Diekmann, O. (Odo)

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stichting mathematisch centrum AFDEL1NG TOEGEPASTE WISKUNDE (DEPARTMENT OF APPLIED MATHEMATICS) O. DIEKMANN A DUALITY PRINCIPLE FOR DELAY EQUATIONS F'repr int ~ MC TN 100/81 AUGUSTUS kruislaan 413 1098 SJ amsterdam PJunte.d a.:t.: :the. Ma:th.e.ma.:tJ..c.al Ce.n.tJLe., · 413 KJr.t.U6laa.n, Am6:tvui.am. The. Mathe.rn1a.:tJ..c.ai. Ce.n.tJLe. , 6ou.nde..d :the. 11-:th 06 Fe.bll.U..aJl.y 1946, b.i a non-p11,06U .ln6.tliu;Uon a,uni,ng at :the. p11,omo:Uon 06 pU11,e, mathe.ma.:tJ..C-6 and Lto a.ppUc.a.:tJ..ow.,. 1:t b.i 1.,pon6011,e.d by :the. Ne.:the,Jr,R.and.J.i Gove,11,nme.nt :thMu.gh :the. Ne.:thma.nd1.i 011,gan.lza.:tJ..on 6011, :the. Advanc.e.me.nt 06 PU/1,e. Re,1.,e.a.11,c.h (Z.W.O.) • . / 1980 Mathematics subject classification: 34K0S, 47D05 A duality principle for delay equations*) by O. Diekmann ABSTRACT In this note we point at the duality relation between two different semigroups of operators associated with a linear retarded functional differ-ential equations. KEY WORDS & PHRASES: Zinear retarded functiona.i differentiai equation, semi-group of operators, adjoints *) To appear in the Proceedings of Equadiff 5, Bratislava, 1981. I • INTRODUCTION A semigroup associated with a first order differential equation describes how the "state" evolves when time proceeds. Here "state" means any object which singles out a unique solution. In the theory of functional differential equations with finite delay it is usage to take initial functions (describing past history) for such objects. This note calls attention to an alternative: forcing functions of a special type (the idea is due to Miller [7]). Here we concentrate on the natural duality relation be-tween the two constructions in the linear case. This relation was discovered by Burns & Herdman [I] in a special case of infinite delay. It seems that the result presented here as well as the general validity of the basic idea have not been no-ticed before. Elsewhere we show that the study of semigroups acting on forcing func-tions is interesting and useful by itself [4]. 2. THE USUAL SEMIGROUP Lets be a given nxn-real-matrix-valued function of bounded variation such that s(8) =' O for 8 s; 0 and s(8) = s(r) for 8 ~ r. With the retarded functional differential equation (I) ~(t) r f ds (8) x (t-0) 0 one can associate a strongly continuous semigroup of bounded linear operators on C = C([-r,O];JRn) as follows: (2) .(T 1 (s)<jl) (8) X (8;<jl), s -r s; 8 s; 0. Here x(t;<jl) denotes the solution of (1) which satisfies the initial condition (3) x(8) = ¢(8), -rs; 8 s; 0. 3. THE UNUSUAL SEMIGROUP (I) and (3) jointly lead to the Volterra integral equation (4) X where f F¢ with (5) (F¢) (t) t r ¢(0) +ff ds(S)¢(T-8)dT. 0 T 2 Note that F¢ is constant fort~ r. With (4) one can associate a semigroup of bound-ed linear operators on several spaces of functions, which are defined on [0, 00 ) and which are constant on [r, 00), as follows: The motivation for this definition stems from the fact that T2 (s)f is precisely the forcing function in the translated equation: x = s*X + T2(s)f s s 4. DUALITY Let NBV NBV([ 0, r] ;lRn) denote the space of bounded variation functions de-fined on [0, 00 ) and such that (i) f(0) = 0; (ii) f is constant fort~ r; (iii) f is continuous from the right on (0,r). We norm NEV by the total variation. Then NEV re-alizes the dual space of C, the pairing being given by r (7) <f,¢> = I df(t)¢(-t) 0 T Lets denote the transpose of s• An explicit representation of solutions and some calculations lead to Theorem. Lack of space prevents the formulation of the corresponding result for the infinite-simal _generatoI's. 5. REMARKS i) There are (at least) two ways to keep books of the information which fixes the future evolution of a delay system: initial functions and forcing functions. Each way induces a natural semigroup construction which we call T1 and T2, re-spectively. Taking adjoints involves two actions: (i) the structure of the target space lRn requires changing over to the transposed matrix; (ii) the structure of time-dependence requires changing over from the T1 point of view to the T2 point of view or vice versa. This is, we believe, a general principle. ii) Let p denote the reflection: (p~)(t) ~(-t). Then <Fp~,¢> = (~,¢), where (•,·) denotes the well-known bilinear form associated with(!). This observation ex-plains the prominent role of this form in the adjoint theory of [5,6] which ignores the T2 semigroup construction. iii) The reflection implicit in (7) is convenient for autonomous equations: one can 3 integrate all equations in the forward direction. In the nonautonomous case one should avoid it and instead integrate the adjoint equation in the backward direction. iv) With (1) one can associate as well a semigroup on the Hilbert space M2 = n n O 1 ~ lR x 1 2 ([-r,0];lR ) by T1 (s) (<f> ,</> ) = (x(s) ,xs). The Sobolev space H1 of func-tions with distributional derivative in 1 2 and constant on [r,~) realizes the dual space of M2 (the pairing being defined by <f,(</> 0 ,</> 1)> = f(O)</>O + J~ f'(t)<f> 1(-t)dt). In this reflexive situation T1 (s;s) and T2 (s;sT) are ad-joints of each other and T2 is strongly continuous as well. F extends to a map-. ~ 0 I O I ping of M2 into H1 [2, Thm. 2.1] and FT 1(s;s)(<f> ,</>) = T2 (s;s)F(<t> ,</>).The structural operator studied by Delfour & Manitius [2] is precisely Fin dis-guise. The formulation of their results can be simplified using duality as de-scribed above. v) In (4) one need not to require thats is of bounded variation. So the T2-approach also covers some neutral equations. Ifs vanishes fort~ r, the space of forcing functions with this property is invariant under T2 and one can re-strict T2 to this space. This amounts to studying certain integral equations. The duality principle still holds in this situation [3]. vi) The T2-approach has a non-linear analogue and, starting from· a variation-of-constants formula, one can build a qualitative theory in this frame-work [4]. 6. REFERENCES [I] Bums, J.A. & T.L. Herdman, Adjoint semigroup theo'I']f for a aZass of functional differential equations, SIAM J. Math. Anal. 7 (1976) 729-745. [2] Delfour, MLC. & A. Manitius, The structural operator F and its role in the theo'I']f of retarded systems, J. Math. Anal. Appl. I 73 (1980) 466-490 and II J..!!.. (1980) 359-381. [3] Diekmann, o., Volterra integral equations and semigroups of operators, Math. Centre Report TW 197, 1980. [4] Diekmann, O. & S.A. van Gils, Invariant manifolds for Volterra integral equa-tions of convolution -type, to appear. [SJ Hale, J.K., Theo'I']f of Functional Differential Equations, Springer, 1977. [6] Henry, D., The adjoint of a linear functional differential equation and bound-a'I']f value problems, JDE -2_ (1971) 55-66. [7] Miller, R.K., Linear· Volterra integro - differential equations as semigroups, Funk. Ekv. 17 (1974) 39-55. 

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