Distributional and entire solutions of ordinary differential and functional differential equations

International Journal of Mathematics and Mathematical Sciences, Jul 2018

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

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Distributional and entire solutions of ordinary differential and functional differential equations

I nternat. J. Math. Math. Vol. S.M. SHAH 0 1 Equations, Integral Equations, Distributional Solutions, Entire Solutions. 0 Department of Mathematics Pan American University Edlnburg , Texas 78539 , USA 1 Department of Mathematics University of Kentucky Lexlngton , Kentucky 40502 , USA 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. AND PHRASES; Ordinary Differential Equations; Functional Differential - DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL AND FUNCTIONAL DIFFERENTIAL EQUATIONS I. INTRODUCTION AND PRELIMINARIES. This paper may be considered as a continuation of [I] which contains, in partfcular, 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 investigat 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) w0 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]. 2. DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF ODE The number m is called the order of the distribution ORDINARY AND FUNCTIONAL DIFFERENTIAL EOUATIONS 0 has a solution of order x 1mr-. 0x () (t), x ’ o, 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. THEOREM 2.1. (Wiener [ 16 ]). If the equation nY. ai(t)x (n-i) (t) 0 i--O with coefficients ai(t)_ C (re+n-i in a neighborhood of t m concentrated on t 0, then: (I) a0(0) 0, (2) m satisfies the relation -(m + n)a 0’(0) + al(O) O, 3) there exists a nontrlvial solution (x0, xm) of the system mj-+E-n0Xk+j -mnlni=(jo ,n)(-l)J-laj-i)(0)(k + J i)! 0 (k 0, 1, m + n). THEOREM 2.2. (Wiener [ 16 ]). Eq. (2.2) has an m order solution with support t 0, if the following hypotheses are satisfied: (i) For some natural N(0 < N < m + n), a (0) O, i 0, rain(N, n); (2.1) (2.2) rn. (-1)iai(O) (m + i) O. (2.4) i=O 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 THEOREM 2.3. The equation Zn (air + bi)x(n-i)(t) 0 i=O with constant coefficients al, bi and a0 i, b0 0 has a finiteorder solution if all poles si of the function R(s) inZ=0 (.is (n- i)ai)sn-i-I/ inI=0aisn-i are real distinct and all residues r i res R(s) are nonnegative integral. S--S i This solution is given by the formula n x C -I (d /dr sl)ri(t), C i-i const md its order is n m= I-E-Ir i. If an 0 there exists also a solution s:t) r I t-1. x- C H (d!dt t--I Polynomial and rational solutions of ODE have been studied extensively [17 25]. In [ 17 ] the author deals with the equation ir-. Oalx (.t) 0 where aS ai{t) have mth order derivatives in [a, b]. Let aio ai / a0 THEOREM 2.4. (Sapkarev [ 17 ]). Eq. (2.5) has a polynomial solution of degree m if and only if P(z, 8 /z) m kE;OPk(Z) k i zk, / z 1 ( / 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 obtainTeHdE.ROEMIn2.t7h.e la(sNtovpaa[r2t5,]).diffLeetrentibaelaeqsuiamtpiloynscoonfne’cEtuelder opteynpeseatreicnonsiadnedreud.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. It is reduced to the form P0(z w, --) -= r. QI(z --W-) 0 (2.7) where Qi(z, n) are polynomials In z and n. Let w(z) be an entire transcendental solution of (2.6) and let be a point on the circle Izl r such that Substituting w w(z), z in (2.7) and, dividing its terms by wn() gives, with regard to Maclntyre’s formula [27] f’ ()/f() rM’ (r)/M(r) K(r), the equation n Q0({, K(r)) E Qi({, K(r))ji({). From here it follows that Q0 (’ Z(r)) o(.i). (2.8) The polynomial Q0(, K) is called the principal polynomial of Eq. (2.7), and (2.8) is called the determining equation. 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 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 transcendental solutions of order O < i In general, i cannot be replaced by a larger number: there are equations of the form (2.6) that have entire transcendental solutions of order I EXAMPLE 2.1. (Strelltz[ 6 ]). The equation w2 + 4zw’ 2 i has an entire transcendental solution w cos / of order 0 i The following result is of interest in this connection. THEOREM 2.10. (Wittlch [ 26 ]). Let R(z, w) be a rational function of z and w. A meromorphlc solution of the equation w’ R(z, w) which is of" order < 1 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) are of the order I + p go" If p _< qI, all transcendental solutions are of the order i + qi go" Deviation from this pattern can occur only if qi < 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 wI and w2 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 w (n) + 7n. Pi(Z)W (n-i) 0 i=l the coefficients pi(z)(i i, 2, 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’ w (n) 0 are investigated, too. A necessary condition 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. THEOREM 2.14. (Zimogliad [ 33 ]). A second order algebraic differential equation P(z, w, w’, w") 0 (P is a polynomial of all its variables) cannot have entire transcendental solutions of zero order. THEOREM 2.15. (Shah [ 34 ]). Let f(z) be an entire solution of an nth order linear homogeneous equation P0 (z)w(n) + + Pn(Z)W(Z) 0 and aP zli-roa PP (z) /zd p 0, n. For cases when the condition on the degree of P is not satisfied, see ([34, Th. 1.6]). The Bessel function of integer order n, Jn(Z), satisfies the ODE z2w,, + zw’ + (z 2 n2 )w 0, and the Coulomb wave function FL(D z) satisfies the ODE z2w + (z2- 2Nz L(L + l))w 0 (N a real constant, L a nonnegative integer). For these functions we have log M(r, Jn r log M(r, FL) as r + . Consider now vector-vM.ued functions F: 1/ I;m. Suppose that the components fk(l <_ k <_ m) are all entire functions. Write llFCz) [l max {Ifk(z) J, I < k <m}, MCr, F) max !! F(z) lJIzJ=r DEFINITION. A vector-valued entire function F is said to be of bounded index (BI) if there exists an integer N such that II F (i) (z) I[ > !1 F(k)(z) Ii i! k! max O< i<N 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 0m<akx<N 11 F((Nk)+(01)) k!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) (apq,i (z)), lira lapq i (z) Z-Oo ]Apq l] sup (IApq,i], 1 <_ p, q THEOREM 2.17. (Roy and Shah [ 35 ]) Let F: + be a ve-tor-valued function whose components fl’ fm are all entire functions Suppose that F satisfies the ODE Ln(W, z, Q) --w (n) (z) + Ql(Z)- (n-l) (z) + + Qn(Z)W(Z) g(z) where g(z) is a vector-valued entire function of BI. Then each fk satisfies an ODE of this form (with possibly different n and coefficients), and F, fl’ fm are 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 Ln (w, z Q) 0 Then w(ei) havlei:m sup log M(r w) < max { i, m n ro r iffil where the numbers Ai are defined above (ii) If the elements of Qi(l <_ i < m) are constant, and p _> 0 is any integer such that m FIn+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 coefficients. 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 such that [u(t lira inf e2 (At) t If u(t) is monotonic for t _> to, then max I =rg(xre f) f(z) I f(i)()(z where (r, f) is the central index of the Taylor expansion 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) /rp < Up. In [ 42 ] the author considers the equation p0(z)w" + pl(z)w’ + p2(z)w 0, (2.10) where p0(z) # 0, pl(z) and p2(z) are entire and have real Taylor coefficients about ny real point. THEOREM 2.22. (Lopusans’kll [ 42 ]). Oscillatory real solutions of (2.10) have only real zeros. THEOREM 2.23. (LopusHans 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 m0 ran_2, respectively, 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 u<l<mn-aZx (m. i +n) /(n i), and c is some positive constant. Asymptotic properties of the solutions of linear ODE with entire coefficients are studied in [ 44 ]. Consider the equation w (n) + an_lW (n-l) + 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" 6w2 + z whose solutions are meromorphic of the form w i/ (z z0) 2 (z 0 /10)(z z 0) 2 i (z z0) 3 + I / (z z0)2 ,, (z), where ,, (z) Yn--O n+2 (z zo)n" She represents w as a quotient of two entire functions: w-- (u’2 uu") /u2 (2.12) where u exp (- I dz Jw dz), and then obtalns recursion relations for the coefflclents of the power series expansions of the numerator and denominator. In conclusion, we note that in some recent works [ 46-50 ] 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 righthand 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 ]. THEOREM 2.26. (Napalkov [ 47 ]). Each entire solution w(z) of the equation MkW=0 is representable in the form w(z) zk-IWl(Z + + w(z), where Mwi 0 (i i, k), if and only if I(z) + I (1)(z) > cle for all z e % with some constants c I, c 2 > 0. In [ 48 ] the author studies the operator k7=P0aikz k p > 0. LPw i7=0p i(z)w(1)(z), pi(z) . (I) applicable to the set H of entire functions at the point z 0 if the series The operator LP is said to be 7 i= 0pi(Zo)W(1)(z0) converges for any function w from H; (2) applicable to H in the domain Iz < if LP is applicable to H at finite point (3) strongly regularly applicable to H inside the domain Iz < w e H and R < oo, if, for any where a--sup { Izl: z e Q}. Conversely, if (2.12) holds, then LP is strongly regularly 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 {w(z) g Aoo llw II B(x,y z=xs+uipye Here #(x,y) is a locally bounded function in R2 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 homogeneous equation is investigated. lw(z lexp(_(x, y)) < o. tx’(t) in=0Ai(t)x(%it) with matrices A (t) CTM in a neighborhood of t roots of the equation det inI=0 I%iI-1% Ai(0 + ( + I)E) 0 be nonpositive integers. If m is the smallest of their absolute values there exists a solution of order m. From here it follows that the system tx’(t) A(t)x(t) + in=l Ai (t)x(%it) 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 important equations of mathematical physics. For equations with more general argument delays we have THEOR 3.2. (Wiener 15] ). The system 0 and constants %i # 0 is that some (3.1) tx’(t) i=0Ai (t)x( i (t)) in which Ai (t) c Cm i (t) g C has a solution (2.1) or order m, if the following 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) n=Z0xntn)(t) in the generalized-function space (S)’ conjugate to the space S of testing functions (3.2) (t) that satisfy the restriction [2] ’’l(n)(t)l < acnnn > i. To ensure the convergence of series (3.2), it is sufficient to require that for n/ the vectors xn satisfy the inequalities since II xn II < bdn n-n0, 0 > Z < xn(n)(t), (t)>ll II Z (-l)n (n) (0)xn [1 < n=O n=O < l l(n)(o)l 11 xn I] < ab Z (cdn-O) n < n=O n=O for < 0. If series (3.2) converges, its sum represents the general form of a linear functional in (So)’ with the support t 0 [ 53 ]. Solutions in (So)’ of some linear ODE with polynomial coefficients were studied in [ 54 ], [ 55 ], [ 56 ]. The particular importance of the system il=0 jm=I0 (Aij + tB1.3.)x(J)(%it tx(%t) 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) (3.3) 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 li jm=IOA.lj (t)x (j) (%ij (t)) 0 (3.4) 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: Aij(t k=PY. 0Ai.ktk A00(t) Atp p > i. (ii) The real-valued functions %.. (t) E C in a neighborhood of the origin, 0 and o< Ioo < t, al >_, + >_t, a (0). (iii) The matrix A is nonsingular and c IO0 I-p-1 ]l AII Iol-P-Xll Aiop I! > O. have been studied in [ 56 ] and [ 57 ], respectively. THEOREM 3.4. (Cooke and Wiener [ 3 ]). Suppose that system (3.5), in which x is tPx ’(t) i=E0 ]m=EoAij(t)x(j)(%ij(t)) the particular cases of which tPx ’(t) A(t)x(t) and tPx ’(t) iI=OAi (t)x(%it) (3.5) (ili) The series Z i=O .-i A. converges, where with constant coefficients ajk. Its formal solutions are obtained in the form of Mellin or Laplace integrals. The functions occuring in the inte.rands satisfy linear difference equations of the form Zvn--0 P (qt)G(t + v) 0 (P(y) polynomials). 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 represented 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 k-I-0akDkw(%m-kz) 0, D d/dz 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 particular 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 Lw(z) km=IOakDkw(%m-kz f(z), where ak 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 ]. Introducing a generating function (3.6) G(z) n=IOGnzn Gn %n(n-l)/2/n! the author shows that the general solution of (3.6) for f 0 is given by w(z) 2z-- G(tz)(t) dt, with (t) q(t)/A(t), where m A(t) k=l0ak%k(2m_k_ I)/2 tk q is a polynomial of degree <_ m and F is a contour enclosing all the zeros of A. S.. SHAH and J. WIENER A similar integral representation is given for a solution of (3.6) with f # 0 in terms of the generalized Borel transform when (t) n=7.O f n / Gntn+l f(z) n=YO fnz In [ 70 ] the author discusses the system w’(z) Aw(lz), 0 < I < i, where A is a cornplex 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 asymptotic behavior. Given the equation m n w" (z) + k=El ak (z)w’(IkZ) + j=7.l bj (z)w(jz) 0, i=nIOa.1w (i)(z) exp(ez) in=YOb.1w (i)(%z), w (i) (0) wi, 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 > bn its solution is an entire function. If III < i, # and II C II < I, the solution of the matrix problem W’(z) AW(z) + exp (z)[BW(lz) + CW’(lz)], W(0) --W0 is an entire function of exponential type [ 76 ]. These results were extended to linear 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 ]). Suppose the system W (p)(z) Y. Yp. Qij(z)W(J)(lijz)’ i=0 j=0 W (j)(0) Wj, j 0 p in which O and W are r r matrices, satisfies the following conditions: (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 II Qijk II and Qijk are the j,k coefficients of Qij(z), and E II 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. 3.6 the parameters lij(0 _< j _< p- I) are separated from unity: 0 < ql < --llijl --< q4 < i, the solution of (3.8) is an entire function of zero order. THEOREM 3.8. (Cooke and Wiener [ 3 ]). Under the assumptions of Th. 3.3 there exists a polynomial Q(z) of degree p such that the system ik=EO jm=IOAij (z)W (j) (ij z) Q(z) linear neutral FDE matrices A, B, W has a unique holomorphic solution which is an entire (1) Ai (z) m 7. AikekZ Bi(z) k=l m k=Y.OBikekz (ii) ai, bi are complex numbers such that 0 _< Rea.l < MI < oo, 0 < M2 _< Rebi _< M3< oo; (iii) the series I A (i) and Z B (i) e-Rebo1 converge where A (1) max II Bik II, and Y. II Bi(0) I[ e k i=O _Rebi < I. The authors [ 78 ] propose a method for finding polynomial solutions of the max II Aik II, k .. with positive constants has a solution W(z) regular at z entire function of zero order. THEOREM 3.9. (Wiener [ 51 ]). The problem W’(z) where b, ao and r. > 0 are given constants. Meromorphic solutions of a class of 1 linear differential-difference equations with constant coefficients are investigated in [ 79 ]. Numerous examples of FDE admitting entire solutions may be found in [40] and [ 80 ]. In conclusion, we mention papers [ 81 ] and [ 82 ], where singular integral equations have been studied in spaces of generalized functions. However, it should be noted that, perhaps, the first work of this kind was [ 83 ]. 2. Gel’fand, I.M. and Shilov, G.E. Generalized Functions, Vol. 2, Academic Press, New York, 1968. 4. Gel’fand, I.M. and Shilov, G.E. Generalized Functions, Vol. I, Academic Press, New York, 1968. S.M. SHAB and J. WIENER 27. Macintyre, A.J. On Bloch’s theorem. Math. Z. 44(1939), 536-540. 28. Wittich, Hans. gleichungen. Ganze transzendente Lsungen algebraischer Differential Gtt Nachrichten, (1952), 277-288. 40. Bellman, Richard; Cooke, Kenneth L. Differentlal-Difference Equations, Acadic Press, New York, 1963. 66. Pandolfi, L. Some observations on the asymptotic behavior of the solutions of the equation (t)=A(t)x(%t)+B(t)x(t), %>0. J. Math. Anal. Appl. 67 (2), (1979), 483-489. ns Research Hindawi Publishing Corporation ht p:/ www.hindawi.com Journal of Algebra Hindawi Publishing Corporation ht p:/ www.hindawi.com Pro bability and Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation ht p:/ www.hindawi.com Combinatorics Hindawi Publishing Corporation ht p:/ www.hindawi.com Submit your manuscr ipts Mathematics Volume 2014 Mathematical Pro blems gineering Discrete Nature International Mathematics and Mathematical Sciences Journal of Stochastic Analysis Journal of Optimization I. Fricke , G.H. ; Roy, Ranjan and Shah, S.M. Bounded index, entire solutions of ordinary differential equations and summability methods , Inter. J. Math. & Math. Sci. 4 ( 3 ), ( 1981 ), 417 - 434 . 3. Cooke , Kenneth and Wiener, Joseph. Distributional and analytic solutions of functional differential equations , J. Math. Anal. and Appl . (to appear). 5. Wiener , Joseph. A retarded type system with infinitely differentiable coefficients has solutions in generalized functions , Uspehi Mat. Nauk 31 ( 5 ), ( 1976 ), 227 - 228 . 6. Strelitz , Sh. Asymptotic Properties of Analytic Solutions of Differential Equations, Izdat "Mintis" Vilnius 1972 7. Wiener , Joseph. Entire solutiormof a linear functional differential system , Differencial'nye Uravnenia 13 ( 3 ), ( 1977 ), 552 - 555 . 8. Golubev , V.V. Lectures on Analytic Theory . of Differential Equations, GITTL , Moscow, 1950 . 9. Izumi , S. On the theory of linear functional differential equations , Thoku Math. J . 30 ( 1929 ), 10 - 18 . I0. Robinson , L.B. Une pseudo-fonction et l'quat+/-on d'Izumi, Bull. Soc . Math. France, 64 ( 1936 ), 66 - 70 . II. Robinson , L.B. Complmente une tude sur l'equation fonctionelle d'Izumi, Bull Soc . Math. France, 64 ( 1936 ), 213 - 215 . 12. Aliev , F.S. The general solution of Euler's equation in generalized functions, Dokl . Akad, Nauk AzSSR, 20 ( 1 ) ( 1964 ), 9 - 13 . 13. Aliev , F.S. The fundamental system of solutions of Euler's equation in generalized functions , Vestn. Mosk. Gos. Univ. Ser. Mat. Mekh. _5 ( 1964 ), 7 - 14 . 14. Wiener , Joseph. Solutions of linear systems in generalized functions, Differencial'nye Uravnenia II(6)( 1975 ), 1128 - 1130 . 15. Wiener , Joseph. Generalized-function solutions of linear systems , J. Differ. Equat . 38 ( 2 ), ( 1980 ), 301 - 315 . 16. Wiener , Joseph. Generalized-function solutions of differential and functional differential equaZlons , J. Math. Anal. and App .l.. 8_8(I), ( 1982 ), 170 - 182 . 17. apkarev, llija. Eine Bemerkung ber Polynomlsungen der homogenen linearen Differentialgleichungen, Bull. Soc. Math. Phys. Macdolne 26 ( 1975 / 1976 ), 5 - 8 ( 1977 ). 18. Lazov , Petard; Dimitrovski, Dragan. Conditions for the existence of a maximal number of polynomial solutions for nonlinear differential equations , Prirod.- Math. Fak. Univ Kiril i Metodi$ SkopOe Godien Zb . 25 /26( 1975 / 1976 ), 101 - 106 . 19. Lazov , Petar; Dimitrovski, Dragan. Conditions for the existence of the maximal number of polynomial solutions of algebraic differential equations , Differencial'nye Uravnenia 13 ( 6 ), ( 1977 ), 1131 - 1134 . 20. Prolisko , E.G. Polynomial solutions of an equation of Li6nard type, Differential equations and their application, D.nepropetrovsk . Gos. Univ. ( 1976 ), 105 - 112 . 21. Pisarenok , V.P. The behavior of solutions of a class of first-order differential equations in the complex plane , Differencial' nye -- Uravnenia 17 ( 5 ), ( 1981 ), 930 - 932 . 22. KecVki, Jovan. Additions to Kamke's treatise . Vlll. On singular solutions of generalized Clairaut's equation . Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz . 577 - 598 ( 1977 ), 30 - 32 . 23. Gromak , V.I. Algebraic solutions of the third Painlev equation , Dokl. Akad. Nauk BSSR 23 ( 6 ), ( 1979 ), 499 - 502 . 24. Gromak , V.I. ; Cedel'nik, V.V. Rational solutions of Panlev's fifth equation, Vesci Akad . Navuk BSSR. Ser Fiz-Mat. Navuk 6 ( 1978 ), 114 - 115 . 25. Nova G. , Lucimar. Certain properties of complex differential equations with polynomial coefficients , Rev. Colombiana Mat. 12(i-2) , ( 1978 ), 13 - 58 . 26. Wittich , Hans. Neure Untersuchungen ber Eindeuti.ge Analytische Funktionen , Springer-Verlag, Berlin, 1955 . 29. Pschl , Klaus. Zur Frage des Maximalbetrages der Lsungen linearer Differentialgleichungen zweiter Ordnung mit Polynomkoeffizienten , Math. Ann. 125 ( 1953 ), 344 - 349 . 30. Hille , Einar. Ordinary Differential Equations in the Complex Domain , John Wiley & Sons, New York, 1976 . 31. Frei , M. Uber die Lsungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten, Comment . Math. He!v. 35 ( 1961 ), 201 - 222 . 32. Valiron , Georges. Analytic Functions (Russian translation) , Moscow, 1957 . 33. Zimogliad , V.V. On the order of growth of entire transcendental solutions of second-order algebraic differential equations , Mat. Sb . 85 ( 127 ), 2 ( 6 ), ( 1971 ), 286 - 302 . 34. Shah , S.M. Entire solutions of linear differential equations and bounds for growth and index numbers , Proc. Royal Soc. of Edinburgh 94 __A( 1983 ), (to appear). 35. Roy , Ranjan and Shah, S.M. Vector-valued entire functions satisfying a differential equation (to appear). 36. Shah , S.M. On real continuous solutions of algebraic difference equations , Bull. Amer. Math. Soc . 53 ( 1947 ), 548 - 558 . 37. Shah , S.M. On real continuous solutions of algebraic difference equations, II, Proc . Nat. Inst. Sci. India 16 ( 1950 ), 11 - 17 . 38. Cooke , K.L. The rate of increase of real continuous solutions of certain algebraic functional equations , Trans. Amer. Math. Soc . 92 ( 1959 ), 106 - 124 . 39. Cooke , K.L. The rate of increase of real continuous solutions of algebraic differential-difference equations of the first order , Pacific J. Math. 4 ( 1954 ), 483 - 501 . 41. Knab , Otto. Zur Werteverteilung der Lzungen linearer Differentialgleichungen, Co.mplex Analysis Joensuu 1978 (Proc. Colloq. Univ. Joensuu, Joensuu , 1978 ), Lecture Notes in Math. 747 , Springer, Berlin, 1979 , 189 - 204 . 42. Lopuans'kii , O.V. The connection between functions of class HB and the solutions of second order linear differential equations, D.opovidi Akad. Nauk Ukra'n. RSR Ser. A(9 ), ( 1975 ), 783 - 785 . 43. Tkaceenko , V.A. The growth of the solutions of a linear differential equation with polynomial coefficients , Mat. Fiz. i Functional. Anal. Vyp . 3 ( 1972 ), 71 - 76 . 44. Boiko , S.S. ; Petrenko , V.P. Asymptotic properties of the solutions of linear differential equations with entire coefficients , Differencial'nye Uravneniia 14 ( 11 ), ( 1978 ), 1923 - 1934 . 45. Fil'akova , V.P. The representation of meromorphic solutions of Painlev's equation by entire functions, Dopovidi Akad . Nauk Ukraln RSR Ser. A , ( 1975 ), 208 - 211 . 46. Kubrak , V.K. The growth of particular solutions of a differential equation of infinite order , Izv. Severo-Kavkaz. Naun. Centra Vys. koly Ser. Estestv. Nauk I ( 1977 ), 5 - 7 . 47. Napalkov , V.V. A property of the solutions of differential equations of infinite order with constant coefficients , Izv. Vys. Uebn. Zaved. Matematika 7 ( 182 ), ( 1977 ), 61 - 65 . 48. Korobeinik Ju . F. Criteria for the applicability of differential operato_r.s of infinite rder to certain classes of exponential functions . Godisnik Viss. Tehn. Uebn. Zaved. Mat. 8 ( 1972 ), ( 3 ), 9 - 18 , ( 1973 ). 49. Korobeinik , Ju . F.; Bogacev , V.A. The solvability of linear differential equations in weighted spaces of entire functions , Differencial'nye Uravnenija 13 ( 12 ), ( 1977 ), 2158 - 2167 . 50. Epifanov , O.V. A differential operator with polynomial coefficients in classes of entire functions with a given estimator of the indicator , Mat. Sb . (N.S.) 114 ( 156 ), ( 1981 ), 85 - 109 . 51. Wiener , Joseph, Solutions of functional differential equations in generalized functions , Differencial'nye Uravnenija , Ryazan', 3 , ( 1974 ), 27 - 33 . 52. Wiener , Joseph. Existence of solutions of differential equations with deviating argument in the space of generalized functions, Sibir .sk, Mt.__Z_. 6 _( 1976 ), 1403 - 1405 . 53. Mityagin , B.S. On the infinitely differentiable function with the given values of the derivatives at a point, Dokl . Akad. NaukSSSR138(2) , ( 1961 ), 289 - 292 . 54. Alley , F.S. On the solutions in generalized functions of some ordinary differential equations with polynomial coefficients , Dokl. Akad. NaukSSSR 167 ( 2 ), ( 1966 ), 259 - 262 . 55. Aliev , F.S. On the solutions in generalized functions of ordinary differential equations with polynomial coefficients , Dokl. Akad. Nauk SSSR 169 ( 5 ), ( 1966 ), 991 - 994 . 56. Aliev , F.S. On the solutions of certain systems of ordinary differential equations in the space of generalized functions , Vestn. Mosk. Gos. Univ. Ser. Mat. Mekh . 5 ( 1973 ), 3 - 10 . 57. Wiener , Joseph. Generalized-function solutions of differential equations with a countable number of argument deviations , Differencial'nye Uravnenia 14 ( 2 ), ( 1978 ), 355 - 358 . 58. Wiener , Joseph. Distributional and entire solutions of linear functional differential equations, Internat . J. Math. & Math. Sci5(4) ,-( 1982 ), 729 - 736 . 59. Kato , T. ; McLeod J.B. The functional differential equation y'(x)--ay(%x)+by(x) , Bull. Amer. Math. Soc. 77 , ( 1971 ), 891 - 937 . 60. McLeod , J.B. The functional-differential equation y'(x)=ay(%x)+by(x) and generalizations , Conference on the Theory of Ordinary and Partial Differential Equations (Univ. Dundee, Dundee , 1972 ), Lecture Notes in Math. 280 , Springer, Berlin, 1972 . 61. Chambers , L.G. Some functional differential equations, Quart . Appl. Math. 32 ( 1974 /75), 445 - 456 . 62. Carr , Jack; Dyson, Janet. The functional differential equation y' (x)--ay(%x) +by(x) . Proc. Roy. Soc. Edinburgh Sect. A 74 ( 1974 / 1975 ), 165 - 174 ( 1976 ). 63. Carr , Jack; Dyson, Janet. The matrix functional differential equation y' (x)= Ay(%x)+By(x) , Proc. Roy. Soc. Edinburgh Sect. A 75 ( 1975 / 1976 ),( 1 ), 5 - 22 . 64. Lim , Eng-Bin. Asymptotic behavior of solutions of the functional differential equation x'(t)=Ax(%t)+Bx(t), >0 . J. Math. Anal. Appl . 55 ( 3 ), ( 1976 ), 794 - 806 . 65. Lim , Eng-Bin. Asymptotic bounds of solutions of the functional differential equation x'(t)=ax(%t)+bx(t)+f(t), 0<%<I. SlAM J . Math. Anal. 9 ( 5 ), ( 1978 ), 915 - 920 . 67. Hahn , Wolfgang. ber die Funktlonal-Differentialglelchung f' (z)--f(qz) und verwandte Funktionalglelchungen , Ann. Univ. Scl. Budapest. Etvs Sect. Math. 16 , ( 1973 ), 3 - 21 , ( 1974 ). 68. Vogl , Fritz. Das Wachstum ganzen Lsungen gewisser linearer Funktlonal-Differentialgleichungen, Monatsh . Math. 86 ( 3 ), ( 1978 / 1979 ), 239 - 250 . 69. Vogl , Fritz. ber eine Integraldarstellung der ganzen Lsungen der FunktlonalDifferentialgleichungen Y.=0akDky(m-kz) --f (z), Demonstratlo Math . 12 ( 3 ), ( 1979 ), 645 - 655 . 70. Vogl . Fritz. Uber ein System linearer Funktlonal-Differentlalglelchungen, Angew . Math. Mech. 60(I) , ( 1980 ), 7 - 17 . 71. Blair , Jacques. Sur une quation differentlelle fonctionnelle analytlque , Canad. Math. Bull . 24 ( 1 ), ( 1981 ), 43 - 46 . 72. Mohon 'ko, A.Z. Differential and functional equations with factors , Differencial'ne Uravnenia 15 ( 9 ), ( 1979 ), 1713 - 1715 . 73. Flamant , P. Sur une uation differentlelle fonctionnelle linalre , Rend. Circ. Mat. Palermo 48 , ( 1924 ), 135 - 208 . 74. Bruwler , L. Sur l'application du calcul cymbolique la iresolution d'equatlons fonctlonneles , Bull. Soc. R. Scl. Liege 17 , ( 1948 ), 220 - 245 . 75. Valeev , K.G. On solutions of some functional equations, Issl . po l. ntegro-Diff. Uravn. v Kirgizli 5 , ( 1968 ), 85 - 89 . 76. Wiener , Joseph. Differential equations with periodic transformations of the argument, Izv . VTss%*. UcWebn. Zaved. Radloflzlka 16 ( 3 ), ( 1973 ), 481 - 484 . 77. Wiener , Joseph. Periodic maps in the study of functional differential equations, Differencial'nye Uravnenia, Ryazan' 3( 1974 ), 34 - 45 78. FodcWuk , V.I. ; Holmatov , A. Polynomial solutions of dlfferentlal-dlfference equations of neutral type. Aproxlmate and Qualitative Methods in the Theor[ of Differential and Functlonal-Differential Equations , Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1979 , 98 - 103 . 79. Naftalevlch , A. ; Gylys , A. On meromorphic solutions of a linear differentialdifference equation with constant coefficients , Michigan Math. J . 27 ( 2 )( 1980 ), 195 - 213 . 80. Przeworska-Rolewicz , Danuta. Equations with Transformed Ar.gument. An Algebraic Approach , PWN, Warszawa, 1973 . 81. Kosulin , A.E. One-dimensional singular equations in generalized functions , Dokl. Akad. NaukSSSRI63(5) , ( 1965 ), 1054 - 1057 . 82. Rogozin , V.S. A general theory of solving boundary problems in the space of generalized functions , Dokl. Akad. Nauk SSSR 164 ( 2 ), ( 1965 ), 277 - 280 . 83. Horvth , J. Sur l'itration de la transforme de Hilbert d'une distribution complexe, C.R. Acad . Sci. 237 ( 23 ), ( 1953 ), 1480 - 1482 . Volume 2014


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S. M. Shah, Joseph Wiener. Distributional and entire solutions of ordinary differential and functional differential equations, International Journal of Mathematics and Mathematical Sciences, DOI: 10.1155/S0161171283000216