Publications: The Entire List

[1]

F. DiPasquantonio and F. Kappel. Applications to nuclear reactor kinetics of an extension of Liapunov's direct method to functional-differential equations. Energia Nucleare, 15:761–770, 1968.

[2]

F. DiPasquantonio and F. Kappel. Deduction of Krasovskii's instability theorem from LaSalle's extension of Liapunov's direct method and an application to nuclear reactor kinetics. Nukleonik, 12:288–289, 1969.

[3]

F. Kappel. Die Stabilität von Bewegungen. In Steiermärkisches Mathematisches Symposium, Grottendorf-Hardt, 6. 10. –9. 10. 1969, pages 1–33. Institute für Mathematik, Universität Graz–Technische Universität Graz, 1969.

[4]

F. DiPasquantonio and F. Kappel. Comments on "Theoretical and experimental criteria for reactor stability". Nuclear Science and Engineering, 39:133–134, 1970.

[5]

F. Kappel. The invariance of limit sets for autonomous functional-differential equations. SIAM J. Appl. Math., 19:408–420, 1970.

[6]

F. Kappel. An invariance principle for functional-differential equations of retarded type. In Symposium on Differential-Delay and Functional Equations: Control and Stability, Warwick, June 26–July 7, 1972, volume 12 of Control Theory Centre Reports, pages 37–39. University of Warwick, 1972.

[7]

F. Kappel. Stability of nuclear reactors. In Symposium on Differential-Delay and Functional Equations: Control and Stability, Warwick, June 26–July 7, 1972, volume 12 of Control Theory Centre Reports, pages 84–87. University of Warwick, 1972.

[8]

F. Kappel. Systeme with linearem Hauptteil und linearen Abschätzungen für die Störfunktion. ZAMM, 52:213–225, 1972.

[9]

F. Kappel. Differenzen-Differentialgleichungen. In Proc. 5. Steiermärkisches Mathematisches Symposium, Stift Rein, September 24–27, 1973, number 1/1973 in Berichte der Mathematisch-Statistischen Sektion, Research Center Graz, pages 1–77, 1973.

[10]

F. Kappel. Some remarks to the problem of degeneracy for functional-differential equations. In Equa-Diff 73, Bruxelles et Louvaine-la-Neuve, September 3–8, 1973, pages 463–471, 1973.

[11]

F. DiPasquantonio and F. Kappel. Stability criteria for kinetic reactor equations. Arch. Rational Mech. Analysis, 58:317–338, 1975.

[12]

F. DiPasquantonio and F. Kappel. A stability criterion for the kinetic reactor equations with linear feedback. Energia Nucleare, 19:162–175, 1975.

[13]

F. Kappel. Degeneracy of functional-differential equations. In H. A. Antosiewics, editor, International Conference on Differential Equations, Proc. Internatl. Symp. Diff. Eqs., Los Angeles, September 3–7, 1974, pages 434–448, New York, 1975. Academic Press.

[14]

F. Kappel and H. K. Wimmer. An elementary divisor theory for linear autonomous functional differential equations. J. Differential Eqs., 21:134–147, 1976.

[15]

F. Kappel. Laplace-transform methods and linear autonomous functional-differential equations. Ber. Math.-Statist. Sekt. 64, Research Center Graz, 1976.

[16]

F. Kappel. On degeneracy of functional-differential equations. J. Differential Eqs., 22:250–267, 1976.

[17]

F. Kappel. A stability criterion for linear autonomous functional differential equations. In L. Cesari, J. K. Hale, and J. P. LaSalle, editors, Dynamical Systems, An International Symposium, Proc. Internatl. Symp. Dynamical Systems, Providence, August 12–16, 1974, pages 103–107, New York, 1976. Academic Press.

[18]

F. Kappel. Degenerate difference-differential equations: Algebraic theory. J. Differential Eqs., 24:99–126, 1977.

[19]

F. Kappel and W. Schappacher. Approximation of functional differential equations by finite dimensional systems. In R. Conti, G. Sestini, and G. Villari, editors, Equazioni differenziali ordinarie ed equazioni funzionali, Proc. Equa-Diff 78, Florence, May 24–30, 1978, pages 479–484, 1978.

[20]

F. Kappel and W. Schappacher. Autonomous nonlinear functional differential equations and averaging approximations. Nonlinear Analysis: TMA, 2:391–422, 1978.

[21]

F. Kappel. Approximation of functional-differential equations by ordinary differential equations and hereditary control problems. In J. Stoer, editor, Optimization Techniques, Vol. I, Proc. 8th IFIP Conf. Optimization Techniques, Würzburg, September 5–9, 1977, volume 6 of Lecture Notes in Control and Information Sciences, pages 103–108, Berlin, 1978. Springer Verlag.

[22]

H. T. Banks and F. Kappel. Spline approximations for functional differential equations. J. Differential Eqs., 34:496–522, 1979.

[23]

F. Kappel and W. Schappacher. Nonlinear functional differential equations and abstract integral equations. Proc. Royal Soc. Edinburgh, 84A:71–91, 1979.

[24]

F. Kappel and W. Schappacher. Some considerations to the fundamental theory of infinite delay equations. J. Differential Eqs., 37:141–183, 1980.

[25]

F. Kappel and K. Kunisch. Spline approximations for neutral functional differential equations. SIAM J. Numerical Analysis, 18:1058–1080, 1981.

[26]

F. Kappel. Approximation of neutral functional differential equations in the state space Rⁿ times L^p. In M. Farkas, editor, Qualitative Theory of Differential Equations, Vol. I, Colloquia Mathematica Societatis Janos Bolyai, Vol. 30, pages 463–506. Janos Bolyai Math. Soc. and North Holland Publ. Comp., 1982.

[27]

F. Kappel. An approximation scheme for delay equations. In V. Lakshmikantham, editor, Nonlinear Phenomena in Mathematical Sciences, Proc. Internat. Conf. Nonl. Phenomena Math. Sci., Arlington (Texas), June 16–20, 1980, pages 585–595, New York, 1982. Academic Press.

[28]

F. Kappel and K. Kunisch. Approximation of the state of infinite delay and Volterra-type equations. In L. Collatz, G. Meinardus, and W. Wetterling, editors, Differenzen-Differentialgleichungen, Anwendungen und numerische Probleme, Proc. Workshop in Oberwolfach, June 6–12, 1982, pages 149–168, Basel, 1983. Birkhäuser.

[29]

F. Kappel, K. Kunisch, and G. Moyschewitz. An approximation scheme for infinite delay equations of Volterra type: Numerical results. Technical Report 11, Institutes for Mathematics–Technical University and University of Graz, 1983.

[30]

F. Kappel. Finite dimensional approximation to systems with infinite dimensional state space. In H.-W. Knobloch and K. Schmitt, editors, Equadiff 82, Proc. Equa-Diff 82, Würzburg, August 23–28, 1982, volume 1017, pages 287–299, Berlin, 1983. Springer Verlag.

[31]

F. Kappel and R. Grimmer. Series expansions for resolvents of Volterra integrodifferential equations in Banach spaces. SIAM J. Math. Analysis, 15:595–604, 1984.

[32]

F. Kappel and G. Propst. Approximation of feedback controls for delay systems using Legendre polynomials. Confer. Mat. Sem. Mat. Univ. Bari, 201:1–36, 1984.

[33]

F. Kappel, K. Kunisch, and G. Moyschewitz. An approximation scheme for parameter estimation in infinite delay equations of Volterra type: Numerical results. Technical Report 51, Institutes for Mathematics – Technical University and University of Graz, 1984.

[34]

F. Kappel. Linear autonomous functional differential equations in the state space C. Technical Report 34, Institutes for Mathematics–Technical University and University of Graz, 1984.

[35]

F. Kappel. Galerkin type approximation schemes for delay systems. Annales of Differential Equations, 1:57–82., 1985.

[36]

F. Kappel and Kangpei Zhang. Equivalence of functional-differential equations of neutral type and abstract Cauchy problems. Monatshefte für Mathematik, 101:115–133, 1986.

[37]

F. Kappel and Kangpei Zhang. On neutral functional differential equations with nonatomic D-operator. J. Math. Analysis Appl., 113:311–343, 1986.

[38]

F. Kappel. Semigroups and delay equations. In H. Brezis, M. G. Crandall, and F. Kappel, editors, Semigroups, Theory and Applications, Vol II, volume 152, pages 136–176, Harlow, 1986. Longman.

[39]

F. Kappel. Spline approximation for autonomous nonlinear functional differential equations. Nonlinear Analysis: TMA, 10:503–513, 1986.

[40]

F. Kappel and K. Kunisch. Invariance results for delay and Volterra equations in fractional order Sobolev spaces. Trans. AMS, 304:1–51, 1987.

[41]

F. Kappel and D. Salamon. Spline approximation for retarded systems and the Riccati equation. SIAM J. Control and Optimization, 25:1082–1117, 1987.

[42]

H. T. Banks, L. W. Botsford, F. Kappel, and C. Wang. Modeling and estimation in size structured population models. In T. G. Hallam, L. J. Gross, and S. A. Levin, editors, Mathematical Ecology, pages 521–541, Singapoore, 1988. World Scientific Publ.

[43]

F. Kappel and D. Salamon. Approximation of the algebraic Riccati operator equation. In Proc. 27th IEEE Conference on Decision and Control, December 7–9, 1988, Austin, Texas, pages 2001–2002, 1988.

[44]

F. Kappel and K. Schmitt. Periodic solutions of systems of ordinary differential equations which approximate delay equations. Differential and Integral Eqs., 1:183–212, 1988.

[45]

H. T. Banks and F. Kappel. Transformation semigroups and L¹-approximation for size structured population models. Semigroup Forum, 38:141–155, 1989.

[46]

K. Ito and F. Kappel. Approximation of infinite delay and Volterra type equations. Numerische Math., 54:405–444, 1989.

[47]

F. Kappel and G. Peichl. Preservation of controllability under approximation for delay systems. Matematica Aplicada e Computational, 8:23–47, 1989.

 

This paper deals with controllability of differential equations with a single delay and its finite dimensional approximations. A set of hypotheses is presented which allow a unified treatment of various approximation schemes. Controllability of the delay equation as well as of the approximating ordinary differential equations is characterized by a finite number of linear homogeneous equations which in both cases are closely related. For the approximating systems the size of the linear system of equations which characterizes controllability is independent of the order of approximation.

[48]

F. Kappel and D. Salamon. On the stability properties of spline approximation for retarded systems. SIAM J. Control and Optimization, 27:407–431, 1989.

[49]

H. T. Banks, L. W. Botsford, F. Kappel, and C. Wang. Estimation of parameters in age/size structured population models. In M. Amouroux and A. El Jai, editors, Control of Distributed Parameter Systems 1989, pages 383–388, ..., 1990.

[50]

F. Kappel and D. Salamon. An approximation theorem for the algebraic Riccati equation. SIAM J. Control and Optimization, 28:1136–1147, 1990.

 

For an infinite dimensional linear quadratic control problem in Hilbert space, approximation of the solution of the algebraic Riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems. As an application strong convergence of the approximating Riccati operators in case of a previously developed spline approximation scheme for delay systems is established. Finally we investigate convergence of the transfer-functions of the approximating systems.

[51]

H. T. Banks, L. W. Botsford, and F. Kappel. Estimation of growth and survival in size-structured cohort data: An application to larval striped bass (morone saxatilis). J. Math. Biology, 30:125–150, 1991.

[52]

H. T. Banks, F. Kappel, and C. Wang. Weak solutions and differentiability for size structured population models. In W. Desch, F. Kappel, and K. Kunisch, editors, Estimation and Control of Distributed Parameter Systems, volume 100 of ISNM (International Series of Numerical Mathematics), pages 35–50, Basel, 1991. Birkhäuser.

[53]

K. Ito and F. Kappel. On integro-differential equations with weakly singular kernels. In J. Goldstein, F. Kappel, and W. Schappacher, editors, Differential Equations with Applications in Biology, Physics and Engineering, pages 209–218, New York, 1991. Marcel-Dekker.

 

In this paper we consider a class of integro-differential equations with weakly singular kernels. Such an equation is treated using the theory of linear C₀-semigroups. Differentiability of the solution semigroup and well-posedness in case of non-integrable kernels on (- infty ,0) are established.

[54]

K. Ito and F. Kappel. A uniformly differentiable approximation scheme for delay systems using splines. Appl. Math. Optim., 23:217–262, 1991.

 

A new spline-based scheme is developed for linear retarded functional differential equations within the framework of semigroups on the Hilbert space bold Rⁿ times L². The approximating semigroups inherit in a uniform way the characterization for differentiable semigroups from the solution semigroup of the delay system (e.g. among other things the logarithmic sectorial property for the spectrum). We prove convergence of the scheme in the state spaces bold Rⁿ times L² and H¹. The uniform differentiability of the approximating semigroups enables us to establish error estimates including quadratic convergence for certain classes of initial data. We also apply the scheme for computing the feedback solutions to linear quadratic optimal control problems.

[55]

K. Ito, F. Kappel, and G. Peichl. A fully discretized approximation scheme for size structured population models. SIAM J. Numerical Analysis, 28:923–954, 1991.

 

An efficient algorithm for computing solutions to a class of models for size structured populations is presented. Furthermore, some numerical examples are discussed.

[56]

K. Ito, F. Kappel, and D. Salamon. A variational approach to approximation of delay systems. Differential and Integral Eqs., 4:51–72, 1991.

 

Using a varational formulation of the Trotter-Kato approximation theorem for strongly continuous semigroups we show that one can construct in a uniform way most of the approximation schemes for delay systems which have been published in recent years. In order to demonstrate that this approach is not restricted to delay systems we also construct two approximation schemes for size structured population models.

[57]

F. Kappel. Approximation of LQR-problems for delay systems: a survey. In K. Bowers and J. Lund, editors, Computation and Control II, pages 187–224, Boston, 1991. Birkhäuser.

[58]

K. Ito and F. Kappel. Two families of approximation schemes for delay systems. Results in Mathematics, 21:93–137, 1992.

 

Based on the Trotter-Kato approximation theorem for strongly continuous semigroups we develop a general framework for the approximation of delay systems. Using this general framework we construct two families of concrete approximation schemes. Approximation of the state is done by functions which are piecewise polynomials on a mesh (m-th order splines of deficiency m). For the two families we also prove convergence of the adjoint semigroups and uniform exponential stability, properties which are essential for approximation of linear quadratic control problems involving delay systems. The characteristic matrix of the delay system is in both cases approximated by matrices of the same structure but with the exponential function replaced by approximations where Padé fractions in the main diagonal resp. in the diagonal below the main diagonal of the Padé table for the exponential function play an essential role.

[59]

F. Kappel and R. O. Peer. Ein mathematisches Modell für grundlegende Regelungsmechnismen im Herz-Kreislaufsystem. Biomedizinische Technik, 37, Ergänzungsband 1:58–60, 1992.

[60]

F. Kappel and Kangpei Zhang. Approximation of linear age-structured population models using Legendre polynomials. J. Math. Anal. Appl., 180:518–549, 1993.

 

We develop a numerical algorithm for approximation of solutions for linear age–structured population models. The construction is based on approximation of age distributions by modified Legendre polynomials and uses the Trotter-Kato theorem of semigroup theory for the corresponding abstract Cauchy problem. Unbounded resp. nonintegrable mortality rates are admissible.

[61]

F. Kappel and R. O. Peer. A mathematical model for fundamental regulation processes in the cardiovascular system. J. Math. Biology, 31:611–631, 1993.

 

Based on the four compartment model by Grodins we develop a model for the response of the cardiovascular system to a short term submaximal workload. Basic mechanisms included in the model are Starling's law of the heart, the Bowditch effect and autoregulation in the peripheral regions. A fundamental assumption is that the action of the feedback control is represented by the baroreceptor loop and minimizes a quadratic cost functional. Simulation results show that the model provides a satisfactory description of data obtained in bicycle ergometer tests.

[62]

F. Kappel and J. Turi. Singular integro-differential equations and product spaces. J. Math. Analysis Appl., 178:450–469, 1993.

 

For certain types of integro-differential equations with weakly singular kernels we investigate well-posedness in a state space with product space structure. This state space is obtained as the extrapolation space corresponding to the solution semigroup of the problem on a weighted L²-space, where well-posedness has been established recently.

[63]

F. Kappel. Parameteridentification for state dependent delays originating from threshold conditions. In Proc. IEEE Mediterranean Symposium on New Directions in Control Theory and Applications, June 21 - 23, 1993, Crete Chandris Hotel, Maleme, Crete, Chania, 1993. Technical University of Crete.

[64]

H. T. Banks, F. Kappel, and C. Wang. A semigroup formulation of a nonlinear size-structured distributed rate population model. In W. Desch, F. Kappel, and K. Kunisch, editors, Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena, volume 118 of ISNM (International Series of Numerical Mathematics), pages 1–19, Basel, 1994. Birkhäuser.

[65]

K. Ito and F. Kappel. Approximation of semilinear equations. Nonlinear Analysis: Theory, Methods and Applications, 24:51–80, 1995.

 

In this paper we prove well-posedness and approximation results for semilinear evolution equations in Banach spaces. These results are applicable to nonlinear delay equations with state and time dependent delays. As an example we consider a model in epidemics with threshold effects in the mechanism of infection.

[66]

F. Kappel and R. O. Peer. Implementation of a cardiovascular model and algorithms for parameter identification. Bericht 26, Spezialforschungsbereich F003 "Optimierung und Kontrolle", 1995.

[67]

P. Bachhiesl, H. Hutten, F. Kappel, K. Köpke, and H. Scharfetter. Dynamical control of the dialysis process. Part I: Structural considerations and first mathematical approach. Biomedizinische Technik, 41:196–202, 1996.

[68]

P. Bachhiesl, H. Hutten, F. Kappel, and H. Scharfetter. Dynamical control of the dialysis process. Part II: An improved algorithm for the solution of a tracking problem. Biomedizinische Technik, 41:228–235, 1996.

[69]

P. Bachhiesl, H. Hutten, F. Kappel, and H. Scharfetter. Dynamische Kontrolle von Dialyseparametern: ein effizienter Algorithmus zur Lösung eines Tracking Problems. Biomedizinische Technik, 41, Ergänzungsband 1:86–87, 1996.

[70]

K. Ito, F. Kappel, and J. Turi. On well-posedness of singular neutral equations in the state space C. J. Differential Eqs., 125:40–72, 1996.

 

In this paper we study necessary und sufficient conditions for well-posedness of scalar, singular (i.e., with a difference operator which is nonatomic at zero) neutral functional differential equations (SNFDEs) in the state space C . We also discuss an ill-posed SNFDE in order to illustrate how the addition of a singularity away from zero (in our case we have a perturbing atom at -r) can destroy well-posedness.

[71]

F. Kappel. The Trotter-Kato theorem and approximation of abstract Cauchy problems. In Bambang Soedijono Subanar and Sri Wahyuni, editors, Proceedings of the Mathematical Analysis and Statistics Conference, Yogyakarta 1995, Part A: Theory and Methods, pages 50–58, 1996. ISBN 979-95118-0-1.

[72]

P. Bachhiesl, H. Hutten, F. Kappel, and H. Scharfetter. Optimale Prozeßkontrolle der Dialysetherapie. Biomedizinische Technik, 42, Ergänzungsband 2:7–8, 1997.

[73]

P. Bachhiesl, H. Hutten, F. Kappel, and H. Scharfetter. Optimierungsverfahren zur Prozeßkontrolle bei der Dialyse. In 42. Internationales Wissenschaftliches Kolloquium, Vol. 2, pages 79–84, 1997. ISSN 0943-7207.

[74]

F. Kappel, S. Lafer, and R. O. Peer. A model for the cardiovascular system under an ergometric workload. Surveys on Mathematics for Industry, 7:239–250, 1997.

[75]

K. Ito and F. Kappel. The Trotter-Kato theorem and approximation of PDEs. Math. Computation, 67:21–44, 1998.

 

We present formulations of the Trotter-Kato theorem for approximation of linear C₀-semigroups which provide very useful framework when convergence of numerical approximations to solutions of PDEs are studied. Applicability of our results is demonstrated using a first order hyperbolic equation, a wave equation and Stokes' equation as illustrative examples.

[76]

P. Bachhiesl, H. Hutten, F. Kappel, B. Puswald, and H. Scharfetter. Patient individual analysis and control of exchange processes during hemodialysis. Med. Biol. Eng. Comput., 37, Suppl. 2:1162–1163, 1999.

[77]

P. Bachhiesl, H. Hutten, F. Kappel, and H. Scharfetter. Ein Ansatz zur Optimierung der Prozeßsteuerung bei der Hämodialyse. Automatisierungstechnik, 47:38–48, 1999.

[78]

H. Hutten, F. Kappel, R. Merwa, and H. Scharfetter. Modular programming environment for simulation, identification and validation of complex nonlinear compartment models. Med. Biol. Eng. Comput., 37, Suppl 2:1204–1205, 1999.

[79]

Abd El Rahman Aly Hussein and F. Kappel. The Trotter-Kato theorem and collocation methods for parabolic equations. In Sri Wahyuni, Retantyo Wardoyo, CH. Rini Indrati, and Supama, editors, Proc. of the SEAMS–GMU Internatl. Conf. Math. and its Appl., Yogyakarta, July 26–29, 1999, pages 22–36, Yogyakarta, Indonesia, 2000. Department of Mathematics, Gadjah Mada University. ISBN 979–95118–2--8.

 

Convergence of a C¹ collocation scheme for a parabolic PDE in one space dimension is established for any initial data in the state space L²(0,1). The proof of converngence is based on the Trotter-Kato approximation scheme for C₀-semigroups.

[80]

Abd El Rahman Aly Hussein, K. Ito, and F. Kappel. Some aspects of the Trotter-Kato approximation theorem for semigroups. In I. I. Eremin, I. Lasicka, and V. I. Maksimov, editors, Distributed Systems: Optimization and Economic-Environmental Applications (DSO'2000), Proceedings of an Internatl. Conf., May 30 – June 2, 2000, Ekaterinburg, Russia, pages 36–37, 2000. ISBN 5–7691–1047–3.

 

The classical proof of the Trotter-Kato theorems also provides rate estimates for smooth initial data. We discuss the possibilities to reduce the socalled smoothness gap for special types of semigroups as for instance analytic semigroups. Furthermore, we show that the assumption of stability of the approximating semigroups can be relaxed if at the same time the consistency requirement is replaced by a stronger assumption. Finally, we give a proof for a `folk' theorem for nonhomogeneous problems which repeatedly was proved under stronger assumptions than necessary.

[81]

F. Kappel and A. V. Kuntsevich. An implementation of Shor's r-algorithm. Computational Optimization and Applications, 15:193–205, 2000.

 

Here we introduce a new implementation of well-known Shor's r-algorithm with space dilations along the difference of two successive (sub)gradients for minimization of a nonlinear (non-smooth) function [Sh85]. The modifications made to Shor's algorithm are heuristicand concern the termination criterion and the line search strategy. A large number of test runs indicate that this implementation is to some degree robust, efficient and accurate. We hope that this implementation of Shor's r-algorithm will prove to be useful for solving a wide class of non-smooth optimization problems.

[82]

F. Kappel and V. I. Maksimov. Dinamiceskaja rekonstrukcija sostojanii i garantirujuscee upravlenie sistemoi reakcii-diffuzii. Doklady Akad. Nauk, 370:599–601, 2000.

[83]

F. Kappel and V. I. Maksimov. Robust dynamical input reconstruction for delay systems. Int. J. Appl. Math. Comp. Sci., 10:283–307, 2000.

[84]

F. Kappel, V. I. Maksimov, and E. N. Skuratov. On dynamical reconstruction of control in a system with time delay: Finite-dimensional models. In I. I. Eremin, I. Lasicka, and V. I. Maksimov, editors, Distributed Systems: Optimization and Economic-Environmental Applications (DSO'2000), Proceedings of an Internatl. Conf., May 30 – June 2, 2000, Ekaterinburg, Russia, pages 262–264, 2000. ISBN 5–7691–1047–3.

 

The problem of dynamical reconstruction of an unknown control acting upon a linear system with time delay through inaccurate observation of phase states is discussed. Regularized solving algorithm functioning in real time mode is proposed. This algorithm is stable with respect to informational noises and computational errors.

[85]

P. Bachhiesl, M. Hintermüller, H. Hutten, F. Kappel, and H. Scharfetter. Efficient computation of optimal controls for the exchange processes during the dialysis therapy. Computational Optimization and Applications, 18:161–174, 2001.

[86]

K. Ito and F. Kappel. Locally quasi-dissipative evolution equations and applications to delay equations. submitted.

 

In this paper we discuss applications of the generation theory of nonlinear semigroups for evolution equations with locally quasi-dissipative operators in the sense of Kobayashi and Oharu to delay differential equations.