DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL AND FUNCTIONAL DIFFERENTIAL EQUATIONS

A brief survey of recent results on distributional and entire solutlosof ordinary differential equations (ODE) and functional differential equations (FDE) is given. Emphasis is made on lnear equations with polynomial coefficients. Some work on generallzed-functlon solutions of integral equations is also mentioned.


I. INTRODUCTION AND PRELIMINARIES.
This paper may be considered as a continuation of [I] which contains, in partf- cular, a survey of recent results on entire solutions of ODE with polynomial coefffclents.Integral transformations establish close links between entire and generalized functions [2].Therefore, a unified approach may be used fn the study of both dlstrfb-utional and entire solutions to some classes of linear ODE and, especially, FDE with linear transformations of the argument [3].It fs well known [4] that normal linear homogeneous systems of ODE with fnffnltely dffferentlable coefficients have no generalized-function solutions other than the classical solutions.In contrast to this case, for equations with singularities in the coefficients, new solutions in generalized functions may appear as well as some classical solutions may disappear.In Section 2 results on distributional and entire solutions of ODE are discussed.In Section 3 we study analogous problems for FDE.
Research in this direction, still developed insufficiently, discovers new aspects and properties in the theory of ODE and FDE.In fact, there are some striking dissimilarities between the behavior of ODE and FDE which deserve further investi- gat ion.
I. Distributional solutions to linear homogeneous FDE may be originated either by singularities of their coefficients or by deviations of argument.In [5] it has been proved that the system x'(t) Ax(t) + tBx(%t), -I % < I has a solution in the class of distributions an impossible phenomenon for ODE without singularities. 2. In [6] it was shown that a first-order algebraic ODE has no entire transcendental solutions of order less than 1/2, whereas even linear first-order FDE may possess such solutions of zero order [3], [7].
3. It is well known [8] that the solution of the initial-value problem for a normal linear ODE wlth entire coefficients is an entire function.Let in the linear FDE w'Cz) a(z)wClCz)) + BCz), w(0) w 0 the functions a(z), b(z), %(z) Be regular in the disk Izl < I, and %(0) O, I%(z) < i for Izl < i.Then there is a unique solution of the problem regular in Izl < i [9].In general, this solution cannot be extended beyond the circle Izl i, if even a(z), b(z), and %(z) are entire functions.Thus, the solution of the eqa_tiOn w'(z) a(z)w(z2), where a(z) is an entire function wlth positive coefficients, has the circle Izl I as the natural boundary [i0], [II].

DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF ODE
The number m is called the order of the distribution m x r. x () (t), x ' o, 1-0 (2.1) where (k) denotes the kth derivative of the Dirac measure, and the variable t is real.Finite order solutions of linear ODE have been studied mainly for equations with regular singular points [12  16].In [16] for the first time an existence criterion of solutions (2.1) to any linear ODE was established.
From these theorems it follows that if the equation n with coefficients ai(t) E C m and an(0) @ 0 has a solution (2.1) of order m, then n r. (-1)ia Conversely, if m is the smallest nonnegative integer root of (2.4), there exists an m order solution of (2.3) concentrated on t 0 [16].This proposition constitutes the basis for the study of finite order solutions to equations with regular singular points.The stated results can be used also in the search of polynomial and rational solutions to linear ODE with polynomial coefficients.
Thus, we formulate x-C H (d!dt s:t) t-1.
Necessary and sufficient conditions for the existence of a maximal number of polynomial solutions to algebraic differential equations are given in [18] and [19].
Existence of polynomial solutions of an equation of Linard type is studied in [20].
The equation w' a0(z) + al(z)w + + a (z)w n with n polynomials as solutions is n considered in [21].In a number of papers additions to Kamke's treatise are made.
Thus, in [22] it is proved that for the generalization An_ I are arbitrary constants and f,(y(n)) (_t)n/n!.In [23] and [24] rational solutions of Palnlev's third and fifth equations are studied.
The work [25] concerns the study of properties of solutions to the complex equation (I) Pf g, where m P(z, 8 /z) E Pk(Z) k;O 1 k i zk, / z ( / x i / y) and g is a given holomorphlc or rational function.Various conditions guaranteeing that the solutions of (i) are polynomial or rational functions of a certain type are obtained.In the last part, differential equations of 'Euler type are considered.
THEROEM 2.7.(Nova [25]).Let be a simply connected open set in and u E .
If u is a regular singular point of P and every solution of Pf g in 0( {}), with g e Ru(), is rational in with a pole at u, then P is normal.
Significant contributions to the study of asymptotic properties of the analytic solutions of algebraic ordinary and partial differential equations are made in [6].
The main properties are the growth of an entire solution, the order of a meromorphic solution and its exceptional values.In a certain sense, thls book completes the fundamental monograph [26].In the second chapter of [6], the author studies the algebraic DE P(z, w, w') 0.
THEOREM 2.8. (Strelitz[6]).The order and type of an entire transcendental solution of (2.6) are equal, respectively, to the positive order Oj > 0 and type of one of the solutions of the determining equation (2.8).Furthermore, rllm K(r) /r ojpj, rllm In M(r) /r j.
The following proposition shows that not all of the numbers 01 indicated in Th. 2.8 may be the orders of the entire solutions of first order algebraic DE.
THEOREM 2.9.(Strelltz [6]).Algebraic DE(2.6) cannot have entire transcenden- i i tal solutions of order O < In general, cannot be replaced by a larger number: there are equations of the form (2.6) that have entire transcendental solutions of I order EXAMPLE 2.1. (Strelltz [6]).The equation has an entire transcendental solution w cos / of order 0 The following result is of interest in this connection.

1
A meromorphlc solution of the equation w' R(z, w) which is of" order < is a rational function.
In the second chapter of [6] it is also proved that the order of any meromorphlc solution of a first order algebraic DE is finite.The orders of the transcendental entire solutions of second order linear DE with polynomial coefficients have been investigated in [28], [29], [30].Suppose that P(z) and O(z) are polynomials of degree p and q, respectively.Set gO i + max(p, q).Let p _> q + I. Then all transcendental solutions of the equation w" + P(z)w' + Q(z)w--0 (2.9)I are of the order I + p go" If p _< q, all transcendental solutions are of the i i order i + q go" Deviation from this pattern can occur only if q < p _< q.
Here go I + p, and there are always solutions of this order; under certain circumstances, however, a lower order q p + I may also be present.
THEOREM 2.11. (Hille [30]).If in (2.9) either P or Q is an entire transcendental function while the other is a polynomial, then every transcendental solution of (2.9) is an entire function of Inflnlteorder.This is not necessarily true, however, if both P and Q are entire.
THEOREM 2.12.(Wittich [30]).In (2.9) suppose that P and Q are entire functions and suppose that the equation has a fundamental system Wl(Z), w2(z), where w I and w 2 are entire functions of order 01 and 02, respectively.Then P and Q are po lynomials.
Th. 2.12 may be regarded as a converse of Th. 2.11.THEOREM 2.13. (Frei [31]).Suppose that in the equation n (n-i) k) are polynomials, and Pk+l(Z) is an entire transcendental function.Under these conditions the equation can have no more than k linear independent entire transcendental solutions of finite order, whereas all other solutions of the fundamental system are of infinite order.
The results by Frei, Pschl, and Wlttich on the growth of solutions of linear DE are generalized in the third chapter of [6].The main tool is the Wlman-Vallron method, but the case when this method fails is also studied.Nonlinear algebraic DE of the form P(z, w, w' (n) w 0 are investigated, too.A necessary con- dition for some complex number a to be a defect value of a meromorphic solution of finite order is P(z, a, 0, 0) 0. We already know that first order algebraic DE have no entire transcendental solutions of zero order.In [32] it is shown that there are algebraic DE of third order that have entire transcendental solutions of zero order.
For these functions we have log M(r, Jn r log M(r, F L) as r + .
Consider now vector-vM.uedfunctions F: 1/ I;m.Suppose that the components fk(l <_ k <_ m) are all entire functions.Write max !! F(z) lJ- llFCz) [l max {If k(z) J, I < k <m}, MCr, F) IzJ=r DEFINITION.A vector-valued entire function F is said to be of bounded index (BI) if there exists an integer N such that max for all z e I and k O, i, The least such integer N is called the index of F. THEOREM 2.16.(Roy and Shah [35]).Let F: i / m he a vector-valued entire function of BIN.Then llF(z) II < A exp((N + i) Izl)   where A max 11 F(k)(0) !1 The result is sharp.
The function F may be of BI but the components fk may not be of BI.In the next theorem, it is shon that if F satisfies an ODE then F and each fk are of BI.Let R denote the class of all rational functions r(z) bounded at infinity and Qi(z) (1 < i < m) denote an m m matrix with entries in R. Write  and Qi(z) (a (z)), lira lapq pq,i i (z) ]Apq l] Z-Oo sup (IApq,i], 1 <_ p, q THEOREM 2.17.(Roy and Shah [35]) Ln(W, z, Q) --w (z) + Ql(Z)- where g(z) is a vector-valued entire function of BI.Then each fk satisfies an ODE are of this form (with possibly different n and coefficients), and F, fl' fm all of BI.If the entries of Qi are not in R then F may not be of BI.
THEOREM 2.18.(Roy and Shah [35]).Let w(z) 0 be a vector-valued entire function satisfying the ODE L (w, z Q) 0 n Then we have: where the numbers A i are defined above (ii) If the elements of Qi(l <_ i < m) are constant, and p _> 0 is any integer such that F m In+P (n + p) (n +'p i) (.n+p Cp+ <_ i, then the index N, of F(z), is less than or equal to n + p I. The bound on N is best possible.
Next we compare these growth results with the corresponding ones for solutions of algebraic difference equations.
THEOREM 2.19. (Shah [36]).Let P(t, u, v) be a polynomial with real coeffi- cients.Let u(t) be a real continuous solution of a frst order algebraic difference equation P(t, u(t), u(t + i)) 0 for t _> t o Then there exists a positive number A which depends only on the polynomial P Here e2(x) denotes exp(exp x).
The function e 2 (At) cannot be replaced by a function of slower rate of growth, in general.
THEOREM 2.20. (Shah [36]).Let (t) be an arbitrary increasing function which tends to + as t /+=o There exists an equation P(t, u(t), u(t + i)) 0 with a real solution u(t) which is continuous for t _> t O and which exceeds (t) at each point of a sequence {t such that t -+oo as n /.
Let f(z) be an entire transcendental function.The (, x) index is defined as where (r, f) is the central index of the Taylor expansion f(z) The author of [41] evaluates the (e, x) indices of entire transcendental solutions of linear ODE with polynomial coefficients.On the basis of these results some theorems concerning the distribution of values of these solutions are proved.
2.21.(Knab [41]).Let w(z) be an entire transcendental solution of order 0 and type of an ordinary linear differential equation with polynomials as coefficients.Let n(r, w c) be the counting function of the zeros of the function w c (c const).Then L llm SUPr_=n(r w c) /r p < Up.
H THEOREM 2.23.(Lopusans kli [42]).Solutions of (2.10) are oscillatory if and only if the function (z) w(z) /(z) maps the upper half-plane conformally onto the unit disk, where w(z) Wl(Z) + iw2(z) and wj (z)(j I, 2) are two indenpendent real solutions of (2.10), and their Wronskian is positive on the real axis.
The following characterization of the class HB(Hermlte-Biehler) of entire functions having all their zeros within the upper half-plane is given in THEOREM 2.24.(Lopusans'kii [42]).An entire function F(z) is of class HB if and only if on the real axis it is a complex solution of an oscillatory equation of the form (2.10).
The ODE w(n) (z) + Pn-2(z)w (n-2) (z) + + p0(z)w(z) %nw(z) is studied in [43], where p0(z), Pn-2(z) are polynomials of degrees m 0 ran_2, respecti- vely, and % is a complex parameter.It is proved that the fundamental system of solutions of the equation, determined by the identity matrix as initial conditions at z 0, satlsifles the estimates lw+/-(z, )I < ]I Iz lOexp=l'Xzi, for all sufficiently large values of I%1 and Iz I.The value of 0 is defined by max (m.i +n) /(n i), u<l<n-Z and c is some positive constant.
Asymptotic properties of the solutions of linear ODE with entire coefficients are studied in [44].Consider the equation where all the coefficients a i ai(z)(i 0, I, n i) are entire functions.
Let f(z) be a meromorphic function in the z-plane.Denote: L(r, a, f) Z+(If(z) al -I) if a @ L(r a f) maXlzl=rZn +If(z) if a oo.
The function 8(a, f) is defined as 8(a, f) lira infr_oL(r a, f)/T(r, f), T(r, f) is the usual Nevanlinna characteristic function of f.The authors call solution w(z) of (2.11) a standard solution if 8(a, f) 0 for all complex a # 0, TIEOREM 2.25. (Boiko and Petrenko [44]).Each fundamental system of solutions of Eq. (2.11) contains at least one standard solution.
In [45] the author considers the first Palnlev equation w" 6w 2 + z whose 2 2 solutions are meromorphic of the form w i/ (z z0) (z 0 /10)(z z 0) Yn--O n+2 (z She represents w as a quotient of two entire functions: where u exp (-I dz Jw dz), and then obtalns recursion relations for the coeffl- clents of the power series expansions of the numerator and denominator. (2.12) In conclusion, we note that in some recent works [46][47][48][49][50] entire solutions to DE of infinite order are discussed as well as properties of differential operators in spaces of entire functions.In [46] the author studies the existence of a solution to the equation 7n=0 anw(n)(z) f(z) whose growth equals that of the right- hand side, in the case when f(z) belongs to the class B, of entire functions g(z) such that Ig(x + iy) < cexp [(x) + (y)], for any x, y; here the functions (x), (y) satisfy Hider conditions.Let (z) be an entire function on of exponential type without multiple roots.Let My be the operator of convolution with (), where is a -functlon.The following result is proved in [47].

P i=0 k=0 for all
The operator L is said to be P (I) applicable to the set H of entire functions at the point z 0 if the series 7. 0pi(Zo)W(1)(z 0) converges for any function w from H; i= applicable to H in the domain Iz < if L is applicable to H at (2) P finite point (3) strongly regularly applicable to H inside the domain Iz < if, for any w e H and R < oo, where M(f, R)   sup { [fez)I: Iz I< R).
Let Q be a bounded simply connected region.In this remarkable paper the author gives necessary and sufficient conditions for L to be applicable to the set P R(Q) of exponential functions whose Borel transforms are regular on C(Q).He proves that if the operator L is applicable to R(Q) at p + i distinct points then where a--sup { Izl: z e Q}.Conversely, if (2.12) holds, then L is strongly regularly P applicable to R(Q) inside Iz < =o, and maps R(Q) into itself.
In [49] the authors investigate the solvability of a class of functional equations, containing as a particular case differential equations of finite and of infinite order with constant coefficients, in the Banach space with weight of entire functions B(x,y {w(z) g Aoo llw I I sup lw(z lexp(_(x, y)) < o.
z=x+iye Here #(x,y) is a locally bounded function in R 2 with a certain growth for Izl The author [50] treats an equation Lw f with L Z i>0Pi(z)di/dz i, where the pi(z) are polynomials, deg Pi ni' lim sup(n i / i) < I, in a space [0, g (8)] of all entire functions satisfying lim SUPr_ (Znlw(reiS) /ro) < g(8).Here g (8)   is a trigonometrically 0-convex function, 0 > 0. It is proved that L is a Noetherian operator, its index is found and the space of solutions of the corresponding homo- geneous equation is investigated.

DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF FDE
Finite order distributional solutions (2.1) of linear FDE have been studied in [15] and [51].
THEOREM 3.1. (Wiener [15]).The criterion for the existence of solutions (2.1)   to the system From here it follows that the system tx'(t) has a solution of order m with support t 0, if Ai(0) 0(i _> I) and m + is the smallest modulus of the negative integer eigenvalues of the matrix A(0).This and similar results were used in [15] to investigate finite order solutions of some im- portant equations of mathematical physics.For equations with more general argument delays we have THEOR 3.2.(Wiener 15] ).The system tx'(t) A (t)x( (t)) has a solution (2.1) or order m, if the following i i hypotheses are satisfied: (i) the real zeros tij of the functions i(t) are simple and form a finite or countable set; (2) A (k)(tij) 0(k 0 m), for tij # 0; (3) m is the smallest modulus of the nonDositive integer roots of Eq. (3.1) with '(0).
In [52] it was shown that, under certain conditions, the system x'(t) Z A.(t)x(%.t) n=0 has a solution x(t) Z x tn)(t) n n=0 (3.2) in the generalized-function space (S)' conjugate to the space S of testing functions (t) that satisfy the restriction [2]   n n ''l(n)(t)l < ac n > i.
To ensure the convergence of series (3.2), it is sufficient to require that for n/ the vectors x satisfy the inequalities n since 2) converges, its sum represents the general form of a linear (So)' with the support t 0 [53].Solutions in (So)' of some linear functional in ODE with polynomial coefficients were studied in [54], [55], [56].The particular importance of the system m l I (Aij + tB..)x(J)(%it tx(%t) i=0 j=0 13 which was considered in [15] is that depending on the coefficients it combines either equations with a singular or regular point at t 0 and in both cases there exists a solution of the form (3.2).The equation tx' (t) Ax(t) + tBx(%t) ( provides an interesting example of a system that may have two essentially different solutions in (SOB) concentrated on t 0. If the matrix A assumes negative integer eigenvalues, (3.3) has a finite order solution (2.1).At the same time there exists an infinite order solution (3.2), if A @ -nE for all n > I.In [3], [16], [57], and [58] the foregoing conclusions were extended to comprehensive systems of any order with countable sets of variable argument deviations.The basic ideas in the method of proof are applied to investigate entire solutions of linear FDE.
THEOREM 3.3. (Cooke and Wiener [3]).Let the system m l I A. (t)x (j) (t)) with a finite number of argument deviations, in which x is an r-vector and Aij are r r matrices, satisfy the following hypotheses.
(i) The coefficients A..(t) are polynomials in t of degree not exceeding p: P Aij(t Y.A .k tk A00(t) At p p > i.
i>l Then in the space of generalized functions (S)' with arbitrary > there exists a solution x(t) supported on t 0.
In [3] it is also proved that system (3.4) with a countable set of argument deviations has a solution (3.2) if, in addition to the conditions of Th. 3.3, there exists a neighborhood of the origin in which each function %ij (t) has the only zero -I t 0 and the series Y.A i converges, where i= i A.
max I I Aijk II inf ij i + j > I.
j,k i j The choice of the coefficients in (3.4) enables us to consider both equations with a singular or regular point and to show that distributional solutions of FDE may be originated by deviations of the argument.The authors of [3] also investigate the system m tPx '(t) E EoAi j(t)x(j)(%ij(t)) i=0 ]= (3.5) the particular cases of which tPx'(t) A(t)x(t) and tPx '(t) I A (t)x(%it) i=O i have been studied in [56] and [57], respectively.THEOREM 3.4. (Cooke and Wiener [3]).Suppose that system (3.5), in which x is an r-vector and Aij are r r -matrices, satisfies the following conditions.
(i) The A..(t) are polynomials in t of degree not exceeding p + j 2: p+j -2 Ai-'3(t) Z Aijktk p __> 2. k=0 (ii) There exists a neighborhood of the origin in which the real-valued functions %ij g C1 have the only zero t 0 and lio _> I, inf lij > i, for i _> 0, j _> i, ij ij (0).
-i (ili) The series Z .A. converges, where i=O a.

inf [aijl'
Then there is a solution of (3.5) in (So)' with some > supported on t O.
The deep study of narrow classes of FDE, and even individual FDE, continues to remain one of the main problems.First of all, such equations can have some special, for example applied, interest.In addition, we can work out on them in the first instance methods of studying properties that are similar to properties of equations without deviation of the argument and are essentially new for equations with devia- tlon, and then try to extend these methods, and the results obtained, to a broader class of FDE.In a number of papers [60][61][62][63][64][65][66] various authors have continued the study (originated in [59]) of the solutions, especially their asymptotic behavior as t 0 or t / , of the equation x' (t) ax(%t) + bx(t), which arises in certain technical problems, and also of systems and some more eneral equations of similar form.These works concern principally real solutions.
The author of ['67] attacks complicated equations with elegant analytical tools.
He investigates analytic solutions of the FDE r s (j) (zqk) Z Z a kW 0 0 < q < I, j--O k=O j with constant coefficients ajk.Its formal solutions are obtained in the form of Mellin or Laplace integrals.The functions occuring in the inte.randssatisfy linear Z n of the form v--0 P (qt)G(t + v) 0 (P(y) polynomials).
difference equations Properties of solutions of such difference equations, in particular the location oF singularities and the asymptotic behavior for absolutely large values of t, are studied.Conditions are derived for formal solutions of Mellin integral type to be actual solutions and these are shown to be often expressible as power or Laurent series.Solutions of Laplace integral type are shown to be representable as Dirichlet series under certain conditions.Finally, questions as to when the llne of convergence of the Dirichlet series is the natural boundary of the function re- presented are discussed.The author asserts that the methods used can be extended to the case when the coefficients ajk are polynomials in z, and to some more general equations.
In [68] the growth of entire solutions of the FDE m I akDkw(%m-kz) 0, D d/dz k--0 is estimated by means of a suitably constructed comparison function.9urthermore, an expllcie representation of all entire solutions is given which in certain cases leads to conclusions concerning loations and multiplicity of the zeros of particu- lar solutions.Finally, the growth of the maximum and minimum modulus of the solutions is compared which implies an estimate of the number of zeros.The FDE m Lw(z) I akDkw(%m-kz f(z), k=O where a k are complex numbers, is a fixed parameter, 0 < < i, and the unknown w and the right member f are entire functions, is considered in [69].2z---G(tz)(t) dt, with (t) q(t)/A(t), where A(t) m ak%k(2m_k_ I)/2 k l t k=0 q is a polynomial of degree <_ m and F is a contour enclosing all the zeros of A.
A similar integral representation is given for a solution of (3.6) with f # 0 in terms of the generalized Borel transform when n+l In [70] the author discusses the system w'(z) Aw(lz), 0 < I < i, where A is a corn- plex constant matrix.First, the form of all entire solutions is given.Subsequently, for z # 0 a special system of particular analytic solutions is constructedhymeans of which all other solutions may be represented.The asymptotic properties as z of all solutions are investigated.Furthermore, it is shown that given a specific asymptotic behavior, there is one and only one solution which possesses that asymp- totic behavior.
Given the equation m w" (z) + E a k k=l n (z)w'(IkZ) + 7. b (z)w(jz) 0, j=l j where ak(z), bj(z) are entire functions of finite order < 0 and the constants I k, j are in the set 0 < zl < I, the author [71] shows that any solution w(z) is also of finite order < 0. As a special case he discusses the equation w"(z) + p(z)w(Iz) O, where 0 < I < and p(z) is an entire function of finite order taking real values on the real axis, and derives an estimate on the type of a solution w(z).
The author of [72] studies properties of solutions to equations of the form m w'(Iz) 7. ak(z)wk(z), k=O where ak(z) are entire functions such that T(r, a k) o(T(r, w)) as r , and I is a complex number, Ill o, where T(r, w) is the Nevanlinna function.He establishes the following THEOREM 3.5.(Mohon'ko [72]).Let w(z) be an entire solution of (3.7).If w i, i 0 n-i, in which ai, bi, e and I are complex numbers, has been studied with various assumptions concerning parameters [73-77].It is proved in [75] that, if III I, I # and anl > b its solution is an entire function.If III < i, # n and I I C II < I, the solution of the matrix problem W'(z) AW(z) + exp (z)[BW(lz) + CW'(lz)], W(0) --W 0 is an entire function of exponential type [76].These results were extended to lin- ear FDE with polynomial coefficients and countable sets of argument delays in [7], [3] and [58].The method of proof employs the ideas developed in the theory of distributional solutions.
THEOREM 3.6.(Wiener[58]). (i) QiJ (z) are polynomials of degree not exceeding m; (ii) lij are complex numbers such that (3.8) 0 < ql < llijl < i, (j 0 p i), 0 < q2 < llipl < q3 < I; (iii) the series I o(i) converges, where Q(i) max I I Qijk II and Qijk are the j,k coefficients of Qij(z), and E I I Qip(O) II < i. i=O Then the problem has a unique holomorphic solution, which is an entire function of order not exceeding m + p. THEOREM 3.7.(Cooke and Wiener [3]).If, in addition to the hypotheses of Th.

THEOREM 2 . 3 .
The equation n Z (air + bi)x(n-i)(t) 0 i=O with constant coefficients al, b i and a 0 i, b 0 0 has a finiteorder solution if all i s (n-i)ai)sn-i-I/ I ais i=0 i=0 are real distinct and all residues r i res R(s) are nonnegative integral.S--S iThis solution is given by the formula n x C -I (d /dr sl)ri(t),

d and suppose that 3 0<j<n 3 deg
PO deg Pn > max deg Pj, (1 _< j _< n 1).Thenf(.) is of order 1 z) /z p 0, n.For cases when the condition on the degree of P P z-o P is not satisfied, see ([34, Th. 1.6]).
-tor-valued function whose components fl' f are all entire functions Suppose that F satisfies the m ODE (n) (n-l) nonpositive integers.If m is the smallest of their absolute values there exists a solution of order m.
m= i, then w(z) is of zero order, and if m > then In T(r, w) (Zn m/ In o) Zn r asr .w (i)(z) exp(ez) Y b.w (i)(%z), 3.3 there   exists a polynomial Q(z) of degree p such that the systemk m E I A (z)W (j) i=O j=O ij (ij z) Q(z) u(t) is monotonic for t _> to, then such that lira inf [u(t O. t e 2 (At) If