Notes on solution maps of abstract FDEs 7 1 0 2 Xiao-Qiang Zhao∗ n a Department of Mathematics and Statistics J 3 Memorial University of Newfoundland ] St. John’s, NL A1C 5S7, Canada P A E-mail: [email protected] . h t a m Let τ be a positive real number, X be a Banach space, and C := [ C([−τ,0],X). For any φ∈ C, define kφk = sup kφ(θ)kX. Then (C,k·k) 1 −τ≤θ≤0 v is a Banach space. Let A be the infinitesimal generator of a C -semigroup 0 0 3 {T(t)} on X. Assume that T(t) is compact for each t > 0, and there 7 t≥0 0 exists M > 0 such that kT(t)k ≤ M for all t ≥ 0. 0 . 1 We consider the abstract functional differential equation 0 7 du(t) 1 = Au(t)+F(t,ut), t > 0, dt : (0.1) v Xi u0 = φ ∈ C. r a HereF : [0,∞)×C → X iscontinuous andmapsboundedsetsintobounded sets, and u ∈ C is defined by u (θ)= u(t+θ), ∀θ ∈ [−τ,0]. t t Theorem A. Assume that for each φ ∈ C, equation (0.1) has a unique solution u(t,φ) on [0,∞), and solutions of (0.1) are uniformly bounded in thesensethatforanyboundedsubsetB ofC,thereexistsaboundedsubset 0 ∗Research supported in part bythe NSERCof Canada. 1 B = B (B ) of C such that u (φ) ∈ B for all φ ∈ B and t ≥ 0. Then for 1 1 0 t 1 0 any given r > 0, there exists an equivalent norm k·k∗ on C such that the r solution maps Q(t) := u of equation (0.1) satisfy α(Q(t)B) ≤ e−rtα(B) for t any bounded subset B of C and t ≥ 0, where α is the Kuratowski measure of noncompactness in (C,k·k∗). r Proof. Define kxk∗ = sup kT(t)xk, ∀x ∈ X. Then kxk ≤ kxk∗ ≤ Mkxk, t≥0 and hence, kxk∗ is an equivalent norm on X. It is easy to see that kT(t)xk∗ = supkT(s)T(t)xk =supkT(s+t)xk ≤ kxk∗, ∀x∈ X, t ≥0, s≥0 s≥0 whichimpliesthatkT(t)k∗ ≤ 1forallt ≥ 0. Thus,withoutlossofgenerality, we assume that M = 1. Let r > 0 be given. Note that for each φ ∈ C, the solution u(t,φ) of (0.1) satisfies the following integral equation t u(t)= Tˆ(t)φ(0)+ Tˆ(t−s)Fˆ(s,u )ds, t ≥ 0, s Z0 (0.2) u = φ ∈ C, 0 where Tˆ(t) = e−rtT(t) and Fˆ(t,ϕ) = rϕ(0) + F(t,ϕ), ∀t ≥ 0, ϕ ∈ C. Then Tˆ(t) is also a C -semigroup on X and kTˆ(t)k ≤ e−rt, ∀t ≥ 0. Let 0 h(θ) = e−rθ,∀ θ ∈[−τ,0], and define kφ(θ)k kφk∗ = sup X, ∀ φ∈ C. r h(θ) −τ≤θ≤0 Then 1 kφk ≤ kφk∗ ≤ kφk , and hence k·k∗ is equivalent to k·k . h(−τ) C r C r C Clearly, kφ(0)k ≤ kφk∗, ∀φ∈ C. Define X r Tˆ(t+θ)φ(0), t+θ > 0, (L(t)φ)(θ) =  φ(t+θ), t+θ ≤ 0,   2 and t+θTˆ(t+θ−s)Fˆ(s,u (φ))ds, t+θ > 0, (Q¯(t)φ)(θ) = 0 s  R0, t+θ ≤ 0.  Thus, Q(t)φ = L(t)φ + Q¯(t)φ,∀ t ≥ 0,φ ∈ C, that is, Q(t) = L(t) + Q¯(t),∀ t ≥ 0. We firstshowthatL(t)isanα-contraction on(C,k·k∗)foreach t > 0. It r is easy to seethat L(t) is compact for each t > τ. Without loss of generality, we may assume that t ∈ (0,τ] is fixed. For any φ ∈ C, we have k(L(t)φ)k kL(t)φk∗ = sup X r h(θ) −τ≤θ≤0 kφ(t+θ)k h(t+θ) kTˆ(t+θ)φ(0)k X X ≤ max sup , sup h(t+θ) h(θ) h(θ) (−τ≤θ≤−t −t≤θ≤0 ) e−r(t+θ)kφ(0)k ≤ max e−rtkφk∗, sup X r h(θ) ( −t≤θ≤0 ) = max e−rtkφk∗, e−rtkφ(0)k ≤ e−rtkφk∗, r X r (cid:8) (cid:9) which implies that α(L(t)B) ≤ e−rtα(B) for any bounded subset B of C. Thus, this contraction property holds true for all t > 0. Next we prove that Q¯(t) : C → C is compact for each t > 0. Let t > 0 and the bounded subset B of C be given. By the uniform boundedness of solutions, there exists a real number K > 0 such that kFˆ(s,u (φ))k ≤ s X K, ∀s∈ [0,t], φ∈ B. It then follows that Q¯(t)B is bounded in C. We only need to show that Q¯(t)B is precompact in C. In view of the Arzela–Ascoli theorem for the space C := C([−τ,0],X), it suffices to prove that (i) for each θ ∈ [−τ,0], the set {(Q¯(t)φ)(θ) : φ ∈ B} is precompact in X; and (ii) the set Q¯(t)B is equi-continuous in θ ∈ [−τ,0]. Clearly, statement (i) holds true if t+θ ≤ 0. In the case where t+θ > 0, for any given ǫ ∈ (0,t+θ), we 3 have t+θ−ǫ t+θ (Q¯(t)φ)(θ) = Tˆ(t+θ−s)Fˆ(s,u (φ))ds+ Tˆ(t+θ−s)Fˆ(s,u (φ))ds s s Z0 Zt+θ−ǫ t+θ−ǫ = Tˆ(ǫ) Tˆ(t+θ−ǫ−s)Fˆ(s,u (φ))ds s Z0 t+θ + Tˆ(t+θ−s)Fˆ(s,u (φ))ds. s Zt+θ−ǫ Define t+θ−ǫ S := Tˆ(ǫ) Tˆ(t+θ−ǫ−s)Fˆ(s,u (φ))ds : φ ∈B 1 s (cid:26) Z0 (cid:27) and t+θ S := Tˆ(t+θ−s)Fˆ(s,u (φ))ds : φ∈ B . 2 s (cid:26)Zt+θ−ǫ (cid:27) Let αˆ be the Kuratowski measure of noncompactness in X. Since Tˆ(ǫ) is compact, it follows that αˆ {(Q¯(t)φ)(θ) : φ∈ B} ≤ αˆ(S )+αˆ(S )≤ 0+2Kǫ = 2Kǫ. 1 2 (cid:0) (cid:1) Letting ǫ → 0+, we obtain αˆ {(Q¯(t)φ)(θ) : φ∈ B} = 0, which implies that the set {(Q¯(t)φ)(θ) : φ ∈ B(cid:0)} is precompact in X(cid:1). It remains to verify statement (ii). Since Tˆ(s) is compact for each s > 0, Tˆ(s) is continuous in the uniform operator topology for s > 0 (see [2, Theorem 2.3.2]). It then follows that for any ǫ ∈ (0,t), there exists a δ = δ(ǫ) < ǫ such that kTˆ(s )−Tˆ(s )k < ǫ, ∀s ,s ∈ [ǫ,t] with |s −s |< δ. (0.3) 1 2 1 2 1 2 We first consider the case where t ∈ (0,τ]. It is easy to see that k(Q¯(t)φ)(θ)k ≤ K(t+θ)≤ Kǫ, ∀θ ∈ [−t,−t+ǫ], φ ∈ B. (0.4) X 4 For any φ ∈ B and θ ,θ ∈ [−t+ǫ,0] with 0 < θ −θ < δ, it follows from 1 2 2 1 (0.3) that (Q¯(t)φ)(θ )−(Q¯(t)φ)(θ ) 2 1 X t−ǫ+θ1 =(cid:13)(cid:13) (Tˆ(t+θ2−s)−(cid:13)(cid:13)Tˆ(t+θ1−s))Fˆ(s,us(φ))ds (cid:13)Z0 (cid:13)X (cid:13) t+θ2 (cid:13) +(cid:13)(cid:13) Tˆ(t+θ2−s)Fˆ(s,us(φ))ds (cid:13)(cid:13) (cid:13)Zt−ǫ+θ1 (cid:13)X (cid:13)(cid:13) t+θ1 (cid:13)(cid:13) +(cid:13)− Tˆ(t+θ1−s)Fˆ(s,us(φ))d(cid:13)s (cid:13) Zt−ǫ+θ1 (cid:13)X ≤ ǫK(cid:13)t+K(θ −θ +ǫ)+Kǫ (cid:13) (cid:13) 2 1 (cid:13) (cid:13) (cid:13) < (t+3)Kǫ. (0.5) Combining (0.4) and (0.5), we then obtain k(Q¯(t)φ)(θ )−(Q¯(t)φ)(θ )k < 2Kǫ+(t+3)Kǫ = (t+5)Kǫ, 2 1 X for all θ ,θ ∈ [−t,0] with 0 ≤ θ − θ < δ. Since (Q¯(t)φ)(θ) = 0,∀θ ∈ 1 2 2 1 [−τ,−t], it follows that Q¯(t)B is equi-continuous in θ ∈ [−τ,0]. In the case where t > τ, for any ǫ ∈ (0,t−τ), the estimate in (0.5) implies that k(Q¯(t)φ)(θ )−(Q¯(t)φ)(θ )k < (t+3)Kǫ, 2 1 X for all θ ,θ ∈ [−τ,0] with 0 ≤ θ − θ < δ, and φ ∈ B. Thus, Q¯(t)B is 1 2 2 1 equi-continuous in θ ∈[−τ,0]. It then follows that Q¯(t) :C → C is compact for each t > 0. Consequently, for any t > 0 and any bounded subset B of C, we have α(Q(t)B) ≤ α(L(t)B)+α(Q¯(t)B)≤ e−rtα(B). This completes the proof. 5 Asanapplicationexample,weconsiderthefollowingω-periodicreaction- diffusion system ∂u = D△u+f(t,u ), x ∈ Ω, t > 0, ∂t t (0.6)  ∂u = 0, x ∈ ∂Ω, t > 0,  ∂ν where D = diag(d1,...,dm) with each di > 0, Ω ⊂ Rn is a boundeddomain with the smooth boundary ∂Ω, and f(t,φ) is ω-periodic in t ∈ [0,∞) for some ω >0. Let Y := C(Ω¯,Rm) and assume that f is continuous and maps bounded subsets of [0,∞) ×C([−τ,0],Y) into bounded subsets of Y. Let T(t) be the semigroup on Y generated by ∂u(t,x) = D△u(t,x) subjectto the ∂t boundary condition ∂u = 0. It is easy to see that kT(t)k ≤ 1,∀t ≥ 0. By ∂ν Theorem A, we then have the following result. TheoremB. Assumethatsolutionsofsystem(0.6)existuniquelyon[0,∞) for any initial data in C := C([−τ,0],Y) and are uniformly bounded. Then for each r > 0, there exists an equivalent norm k ·k∗ on C such that for r each t > 0, the solution map Q(t) = u of system (0.6) is an α-contraction t on (C,k·k∗) with the contraction constant being e−rt. r Remark. By using the theory of evolution operators (see, e.g., [1, Section II.11] and [2, Section 5.6]), one may extend Theorem A to the abstract functional differential equation du(t) = A(t)u(t)+F(t,u ) with u = φ ∈ C dt t 0 under appropriate assumptions. References [1] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Math., Series 247, Longman Scientific and 6 Technical, 1991. [2] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. 7