On the Cauchy Problem for a Linear Harmonic Oscillator with Pure Delay
aa r X i v : . [ m a t h . D S ] D ec On the Cauchy Problem for a Linear HarmonicOscillator with Pure Delay
Denys Khusainov ∗ Michael Pokojovy † Elvin Azizbayov ‡ July 5, 2017
Abstract
In the present paper, we consider a Cauchy problem for a linear second orderin time abstract differential equation with pure delay. In the absence of delay, thisproblem, known as the harmonic oscillator, has a two-dimensional eigenspace so thatthe solution of the homogeneous problem can be written as a linear combinationof these two eigenfunctions. As opposed to that, in the presence even of a smalldelay, the spectrum is infinite and a finite sum representation is not possible. Usinga special function referred to as the delay exponential function, we give an explicitsolution representation for the Cauchy problem associated with the linear oscillatorwith pure delay. In contrast to earlier works, no positivity conditions are imposed.
Keywords: functional-differential equations, harmonic oscillator, pure delay, well-posedness, solution representation
AMS:
Let X be a (real or complex) Banach space and let x ( t ) ∈ X describe the state of aphysical system at time t ≥
0. With a ( t ) = ¨ x ( t ) denoting the acceleration of system, theNewton’s second law of motion states that F ( t ) = M a ( t ) for t ≥ , (1.1)where M : D ( M ) ⊂ X → X is a linear, continuously invertible, accretive operator repre-senting the “mass” of the system. When being displaced from its equilibrium situated in ∗ Faculty of Cybernetics, Kyiv National Taras Shevchenko University, Kyiv, Ukraine [email protected] † Department of Mathematics and Statistics, University of Konstanz, Konstanz, Germany [email protected] ‡ Faculty of Mechanics and Mathematics, Baku State University, Azerbaijan [email protected] inear Harmonic Oscillator with Pure Delay F ( t ). In classical mechanics, thisforce is postulated to be proportional to the instantaneous displacement, i.e., F ( t ) = Kx ( t ) for t ≥ K : D ( K ) ⊂ X → X . When M − K is a bounded linearoperator, plugging Equation (1.2) into (1.1), we arrive at the classical harmonic oscillatormodel ¨ x ( t ) = M − Kx ( t ) for t ≥ . (1.3)Assuming now that the restoring force is proportional to the value of the system at somepast time t − τ , Equation (1.2) is replaced with the relation F ( t ) = Kx ( t − τ ) for t ≥ , (1.4)where τ > x ( t ) = M − Kx ( t − τ ) for t ≥ . (1.5)Problems similar to Equation (1.5) also arise when modeling systems with distributedparameters such as general wave phenomena (cf. [14]).Equations similar to (1.5) are often referred to as delay or retarted differential equations.After being transformed to a first order in time system on a Banach space X , a generalequation with constant delay can be written as˙ u ( t ) = H ( t, u ( t ) , u t ) for t > , u (0) = u , u = ϕ. (1.6)Here, τ > u t := u ( t + · ) ∈ L ( − τ, X ), t ≥
0, denotes thehistory variable, H is an X -valued operator defined on a subset of [0 , ∞ ) × X × L ( − τ, X )and u ∈ X , ϕ ∈ L ( − τ, X ) are appropriate initial data. Equations of type (1.6) havebeen intensively studied in the literature. We refer the reader to the monographs byEls’gol’ts & Norkin [7] and Hale & Lunel [8] for a detailed treatment of Equations (1.6)in finite-dimensional spaces X . In contrast to this, results on Equation (1.6) in infinite-dimensional spaces X are less numerous. A good overview can be found in the monographof B´atkai & Piazzera [2].Khusainov et al. considered in [9] Equation (1.6) in R n with H ( t, u ( t ) , u t ) = A u ( t ) + A u ( t − τ )+ (cid:0) u T ( t ) ⊗ b (cid:1) u ( t ) + (cid:0) u T ( t ) ⊗ b (cid:1) u ( t − τ ) + (cid:0) u T ( t − τ ) ⊗ b (cid:1) u ( t − τ ) (cid:1) for symmetric matrices A , A ∈ R n × n and column vectors b , b , b ∈ R n and proposed arational Lyapunov function to study the asymptotic stability of solutions to this system.In their work [10], Khusainov, Agarwal et al. studied a modal, or spectrum, controlproblem for a linear delay equation on R n reading as˙ x ( t ) = Ax ( t ) + bu ( t ) for t > inear Harmonic Oscillator with Pure Delay u ( t ) = m P j =0 c Tj x ( t − jτ ) for some delay time τ > c j ∈ R n . For canonical systems, they developed a method to compute theunknown parameters such that the closed-loop system possesses the spectrum prescribedbeforehand. Under appropriate “concordance” conditions, they were able to carry overtheir considerations for a rather broad class of non-canonical systems.In the infite-dimensional situation, a rather general particular case of (1.6) with H ( t, v, ψ ) = Av + F ( ψ ) where A generates a C -semigroup ( S ( t )) t ≥ on X and F is a nonlinear operatoron L ( − τ, X ) was studied by Travies & Webb in their work [21]. Under appropriate as-sumptions on F , they proved the integral equation corresponding to the weak formulationof the delay equation given by u ( t ) = S ( t ) ϕ (0) + Z t S ( t − s ) F ( u s )d s for t > H (0 , ∞ ; X ).Di Blasio et al. addressed in [4] a similar problem˙ u ( t ) = (cid:0) A + B (cid:1) u ( t ) L u ( t − r ) + L u t , for t > , u (0) = u , u = ϕ (1.8)where A generates a holomorphic C -semigroup on a Hilbert space H , B is a perturbationof A and L , L are appropriate linear operators. If u and ϕ possess a certain regularity,they proved the existence of a unique strong solution in H (0 , ∞ ; X ) ∩ L (cid:0) , ∞ ; D ( A ) (cid:1) by analyzing the C -semigroup inducing the the semiflow t ( u ( t ) , u t ). These resultswere elaborated on by Di Blasio et al. in [5] leading to a generalization for the case ofweighted and interpolation spaces and including a desription of the associated infinitesimalgenerator. Finally, the general L p -case for p ∈ (0 , ∞ ) was investigated by Di Blasio in [3].Recently, in their work [15], Khusainov et al. proposed an explicit L -solution theory fora non-homogeneous initial-boundary value problem for an isotropic heat equation withconstant delay u t ( t, x ) = ∂ i (cid:0) a ij ( x ) ∂ j u ( t, x ) (cid:1) + b i ( x ) ∂ i u ( t, x ) + c ( x ) u ( t, x )+ ∂ i (cid:0) ˜ a ij ( x ) ∂ j u ( t − τ, x ) (cid:1) + ˜ b i ( x ) ∂ i u ( t − τ, x ) + ˜ c ( x ) u ( t − τ, x )++ f ( t, x ) for ( t, x ) ∈ (0 , ∞ ) × Ω ,u ( t, x ) = γ ( t, x ) for ( t, x ) ∈ (0 , ∞ ) × ∂ Ω ,u (0 , x ) = u ( x ) for x ∈ Ω ,u ( t, x ) = ϕ ( t, x ) for ( t, x ) ∈ ( − τ, × Ω . where Ω ⊂ R d is a regular bounded domain and the coefficient functions are appropriate.Conditions assuring for exponential stability were also given.Over the past decade, hyperbolic partial differential equations have attracted a consider-able amound of attention, too. In [17], Nicaise & Pignotti studied a homogeneous isotropic inear Harmonic Oscillator with Pure Delay ∂ tt u ( t, x ) − △ u ( t, x ) + a ∂ t u ( t, x ) + a∂ t u ( t − τ, x ) = 0 for ( t, x ) ∈ (0 , ∞ ) × Ω ,u ( t, x ) = 0 for ( t, x ) ∈ (0 , ∞ ) × Γ ,∂u∂ν ( t, x ) = 0 for ( t, x ) ∈ (0 , ∞ ) × Γ under usual initial conditions where Γ , Γ ⊂ ∂ Ω are relatively open in ∂ Ω with ¯Γ ∩ ¯Γ = ∅ and ν denotes the outer unit normal vector of a smooth bounded domain Ω ⊂ R d . Theyshowed the problem to possess a unique global classical solution and proved the latterto be exponentially stable if a > a > ∂ tt u ( t, x ) = a ∂ xx u ( t − τ, x ) + b∂ x u ( t − τ, x ) + cu ( t − τ, x )+ f ( t, x ) for ( t, x ) ∈ (0 , T ) × (0 , l ) ,u ( t, x ) = γ ( t, x ) for ( t, x ) ∈ (0 , T ) × { , } ,u (0 , x ) = u ( x ) for x ∈ (0 , ,u ( t, x ) = ϕ ( t, x ) for t ∈ ( − τ, , x ∈ (0 , . Under appropriate regularity and compatibility assumptions, they proved the problem topossess a unique C -solution for any finite T >
0. Their proof was based on extrapolationmethods for C -semigroups and an explicit solution representation formula.Recently, Khusainov & Pokojovy presented in [13] a Hilbert-space treatment of the initial-boundary value problem for the equations of thermoelasticity with pure delay ∂ tt u ( x, t ) − a∂ xx u ( x, t − τ ) + b∂ x θ ( x, t − τ ) = f ( x, t ) for x ∈ Ω , t > ,∂ t θ ( x, t ) − c∂ xx θ ( x, t − τ ) + d∂ tx u ( x, t − τ ) = g ( x, t ) for x ∈ Ω , t > ,u (0 , t ) = u ( l, t ) = 0 , ∂ x θ (0 , t ) = ∂ x θ ( l, t ) = 0 for t > ,u ( x,
0) = u ( x ) , u ( x, t ) = u ( x, t ) for x ∈ Ω , t ∈ ( − τ, ,∂ t u ( x,
0) = u ( x ) , ∂ t u ( x, t ) = u ( x, t ) for x ∈ Ω , t ∈ ( − τ, ,θ ( x,
0) = θ ( x ) , θ ( x, t ) = θ ( x, t ) for x ∈ Ω , t ∈ ( − τ, . Their proof exploited extrapolation techniques for strongly continuous semigroups and anexplicit solution representation formula.In the present paper, we give a Banach space solution theory for Equation (1.5) subjectto appropriate initial conditions. Our approach is solely based on the step method anddoes not incorporate any semigroup techniques. In contrast to earlier works by Khusainovet al. [11, 12, 14], we only require the invertibility and not the positivity of M − K inEquation (1.5).In Section 2, we briefly outline some seminal results on second-order abstract Cauchyproblems. In our main Section 3, we prove the existence and uniqueness of solutions to inear Harmonic Oscillator with Pure Delay τ goes to zero. For the sake of completeness, we briefly discuss the initial value problem for the harmonicoscillator being a second order in time abstact differential equation¨ x ( t ) − Ω x ( t ) = f ( t ) for t ≥ x (0) = x ∈ D (Ω) , ˙ x (0) = x ∈ X. (2.2)Here, we assume the linear operator Ω : D (Ω) ⊂ X → X to be continuously invertibleand generate a C -group ( e t Ω ) t ∈ R ⊂ L ( X ) on a (real or complex) Banach space X with L ( X ) denoting the space of bounded, linear operators on X equipped with the norm k A k L ( X ) := sup (cid:8) k Ax k X | x ∈ X, k x k X ≤ (cid:9) . A more rigorous treatment of this problemcan be found in [1, Section 3.14].The general solution to the homogeneous equation is known to read as x h ( t ) = e Ω t c + e − Ω t c for t ≥ c , c ∈ D (Ω). Vectors c , c can be computed using the initial conditions fromEquation (2.2) leading to a system of linear operator equations c + c = x , Ω c − Ω c = x . The latter is uniquely solved by c = Ω − (Ω x + x ) , c = Ω − (Ω x − x ) . Thus, the unique solution of the homogeneous equation with the initial conditions (2.2)is given by x h ( t ) = Ω − e Ω t (Ω x + x ) + Ω − e − Ω t (Ω x − x ) for t ≥ x h ( t ) = ( e Ω t + e − Ω t ) x + Ω − ( e Ω t − e − Ω t ) x for t ≥ . (2.4)A particular solution to the non-homogeneous equation with zero initial conditions willbe determined in the Cauchy form x p ( t ) = Z t K ( t, s ) f ( s )d s for t ≥ . (2.5) inear Harmonic Oscillator with Pure Delay X -valuedfunctions. In Equation (2.5), the function K ∈ C ([0 , ∞ ) × [0 , ∞ ) , L ( X )) is the Cauchykernel, i.e., for any fixed s ≥
0, the function K ( · , s ) is the solution of the homogeneousproblem satisfying the initial conditions K ( t, s ) (cid:12)(cid:12) t = s = 0 L ( X ) , ∂ t K ( t, s ) (cid:12)(cid:12) t = s = id X . Using the ansatz K ( t, s ) = e Ω t c ( s ) + e − Ω t c ( s ) for t, s ≥ c , c ∈ C ([0 , ∞ ) , L ( X )) and taking into account the initial conditions, we arriveat K ( t, s ) (cid:12)(cid:12) t = s = e Ω t c ( s )+ e − Ω t c ( s ) = 0 L ( X ) , ∂ t K ( t, s ) (cid:12)(cid:12) t = s = Ω e Ω s c ( s ) − Ω e − Ω s c ( s ) = id X . Solving this system with generalized Cramer’s rule, we obtain for s ≥ c ( s ) = (cid:18) det L ( X ) (cid:18) e Ω s e − Ω s Ω e Ω s − Ω e − Ω s (cid:19)(cid:19) − det L ( X ) (cid:18) L ( X ) e − Ω s id X − Ω e − Ω s (cid:19) = Ω − e − Ω s ,c ( s ) = (cid:18) det L ( X ) (cid:18) e Ω s e − Ω s Ω e Ω s − Ω e − Ω s (cid:19)(cid:19) − det L ( X ) (cid:18) e Ω s L ( X ) Ω e Ω s id X (cid:19) = Ω − e − Ω s . Thus, the Cauchy kernel is given by K ( t, s ) = Ω − ( e Ω( t − s ) − e − Ω( t − s ) ) for t, s ≥ , whereas the particular solution satisfying zero initial conditions reads as x p ( t ) = 12 Ω − Z t ( e Ω( t − s ) − e − Ω( t − s ) ) f ( s )d s for t ≥ . Hence, for x ∈ D (Ω), x ∈ X and f ∈ L (0 , ∞ ; X ), the unique mild solution x ∈ W , (0 , ∞ ; X ) to the Cauchy problem (2.1)–(2.2) can be written as x ( t ) = ( e Ω t + e − Ω t ) x + Ω − ( e Ω t − e − Ω t ) x + Ω − Z t ( e Ω( t − s ) − e − Ω( t − s ) ) f ( s )d s for t ≥ . (2.6)If the data additionally satisfy x ∈ D (Ω ), x ∈ D (Ω) and f ∈ W , (0 , ∞ ; X ) ∪ C (cid:0) [0 , ∞ ) , D (Ω ) (cid:1) , then the mild solution x given in Equation (2.6) is a classical solutionsatisfying x ∈ C (cid:0) [0 , ∞ ) , X (cid:1) ∩ C (cid:0) [0 , ∞ ) , D (Ω) (cid:1) ∩ C (cid:0) [0 , ∞ ) , D (Ω ) (cid:1) . In this section, we consider a Cauchy problem for the linear oscillator with a single puredelay ¨ x ( t ) − Ω x ( t − τ ) = f ( t ) for t ≥ inear Harmonic Oscillator with Pure Delay x ( t ) = ϕ ( t ) for t ∈ [ − τ, . (3.2)Here, X is a Banach space, Ω ∈ L ( X ) is a bounded, linear operator and ϕ ∈ C (cid:0) [ − τ, , X (cid:1) , f ∈ L (0 , ∞ ; X ) are given functions. In contrast to Section 2, the boundedness of Ω isindespensable here. Indeed, Dreher et al. proved in [6] that Equations (3.1)–(3.2) are ill-posed even if X is a Hilbert space and Ω possesses a sequence of eigenvalues ( λ n ) n ∈ N ⊂ R with λ n → ∞ or λ n → −∞ as n → ∞ . The necessity for Ω being bounded has also beenpointed out by Rodrigues et al. in [20] when treating a linear heat equation with puredelay. Definition 3.1.
A function x ∈ C (cid:0) [ − τ, ∞ ) , X (cid:1) ∩ C (cid:0) [ − τ, , X (cid:1) ∩ C (cid:0) [0 , ∞ ) , X (cid:1) sat-isfying Equations (3.1)–(3.2) pointwise is called a classical solution to the Cauchy problem(3.1)–(3.2). A mild formulation of (3.1)–(3.2) is given by˙ x ( t ) = ˙ x (0) + Ω Z t x ( s − τ )d s + Z t f ( s )d s for t ≥ , (3.3) x ( t ) = ϕ ( t ) for t ∈ [ − τ, . (3.4) Definition 3.2.
A function x ∈ C (cid:0) [ − τ, ∞ ) , X (cid:1) satisfying Equations (3.3)–(3.4) iscalled a mild solution to the Cauchy problem (3.1)–(3.2). By the virtue of fundamental theorem of calculus, any mild solution x to (3.1)–(3.2)with x ∈ C (cid:0) [ − τ, ∞ ) , X (cid:1) ∩ C (cid:0) [ − τ, , X (cid:1) ∩ C (cid:0) [0 , ∞ ) , X (cid:1) is also a classical solution.Obviously, for the problem (3.1)–(3.2) to possess a classical solution, one necessarilyrequires ϕ ∈ C (cid:0) [ − τ, , X (cid:1) .In the following subsection, we want to study the existence and uniquess of mild and clas-sical solutions to the Cauchy problem (3.1)–(3.2) as well as their continuous dependenceon the data. Rather then using the semigroup approach (cf. [8, Chapter 2]), we decided to use themore straightforward step method here reducing (3.3)–(3.4) to a difference equation onthe functional vector space ˆ C τ ( N , X ) defined as follows. Definition 3.3.
Let X be a Banach space, τ > and s ∈ N . We introduce the metricvector space ˆ C sτ ( N , X ) := l ∞ loc (cid:0) N , C s (cid:0) [ − τ, , X ) (cid:1)(cid:1) := n x = ( x n ) n ∈ N (cid:12)(cid:12) x n ∈ C s (cid:0) [ − τ, , X (cid:1) for n ∈ N , d j d t j x n ( − τ ) = d j d t j x n − (0) for j = 0 , . . . , s − , n ∈ N o inear Harmonic Oscillator with Pure Delay equipped with the distance function d ˆ C sτ ( N ,X ) ( x, y ) := X n ∈ N − n max k =0 ,...,n k x k − y k k C s ([ − τ, ,X ) k =0 ,...,n k x k − y k k C s ([ − τ, ,X ) for x, y ∈ ˆ C sτ ( N , X ) . Obviously, ˆ C sτ ( N , X ) is a complete metric space which is isometrically isomorphic to themetric space C sτ (cid:0) [ − τ, ∞ ) , X (cid:1) := C s (cid:0) [ − τ, ∞ ) , X (cid:1) equipped with the distance d C sτ ([0 , ∞ ) ,X ) ( x, y ) := X n ∈ N − n k x − y k C s ([ − τ,τn ] ,X ) k x − y k C s ([ − τ,τn ] ,X ) for x, y ∈ C s (cid:0) [ − τ, ∞ ) , X (cid:1) . For any x : [ − τ, ∞ ) → X , we define for n ∈ N the n -th segment of x by means of x n := x ( nτ + s ) for s ∈ [ − τ, . By induction, x is a mild solution of (3.1)–(3.2) if and only if ( x n ) n ∈ N ∈ ˆ C τ ( N , X ) solves˙ x n ( s ) = ˙ x n − (0) + Ω x n − ( s ) + Z n − τ + s n − τ f ( σ )d σ for s ∈ [ − τ, , n ∈ N ,x ( s ) = ϕ ( s ) for s ∈ [ − τ, . (3.5) Theorem 3.4.
Equation (3.5) has a unique solution ( x n ) n ∈ N ∈ ˆ C τ ( N , X ) . Moreover, x continuously depends on the data in sense of the estimate k x n k C ([ − τ, ,X ) ≤ κ n (cid:16) k ϕ k C ([ − τ, ,X ) + k f k L (0 , τn ; X ) (cid:17) for any n ∈ N with κ := 1 + (1 + 2 τ ) (cid:0) k Ω k L ( X ) (cid:1) .Proof. By the virtue of the fundamental theorem of calculus, Equation (3.5) is satisfiedif and only if x n ( s ) = x n − (0) + ( s − τ ) ˙ x n − (0) + Ω Z s − τ x n − ( σ )d σ (3.6)+ Z s − τ Z n − τ + σ n − τ f ( ξ )d ξ d σ for s ∈ [ − τ, , n ∈ N , (3.7) x n ( − τ ) = x n − (0) , ˙ x n ( − τ ) = ˙ x n − (0) for n ∈ N , (3.8) x ( s ) = ϕ ( s ) for s ∈ [ − τ, . (3.9)By induction, we can easily show that for any n ∈ N there exists a unique local solution( x , x , . . . , x n ) ∈ (cid:16) C (cid:0) [ − τ, , X (cid:1)(cid:17) n +1 to (3.7)–(3.9) up to the index n . Here, we usedthe Sobolev embedding theorem stating W , (0 , T ; X ) ֒ → C (cid:0) [0 , T ] , X (cid:1) for any T > . inear Harmonic Oscillator with Pure Delay k x n k C ([ − τ, ,X ) ≤ (cid:16) τ (cid:0) k Ω k L ( X ) (cid:1)(cid:17) k x n − k C ([ − τ, ,X ) + 2 τ k f k L (2( n − τ, nτ ; X ) . (3.10)Similarly, Equation (3.5) yields k ˙ x n k C ([ − τ, ,X ) ≤ (cid:0) k Ω k L ( X ) (cid:1) k x n − k C ([ − τ, ,X ) + k f k L (2( n − τ, nτ ; X ) . (3.11)Equations (3.10) and (3.11) imply together k x n k C ([ − τ, ,X ) ≤ κ (cid:0) k ϕ k C ([ − τ, ,X ) + k f k L (2( n − τ,nτ : X ) (cid:1) . By induction, we then get for any n ∈ N k x n k C ([ − τ, ,X ) ≤ κ n (cid:0) k ϕ k C ([ − τ, ,X ) + k f k L (0 , τn,X ) (cid:1) which finishes the proof.Letting x ( t ) := x k ( t − k + 1) τ ) for t ≥ k := ⌊ t τ ⌋ ∈ N , we obtain the unique mildsolution x of Equations (3.1)–(3.2). Corollary 3.5.
Equations (3.1)–(3.2) possess a unique mild solution x satisfying for any T := 2 nτ , n ∈ N , k x k C ([ − τ,T ] ,X ) ≤ κ n (cid:16) k ϕ k C ([ − τ,T ] ,X ) + k f k L (0 ,T ; X ) (cid:17) for any n ∈ N . with κ := 1 + (1 + 2 τ ) (cid:0) k Ω k L ( X ) (cid:1) . Theorem 3.6.
Under additional conditions ϕ ∈ C (cid:0) [ − τ, , X (cid:1) and f ∈ C (cid:0) [0 , ∞ ) , X (cid:1) ,the unique mild solution given in Corollary 3.5 is a classical solution.Proof. Differentiating Equation (3.5) with respect to t , using the assumptions and thefact that x ∈ C (cid:0) [ − τ, ∞ ) , X (cid:1) , we deduce that x | [ − τ, ≡ ϕ ∈ C (cid:0) [ − τ, , X (cid:1) and¨ x = Ω x ( · − τ ) + f ∈ C (cid:0) [0 , ∞ ) , X (cid:1) . Hence, x ∈ C (cid:0) [ − τ, ∞ ) , X (cid:1) ∩ C (cid:0) [ − τ, , X (cid:1) ∩ C (cid:0) [0 , ∞ ) , X (cid:1) and is thus a classicalsolution of Equations (3.1)–(3.2). Following Khusainov & Shuklin [16] and Khusainov et al. [15], we define for t ∈ R theoperator-valued delayed exponential functionexp τ ( t ; Ω) := L ( X ) , −∞ < t < − τ, id X , − τ ≤ t < , id X + Ω t , ≤ t < τ, id X + Ω t + Ω t − τ ) , τ ≤ t < τ,. . . . . . id X + Ω t + Ω t − τ ) + · · · + Ω k ( t − ( k − τ ) k k ! , ( k − τ ≤ t < kτ,. . . . . . . (3.12) inear Harmonic Oscillator with Pure Delay X → X is an isomorphism fromthe Banach space X onto itself. Theorem 3.7.
The delayed exponential function exp τ ( · ; Ω) lies in C (cid:0) [ − τ, ∞ ) , X (cid:1) ∩ C (cid:0) [0 , ∞ ) , X (cid:1) ∩ C (cid:0) [ τ, ∞ ) , X (cid:1) and solves the Cauchy problem ¨ x ( t ) − Ω x ( t − τ ) = 0 X for t ≥ τ, (3.13) x ( t ) = ϕ ( t ) for t ∈ [ − τ, τ ] (3.14) where ϕ ( t ) = (cid:26) id X , − τ ≤ t < , id X + Ω t, ≤ t ≤ τ. Proof.
To prove the smoothness of x , we first note that x is an operator-valued polynomialand thus analytic on each of the intervals [( k − τ, kτ ] for k ∈ Z . By the definition ofexp τ ( · ; Ω), we further findd j d t j x ( kτ −
0) = d j d t j x ( kτ + 0) for j = 0 , . . . , k, k ∈ N . Hence, x ∈ C (cid:0) [ − τ, ∞ ) , X (cid:1) ∩ C (cid:0) [0 , ∞ ) , X (cid:1) ∩ C (cid:0) [ τ, ∞ ) , X (cid:1) .For k ∈ N , k ≥
2, we have x ( t ) = id X + Ω t
1! + Ω ( t − τ )
2! + Ω ( t − τ )
4! + Ω ( t − τ )
4! + · · · + Ω k ( t − ( k − τ ) k k ! . For t ≥ τ , differentiation yields˙ x ( t ) = Ω + Ω t − τ
2! + Ω ( t − τ )
4! + Ω ( t − τ )
3! + · · · + Ω k ( t − ( k − τ ) k − ( k − (cid:16) id X + Ω t − τ
2! + Ω ( t − τ )
4! + Ω ( t − τ )
3! + · · · + Ω k − ( t − ( k − τ ) k − ( k − (cid:17) = Ω exp τ ( t − τ ; Ω) = Ω x ( t − τ )and therefore¨ x ( t ) = Ω + Ω t − τ
1! + Ω ( t − τ )
2! + · · · + Ω k ( t − ( k − τ ) k − ( k − (cid:16) id X + Ω t − τ
1! + Ω ( t − τ )
2! + · · · + Ω k − ( t − ( k − τ ) k − ( k − (cid:17) = Ω exp τ ( t − τ ; Ω) = Ω x ( t − τ ) . Hence, x satisfies Equation (3.13). Finally, by the definition of exp τ ( · ; Ω), x satisfiesEquation (3.14), too. Corollary 3.8.
The delayed exponential function exp τ ( · ; − Ω) lies in C ([ − τ, ∞ ) , X ) ∩ C (cid:0) [0 , ∞ ) , X (cid:1) ∩ C (cid:0) [ τ, ∞ ) , X ) ∩ and solves the Cauchy problem (3.13)–(3.14) with theinitial data ϕ ( t ) = (cid:26) id X , − τ ≤ t < , id X − Ω t, ≤ t ≤ τ. inear Harmonic Oscillator with Pure Delay x ( t ; Ω) := 12 (cid:0) exp τ ( t ; Ω) + exp τ ( t ; − Ω) (cid:1) for t ≥ − τ,x ( t ; Ω) := 12 Ω − (cid:0) exp τ ( t ; Ω) − exp τ ( t ; − Ω) (cid:1) for t ≥ − τ. (3.15)From Equation (3.12), we explicitly obtain x ( t ; Ω) = id X , − τ ≤ t < τ, id X + Ω t − τ ) , τ ≤ t < τ, id X + Ω t − τ ) + Ω t − τ ) , τ ≤ t < τ,. . . . . . id X + Ω t − τ ) + · · · + Ω k ( t − (2 k − τ ) k (2 k )! , (2 k − τ ≤ t < (2 k + 1) τ and x ( t ; Ω) = L ( X ) , − τ ≤ t < , id X t , ≤ t < τ, id X t + Ω t − τ ) , τ ≤ t < τ, id X t + Ω t − τ ) + Ω t − τ ) , τ ≤ t < τ,. . . . . . id X t + Ω t − τ ) + · · · + Ω k ( t − (2 k ) τ ) k +1 (2 k +1)! , kτ ≤ t < k + 1) τ. Obviously, x and x are even functions with respect to Ω. Figure 1 displays the functions x ( · ; Ω) and x ( · ; Ω) for various values of τ and Ω. Theorem 3.9.
The functions x ( · ; Ω) , x ( · ; Ω) satisfy x ( · ; Ω) , x ( · ; Ω) ∈ C (cid:0) [ − τ, ∞ ) , X (cid:1) ∩ C (cid:0) [ − τ, , X ) ∩ C (cid:0) [ τ, ∞ ) , X (cid:1) . Further, x ( · ; Ω) and x ( · ; Ω) are solutions to the Cauchyproblem (3.13)–(3.14) with the initial data ϕ ( t ) = id X , − τ ≤ t ≤ τ , and ϕ ( t ) = id X t , − τ ≤ t ≤ τ , respectively. First, assuming f ≡ X , Equations (3.1)–(3.2) reduce to¨ x ( t ) − Ω x ( t − τ ) = 0 for t ≥ , (3.16) x ( t ) = ϕ ( t ) for t ∈ [ − τ, , (3.17) Theorem 3.10.
Let ϕ ∈ C (cid:0) [ − τ, , X (cid:1) . Then the unique classical solution x to Cauchyproblem (3.16)–(3.17) is given by x ( t ) = x τ ( t + τ ; Ω) ϕ ( − τ ) + x τ ( t + 2 τ ; Ω) ˙ ϕ ( − τ ) + Z − τ x τ ( t − s ; Ω) ¨ ϕ ( s )d s. Proof.
To solve Equations (3.1)–(3.2), we use the ansatz x ( t ) = x τ ( t + τ ; Ω) c + x τ ( t + 2 τ ; Ω) c + Z − τ x τ ( t − s ; Ω)¨ c ( s )d s (3.18) inear Harmonic Oscillator with Pure Delay −1 −0.5 0 0.5 1 1.5 201234567 t x τ ( t , . ) τ = 0 . τ = 0 . τ = 0 . −1 −0.5 0 0.5 1 1.5 201234567 t x τ ( t , ) τ = 0 . τ = 0 . τ = 0 . −1 −0.5 0 0.5 1 1.5 201234567 t x τ ( t , . ) τ = 0 . τ = 0 . τ = 0 . −1 −0.5 0 0.5 1 1.5 201234567 t x τ ( t ; . ) τ = 0 . τ = 0 . τ = 0 . −1 −0.5 0 0.5 1 1.5 201234567 t x τ ( t ; ) τ = 0 . τ = 0 . τ = 0 . −1 −0.5 0 0.5 1 1.5 201234567 t x τ ( t ; . ) τ = 0 . τ = 0 . τ = 0 . Figure 1: Functions x τ ( · ; Ω) and x τ ( · ; Ω).for some c , c ∈ X and c ∈ C (cid:0) [ − τ, , X (cid:1) .Plugging the ansatz from Equation (3.18) into Equation (3.16), we obtain for t ≥ d t (cid:16) x τ ( t + τ ; Ω) c + x τ ( t + 2 τ ; Ω) c + Z − τ x τ ( t − s ; Ω)¨ c ( s )d s (cid:17) − Ω (cid:16) x τ (( t + τ ) − τ ; Ω) c + x τ (( t + 2 τ ) − τ ; Ω) c + Z − τ x τ (( t − τ ) − s ; Ω)¨ c ( s )d s = 0or, equivalently, (cid:16) d d t x τ ( t + τ ; Ω) − Ω x τ (( t + τ ) − τ ; Ω) (cid:17) c + (cid:16) d d t x τ ( t + 2 τ ; Ω) − Ω x τ (( t + 2 τ ) − τ ; Ω) (cid:17) c + Z τ (cid:16) d d t x τ ( t − s ; Ω) − Ω x τ (( t − τ ) − s ; Ω) (cid:17) ¨ c ( s )d s ≡ X . inear Harmonic Oscillator with Pure Delay x τ ( · ; Ω) and x τ ( · ; Ω) solve the homogeneous equation, all three coefficients at c , c and ¨ c vanish implying that the function x in Equation (3.18) is a solution of Equation(3.16).Now, we show that selecting c := ϕ ( − τ ), c := ˙ ϕ ( − τ ) and c := ϕ , the function x inEquation (3.18) satisfies the initial condition (3.17). Letting for t ∈ [ − τ, (cid:2) Iϕ (cid:3) ( t ) := Z − τ x τ ( t − s ; Ω) ¨ ϕ ( s )d s and performing a change of variables σ := t − s , we find (cid:2) Iϕ (cid:3) ( t ) = Z tt +2 τ x τ ( σ ; Ω) ¨ ϕ ( t − σ )d σ = − Z t +2 τt x τ ( σ ; Ω) ¨ ϕ ( t − σ )d σ. Since x can continuously be extended by 0 L ( X ) onto ( −∞ , − τ ], we get (cid:2) Iϕ (cid:3) ( t ) = − Z t +2 τ x ( σ ; Ω) ¨ ϕ ( t − σ )d σ. Integrating by parts, we further get (cid:2) Iϕ (cid:3) ( t ) = − Z t +2 τ x τ ( σ ; Ω) ¨ ϕ ( t − σ )d σ = − x τ ( σ ; Ω) ˙ ϕ ( t − σ ) (cid:12)(cid:12) σ = t +2 τσ =0 + Z t +2 τ ˙ x τ ( σ ; Ω) ˙ ϕ ( t − σ )d σ. Now, taking into account x τ ( t ; Ω) = t id X , ≤ t ≤ τ, (3.19)we obtain (cid:2) Iϕ (cid:3) ( t ) = − x τ ( t + 2 τ ; Ω) ˙ ϕ ( − τ ) + Z t +2 τt ˙ x τ ( σ ; Ω) ˙ ϕ ( t − σ )d σ. Again, using Equation (3.19), we compute (cid:2) Iϕ ]( t ) = − t ˙ ϕ ( − τ ) − ϕ ( t − σ ) (cid:12)(cid:12) σ = t +2 τσ = t = − x τ ( t ; Ω) ˙ ϕ ( − τ ) − ϕ ( − τ ) + ϕ ( t ) . Hence, for t ∈ [ − τ, x ( t ) = x τ ( t + τ ; Ω) ϕ ( − τ ) + x τ ( t + 2 τ ; Ω) ˙ ϕ ( − τ ) + Z − τ x τ ( t − s ; Ω) ¨ ϕ ( s )d s = ϕ ( t )as claimed.Next, we consider Equations (3.1)–(3.2) for the trivial initial data, i.e.,¨ x ( t ) − Ω x ( t − τ ) = f ( t ) for t ≥ , (3.20) x ( t ) = 0 for t ∈ [ − τ, , (3.21) inear Harmonic Oscillator with Pure Delay Theorem 3.11.
Let f ∈ C (cid:0) [0 , ∞ ) , X (cid:1) . The unique classical solution x to Cauchyproblem (3.20)–(3.21) is given by x ( t ) = Z t x τ ( t − s ; Ω) f ( s )d s. Proof.
To find an explicit solution representation, we use the ansatz x ( t ) = Z t x ( t − s ; Ω) c ( s )d s for t ≥ τ for some function c ∈ C (cid:0) [0 , ∞ ) , X (cid:1) . Differentiating this expression with respect to t andexploiting the initial conditions for x τ ( · ; Ω), we get˙ x ( t ) = Z t ˙ x τ ( t − s ; Ω) c ( s )d s + x τ ( t − s ; Ω) c ( s ) (cid:12)(cid:12) s = t = Z t ˙ x τ ( t − s ; Ω) c ( s )d s + x (0) c ( t )= Z t ˙ x τ ( t − s ; Ω) c ( s )d s. Differentiating again, we find¨ x ( t ) = Z t ¨ x τ ( t − s ; Ω) c ( s )d s + ˙ x τ ( t − s ; Ω) c ( s ) (cid:12)(cid:12) s = t = Z t ¨ x τ ( t − τ − s ; Ω) c ( s )d s + ˙ x τ (0+; Ω) c ( t )= Z t ¨ x τ ( t − s ; Ω) c ( s )d s + c ( t ) . Plugging this into Equation (3.20) and recalling that x τ (Ω; Ω) is a solution of the homo-geneous equation, we get c ( t ) Z t (cid:0) ¨ x τ ( t − s ; Ω) − Ω x τ ( t − τ − s ; Ω) (cid:1) c ( s )d s = f ( t )and therefore c ≡ f .As a consequence from Theorems 3.10 and 3.11, we obtain using the linearity property ofEquations (3.1)–(3.2): Theorem 3.12.
Let ϕ ∈ C (cid:0) [ − τ, , X (cid:1) and f ∈ C (cid:0) [0 , ∞ ) , X (cid:1) . The unique classicalsolution to Equations (3.1)–(3.2) is given by x ( t ) = x τ ( t + τ ; Ω) ϕ ( − τ ) + x τ ( t + 2 τ ; Ω) ˙ ϕ ( − τ ) + Z − τ x τ ( t − s ; Ω) ¨ ϕ ( s )d s + (cid:26) , t ∈ [ − τ, , R t x τ ( t − s ; Ω) f ( s )d s, t ≥ for t ∈ [ − τ, ∞ ) . inear Harmonic Oscillator with Pure Delay Theorem 3.13.
Let ϕ ∈ C (cid:0) [ − τ, , X (cid:1) and f ∈ L (0 , ∞ ; X ) . The unique mild solu-tion to Equations (3.1)–(3.2) is given by x ( t ) = x τ ( t + τ ; Ω) ϕ ( − τ ) + x τ ( t + 2 τ ; Ω) ˙ ϕ (0) − Z − τ ˙ x τ ( t − s ; Ω) ˙ ϕ ( s )d s + (cid:26) , t ∈ [ − τ, , R t x τ ( t − s ; Ω) f ( s )d s, t ≥ for t ∈ [ − τ, ∞ ) .Proof. Approximating ϕ in C (cid:0) [ − τ, , X (cid:1) with ( ϕ n ) n ∈ N ⊂ C (cid:0) [ − τ, , X (cid:1) and f in L (0 , ∞ ; X ) with ( f n ) n ∈ N ⊂ C (cid:0) [0 , ∞ ) , X (cid:1) , applying Theorem 3.12 to solve the Cauchyproblem (3.1)–(3.2) for the right-hand side f and the initial data ϕ n , performing a partialintegration for the integral involving ¨ ϕ n and passing to the limit as n → ∞ , the claimfollows. τ → Again, we assume X to be a Banach space and prove the following generalization of [13,Lemma 4]. Lemma 3.14.
Let Ω ∈ L ( X ) , T > , τ > and let α := 1 + k Ω k L ( X ) exp (cid:0) τ k Ω k L ( X ) (cid:1) . Then for any τ ∈ (0 , τ ] , k exp τ ( t − τ ; Ω) − exp(Ω t ) k L ( X ) ≤ τ exp( αT k Ω k L ( X ) ) for t ∈ [0 , T ] . Proof.
Let τ ∈ (0 , τ ]. For t ∈ [0 , τ ], the claim easily follows from the mean value theoremfor Bochner integration. Next, we want to exploit the mathematical induction to showfor any k ∈ N . k exp τ ( t − τ ; Ω) − exp( t Ω) k L ( X ) ≤ τ exp (cid:0) αkτ k Ω k L ( X ) (cid:1) for t ∈ (( k − τ, kτ ] . Indeed, assuming that the claim is true for some k ∈ N , we use the fundamental theoremof calculus and find for t ∈ ( kτ, ( k + 1) τ ] k exp τ ( t − τ ; Ω) − exp( t Ω) k L ( X ) ≤ τ exp (cid:0) αkτ k Ω k L ( X ) (cid:1) + Z ( k +1) τkτ (cid:13)(cid:13)(cid:13) dd s exp τ ( s − τ ; Ω) − dd s exp( s Ω) (cid:13)(cid:13)(cid:13) L ( X ) d s ≤ τ exp (cid:0) αkτ k Ω k L ( X ) (cid:1) + k Ω k L ( X ) Z ( k +1) τkτ k exp τ ( s − τ ; Ω) − exp( s Ω) k L ( X ) d s ≤ τ exp (cid:0) αkτ k Ω k L ( X ) (cid:1) + k Ω k L ( X ) Z ( k +1) τkτ (cid:13)(cid:13) exp τ ( s − τ ; Ω) − exp (cid:0) ( s − τ )Ω (cid:1)(cid:13)(cid:13) L ( X ) d s + k Ω k L ( X ) Z ( k +1) τkτ (cid:13)(cid:13) exp( s Ω) − exp (cid:0) ( s − τ )Ω (cid:1)(cid:13)(cid:13) L ( X ) d s inear Harmonic Oscillator with Pure Delay ≤ τ exp (cid:0) αkτ k Ω k L ( X ) (cid:1) + k Ω k L ( X ) Z kτ ( k − τ (cid:13)(cid:13) exp τ ( s − τ ; Ω) − exp( s Ω) (cid:13)(cid:13) L ( X ) d s + k Ω k L ( X ) Z ( k +1) τkτ Z ss − τ (cid:13)(cid:13)(cid:13) dd σ exp( σ Ω) (cid:13)(cid:13)(cid:13) L ( X ) d σ d s ≤ τ exp (cid:0) αkτ k Ω k L ( X ) (cid:1) + τ k Ω k L ( X ) exp (cid:0) αkτ k Ω k L ( X ) (cid:1) + τ k Ω k L ( X ) exp (cid:0) ( k + 1) τ k Ω k L ( X ) (cid:1) ≤ τ exp (cid:0) αkτ k Ω k L ( X ) (cid:1)(cid:16) τ k Ω k L ( X ) + τ k Ω k L ( X ) exp (cid:0) τ k Ω k L ( X ) (cid:1)(cid:17) ≤ τ exp (cid:0) αkτ k Ω k L ( X ) (cid:1)(cid:18) τ k Ω k L ( X ) (cid:16) τ k Ω k L ( X ) exp (cid:0) τ k Ω k L ( X ) (cid:1)(cid:17)(cid:19) ≤ τ exp (cid:0) αkτ k Ω k L ( X ) (cid:1) exp( ατ k Ω k L ( X ) (cid:1) ≤ exp (cid:0) α ( k + 1) τ k Ω k L ( X ) (cid:1) since α ≥
1. The claim follows by induction.
Corollary 3.15.
Let the assumptions of Lemma 3.14 be satisfied and let γ ≥ . Then (cid:13)(cid:13) exp τ ( t + γ ; Ω) − e Ω t (cid:13)(cid:13) L ( X ) ≤ ( γ + τ ) (cid:0) k Ω k L ( X ) (cid:1) exp (cid:0) α ( T + γ + τ ) k Ω k L ( X ) (cid:1) . Proof.
Lemma 3.14 and the mean value theorem for Bochner integration yield (cid:13)(cid:13) exp τ ( t + γ ; Ω) − e Ω t (cid:13)(cid:13) L ( X ) ≤ (cid:13)(cid:13) exp τ ( t + γ ; Ω) − e Ω( t + γ + τ ) (cid:13)(cid:13) L ( X ) + (cid:13)(cid:13) e Ω( t + γ + τ ) − e Ω t (cid:13)(cid:13) L ( X ) ≤ τ exp (cid:0) α ( T + γ + τ ) k Ω k L ( X ) (cid:1) + ( γ + τ ) k Ω k L ( X ) exp (cid:0) ( T + γ + τ ) k Ω k L ( X ) (cid:1) ≤ ( γ + τ ) (cid:0) k Ω k L ( X ) (cid:1) exp (cid:0) α ( T + γ + τ ) k Ω k L ( X ) (cid:1) as we claimed.Let T > τ > x , x ∈ X and f ∈ L (0 , ∞ ; X ) be fixed and let ¯ x ∈ C (cid:0) [0 , ∞ ) , X (cid:1) denote the unique mild solution to the Cauchy problem (2.1)–(2.2) from Section 2. Theorem 3.16.
Let τ > . For any τ ∈ (0 , τ ) , let x ( · ; τ ) denote the unique mild solutionof (3.1)–(3.2) for the initial data ϕ ( · ; τ ) ∈ C (cid:0) [ − τ, , X (cid:1) . Then we have k x ( · ; τ ) − ¯ x k C ([0 ,T ] ,X ) ≤ β (cid:16) k ϕ ( − τ ; τ ) − x k X + k ˙ ϕ (0; τ ) − x k X (cid:17) + 3 βτ (cid:16) k ϕ ( · ; τ ) k C ([ − τ, ,X ) + k f k L (0 ,T ; X ) (cid:17) with β ( T ) := 2 (cid:0) k Ω k L ( X ) (cid:1)(cid:0) k Ω − k L ( X ) (cid:1) exp (cid:0) α ( T + 2 τ ) k Ω k L ( X ) (cid:1) .Proof. Using the explicit representation of ¯ x and x ( · ; τ ) and x from Sections 2 and 3.2,respectively, we can estimate k x ( t ; τ ) − ¯ x ( t ) k X ≤ I , ( t ) + I , ( t ) + I , ( t ) for t ∈ [0 , T ] inear Harmonic Oscillator with Pure Delay I , ( t ) := (cid:13)(cid:13) x τ ( t + τ ; Ω) ϕ ( − τ ; τ ) − ( e Ω t + e − Ω t ) x (cid:13)(cid:13) X + (cid:13)(cid:13) x τ ( t + 2 τ ; Ω) ˙ ϕ (0; τ ) + Ω − ( e Ω t − e − Ω t ) x (cid:13)(cid:13) X ,I , ( t ) := Z t (cid:13)(cid:13) x τ ( t − s ; Ω) − Ω − ( e Ω( t − s ) − e − Ω( t − s ) ) (cid:13)(cid:13) L ( X ) k f ( s ) k X d s,I , ( t ) := Z − τ k x τ ( t − s − τ ; Ω) k L ( X ) k ˙ ϕ ( s ; τ ) k X d s. Corollary 3.15 yields (cid:13)(cid:13) x τ ( t + τ ; Ω) − ( e Ω t + e − Ω t ) (cid:13)(cid:13) L ( X ) ≤ βτ, (cid:13)(cid:13) x τ ( t + τ ; Ω) − Ω − ( e Ω t − e − Ω t ) (cid:13)(cid:13) L ( X ) ≤ βτ and, therefore, I , ( t ) ≤ βτ (cid:0) k ϕ ( − τ ; τ ) k X + k ˙ ϕ (0; τ ) k X (cid:1) + β (cid:0) k ϕ ( − τ ; τ ) − x k X + k ˙ ϕ (0; τ ) − x k X (cid:1) ≤ βτ k ϕ k C ([ − τ, ,X ) + β (cid:0) k ϕ ( − τ ; τ ) − x k X + k ˙ ϕ (0; τ ) − x k X (cid:1) . Similarly, I , ( t ) ≤ βτ k f k L (0 ,T ; X ) and I , ( t ) ≤ βτ k ϕ k C ([0 ,T ] ,X ) . Hence, the claim follows.
Corollary 3.17.
Under conditions of Theorem 3.16, we additionally have k x ( · ; τ ) − ¯ x k C ([0 ,T ] ,X ) ≤ β ( T ))(1 + δ ( T ))(1 + T ) (cid:16) k ϕ ( − τ ; τ ) − x k X + k ˙ ϕ (0; τ ) − x k X + τ (cid:0) k ϕ ( · ; τ ) k C ([ − τ, ,X ) + k f k L (0 ,T ; X ) + k x k X + k x k X (cid:1)(cid:17) with δ ( T ) := k Ω k L ( X ) (cid:0) k Ω − k L ( X ) + k Ω − k L ( X ) T (cid:1) e k Ω k L ( X ) T .Proof. Integrating Equation (2.1) and using Equation (2.2) as well as exploiting Equations(3.3)–(3.4) yields k ˙ x ( t ; τ ) − ˙¯ x ( t ) k ≤ k ˙ ϕ (0; τ ) − x k X + Z t k Ω x ( s − τ ; τ ) − Ω ¯ x ( s ) k X d s ≤ I , ( t ) + I , ( t ) + I , ( t ) for t ∈ [0 , T ]with I , ( t ) := k ˙ ϕ (0; τ ) − x k X , I , := k Ω k L ( X ) Z − τ k ϕ ( s ) − ¯ x ( s + 2 τ ) k X d s,I , ( t ) := k Ω k L ( X ) Z t τ k x ( s − τ ; τ ) − ¯ x ( s ) k X d s inear Harmonic Oscillator with Pure Delay k ¯ x k C ([0 , τ ] ,X ) ≤ (cid:0) k x k + k Ω − k L ( X ) k x k (cid:1) e k Ω k L ( X ) T + k Ω − k L ( X ) T e k Ω k L ( X ) T k f k L (0 ,T ; X ) . Hence, I , ( t ) ≤ δτ (cid:16) k ϕ k C ([0 ,T ] ,X ) + k x k X + k x k X (cid:17) . Applying Theorem 3.16, we further get I , ( t ) ≤ k Ω k L ( X ) T β (cid:16) k ϕ ( − τ ; τ ) − x k X + k ˙ ϕ (0; τ ) − x k X + τ (cid:0) k ϕ ( · ; τ ) k C ([ − τ, ,X ) + k f k L (0 ,T ; X ) (cid:1)(cid:17) . Combining these inequalities and using again Theorem Theorem 3.16, we deduce theestimate asserted.
Acknowledgment
This work has been funded by a research grant from the Young Scholar Fund supportedby the Deutsche Forschungsgemeinschaft (ZUK 52/2) at the University of Konstanz, Kon-stanz, Germany.
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