Reducible functional differential equations

International Journal of Mathematics and Mathematical Sciences, Aug 2018

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.

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Reducible functional differential equations

Journal of REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS S.M. SHAH 0 1 2 0 Functional Differential Equation , Argument Deviation, Involu- 1 Department of Mathematics Pan American University Edinburg , Texas 78539 , USA 2 Department of Mathematics University of Kentucky Lexington , Kentucky 40506 , USA 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. tion. AND PHRASES I. INTRODUCTION. 1980 MATHEt.TICS SUBJECT CLASSIFICATION CODES. 34K05, 34K20, 34K09. In [ 1-4 ] a method has been discovered for the study of a special class of functional differential equations differential equations with involutions. This basi cally algebraic approach was developed also in a number of other works and culminated in the monograph [ 5 ]. Though numerous papers continue to appear in this field [ 6-10 ], some aspects of the theory still require further investigation. In connection with the DurDoses of our article we mention only such topics as hiher-order equations with rotation of the argument, equations in partial derivatives with involutions, influence of the method on the study of systems with deviations of more general nature, and solutions in spaces of generalized and entire functions. In [12] we proved that the solution is obtained very simply by a differentiation of (2.1). As a matter of fact, x"(t) x(), x’(t) -- x’( t t2x"(t) + x(t) whence, Cons equ en t ly, x(t) r[ClCS(--z In t) + C2sin(Z-z in t)]. Substituting x(t) in (2.1), we obtain CI=C2, and finally, x(t) C cos(- In t ). Obviously, the key to the solution is the fact that the function f(t) interval (0, ) one-to-one onto itself and that the relation f(f(t)) t, i/t maps the (2.3) or, equivalently, t E G, and fn (t) t for n functions are involutions. EXAMPLE 2.1. f(t) EXAMPLE 2.2. I, m i. It is easy to check that the following c t on R (_oo, oo), where c is an arbitrary real. f(t) -at for t _> 0, -t/a for t <_ O, on R, where a > 0 is arbitrary [ 5 ]. EXAfP LE 2.3. t k for 0 < t < i, on (0, oo), where k is an arbitrary positive integer [ 5 ]. EXAMPLE 2.4. The function f(z) ez, where E exp(2i/m), is an involution of order m on the complex plane. EXAMPLE 2.5. The function [13] We denote the set of all such functions by I. The graph of each f e I is symmetric 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) t + x c, then f(t) c- t. If we take then S. Mo SHAH AND J. WIENER g(t, x) f(t) 3Jc t 3 Every continuous function f I is strictly decreasing [15]. Hence, t-l_iomo f(t) +oo, tl-i+mo f(t) _o. (2.4) 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 f(t) Ytt+- (2 + > 0) (2.5) which leave two points fixed. We introduce the following definition. DEFINITION 2.2. A relation of the form F(t, x(fl(t)) x(fk(t)) fk(t) are involutions, is called a differential equation with in which fl(t), involutions [i]. THEOM 2.2([1]). Let the equation x’(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 tO. x(n) (fl(t)) x (n) (fk(t)))=O, PROOF. Eq. (2.8) is obtained by differentiating (2.6). x"(t) F + xF(t) x (t) + x(fF(t)) x (f(t))f’(t). Indeed, we have The fixed point of the involution f(t) (2.11) is x() xO; a-t is t o a/2. The initial condition for where x is an unknown function and where the following conditions are fulfilled: p" r 2, n and every t G. THEOREM 2.5 ([]7]). If conditions (1)-(3) are satisfied, then every p-times differentiable solution of Eq. (2.14) is a component of the solution of a system of ordinary differential equations with argument t only. This system is obtained from Fq. (2.14). To investigate the equation x’(t) f(x(t), x(-t)), the author of [6] denotes y(t) x(-t) and obtains M f’ i(t) ddt Then the solution of the linear ordinary differential equation kn=O ak(f(t))MLx(t) x(t) kn=ZOak(f(t))M@(t) + @(f(t)) with the initial conditions x (k) (t O xk, k Mx(t) lt=t0 Xk + Mk*(t)It=to, 0 k n 1, 0 n i is a solution of problem (3.1)-(3,2). PROOF. By successively differentiating (3.1) n times, we obtain x(f(t)) Lx(t) (t) MOLx(t) MO(t), (3.3) (3.4) x’(f(t)) x"(f(t)) f’(t) ddt Lx(t) f’(t) ddt (t) 1 MLx(t) M(t), 1 f,(t) ddt MLx(t) I f’(t) dd- M(t) M2Lx(t) M2(t), x (n) (f(t)) f’ I(t) ddt Mn-ILx(t) f’ i(t) ddt Mn-l(t) These relations are multiplied by a0(f(t)) al(f(t) and the results are added together: n Z ak(f(t))x (k) (f(t)) k=O By virtue of f(f(t)) nZ ak(f(t))Mx(t )- nF. ak(f(t))Mk(t). k=O k--O t, it follows from (3.1) that an (f(t)) MnLx(t) Mn(t). respectively n k=O ak(f(t))x (k) (f(t)) x(t) + (f(t)). 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): values of the function x(t) and of its n 1 derivatives at the point t O should equal Xk, k O, n i, from (3.2), while the values x (n) (tO x (2n-l) (t O are determined from the relations the Mx(t) 0 n i by substituting the values t O and xk for t and x (k) (t). THEOREM 3.2 ([i]), The equation x (t) x( (3.6) 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), (3.7) a and b are real and is a nonnegative integer. PROOF. By an n-fold differentiation Eq. (3.6) is reduced to the Euler equation k=nI2+nl b(kn) tkx (k) (t) x(t). (3.8) 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 M -t 2 ddt On the basis of (3.4) and (3.8) we have Mnx (n) (t) l2n k=n+l b(kn) tkx (k) (t), Then Mnx (n+l (t) 2n I k=n+l b (n) tkx(k+l) (t). k Mn+ix (n+l) (t) -t 2 d 2n k=n+lZ bk(n)tkx(k+l)__ (t) 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 b k(n+l)= -(k-l)"bk(_n)1 bk(_n)2 n + 2 <_ k < 2n + 2, b(n’n’= 0 b(2nn)+l 0 connecting the coefficients of the Euler equations k=n2+nlb kn) t__ (k) (t) x(t) and k=2nnF,++22 bk(n+l)tkx(k)(t): x(t), which correspond to the equations x (n) (t) x( and x (n+l) (t) x(_t)i. 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. EXAMPLE 3.1. The investigation of the nonhomogeneous equation [I] x’(t) x( ) + 9(t), 0 < t < o, 9(t) g C 1 (0, reduces to the problem t2x"(t) + x(t) x(1) The solution is x(1) x0 t29’(t) O, x’(1) -e2 --- A2 > O, (3.9) -a2 XXlo_% 2 [(a-%2) t kl + (%l-a)t%2]. 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), (i) The function f maps an open set G onto G. (2) The function f can be iterated in the following way: fl(t) f(t) fk(t) f(fk_l(t)) fm (t) t (t E G) (3.13) 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 3.4 (3]). Eq. (3.13), for which conditions (I)-(3) hold, can be reduced to a linear differential equation of order m. EXAMPLE 3.3. Consider the equation 1161 x’(t) x(f(t)), f(t) (l-t) -I (3 14) and G (_o% 0)U(0, I)U(lo + oo). For f we have f3(t) t on G. In this case (3.14) is reducible to the equation t2(l-t)2x (t) 2t2(l-t)x"(t) x(t) O. THEOREM 3.5 ([i]). In the system x’(t) Ax(t) + Bx(c-t), x(c/2) x0 (3.15) let A and B be constant commutative r xr matrices, x be an r-dimensional vector, and B be nonsingular. Then the solution of the system x"(t) (A2-B2)x(t) x(cl2) x0, x’(cl2) =(A+B)x0 is the solution of problem (3.15). In [7] it has been proved that the equation has the general solution while the equation t-2) + c4[sin(’] In t) + 3+I/ cos( In t)]. 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 approximat ion. t a-.- [tkx)kt (t)] kt k x (k) (t) 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 functionals 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 dimensionality of the vector space of solutions of functional differential equations, the linear vector space of solutions to (3.17) is of dimension n2. Moreover, the authors give a complete algebraic characterization of these n2 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) is 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) and (3.20) can be rewritten as r(t t)B -I IA (t and q() [kl,, kn,]T, r(-) [k,Tl, kT,n T which, with the obvious correspondence and for simplicity of notation, are denoted as p’(t) Cp(t), p(T/2) PT/2" (3.21) Here p(t) is an 2n2-vector and C is a 2n2 2n 2 constant matrix. (3.21) is used in provinR the followin result: 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) each ., nj,r Zr=s I n. mj, Zj m.=3 2n2" given by where the generalized eigenmatrix pair (Lji ,r’ Mj ,ri)associated with the eigenvalue satisfies the equations T L.2,r0 M.3,r0 0. But this result demonstrates that if %.3 is a solution of (3.22), -%. will also be a solution; moreover, %. and -%. have the same geometric multiplici3 3 3 (-I) q+i the n2 linearly independent solutions of (3 19) given by Zj ,qr(t) W.3 qr(t) exp(j(t )T) q (t i=l Tq-i 1 (q- i): exp(-j (t T q (t )) Y i=l (q )T q-i i): (-I) q+i r L i j r i Mj ,r + Mj r Lj r satisfy the condition T Zj, Wj,r But this is precisely condition (3.20)" it therefore follows that the expressions Zj,rq(t) q (t l i= I )T q-i (q- i)! [exp(%j(t )T)Lj,ri + (-1) q+i exp(-Ej(t -zT) )Mj,riT (3"25) are n 2 linearly independent solutions of (3.17). THEOREM 3.8 ([20]). Eq. (3.17) has n 2 linearly independent solutions given by Eq. (3.25), where the generalized eigenmatrix pairs (Lj, ri Mj,ri satisfy Eq. (3.24) for one of the elements of the pair (j, -j), each of which is a solution of Eq. (3.22). 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 argument 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 initial values at the fixed point t o of the involution f(t) for the equations ut(t, x) auXX (t x) + buXX (f(t) x) (3.26) and Cn T’n(t O -%n (a+b)Cn satisfies Eq. (3.29). The following theorems illustrate striking dissimilarities between equations of the form (3.26) and (3.27) and the corresponding equations without argument deviations. THEOREM 3.9. The solution of the problem ut(t x) auxx (t, x) + huxx (c-t, x), (3.31) u(t, 0) u(t, ) 0, u(cl2, x) (x) is unbounded as t +, if a b # 0. If Ibl < lal, b # 0, expansion (3,28) diverges for all t # c/2. PROOF. By separating the variables, we obtain 2 2 Tn,(t)=_o_zn (aTn (t) + bTn (c-t)), Tn (c/2) Cn (3.32) The initial conditions for equations (3.31) and (3.32) are posed at the fixed point of the involution f(t) c t. In this case, Eq. (3.30) takes the form 4 4 T("t)n n4 ,b 2 a 2) Tn (t) Tn (c/2) 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: 2n 2 2 a2 (t-c) (i) Tn (t) Cn (cos 2 a+b b2_a2 s in (x), ut(O x) Tn (0) An T’n(0) Bn by successive differentiation of which we obtain Tn(3)(t) 22a2n 2 T(t) + T262-n2 T’n(-t) From Eq. (3.33) we find and also T (4) (t) n T"n(-t) r2 a 2n 2 T"n(t) 2 7r 22b2n2 T"n (-t) 2a2n2 2 Tn(_t) 22b2n2 Tn (t) Tn(4)(t) + 22a22n2 Tn"(t) + with the initial conditions Thus, Eq. (3.34) is reduced to the fourth-order ordinary differential equation 4 (a44-b4)n4 Tn (t) 0 (3) Finally, the inequality a 2 < b 2 leads to the result n 2b_a2 Bn sinh n b2-a 2 t. Of some interest is the equation ut(t, x) (t +_____B Auxxyt x) with the hyperbolic involution having two fixed points which is a generalization of Eq. (2.1). Differentiation changes (3.37) to the form (yt )2T"n(t) + A2A244n4 Tn(t) 0. (3.38) o. For the functions Tn (t) are oscillatory. 4. EQUATIONS WITH ROTATION OF THE ARGUMENT An equation that contains, along with the unknown function x(t) and its derivatives, 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-a I) X(mt-am and the corresponding values of the derivatives appear, where gl’ m are mth roots of unity and al’ m are complex numbers, is called a differential equation with rotation. 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 difficult 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 k=En0 akx(k) (t) k=nE0bkX (k) (ct) + lp(t), m__ 1 (4.1) AIAoX (BiBoX)(e2t) + AI@ + (Bo)(et), and act on this relation by A2. From AoX REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS . (B0x)(Et) + Finally, this process leads to the ordinary differential equation (AO(m-l) B 0(m-l))x mj-=ZIO I, and I is the identity operator. Thus, (4.1) is reduced to the ODE (4.2) of order mn. ’I make the initil onditions for (4.2) agree with the riRinal probl,m, it necessary to attach to ’onditions (4.1) the additional relati,,s (A0(j) gk(j+l) B0(j))x(k)(t)] t=0 Y. g ikAJ-i) B i-l)(*k) (t) It__0 i=O (j 0 m-o 2; k 0 n- i). System (4.3) has a unique solution for x (k)(O)(n < k _< mn- I), iff anj # (eibn)J (0 < i < m- i, i < j < m- i) These considerations enable us to formulate THEOREM 4 1 ([ 9 ]) If @EC (m-l)n and inequalities (4.4) are fulfilled, the solution of ordinary differential equation (4.2) with initial conditions (4.1)-(4.3) satisfies problem (4.1). THEOREM 4.2 ([ 9 ]). If g # i, the substitution y x exp(at/l e) exp(ct)(By)(et) + (4.2) (4 3) (4 4) (4.5) with operators A and B defined by (4.1) to Px (Qx)(et) + exp(-c,t/l where P and Q are linear differential operators of order n with constant coefficients 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. The following particular case of Eq. (4.1) has been investigated in [27]. THEOREM 4.4. ([27]). Suppose we are given a differential equation with reflection of order n with constant coefficients kn=7.O [ak-x(k) (t) + bkX (k) (-t)] y(t). (4.8) bj_kb k O, I, n and j k + I k + n, n (c) the polynomial 7. %2jtj has simple roots uq only, where j--O J k=ZO Cjk for 0 < j < n, k=j-n j k for n < j 5_ 2n, n x(t) Then every solution of Eq. (4.8) is of the form Cjk (-l)n+j-k(an2-bn2)(aj_ ka k-bj_kb k). (4.10) (4.11) where the Ck are arbitrary constants and (t) is a solution of the equation THEOREM 4.5 ([ 9 ]). Suppose that the coefficients of the equation n Y. ak(t)x (k) (t) k=O x(et) + (t), x (k) (0) xk, k O, n 1 (4.9) belong to C (m-l)n em i, a (0) # 0 and ej n l E-Jka,_(eJt)dk/dt k, k=O 0, m- 1. Then the solution of the linear ordinary differential equation L (m-l) x(t) 0 m-i x(t) + Z (Lk(m-l) )(ek-lt) + (em-lt) k=l (Lk(m-l) Lm_ILm_2 L k 0 k < m with the initial conditions satisfies problem (4,9), x (k) (0) Xk(k 0 n I) n(m-1) 1 kx (k) (0) + (k) (0) ,k=O PROOF. Applying the operator LI to (4.9) and taking into account that we get (LoX)(et) x(e2t) + (et) LIeOX(t) x(2t) + Ll(t) + (t) and act on this equation by L2 to obtain It is easy to verify the relations In particular, (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 (4.14) with constant coefficients A and B, det A # 0 and em and has a solution X(t) e(t)tA-IB 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 (4.16) 1 is Integrable in closed form (4.17) L. 0 in consequence of which we obtain the relation (Ad/dt t-IB) m X(t) em(m-l)/2X(t). e-J(Ad/dt t-IB), j m Xk(t) exp(ktA-l)tA-IB, k I, mo Their linear combination represents the general solution of (4.15). EXAMPLE 4.3. In accordance with (4.17) to the equation [ 9 ] tx’(t) 3x(t) + tx(-t) (4.18) there correspond two ordinary relations x’(t) (3t-I + i)x(t), x’(t) (3t-I i)x(t). We substitute into (4.18) the linear combination of their solutions x(t) t3(Clexp(it) + C2exp(-it)) and find C 2 IC I. A solution of (4.18) is x(t) Ct3(slnt + cost). 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 frequently 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 ]. I0. 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Time lags in biological models, in "Lecture Notes in Biomathematics, No. 27", Springer-Verlag, Berlin, 1978. 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 0 for k 2 . WIENER , J. Differential equations in partial derivatives with involutions , Differencial'nye Uravnenija 7 ( 1970 ), 1320 - 1322 . 3, WIENER , J. Differential equations with periodic transformations of the argument , Izv. Vys. Uebn. Zaved. Radiofizika 3 _ ( 1973 ), 481 - 484 . 4. WIENER , J. 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On solutions of some functional equations, Isis.led. po Integro-diff . Uravn. v Kirgizii 5 ( 1968 ), 85 - 89 . 30. FARGUE , D.M. Rducibilit des systmes hrditaires a des systmes dynamiques , C. R. Acad . Sci. Paris Ser. B 277 ( 1973 ), 471 - 473 . 31. BUSENBERG , S. and TRAVIS , C. On the use of reducible-functional differential equations in biological models , J. Math. Anal. Appl . 89 ( 1982 ), 46 - 66 . 32. SHAH , S.M. and WIENER , J. Distributional and entire solutions of ordinary differential and functional differential equations, Internat . J. Math. & Math. Sci. 6 ( 2 ), ( 1983 ), 243 - 270 . Volume 2014


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S. M. Shah, Joseph Wiener. Reducible functional differential equations, International Journal of Mathematics and Mathematical Sciences, DOI: 10.1155/S0161171285000011