REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS

This is the first part of a survey on analytic solutions of functional differential equations (FDE). Some classes of FDE that can be reduced to ordinary differential equations are considered since they often provide an insight into the structure of analytic solutions to equations with more general argument deviations. Reducible FDE also find important applications in the study of stability of differential-difference equations and arise in a number of biological models.

In studying equations with a deviating argument, not only the general properties are of interest, but also the selection and analysis of the individual classes of such equations which admit of simple methods of investigation.In this section we consider a special type of functional differential equations that can be transformed into ordinary differential equations and thus provide an abundant source of relations with analytic solutions.
Obviously, the key to the solution is the fact that the function f(t) i/t maps the interval (0, ) one-to-one onto itself and that the relation f(f(t)) t, (2.3) or, equivalently, is satisfied for each t (0, oo).
A function f(t) 7 t that maps a set G onto itself and satlsles on G condition (2.3), is called an involution.In other words, an involution is a mapping which coincides with its own inverse.Let fl(t) f(t) fn+l(t) f(f (t)) n 1 2 n denote the iterations of a function f:G G.A function f:G G is said to be an involution of order m if there exists an integer m _> 2 such that f (t)   t for each m t E G, and f (t)   t for n I, m i.It is easy to check that the following n functions are involutions.
EXAMPLE 2.1.f(t) c t on R (_oo, oo), where c is an arbitrary real.
EXAMPLE 2.4.The function f (z)   ez, where E exp(2i/m), is an involution of order m on the complex plane.
We denote the set of all such functions by I.The graph of each f e I is sym- metric about the line x t in the (t, x) plane.Conversely, if F is the set of points of the (t, x) plane, symmetric about the line x t and which contains for each t a single point with abscissa t, then F is a graph of a function from I. One of the methods for obtaining strong involutions is the following [14].Assume that a real function g(t, x) is defined on the set of all ordered pairs of real numbers and is such that if g(t, x) O, then g(x, t) 0 (in particular, this is fulfilled if g is symmetric, i.e., g(t, x) g(x, t)).If to each t there corresponds a single real x f(t) such that g(t, x) 0, then f g I.For example, g(t, x) (2.4) t-_oo t-+o THEOREM 2.1.A continuous strong involution f(t) has a unique fixed point.
PROOF.The continuous function @(t) f(t) t satisfies relations of the form (2.4) and, therefore, has a zero which is unique by virtue of its strict monotonicity.
We also consider hyperbolic involutary mappings t+ (2 + > 0) (2.5) f(t) Ytwhich leave two points fixed.We introduce the following definition.DEFINITION 2.2.A relation of the form (n) F(t, x(fl(t)) x(fk(t)) F(t, x(t), x(f(t))) (2.6) satisfy the following hypotheses. (i) The function f(t) is a continuously differentiable strong involution with a fixed point t O. (ii) The function F is defined and is continuously differentiable in the whole space of its arguments.
(iii) The given equation is uniquely solvable with respect to x(f(t)): x(f(t)) G(t, x(t), x'(t)). (2.7) Then the solution of the ordinary differential equation F F F x"(t) (2.8) (where x(f(t)) is given by expression (2.7)) with the initial conditions is a solution of Eq. (2.6) with the initial condition x(t O) x O.
The second of the initial conditions (2.8')is a compatibility condition and is found from Eq. (2.6), with regard to (2.9) and f(t 0)  (_oo, o).
Then the solution of the ordinary differential equation x"(t) with the initial conditions x(t O) Xo, x'(t o F(Xo) is a solution of Eq. (2.10) with the initial condition X(to) x O.
COROLLARY.Theorems 2.2 and 2.3 remain valid if f(t) is an involution of the form (2.5), while the equations are considered on one of the intervals (_oo, a/y) or (I, ).
REMARK.Let t O be the fixed point of an involution f(t).For t > to, (2.6) and (2.10) are retarded equations, whereas for t < t O they are of advanced  (2.12) The fixed point of the involution f(t) a-t is t o a/2.The initial condition for (2.11) is x() x O; the corresponding conditions for (2.12) are 1 x() x o, x'() x0 Eq. (2.12) is integrable in quadratures: This is the solution of the original equation (2.11).
The topic of the paper [16] is the equation (2.13) where x is an unknown function.
THEOREM 2.4 ([16]).Let the following conditions be satisfied: (I) The function f maps the open set G into G, G being a subset of the set R of real numbers.
(2) The function f has iterations such that fl(t) f(t)  fk(t) f(fk_l(t)) f (t) t for each t e G, where m is the smallest natural number for which the last expression holds.
(3) The function f has derivatives up to, and inclusive of, the order mn n for each t g G, f'(t) # 0 for each t G.
(4) The function F(t, Ul, u2, Un+ I) is mn n times differentiable of its F arguments for each t g G and Ur R (r I, n + I) and u n+l (5) The unknown function x has derivatives up to, and inclusive of, the order mn on G.
In this case there exists an ordinary differential equation of order mn such that each solution of Eq. (2.13) is simultaneously a solution of this differential equat ion.
Let us consider the functional differential equation [17] (2) The functions x and f (r i n) have derivatives up to the order p, r where p max(k I kn) so that f'(t) # 0 for every t G and r I n.

r
(3) For the function F at least one relation F x (s)  To investigate the equation x'(t) f(x(t), x(-t)), the author of [6] denotes y(t) x(-t) and obtains y'(t Hence, the solutions of the original equation correspond to the solutions of the system of ordinary differential equations d__x f(x y) dy _f(y x) dt dt with the condition x(O) y(O).From the qualitative analysis of the solutions of the associated system he derives qualitative information about the solutions of the equation with transformed argument.The linear case is discussed in some detail.
Several examples of more general equations are also considered.
Boundary-value problems for differential equations with reflection of the argu- ment are studied in [I0].

LINEAR EQUATIONS
In this section we study equations of the form n Lx(t) with an involution f(t).
THEOREM 3.1 ([i]).Suppose that the initial conditions are posed for Eq. (3.1) in which the coefficients ak(t) the function (t), and the strong (or hyperbolic (2.5)) involution f(t) with fixed point t o belong to the class cn(_m, oo) (or cn(/y, oo)).If n > I, then f'(t) # O.We introduce the operator Then the solution of the linear ordinary differential equation PROOF.By successively differentiating (3.1) n times, we obtain These relations are multiplied by a0(f(t)) al(f(t) a (f(t)) respectively n and the results are added together: k=O Thus, we obtain Eq. (3,4).In order that the solution of this equation satisfies problem (3.1)-(3.2),we need to pose the following initial conditions for (3.4): the values of the function x(t) and of its n are determined from the relations Mx(t) x (k)(f(t)) + Mk@(t), k 0 n i (k) by substituting the values t O and x k for t and x (t). is integrable in quadratures and has a fundamental system of solutions of the form ta(In t) sin(b In t), ta(In t) j cos(b In t), a and b are real and is a nonnegative integer. (3.7) PROOF.By an n-fold differentiation Eq. (3.6) is reduced to the Euler equation For n i this follows from (2.2).Let us assume that the assertion is true for n and prove its validity for n + i.In accordance with formula (3.3), we introduce for Eq. (3.6) the operator On the basis of (3.4) and (3.8) we have 2n Mnx (n) Consequently, the equation x (n+l) (t) x() is reduced by an (n+l) -fold differentiation to the Euler equation Mn+ix(n+l)(t) x(t).
At the same time we established the recurrence relation It is well known [18] that the Euler equation has a fundamental system of solutions of the form (3.7), where a + bi is a root of the characteristic equation and j is a nonnegative integer smaller than its multiplicity.The theorem is proved.x'(t) x( + 9(t), 0 < t < 9(t) g (0, x x 0 reduces to the problem t2x"(t) + x(t) t29'(t) x(1) O, x'(1) x 0 + 9(1).
If b d and a c, Eq. (3.10) is equivalent to the system of equations ax(t) + btx'(t) u(t), u() tr-s-lu(t).
If b d and a c, (3.10) reduces to the functional equation x(tl_) t r-s-I x(t).
In the case of b -d and a -c Eq. (3.10) reduces to the system ax(t) + btx'(t) u(t) u() -t r-s-I u(t).
In the case of b -d and a # -c (3.10) reduces to the functional equation x() -t r-s-I x(t).
The equation x'(t) x(f(t)) with an involution f(t) has been studied in [19].
Consider the equation [13] with respect to the unknown function x(t): x'(t) a(t)x(f(t)) + b(t), (3.13) (i) The function f maps an open set G onto G.
(2) The function f can be iterated in the following way: where m is the least natural number for which the last relation holds.
(3) The functions a(t), b(t) and f(t) are m 1 times differentiable on G, and x(t) is m times differentiable on the same set.
THEOREM 3.5 ([i]).In the system x'(t) Ax(t) + Bx(c-t), x(c/2) x 0 (3.15) let A and B be constant commutative r xr matrices, x be an r-dimensional vector, and B be nonsingular.
It follows from here that, by appropriate choice of c I, c2, c3, and c4, we can obtain both oscillating and nonoscillating solutions of the above equations.On the other hand, it is known that, for ordinary second-order equations, all solutions are either simultaneously oscillating or simultaneously nonoscillating.
It has been also proved in [7] that the system x'(t) A(t)x(t) + f(t, x(tl-)) 1 <_ t < II f(t, x())II <--II x()ll q, where 6 > 0 and q _> 1 are constants, is stable with respect to the first approxima- t ion.
For the equation n Z aktkx(k) k=0 (t) x(), 0 < t < (3.16) we prove the following result.s THEOREM 3.6.Eq.(3.16) is reducible by the substitution t e to a linear ordi- nary differential equation with constant coefficients and has a fundamental system of solutions of the form (3.7).s PROOF.Put e and x(e s) y(s), then tx'(t) y'(s).Assume that tkx(k)(t) Ly(s), where L is a linear differential operator with constant coefficients.From the relation we obtain tk+l x (k+l) L[y'(s) ky(s)], which proves the assertion.
The functional differential equation Q'(t) AQ(t) + BQT(T t), < t < (3.17) where A, B are n x n constant matrices, T _> 0, Q(t) is a differentiable n n matrix and QT(t) is its transpose, has been studied in [20].Existence, uniqueness and an algebraic representation of its solutions are given.This equation, of considerable interest in its own right, arises naturally in the construction of Liapunov functio- nals for retarded differential equations of the form x'(t) Cx(t) + Dx(t-I), where C, D are constant n n matrices.The role played by the matrix Q(t) is analogous to the one played by a positive definite matrix in the construction of Liapunov functions for ordinary differential equations.It is shown that, unlike the infinite dimen- sionality of the vector space of solutions of functional differential equations, the linear vector space of solutions to (3.17) is of dimension n 2.Moreover, the authors 2 give a complete algebraic characterization of these n linearly independent solutions which parallels the one for ordinary differential equations, indicate computationally simple methods for obtaining the solutions, and allude to the variation of constants formula for the nonhomogeneous problem.
The initial condition for (3.17 where K is an arbitrary n n matrix.Eq. (3.17) is intimately related to the system Q'(t) For any two n n matrices P, S, let the n x n matrix PS denote the Kronecker (or direct) product [21] and introduce the notation for the n x n matrix Sl* S (sij) (n,) Sn* where si, and s,j are, respectively, the i th row and the j th colun of S; further, let there correspond to the n> n matrix S the n2-vector s (Sl, Sn,)T.With this notation Eqs. (3.19)  THEOREM 3.7 ([20]).Eq. (3.17) with the initial condition (3.18) has a unique solution Q(t) for < t < oo.
Examination of the proof makes it clear that knowledge of the solution to (3.21) immediately yields the sol'ution of (3.17)-(3.18).But (3.21) is a standard initial- value problem in ordinary differential equations; the structure of the solutions of such problems is well known.Furthermore, since the 2n 2 2n 2 matrix C has a very special structure, it is possible to recover the structure of the solutions of Eq.
(3.17).Let I' %p' p 2n2' be the distinct eigenvalues of the matrix C, that is, solutions of the determinantal equation each ., I, p, with algebraic multiplicity m. and geometric multiplicities nj,r Zr =s I n.mj, Zj m.=3 2n2" Then 2n 2 linearly independent solutions of (3.21) are given by T q-i j q (t) exp(% (t-T q (t -) )) Z (q-i)' r i=l where q i, n., and the 2n 2 linearly independent eigenvectors and generalized eigenvectors are given by A change of notation, and a return from the vector to the matrix form, shows that 2n linearly independent solutions of (3.19) are given by r T q (t-r exp(%j(t where the generalized eigenmatrix pair (L i Mj i)associated with the eigenvalue j ,r' ,r satisfies the equations (3.24) The structure of these equations is a most particular one; indeed, if they are multi- plied by -I, transposed, and written in reverse order, they yield BL.
M. -%. will also be a solution; moreover, %. and -%.have the same geometric multiplici- 3 3 3 16 S.M. SHAH AND J. WIENER ties and the same algebraic multiplicity.Hence, the distinct eigenvalues always appear in pairs (%.%j), and if the generalized eigenmatrix pairs corresponding to 3' i i %. are (L., r, Mj,r), the generalized eigenmatrix pairs corresponding to -%j will be .T .T 1 (-i) i+l Lj I ).These remarks imply that if the solution (3.23) cor- responding to %. is added to the solution (3.23) corresponding to -%. multiplied by (-I) q+i the n 2 linearly independent solutions of (3 19) given by Zj q(t) But this is precisely condition (3.20)" it therefore follows that the expressions r (3"25)   2 are n linearly independent solutions of (3.17).
Eq. (3.17) has been used in [22] for the construction of Liapunov functionals and also encountered in a somewhat different form in [23].
Some problems of mathematical physics lead to the study of initial and boundary value problems for equations in partial derivatives with deviating arguments.Since research in this direction is developed poorly, the investigation of equations with involutions is of certain interest.They can be reduced to equations without argu- ment deviations and, on the other hand, their study discovers essential differences that may appear between the behavior of solutions to functional differential equations and the corresponding equations without argument deviations.
The solution of the mixed problem with homogeneous boundary conditions and ini- tial values at the fixed point t o of the involution f(t) for the equations u t(t, x) au (t x) + bu (f(t) x) Its investigation is carried out by means of Theorem 3.1, according to which the solution of the equation THEOREM 3.9.The solution of the problem ut(t x) au (t, x) + hu (c-t, x), PROOF.By separating the variables, we obtain In this case, Eq. (3.30) takes the form The completion of the proof is a result of simple computations.Depending on the relations between the coefficients a and b, the following possibilities may occur: (i) T (t) C (cos ), (lal < Ibl); 2 2 (2) T (t) (a+b)(t---)), (lal Ibl); (3) T (t) )exp(-c (t-))], (Ib < lt).
IEOREM 3.10.The solution of the equation u ( satisfying the boundary and initial conditions if a 2 b 2. In the case a < expansion (3.28) diverges for all t # 0.
Omitting the calculations, we formulate a qualitative result.
THEOREM 3.11.An equation that contains, along with the unknown function x(t) and its deriva- tives, the value x(-t) and, possibly, the derivatives of x at the point -t, is called a differential equation with reflection.An equation in which as well as the unknown function x(t) and its derivatives, the values x(1t-aI) X(mt-am and the cor- are mth roots of uni- responding values of the derivatives appear, where gl' m ty and al' m are complex numbers, is called a differential equation with rota- tion. For m 2 this last definition includes the previous one.Linear first-order equations with constant coefficients and with reflection have been examined in detail in [5].There is also an indication (p.169) that "the problem is much more diffi- cult in the case of a differential equation with reflection of order greater than one".Meanwhile, general results for systems of any order with rotation appeared in [3], [4], [9], and [24].
Consider the scalar equation Xk, k 0 n-1 with complex constants ak, bk, e, then the method is extended to some systems with variable coefficients.Turning to (4.1) and assuming that is smooth enough, we where P and Q are linear differential operators of order n with constant coefficients Pk' qk and Pn an' qn hn.
COROLLARY.Under assumptions (4.4) and m I, (4.5) is reducible to a linear ordinary differential equation with constant coefficients.
The analysis of the matrix equation X'(t) AX(t) + exp(at) [BX(et) + CX'(et)], (4.6) x(0) E with constant (complex) coefficients was carried out in [3].The norm of a matrix is defined to be lcll max .leij!, (4.7) and E is the identity matrix.
THEOREM 4.3. ([3]).If e is a root of unity (e # I), Icll < 1, and the matrix A is commuting with B and C, then problem (4.6) is reducible to an ordinary linear system with constant coefficients.
Thus, the use of the operator Lm_ I at the conclusive stage yields (4.11).
THEOREM 4.6 ([9]).The system tAX'(t) + BX(t) X(Et) (4.12) with constant matrices A and B is integrable in the closed form if e m I, det A 0.
3 Hence, on the basis of the previous theorem, (4.12) is reducible to the ordinary system (tAdldt + B) m X(t) X(t).(4.13) This is Euler's equation with matrix coefficients.Since its order is higher than that of (4.12) we substitute the general solution of (4.13) in (4.12) and equate the coefficients of the like terms in the corresponding logarithmic sums to find the additional unknown constants.
EXAMPLE 4.2.We connect with the equation [9] tx'(t) 2x(t) x(et), e3 1 with constant coefficients A and B, det A # 0 and e m 1 is Integrable in closed form and has a solution X(t) e(t)t A-IB (4.16) where the matrix P(t) is a finite linear combination of exponential functions.
PROOF.The transition from (4.15) to an ordinary equation is realized by means of the operators L.
Biological models often lead to systems of delay or functional differential equations (FDE) and to questions concerning the stability of equilbrium solutions of such equations.The monographs [28] and [29] discuss a number of examples of such models which describe phenomena from population dynamics, ecology, and physiology.
The work [29] is mainly devoted to the analysis of models leading to reducible FDE.
A necessary and sufficient condition for the reducibility of a FDE to a system of ordinary differential equations is given by the author of [30].His method is fre- quently used to study FDE arising in biological models.We omit these topics and refer to a recent paper [31].For the study of analytic solutions to FDE, which will be the main topic in the next part of our paper, we also mention survey [32]. 1.
16. 17. 18. a (t), x(fl(t))x (fl(t))X(fn(t))x (f (t)))=0 (2 14)where x is an unknown function and where the following conditions are fulfilled:(I) The functions fl'f form a finite group of order n with respect to n superposition of functions, fl(t) t, and map the open set G into G, G being the largest open set wherein all expressions appearing in this paper are defined.
for equations(3.31)and (3.32)  are posed at the fixed point of the involution f(t) c t.
and apply A I to te given equation with operators A and B defined by (4.1) to Px (Qx)(et) + exp(-c,t/l THEOREM 4.4.([27]).Suppose we are given a differential equation with reflec- tion of order n with constant coefficients n [a-x(k) + bkX(-t)] y(t).
of the linear ordinary differential equation m + (0) ,k=O PROOF.Applying the operator L I to (4.9) and taking into account that (LoX)(et) x(e2t)+ (et) t O It is especially clear to see the function f(t) is a continuously differentiable strong involution with a fixed point t o and the function F is defined, continuously differentiable, and strictly monotonic on type.
The solutions of the equations AX'(t) (ekE + t-IB)x(t) are matrices Xk(t) exp(ktA-l)t A-IB, k I, mo