the problem (4.11) u (t) Aun (t) + Fn (t, unt ), n un (t) = pn (un )(t), t [ 0, +), t [ -, 0 ], whose existence and uniqueness is ensured by Burlica and Rosu [8, The¸ orem 3.1]. Then lim un = u in Cb ([ -, +); X) (in Cb ([ -, +); X)), where u is the C 0 -solution of the limiting problem (4.12) u (t) Au(t) + F (t, ut ), u(t) = p(u)(t), t [ 0, +), t [ -, 0 ]. Lemma 4.4. Let us suppose that the hypotheses (HA ), (HB ), (HF ), (HG ), (Hp ), (Hq ) and (Hc ) are satisfied, and let r > 0 be given by (4.7). Let = r/ and K := K × Kr , where K is the closed ball with center 0 and radius in Cb ([ -, +); X) and Kr is the closed ball with center 0 and radius r in L (0, +; Y ) multiplied by [0,1/] . Then the operator defined by (4.6) maps K into itself and has sequentially closed graph with respect to the norm topology on Cb ([ -, +); X) and the locally convex topology on L1 (0, +; Y ). Proof. The operator : Cb ([ -, + ); X)×L (0, +; Y ) is defined by (u, g):= (u, g); g L1 (0, +; Y ), g(t)G (t, ut , vt ) a.e. for t [0, +) where G (t, ut , vt ) = [ 0,1/ ] (t)G(t, ut , vt ) and u is the unique C 0 -solution of the problem (4.5) where v is the unique C 0 -solution of the problem (4.4). If (u, g) K , from Lemma 4.2 we deduce that the pair (u, v) Cb ([ -, ); D(A)) × Cb ([ -, ); D(B)) satisfies (4.13) u Cb ([ -,+);Y ) Cb ([ -, +); X)×L (0, +; Y ) r , Cb ([ -,+);Y ) r . An appeal to (G1 ) shows that maps K into itself. To prove that has sequentially closed graph with respect to the norm topology on Cb ([ -, +); X) and the locally convex topology on L1 (0, +; Y ), let ((un , gn ))n be an arbitrary sequence in K and (un , gn ) (un , gn ) for each n N. That means there exists vn Cb ([ -, ); D(B)), the unique C 0 -solution for the problem (4.14) vn (t) Bvn (t) + gn (t), vn (t) = (1 - )q(un , vn )(t), t [ 0, +), t [ -, 0 ], and un Cb ([ -, ); D(A)), the unique C 0 -solution for the problem (4.15) u (t) Aun (t) + F (t, unt , vnt ), n un (t) = (1 - )p(un , vn )(t), t [ 0, +), t [ -, 0 ], MONICA-DANA BURLICA and DANIELA ROSU ¸ for n N. By the definition of , gn L1 (0, +; Y ) and gn (t) G (t, unt , vnt ) a.e. for t [ 0, +) and for each n N. We suppose that (4.16) and (4.17) lim(un , gn ) = (u, g) in Cb ([ -, +); X) × L1 (0, +; Y ). lim(un , gn ) = (u, g) in Cb ([ -, +); X) × L1 (0, +; Y ) Reasoning as in Burlica, Rosu and Vrabie [10, Lemma 5.3], the set ¸ {vn ; n N} is relatively compact in Cb ([ -, +); Y ). Indeed, from (4.13), r we get vn (0) vn Cb ([ -,+);Y ) for each n N and, by Remark 2.1, the set {gn ; n N } is uniformly integrable, so we are in the hypotheses of Theorem 2.2, wherefrom we obtain that {vn ; n N} is relatively compact in C([ , k ]; Y ) for k = 1, 2, . . . and (0, k). Since {un ; n N} is bounded in Cb ([ -, +); X) and, from (q3 ), {vn (0); n N} is relatively compact in Y, we can apply the second part of Theorem 2.2, wherefrom {vn ; n N} is relatively compact in Cb ([ 0, k ]; Y ) for k = 1, 2, . . . . An appeal to condition (q1 ) shows that {vn ; n N} is relatively compact in Cb ([ -, +); Y ). That means that there exists v Cb ([ -, +); Y ) such that on a subsequence of (vn )n ­denoted for simplicity by (vn )n ­, we have (4.18) lim vn = v in Cb ([ -, +); Y ). In fact, we can prove a stronger condition, i.e. the convergence in Cb ([ -, +); Y ). Since gn Kr which is weakly closed in L1 (0, +; Y ) and limn gn = g weakly in L1 (0, +; Y ), we deduce that g Kr . From this relation combined with (4.18) and with the fact that B is of complete continuous type, we obtain that v is C 0 -solution of the problem v (t) Bv(t)+g(t), t [ 0, +). Next, we fix k N, with k 1 + 1/ and an arbitrary constant > 0. The sequence (vn )n is convergent to v on [ 0, k ], so there exists n () N such that vn (t) - v(t) for n N, n n () and t [ 0, k ]. For a.e. s [ k , t ], gn (s) - g(s) = 0 and, using (2.2), we deduce vn (t) - v(t) e-k vn (k ) - v(k ) + vn (k ) - v(k ) t k e-(t-s) gn (s) - g(s) ds for n n () and t k . From this inequality and taken into account (4.16), we obtain that (4.19) lim vn = v in Cb ([ 0, +); Y ) and so, in Cb ([ -, +); Y ). From (q2 ) we deduce that v(t) = (1 - )q(u, v)(t) for each t [ -, 0 ]. Since for each n N, un is the C 0 -solution of the problem (4.15) and, by (F1 ), (p1 ) and (p2 ), the functions Fn (t, ·) := F (t, ·, vnt ) and pn (·) := (1 - )p(·, vn ) for n N and t [ 0, +), satisfy Lemma 4.3, we deduce that the limit function u = limn un is the unique C 0 -solution of the problem (4.20) u (t) Au(t) + F (t, ut , vt ), t [ 0, +), u(t) = (1 - )p(u, v)(t), t [ -, 0 ]. Finally, let us prove that g(t) G (t, ut , vt ) a.e. t [ 0, ). We have gn (t) G (t, unt , vnt ) a.e. t [ 0, k ] and G is strongly-weakly u.s.c., lim gn = g weakly in L1 ([ 0, k ]; Y ), lim(unt , vnt ) = (ut , vt ) in C([-, 0]; D(A))×C([-, 0]; D(B)) a.e. t [ 0, k ], for each k = 1, 2, .... So, we are in the hypotheses of Vrabie [21, Theorem 3.1.2, p. 88], wherefrom we deduce that g(t) G (t, ut , vt ) a.e. t [ 0, k ], for k = 1, 2, ... and thus a.e. t [ 0, ). Lemma 4.5. Let us suppose that the hypotheses (HA ), (HB ), (HF ), (HG ), (Hp ), (Hq ) and (Hc ) are satisfied. Then, for each (0, 1), the set conv (K ) is compact in the product space Cb ([ -, +); X)×L1 (0, +; Y ). Proof. We begin by proving that the set (K ) is relatively compact in the product space Cb ([ -, +); X) × L1 (0, +; Y ). Let ((un , gn ))n be an arbitrary sequence in (K ) and ((un , gn ))n K such that (un , gn ) (un , gn ) for n N. So, vn is the unique C 0 -solution of the problem vn (t) Bvn (t) + gn (t), vn (t) = (1 - )q(un , vn )(t), t [ 0, +), t [ -, 0 ] and un is the unique C 0 -solution of the problem u (t) Aun (t) + F (t, unt , vnt ), n un (t) = (1 - )p(un , vn )(t), t [ 0, +), t [ -, 0 ], MONICA-DANA BURLICA and DANIELA ROSU ¸ for each n N. Since {gn ; n N } is bounded in L (0, +; Y ) we deduce that there exits g L (0, +; Y ) such that on a subsequence at least, we have limn gn = g weakly in L1 ( 0, ; Y ). Reasoning as in Lemma 4.4, we deduce that there exists v Cb ([ -, +); Y ) and a sub-subsequence of (vn )n ­denoted for simplicity again by (vn )n ­, such that limn vn = v in Cb ([ -, +); Y ). Let u Cb ([ -, +); D(A)) the unique C 0 -solution of the problem (4.21) u (t) Au(t) + F (t, ut , vt ), u(t) = (1 - )p(u, v)(t), t [ 0, +), t [ -, 0 ], with v as above. Reasoning as in Lemma 4.4, we deduce that on a subsequence at least, we have limn un = u in Cb ([ -, +); X). Since gn (t) G (t, unt , vnt ) a. e. for t [ 0, +) and for each n N, by (G1 ), we deduce that, at least on a subsequence­denoted for simplicity by (gn )n ­, we have (4.22) lim gn = g weakly in L1 ( 0, ; Y ). Furthermore, because the set Kr is weakly closed in L1 (0, +; Y ) we deduce that g Kr . Let us observe that we have (4.23) g(t) G (t, ut , vt ) a.e. t [ 0, ). Indeed, let us remind that gn (t) G (t, unt , vnt ) a.e. t [ 0, k ], G is strongly-weakly u.s.c., lim gn = g weakly in L1 ([ 0, k ]; Y ), lim(unt , vnt ) = (ut , vt ) in C([ -, 0 ]; X) × C([ -, 0 ]; Y ) a.e. t [ 0, k ], for each k = 1, 2, ..... So, by Vrabie [21, Theorem 3.1.2, p. 88], we obtain that g(t) G (t, ut , vt ) a.e. t [ 0, k ], for k = 1, 2, ... and thus a.e. t [ 0, ). We conclude that (K ) is relatively compact in Cb ([ -, +); X)× L1 (0, +; Y ) and thus, by Dunford and Schwartz [13, Theorem 6, p. 416], conv (K ) is compact in Cb ([ -, +); X) × L1 (0, +; Y ). Now, we prove Theorem 3.1. Proof. Let (0, 1) be arbitrary but fixed and let K = conv (K ). By Lemma 4.5, it follows that the operator : K K has convex and compact values. Moreover, by Lemma 4.4, the graph of is sequentially closed. Since, in a Banach space, the weak closure of a weakly relatively compact set coincides with its weak sequential closure ­ see Edwards [14, Theorem 8.12.1, p. 549] ­, using a standard diagonal process, we deduce that the graph of is even closed in Cb ([ -, +); X) × L1 (0, +; Y ). From Theorem 2.3, we deduce that has at least on fixed point (u , g ). Clearly this means that the approximate problem (4.3) has at least one solution (u , v ). For each (0, 1), fix such a solution and consider the set {(u , v ); (0, 1)}. At this point, reasoning as in the proof of Burlica, Rosu and Vra¸ bie [10, Theorem 3.1], we deduce that the set {(u , v ); (0, 1)} is relatively compact in Cb ([ -, +); X) × Cb ([ -, +); Y ). Indeed, let n 0 and let ((un , vn ))n be a sequence of solutions for u (t) Aun (t) + F (t, unt , vnt ), n v (t) Bvn (t) + gn (t), n g (t) [ 0,1/n ] (t)G(t, unt , vnt ), n un (t) = (1 - n )p(un , vn )(t), vn (t) = (1 - n )q(un , vn )(t), t R+ , t R+ , t R+ , t [ -, 0 ], t [ -, 0 ]. Arguing as in Lemma 4.5, we deduce that the set {gn ; n N} is weakly relatively compact in L1 (0, k; Y ), for k = 1, 2, . . . and the set {vn ; n N} is relatively compact in Cb ([ -, +); Y ). So, at least on a subsequence, we have both limn gn = g weakly in L1 ( 0, ; Y ), limn vn = v in Cb ([ -, +); Y ) and the function v satisfies v (t) Bv(t) + g(t) for each t [ 0, +). Using (HF ), (p1 ), (p2 ) and the continuity property of p we deduce that we can apply Theorem 4.3, Cb ­continuity part, with Fn (t, ·) := F (t, ·, vnt ) and pn (·) := (1 - )p(·, vn ) for n N and t [ 0, +). So, at least on a subsequence, we have limn un = u in Cb ([ -, +); X) where u (t) Au(t) + F (t, ut , vt ), t R+ , u(t) = p(u, v)(t), t [ -, 0 ]. Since G is strongly-weakly u.s.c. and gn (t) [ 0,1/n ] (t)G(t, unt , unt ) a.e. for t [ 0, +), from Vrabie [21, Theorem 3.1.2, p. 88] we get g(t) G(t, ut , vt ) a.e. t [ 0, ). Finally, from the continuity property of q we get v(t) = q(u, v)(t) for each t [ -, 0 ] and this completes the proof. MONICA-DANA BURLICA and DANIELA ROSU ¸ 5. An example Let be a nonempty, bounded domain in Rd , d 2, with C 1 boundary , let > 0, > 0, let Q = R+ ×, = R+ ×, = [ -, 0]× and let be the Laplace operator in the sense of distributions over . Let : D() R R and : D() R R be maximal-monotone operators with 0 (0), 0 (0), let F : R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) be continuous and gi : R+ ×C([ -, 0 ]; )×C([ -, 0 ]; ) , i = 1, 2, two given functions such that g1 is l.s.c, g2 is u.s.c and g1 (t, u, v)(x) g2 (t, u, v)(x), for each (t, u, v) R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) and a.e. x . Let a > 0, let µ be a positive -finite and complete measure on the class of Borel measurable sets in [ a, +), k L1 (a, +; µ, R) be a nonnegative function with k L1 (a,+;µ,R) 1 and let W : C([ -, 0 ]; ) R+ be nonexpansive with W (0) = 0. We consider the following system: u (t, x) = (u(t, x)) - u(t, x) + F (t, u , v )(x), (t, x) Q, t t t v (t, x) = (v(t, x)) - v(t, x) + g(t)(x), (t, x) Q, t g(t) G(t, ut , vt ), t R+ , (5.1) (u(t, x)) = 0, (v(t, x)) = 0, (t, x) , u(t, x) = k(s)W (v(t + s, x))u(t + s, x) dµ(s), (t, x) , a v(t, x) = v(t + T, x), (t, x) , where G : R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) is defined by G(t, u, v) := { h ; g1 (t, u, v)(x) h(x) g2 (t, u, v)(x), a.e. x }. If : D() R denote by R is monotone with 0 (0) and u : D(), we S (u) = {v ; v(x) (u(x)), a.e. for x }. The first part of the result below was proved by Brezis and Strauss [4] while the second was proved by Badii, D´ and Tesei [1]. iaz Theorem 5.1. Let be a nonempty, bounded and open subset in Rd with C 1 boundary and let : D() R R be maximal monotone with 0 D() and 0 (0). (i) Then the operator : D() , defined by 1,1 D() = {u ; v S (u) W0 (), v } 1,1 (u) = {v; v S (u) W0 ()} for u D(), is m-dissipative on . (ii) If, in addition, : R R is continuous on R and C 1 on R \ {0} and there exist two constants C > 0 and > 0 if d 2 and > (d - 2)/d if d 3 such that (r) C|r|-1 for each r R \ {0}, then generates a compact semigroup. For a sufficient condition in order that the semigroup generated by maps weakly compact sets in into compact sets in for t > 0, see D´ and Vrabie [11]. iaz Before proceeding to the statement of the main result of this section, let us define the multifunction G : R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) by G(t, u, v)(x) := [ g1 (t, u, v)(x), g2 (t, u, v)(x) ], for each t R+ , x and u, v C([ -, 0 ]; ). Theorem 5.2. Let be a nonempty, bounded and open subset in Rd , d 1, with C 1 boundary and let : D() R R and : D() R R be maximal monotone operators with 0 D(), 0 D(), 0 (0) and 0 (0). Let F : R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) be continuous, let gi : R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) , i = 1, 2, be two given functions such that g1 (t, u, v)(x) g2 (t, u, v)(x), for each (t, u, v) R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) and a.e. x . Let µ be a positive -finite and complete measure defined on the class of Borel measurable sets in [ a, +), let k L1 (a, +; µ, R) be nonnegative and let W : C([ -, 0 ]; ) R+ . Let us assume that : (h1 ) : R R is continuous on R and C 1 on R \ {0} and there exist two constants C > 0 and > 0 if d 2 and > (d - 2)/d if d 3 such that (r) C|r|-1 for each r R \ {0} ; MONICA-DANA BURLICA and DANIELA ROSU ¸ (h2 ) there exist > 0 and m > 0 such that F (t, u, v) - F (t, u, v) [ u - u -v C([ -,0 ];) C([ -,0 ];) ], [ u C([ -,0 ];) C([ -,0 ];) ] + m, F (t, 0, 0) m, for each (t, u, v), (t, u, v) R+ ×C([ -, 0 ]; )×C([ -, 0 ]; ) and for each y G(t, u, v). (h3 ) g1 is l.s.c and g2 is u.s.c; (h4 ) there exist 1 > 0 and m1 > 0 such that for i = 1, 2 and for each (t, u, v) R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) we have gi (t, u, v) 1 [ u (h5 ) k L1 (a,+;µ,R) C([ -,0 ];) C([ -,0 ];) ] + m1 ; 1; C([-,0];) , (h6 ) |W (v)-W (v)| v-v (h7 ) W (0) = 0. for each v, v C([ -, 0 ]; ); Let us assume also that (Hc ) is satisfied. Then, (5.1) has at least one C 0 solution. Proof. The problem (5.1) can be rewritten as an abstract one of the form (1.1). Since g1 is l.s.c., g2 is u.s.c. and both have sublinear growth, we conclude that G is strongly-weakly u.s.c. with nonempty, convex and weakly compact values. So, all hypotheses of the Theorem 3.1 are satisfied. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Alexandru Ioan Cuza University - Mathematics de Gruyter

A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

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Abstract

multi-valued reaction-diffusion systems. The paper is divided into 5 sections. In Section 2 we recall some useful results throughout the paper. In Section 3 we formulate the main result, i.e. Theorem 3.1. In Section 4 we give some auxiliary lemmas and we prove the main result. In the last Section 5 we give an example referring to the some nonlinear reaction-diffusion system with delay, subjected to nonlocal initial conditions: one of the unknown function is subjected to a periodic condition and the other one to a mean condition. 2. Preliminaries We recall some basic concepts and results concerning m-dissipative operators and nonlinear evolution equations in Banach spaces. For other definitions, results and details regarding these topics, we refer the reader to Barbu [3] and Vrabie [21]. For more details concerning functional differential equations with delay we refer to Hale [16]. Let X be a Banach space with norm · . An operator A : D(A) X X is called dissipative if for each xi D(A) and yi Axi , i = 1, 2, we 1 have [x1 - x2 , y2 - y1 ]+ 0. Here, [x, y]h := h ( x + hy - x ) and [x, y]+ := lim[x, y]h = inf{[x, y]h ; h > 0}, h0 for each x, y X. We remark that |[x, y]+ | y for each x, y X. For other properties of the mapping (x, y) [x, y]+ , see Barbu [3, Proposition 3.7, p. 100]. The operator A : D(A) X X is called m-dissipative if it is dissipative and the range of I - A is R(I - A) = X, for each > 0. We consider the evolution equation (2.1) u (t) Au(t) + f (t), where f L1 (a, b ; X). A continuous function u C([ a, b ]; D(A)) is called a C 0 -solution, or integral solution of (2.1) on [ a, b ], if it satisfies: u(t) - x u(s) - x + [u( ) - x, f ( ) + y]+ d for each x D(A), y Ax and a s t b. Theorem 2.1. Let > 0 and let A : D(A) X X be an mdissipative operator such that A+I is dissipative. Then, for each 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) = . 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 : (2.2) u(t) - v(t) e-(t-s) u(s) - v(s) + s t e-(t-) f () - g() d MONICA-DANA BURLICA and DANIELA ROSU ¸ for each a s t b. In particular, if x D(A) and y Ax, we have (2.3) u(t) - x e-(t-s) u(s) - x + s t e-(t-) f () + y d for each a s t b. See Barbu [3, Theorem 4.1, p. 128]. For D(A), f L1 (a, b ; X) and [ a, b), we denote by u(·, , , f ) the unique C 0 -solution u : [ , b ] D(A), of the problem (2.1) which satisfies the initial condition u( ) = . The semigroup generated by A on D(A) is denoted by {S(t) : D(A) D(A), t 0} and is defined by S(t) = u(t, 0, , 0) for each D(A) and t 0. We say that the operator A generates a compact semigroup if, for each t > 0, the operator S(t) is a compact one. Definition 2.1. The m-dissipative operator A : D(A) X X is called of complete continuous type if for each (fn )n in L1 (a, b ; X) with limn fn = f weakly in L1 (a, b ; X) and (un )n in C([ a, b ]; D(A)), with un a C 0 -solution on [ a, b ] of u (t) Aun (t) + fn (t), t [ a, b ] and limn un = u n strongly in C([ a, b ]; X), it follows that u is a C 0 -solution on [ a, b ] of the limit problem u (t) Au(t) + f (t), t [ a, b ]. If A generates a compact semigroup and the topological dual of X is uniformly convex or A is linear, then A is of complete continuous type. See Vrabie [21, Corollary 2.3.1, p. 49]. The nonlinear diffusion operator in is an example of a m-dissipative operator which generates a compact semigroup and is of complete continuous type in a Banach space whose dual is not uniformly convex. See Theorem 5.1. Definition 2.2. A subset F L1 (a, b; X) is called uniformly integrable if, for each > 0 there exists () > 0 such that f (s) ds for each f F and each measurable subset E [ a, b ] whose Lebesgue measure satisfies (E) < (). Remark 2.1. One may easily check out that each uniformly integrable subset in L1 (a, b; X) is norm bounded in L1 (a, b; X). We also remark that if F L1 (a, b; X) is bounded in Lp (a, b; X) for some p (1, + ], then F is uniformly integrable. Theorem 2.2. Let A : D(A) X X be m-dissipative operator which 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, , f ) ; B, f F} is relatively compact in C([ c, b ]; X). If, in addition, B is relatively compact in X, then the C 0 -solutions set is relatively compact even in C([ a, b ]; X). See Baras [2] and Vrabie [21, Theorems 2.3.2 and 2.3.3, p. 46-47]. Now, we recall a version of a general fixed point theorem, for multifunctions, due to Glicksberg [15] in a locally convex space. Theorem 2.3. 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 ). 3. The main result Let a R. On the space Cb ([ a, +); Y ) we consider the family of seminorms { · k ; k N, k > a }, defined by v k = v C([ a,k ];Y ) which generates a locally convex and separated topology. This space will be denoted by Cb ([ a, +); Y ). We also denote by L1 (0, +; Y ) the space L1 (0, +; Y ) endowed with loc the family of semi-norms { · k,x ; x L1 (0, k; Y ) , k = 1, 2, . . . }, defined by g k,x = |x (g)|. This space is also locally convex and separated. Moreover, the convergence in L1 (0, +; Y ) is nothing but the weak convergence in L1 (0, k; Y ) for k = 1, 2, . . . . In the sequel we need the next hypotheses: (HA ) the operator A : D(A) X X is m-dissipative, 0 D(A), 0 A0 and there exists > 0 such that A + I is dissipative ; (HB ) the operator B : D(B) Y Y satisfies: (B1 ) B is m-dissipative, 0 D(B), 0 B0 and there exists > 0 such that B + I is dissipative ; MONICA-DANA BURLICA and DANIELA ROSU ¸ (B2 ) B generates a compact semigroup ; (B3 ) B is of complete continuous type ; (HF ) the function F : R+ × C([ -, 0 ]; D(A)) × C([ -, 0 ]; D(B)) X is continuous and satisfies: (F1 ) there exists > 0 such that F (t, u, v) - F (t, u, v) u-u C([-,0];X) + v-v C([-,0];Y ) for each t R+ , each u, u C([ -, 0 ]; D(A)) and each v, v C([ -, 0 ]; D(B)) ; (F2 ) there exists m > 0 such that F (t, 0, 0) [ 0, +) ; m for each t (HG ) the multi-function G : R+ ×C([ -, 0 ]; D(A))×C([ -, 0 ]; D(B)) Y has nonempty, convex and weakly compact values and it is stronglyweakly u.s.c. ; (G1 ) with > 0 and m > 0 given by (F1 ) and (F2 ), we have y u C([ -,0 ];X) C([ -,0 ];Y ) +m for each u C([ -, 0 ]; D(A)), each v C([ -, 0 ]; D(B)), each y G(t, u, v) and each t 0 ; (Hc ) the constants > 0, > 0 and > 0 satisfy the nonresonance condition < ; + (Hp ) p : Cb ([-, +); D(A))×Cb ([-, +); D(B)) C([-, 0]; D(A)) is continuous from the space Cb ([-, +); D(A))×Cb ([-, +); D(B)) to C([ -, 0 ]; D(A)) and satisfies: (p1 ) for each u Cb ([ -, +); D(A)) and each v Cb ([ -, +); D(B)), we have p(u, v) C([ -,0 ];X) u Cb ([ 0,+);X) ; (p2 ) there exists a > 0 such that for each u, u Cb ([ -, +); D(A)) and each v, v Cb ([ -, +); D(B)), we have p(u, v) - p(u, v) C([ -,0 ];X) Cb ([ a,+);X) , max{ u - u v-v Cb ([ 0,+);Y ) } ; (Hq ) q : Cb ([ -, +); X) × Cb ([ -, +); D(B)) C([ -, 0 ]; D(B)) is continuous from Cb ([ -, +); D(A)) × Cb ([ -, +); D(B)) to C([ -, 0 ]; D(B)) and satisfies: (q1 ) for each u Cb ([ -, +); X) and each v Cb ([ -, +); D(B)), we have q(u, v) C([ -,0 ];Y ) v Cb ([ 0,+);Y ) ; (q2 ) for each u, u Cb ([ -, +); X) and v, v Cb ([ -, +); D(B)), we have q(u, v) - q(u, v) C([ -,0 ];Y ) max{ u - u Cb ([ 0,+);X) , v - v Cb ([ 0,+);Y ) } ; (q3 ) for each bounded set U in Cb ([ -, +); X) and each bounded V in Cb ([ -, +); D(B)) which is relatively compact in Cb ([ , +); Y ) for each (0, +), the set q(U, V) is relatively compact in the space C([ -, 0 ]; Y ) . Our main result is: Theorem 3.1. If (HA ), (HB ), (HF ), (HG ), (Hp ), (Hq ) and (Hc ) are satisfied, then (1.1) has at least one C 0 -solution, (u, v) Cb ([ -, +); D(A)) × Cb ([ -, +); D(B)), satisfying u v Cb ([ -,+);X) Cb ([ -,+);Y ) (3.1) m - m . - 4. Proof of Theorem 3.1 Let > 0, let (f, g) Cb ([ 0, + ); X)×L (0, +; Y ) and let us consider the system u (t) Au(t) + f (t), t R+ , v (t) Bv(t) + g(t), t R+ , (4.1) t [ -, 0 ], u(t) = (1 - )p(u, v)(t), v(t) = (1 - )q(u, v)(t), t [ -, 0 ]. Lemma 4.1. If the hypotheses (HA ), (B1 ), (p1 ), (p2 ), (q1 ) and (q2 ) are satisfied, then the system (4.1) has a unique C 0 -solution (u, v) Cb ([ -, ); D(A)) × Cb ([ -, ); D(B)). MONICA-DANA BURLICA and DANIELA ROSU ¸ If, in addition, f (4.2) For the proof, see Burlica, Rosu and Vrabie [10, Lemma 5.1]. ¸ Idea of the proof of Theorem 3.1. Let us consider the approximate problem u (t) Au(t) + F (t, ut , vt ), t R+ , v (t) Bv(t) + g(t), t R+ , (4.3) g(t) [ 0,1/ ] (t)G(t, ut , vt ), t R+ , u(t) = (1 - )p(u, v)(t), t [ -, 0 ], v(t) = (1 - )q(u, v)(t), t [ -, 0 ], as well as the following two auxiliary systems (4.4) and (4.5) u (t) Au(t) + F (t, ut , vt ), u(t) = (1 - )p(u, v)(t), t [ 0, +), t [ -, 0 ], v (t) Bv(t) + g(t), v(t) = (1 - )q(u, v)(t), t [ 0, +), t [ -, 0 ] m and g m, then we have m u Cb ([ -,+);X) m v Cb ([ -,+);Y ) . where (0, 1) is arbitrary fixed. We will use a fixed point device described below. Fix an arbitrary (u, g) Cb ([ -, + ); X) × L (0, +; Y ). Using Vrabie [24, Lemma 4.5], the problem (4.4) has a unique C 0 -solution v Cb ([ -, + ); D(B)). Next, by Burlica and Rosu [8, Theorem 3.1], the ¸ problem (4.5) has a unique C 0 -solution u Cb ([ -, + ); D(A)). Now we define the multifunction : Cb ([ -, + ); X)×L (0, +; Y ) by (u, g) := (u, g); g L1 (0, +; Y ), g(t) G (t, ut , vt ) (4.6) a.e. for t [ 0, +) , Cb ([ -, + ); X)×L (0, +; Y ) for each (u, g) Cb ([ -, + ); X) × L (0, +; Y ), where G (t, u, v) = [ 0,1/ ] (t)G(t, u, v), for each (t, u, v) [ 0, +) × C([ -, 0 ]; D(A)) × C([ -, 0 ]; D(B)), and [ 0,1/ ] is the characteristic function of [ 0, 1/ ]. Obviously (4.3) has a C 0 solution if and only if the multi-function has a fixed point on a suitably defined set. Finally, we consider a family {(u , v ); (0, 1)} of C 0 -solutions for the problem (4.3) and we show that we can pick up a sequence ((un , vn ))n , such that limn n =0, limn (un , vn )=(u, v) in Cb ([ -, +); X)×Cb ([ -, +); Y ) and (u, v) is a C 0 -solution of (1.1). We begin by showing that we can suitably define a nonempty, convex and compact set K in the product space Cb ([ -, +); X)× L (0, +; Y ) such that maps K into itself and has sequentially closed graph with respect to the strong topology on Cb ([ -, +); X) and the locally convex topology on L1 (0, +; Y ). We will do this with the help of the next lemmas. Lemma 4.2. Let (HA ), (B1 ), (B2 ) in (HB ), (HF ), (Hp ), (Hq ) and (Hc ) be satisfied, and let (4.7) r= m . - ( + ) r Then, for each (u, g) Cb ([ -, +); X) × L (0, +; Y ) satisfying (4.8) u g Cb ([ -,+);X) L (0,+;Y ) r, the pair (u, v), where v is the unique C 0 -solution of (4.4) and u the unique C 0 -solution of (4.5), satisfies r u Cb ([ -,+);X) (4.9) r v Cb ([ -,+);Y ) , and (4.10) for each t [ 0, +). F (t, ut , vt ) r, MONICA-DANA BURLICA and DANIELA ROSU ¸ For the proof, see Burlica, Rosu and Vrabie [10, Lemma 5.2]. ¸ The lemma below is a continuity with respect to the data result and it was proved by Burlica, Rosu and Vrabie [9]. ¸ Lemma 4.3. Let {Fn : R+ × C([ -, 0 ]; D(A)) X; n N} be a family of continuous functions satisfying : (h1 ) there exists > 0 such that Fn (t, x) - Fn (t, y) x - y C([ -,0 ];X) for each n N, each t [ 0, +) and x, y C([ -, 0 ]; D(A)) ; (h2 ) there exists m > 0 such that Fn (t, 0) m for each n N and each t [ 0, +) ; (h3 ) limn Fn (t, x) = F (t, x) uniformly for t [ 0, +) (for t in bounded intervals in [ 0, +)) and x in bounded subsets in C([ -, 0 ]; D(A)). Let {pn : Cb ([-, +); D(A)) C([ -, 0 ]; D(A)); n N} be a family of functions satisfying : (h4 ) for each nN and uCb ([ -, +); D(A)), we have pn (u) u Cb ([ 0,+);X) ; C([ -,0 ];X) (h5 ) there exists a > 0 such that for each n N and u, u Cb ([ -, +); D(A)), we have pn (u) - pn (u) C([ -,0 ];X) u - u Cb ([ a,+);X) ; (h6 ) limn pn (u) = p(u) uniformly for u in bounded subsets in Cb ([ -, +); D(A)) (and p is continuous from Cb ([ -, +); D(A)) to C([ -, 0 ]; D(A))). Let us assume further that A satisfies (HA ) and < holds true. Let (un )n be the sequence of C 0 -solutions of the problem (4.11) u (t) Aun (t) + Fn (t, unt ), n un (t) = pn (un )(t), t [ 0, +), t [ -, 0 ], whose existence and uniqueness is ensured by Burlica and Rosu [8, The¸ orem 3.1]. Then lim un = u in Cb ([ -, +); X) (in Cb ([ -, +); X)), where u is the C 0 -solution of the limiting problem (4.12) u (t) Au(t) + F (t, ut ), u(t) = p(u)(t), t [ 0, +), t [ -, 0 ]. Lemma 4.4. Let us suppose that the hypotheses (HA ), (HB ), (HF ), (HG ), (Hp ), (Hq ) and (Hc ) are satisfied, and let r > 0 be given by (4.7). Let = r/ and K := K × Kr , where K is the closed ball with center 0 and radius in Cb ([ -, +); X) and Kr is the closed ball with center 0 and radius r in L (0, +; Y ) multiplied by [0,1/] . Then the operator defined by (4.6) maps K into itself and has sequentially closed graph with respect to the norm topology on Cb ([ -, +); X) and the locally convex topology on L1 (0, +; Y ). Proof. The operator : Cb ([ -, + ); X)×L (0, +; Y ) is defined by (u, g):= (u, g); g L1 (0, +; Y ), g(t)G (t, ut , vt ) a.e. for t [0, +) where G (t, ut , vt ) = [ 0,1/ ] (t)G(t, ut , vt ) and u is the unique C 0 -solution of the problem (4.5) where v is the unique C 0 -solution of the problem (4.4). If (u, g) K , from Lemma 4.2 we deduce that the pair (u, v) Cb ([ -, ); D(A)) × Cb ([ -, ); D(B)) satisfies (4.13) u Cb ([ -,+);Y ) Cb ([ -, +); X)×L (0, +; Y ) r , Cb ([ -,+);Y ) r . An appeal to (G1 ) shows that maps K into itself. To prove that has sequentially closed graph with respect to the norm topology on Cb ([ -, +); X) and the locally convex topology on L1 (0, +; Y ), let ((un , gn ))n be an arbitrary sequence in K and (un , gn ) (un , gn ) for each n N. That means there exists vn Cb ([ -, ); D(B)), the unique C 0 -solution for the problem (4.14) vn (t) Bvn (t) + gn (t), vn (t) = (1 - )q(un , vn )(t), t [ 0, +), t [ -, 0 ], and un Cb ([ -, ); D(A)), the unique C 0 -solution for the problem (4.15) u (t) Aun (t) + F (t, unt , vnt ), n un (t) = (1 - )p(un , vn )(t), t [ 0, +), t [ -, 0 ], MONICA-DANA BURLICA and DANIELA ROSU ¸ for n N. By the definition of , gn L1 (0, +; Y ) and gn (t) G (t, unt , vnt ) a.e. for t [ 0, +) and for each n N. We suppose that (4.16) and (4.17) lim(un , gn ) = (u, g) in Cb ([ -, +); X) × L1 (0, +; Y ). lim(un , gn ) = (u, g) in Cb ([ -, +); X) × L1 (0, +; Y ) Reasoning as in Burlica, Rosu and Vrabie [10, Lemma 5.3], the set ¸ {vn ; n N} is relatively compact in Cb ([ -, +); Y ). Indeed, from (4.13), r we get vn (0) vn Cb ([ -,+);Y ) for each n N and, by Remark 2.1, the set {gn ; n N } is uniformly integrable, so we are in the hypotheses of Theorem 2.2, wherefrom we obtain that {vn ; n N} is relatively compact in C([ , k ]; Y ) for k = 1, 2, . . . and (0, k). Since {un ; n N} is bounded in Cb ([ -, +); X) and, from (q3 ), {vn (0); n N} is relatively compact in Y, we can apply the second part of Theorem 2.2, wherefrom {vn ; n N} is relatively compact in Cb ([ 0, k ]; Y ) for k = 1, 2, . . . . An appeal to condition (q1 ) shows that {vn ; n N} is relatively compact in Cb ([ -, +); Y ). That means that there exists v Cb ([ -, +); Y ) such that on a subsequence of (vn )n ­denoted for simplicity by (vn )n ­, we have (4.18) lim vn = v in Cb ([ -, +); Y ). In fact, we can prove a stronger condition, i.e. the convergence in Cb ([ -, +); Y ). Since gn Kr which is weakly closed in L1 (0, +; Y ) and limn gn = g weakly in L1 (0, +; Y ), we deduce that g Kr . From this relation combined with (4.18) and with the fact that B is of complete continuous type, we obtain that v is C 0 -solution of the problem v (t) Bv(t)+g(t), t [ 0, +). Next, we fix k N, with k 1 + 1/ and an arbitrary constant > 0. The sequence (vn )n is convergent to v on [ 0, k ], so there exists n () N such that vn (t) - v(t) for n N, n n () and t [ 0, k ]. For a.e. s [ k , t ], gn (s) - g(s) = 0 and, using (2.2), we deduce vn (t) - v(t) e-k vn (k ) - v(k ) + vn (k ) - v(k ) t k e-(t-s) gn (s) - g(s) ds for n n () and t k . From this inequality and taken into account (4.16), we obtain that (4.19) lim vn = v in Cb ([ 0, +); Y ) and so, in Cb ([ -, +); Y ). From (q2 ) we deduce that v(t) = (1 - )q(u, v)(t) for each t [ -, 0 ]. Since for each n N, un is the C 0 -solution of the problem (4.15) and, by (F1 ), (p1 ) and (p2 ), the functions Fn (t, ·) := F (t, ·, vnt ) and pn (·) := (1 - )p(·, vn ) for n N and t [ 0, +), satisfy Lemma 4.3, we deduce that the limit function u = limn un is the unique C 0 -solution of the problem (4.20) u (t) Au(t) + F (t, ut , vt ), t [ 0, +), u(t) = (1 - )p(u, v)(t), t [ -, 0 ]. Finally, let us prove that g(t) G (t, ut , vt ) a.e. t [ 0, ). We have gn (t) G (t, unt , vnt ) a.e. t [ 0, k ] and G is strongly-weakly u.s.c., lim gn = g weakly in L1 ([ 0, k ]; Y ), lim(unt , vnt ) = (ut , vt ) in C([-, 0]; D(A))×C([-, 0]; D(B)) a.e. t [ 0, k ], for each k = 1, 2, .... So, we are in the hypotheses of Vrabie [21, Theorem 3.1.2, p. 88], wherefrom we deduce that g(t) G (t, ut , vt ) a.e. t [ 0, k ], for k = 1, 2, ... and thus a.e. t [ 0, ). Lemma 4.5. Let us suppose that the hypotheses (HA ), (HB ), (HF ), (HG ), (Hp ), (Hq ) and (Hc ) are satisfied. Then, for each (0, 1), the set conv (K ) is compact in the product space Cb ([ -, +); X)×L1 (0, +; Y ). Proof. We begin by proving that the set (K ) is relatively compact in the product space Cb ([ -, +); X) × L1 (0, +; Y ). Let ((un , gn ))n be an arbitrary sequence in (K ) and ((un , gn ))n K such that (un , gn ) (un , gn ) for n N. So, vn is the unique C 0 -solution of the problem vn (t) Bvn (t) + gn (t), vn (t) = (1 - )q(un , vn )(t), t [ 0, +), t [ -, 0 ] and un is the unique C 0 -solution of the problem u (t) Aun (t) + F (t, unt , vnt ), n un (t) = (1 - )p(un , vn )(t), t [ 0, +), t [ -, 0 ], MONICA-DANA BURLICA and DANIELA ROSU ¸ for each n N. Since {gn ; n N } is bounded in L (0, +; Y ) we deduce that there exits g L (0, +; Y ) such that on a subsequence at least, we have limn gn = g weakly in L1 ( 0, ; Y ). Reasoning as in Lemma 4.4, we deduce that there exists v Cb ([ -, +); Y ) and a sub-subsequence of (vn )n ­denoted for simplicity again by (vn )n ­, such that limn vn = v in Cb ([ -, +); Y ). Let u Cb ([ -, +); D(A)) the unique C 0 -solution of the problem (4.21) u (t) Au(t) + F (t, ut , vt ), u(t) = (1 - )p(u, v)(t), t [ 0, +), t [ -, 0 ], with v as above. Reasoning as in Lemma 4.4, we deduce that on a subsequence at least, we have limn un = u in Cb ([ -, +); X). Since gn (t) G (t, unt , vnt ) a. e. for t [ 0, +) and for each n N, by (G1 ), we deduce that, at least on a subsequence­denoted for simplicity by (gn )n ­, we have (4.22) lim gn = g weakly in L1 ( 0, ; Y ). Furthermore, because the set Kr is weakly closed in L1 (0, +; Y ) we deduce that g Kr . Let us observe that we have (4.23) g(t) G (t, ut , vt ) a.e. t [ 0, ). Indeed, let us remind that gn (t) G (t, unt , vnt ) a.e. t [ 0, k ], G is strongly-weakly u.s.c., lim gn = g weakly in L1 ([ 0, k ]; Y ), lim(unt , vnt ) = (ut , vt ) in C([ -, 0 ]; X) × C([ -, 0 ]; Y ) a.e. t [ 0, k ], for each k = 1, 2, ..... So, by Vrabie [21, Theorem 3.1.2, p. 88], we obtain that g(t) G (t, ut , vt ) a.e. t [ 0, k ], for k = 1, 2, ... and thus a.e. t [ 0, ). We conclude that (K ) is relatively compact in Cb ([ -, +); X)× L1 (0, +; Y ) and thus, by Dunford and Schwartz [13, Theorem 6, p. 416], conv (K ) is compact in Cb ([ -, +); X) × L1 (0, +; Y ). Now, we prove Theorem 3.1. Proof. Let (0, 1) be arbitrary but fixed and let K = conv (K ). By Lemma 4.5, it follows that the operator : K K has convex and compact values. Moreover, by Lemma 4.4, the graph of is sequentially closed. Since, in a Banach space, the weak closure of a weakly relatively compact set coincides with its weak sequential closure ­ see Edwards [14, Theorem 8.12.1, p. 549] ­, using a standard diagonal process, we deduce that the graph of is even closed in Cb ([ -, +); X) × L1 (0, +; Y ). From Theorem 2.3, we deduce that has at least on fixed point (u , g ). Clearly this means that the approximate problem (4.3) has at least one solution (u , v ). For each (0, 1), fix such a solution and consider the set {(u , v ); (0, 1)}. At this point, reasoning as in the proof of Burlica, Rosu and Vra¸ bie [10, Theorem 3.1], we deduce that the set {(u , v ); (0, 1)} is relatively compact in Cb ([ -, +); X) × Cb ([ -, +); Y ). Indeed, let n 0 and let ((un , vn ))n be a sequence of solutions for u (t) Aun (t) + F (t, unt , vnt ), n v (t) Bvn (t) + gn (t), n g (t) [ 0,1/n ] (t)G(t, unt , vnt ), n un (t) = (1 - n )p(un , vn )(t), vn (t) = (1 - n )q(un , vn )(t), t R+ , t R+ , t R+ , t [ -, 0 ], t [ -, 0 ]. Arguing as in Lemma 4.5, we deduce that the set {gn ; n N} is weakly relatively compact in L1 (0, k; Y ), for k = 1, 2, . . . and the set {vn ; n N} is relatively compact in Cb ([ -, +); Y ). So, at least on a subsequence, we have both limn gn = g weakly in L1 ( 0, ; Y ), limn vn = v in Cb ([ -, +); Y ) and the function v satisfies v (t) Bv(t) + g(t) for each t [ 0, +). Using (HF ), (p1 ), (p2 ) and the continuity property of p we deduce that we can apply Theorem 4.3, Cb ­continuity part, with Fn (t, ·) := F (t, ·, vnt ) and pn (·) := (1 - )p(·, vn ) for n N and t [ 0, +). So, at least on a subsequence, we have limn un = u in Cb ([ -, +); X) where u (t) Au(t) + F (t, ut , vt ), t R+ , u(t) = p(u, v)(t), t [ -, 0 ]. Since G is strongly-weakly u.s.c. and gn (t) [ 0,1/n ] (t)G(t, unt , unt ) a.e. for t [ 0, +), from Vrabie [21, Theorem 3.1.2, p. 88] we get g(t) G(t, ut , vt ) a.e. t [ 0, ). Finally, from the continuity property of q we get v(t) = q(u, v)(t) for each t [ -, 0 ] and this completes the proof. MONICA-DANA BURLICA and DANIELA ROSU ¸ 5. An example Let be a nonempty, bounded domain in Rd , d 2, with C 1 boundary , let > 0, > 0, let Q = R+ ×, = R+ ×, = [ -, 0]× and let be the Laplace operator in the sense of distributions over . Let : D() R R and : D() R R be maximal-monotone operators with 0 (0), 0 (0), let F : R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) be continuous and gi : R+ ×C([ -, 0 ]; )×C([ -, 0 ]; ) , i = 1, 2, two given functions such that g1 is l.s.c, g2 is u.s.c and g1 (t, u, v)(x) g2 (t, u, v)(x), for each (t, u, v) R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) and a.e. x . Let a > 0, let µ be a positive -finite and complete measure on the class of Borel measurable sets in [ a, +), k L1 (a, +; µ, R) be a nonnegative function with k L1 (a,+;µ,R) 1 and let W : C([ -, 0 ]; ) R+ be nonexpansive with W (0) = 0. We consider the following system: u (t, x) = (u(t, x)) - u(t, x) + F (t, u , v )(x), (t, x) Q, t t t v (t, x) = (v(t, x)) - v(t, x) + g(t)(x), (t, x) Q, t g(t) G(t, ut , vt ), t R+ , (5.1) (u(t, x)) = 0, (v(t, x)) = 0, (t, x) , u(t, x) = k(s)W (v(t + s, x))u(t + s, x) dµ(s), (t, x) , a v(t, x) = v(t + T, x), (t, x) , where G : R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) is defined by G(t, u, v) := { h ; g1 (t, u, v)(x) h(x) g2 (t, u, v)(x), a.e. x }. If : D() R denote by R is monotone with 0 (0) and u : D(), we S (u) = {v ; v(x) (u(x)), a.e. for x }. The first part of the result below was proved by Brezis and Strauss [4] while the second was proved by Badii, D´ and Tesei [1]. iaz Theorem 5.1. Let be a nonempty, bounded and open subset in Rd with C 1 boundary and let : D() R R be maximal monotone with 0 D() and 0 (0). (i) Then the operator : D() , defined by 1,1 D() = {u ; v S (u) W0 (), v } 1,1 (u) = {v; v S (u) W0 ()} for u D(), is m-dissipative on . (ii) If, in addition, : R R is continuous on R and C 1 on R \ {0} and there exist two constants C > 0 and > 0 if d 2 and > (d - 2)/d if d 3 such that (r) C|r|-1 for each r R \ {0}, then generates a compact semigroup. For a sufficient condition in order that the semigroup generated by maps weakly compact sets in into compact sets in for t > 0, see D´ and Vrabie [11]. iaz Before proceeding to the statement of the main result of this section, let us define the multifunction G : R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) by G(t, u, v)(x) := [ g1 (t, u, v)(x), g2 (t, u, v)(x) ], for each t R+ , x and u, v C([ -, 0 ]; ). Theorem 5.2. Let be a nonempty, bounded and open subset in Rd , d 1, with C 1 boundary and let : D() R R and : D() R R be maximal monotone operators with 0 D(), 0 D(), 0 (0) and 0 (0). Let F : R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) be continuous, let gi : R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) , i = 1, 2, be two given functions such that g1 (t, u, v)(x) g2 (t, u, v)(x), for each (t, u, v) R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) and a.e. x . Let µ be a positive -finite and complete measure defined on the class of Borel measurable sets in [ a, +), let k L1 (a, +; µ, R) be nonnegative and let W : C([ -, 0 ]; ) R+ . Let us assume that : (h1 ) : R R is continuous on R and C 1 on R \ {0} and there exist two constants C > 0 and > 0 if d 2 and > (d - 2)/d if d 3 such that (r) C|r|-1 for each r R \ {0} ; MONICA-DANA BURLICA and DANIELA ROSU ¸ (h2 ) there exist > 0 and m > 0 such that F (t, u, v) - F (t, u, v) [ u - u -v C([ -,0 ];) C([ -,0 ];) ], [ u C([ -,0 ];) C([ -,0 ];) ] + m, F (t, 0, 0) m, for each (t, u, v), (t, u, v) R+ ×C([ -, 0 ]; )×C([ -, 0 ]; ) and for each y G(t, u, v). (h3 ) g1 is l.s.c and g2 is u.s.c; (h4 ) there exist 1 > 0 and m1 > 0 such that for i = 1, 2 and for each (t, u, v) R+ × C([ -, 0 ]; ) × C([ -, 0 ]; ) we have gi (t, u, v) 1 [ u (h5 ) k L1 (a,+;µ,R) C([ -,0 ];) C([ -,0 ];) ] + m1 ; 1; C([-,0];) , (h6 ) |W (v)-W (v)| v-v (h7 ) W (0) = 0. for each v, v C([ -, 0 ]; ); Let us assume also that (Hc ) is satisfied. Then, (5.1) has at least one C 0 solution. Proof. The problem (5.1) can be rewritten as an abstract one of the form (1.1). Since g1 is l.s.c., g2 is u.s.c. and both have sublinear growth, we conclude that G is strongly-weakly u.s.c. with nonempty, convex and weakly compact values. So, all hypotheses of the Theorem 3.1 are satisfied.

Journal

Annals of the Alexandru Ioan Cuza University - Mathematics – de Gruyter

Published: Jan 1, 2015

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