Džurina, Jozef

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Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaJozef DžurinaProperty (A) of advanced functional-differential equationsMathematica Slovaca, Vol. 45 (1995), No. 2, 129--137Persistent URL: http://dml.cz/dmlcz/129312Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1995Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathematica Slovaca ©1995 . . . _. ._ /^^, , . , -N ., ^ ., _~ ., ~__ Mathematical Institute Math. SlOVaCa, 4 5 ( 1 9 9 5 ) , NO. 2, 1 2 9 - 1 3 7 Slovák Academy of Sciences PROPERTY (A) OF ADVANCED FUNCTIONAL DIFFERENTIAL EQUATIONS J O Z E F DZURINA (Communicated by Milan Medved') ABSTRACT. In this paper, property (A) of the advanced functional differential equation Lnu(t) + F(t,u[g(t)]) =0 (*) is derived from the asymptotic behaviour of a set of ordinary functional equations aiu(t) + F(t,u(t)) = 0 . On the basis of this comparison principle the sufficient conditions for property (A) of equation (*) are deduced. We consider the functional differential equation with advanced argument Lnu(t) + F(t,u[g(t)])=0, (1) where n > 3 and Ln denotes the disconjugate differential operator L 1 d * d A - i - - — (2) n rn(t) dt r n _ i ( t ) dt'" dt n ( t ) dt r0(t) ' [ ) We always assume that (i) ri, g: [to, CXD) —> K are continuous, r^(i) > 0, 0 < i < n , and g(t) > t; (ii) F: [to,oo) x R -> R is continuous and sgnF(t,x) = sgnx for each t G [<o,oo). We will assume that oo / ГІ(S) ds = cxэ for 1 < i < n — 1. (3) We say that operator Ln is in the canonical form if (3) holds. In the sequel we will suppose that operator Ln is in its canonical form. It is well known that any A M S S u b j e c t C l a s s i f i c a t i o n (1991): Pr imary 34C10. K e y w o r d s : Comparison theorem, Advanced argument, Property (A). 129 JOZEF DZURINA differential operator of the form (2) can always be represented in a canonical form in an essentially unique way (see T r e n c h [11]). We introduce the following notation: D0u{t) = D0{U]r0){t) = ^ l , ro{t) Diu{t) = Di(u\r0,...,ri)(t) = — r - A - i ^ ) , 1 < z < n . ri{t) at The domain V{Ln) of Ln is defined to be the set of all functions u: [Tu, oo) —* R such that Diu{t), 0 < i < n , exist and are continuous on [Tu, oo). A nontrivial solution of (1) is called oscillatory if it has arbitrarily large zeros; otherwise it is called nonosdilatory. Equation (1) is said to be oscillatory if all of its solutions are oscillatory. LEMMA 1. If u{t) is a nonoscillatory solution of (1), then there exists a t\ £ [to, oo) and an integer £ G { 0 , 1 , . . . , n — 1} such that £ ^ n (mod 2) and u{t)Diu{t) > 0, 0 < z <£, {-iy-*u{t)Diu{t) > 0 , £+l<i<n, for all t > t\. This lemma is a generalization of a lemma o f K i g u r a d z e ([5; Lemma 3]). A function u{t) satisfying (4) is said to be a function of degree £. The set of all nonoscillatory solutions of degree £ of (1) is denoted by Mt. If we denote by J\f the set of all nonoscillatory solutions of (1), then, by Lemma 1, J\f = Alo U Af2 U • • • U Mn-i if n is odd, and J\f = J\fx u A/3 U • • • l .\fn-i if n is even. DEFINITION 1. Equation (1) is said to have property (A) if for n even (1) is oscillatory (i.e. J\f = 0) and for n odd Af = J\f0. The main purpose of this paper is to establish a comparison principle between advanced equation (1) and the corresponding ordinary equation and to obtain sufficient conditions for equation (1) to have property (A). We remark that for delay equations {g{t) < t) of the form (1), efforts in this direction have been undertaken by several authors, see e.g. M a h f o u d [9], E r b e [1], [2], and K u s a n o and N a i t o [8] in which delay equations of the form (1) are compared with ordinary equations without delay, and on the basis of such a comparison theorem many sufficient conditions for property (A) of delay equation (1) are deduced. 130 PROPERTY (A) OF ADVANCED FUNCTIONAL DIFFERENTIAL EQUATIONS Let us consider the set of the disconjugate differential operators 1 d 1 d d 1 d 1 d &i = rn(t) dt r n _ i ( t ) dt'" dt n(t) dt r.i_i[^(t)]^(t) dt " ' d 1 d _ _ _ _ _ • • " d t r i ^ t M t ) dtr0[g(t)] for i = 1, 2 , . . . , n — 1. THEOREM 1. Suppose that (5) F(t,x) is nondecreasing in x, (6) sWee'dto.oo)), </(*)> o, g(t)>t. Further assume that for i — 1, 3 , . . . , n — 1 z/ n is even and for i = 2 , 4 , . . . , n — 1 if n zs odd. the functional equation aiU(t) + F(t,u(t)) = 0 (_7i) /ms not an?/ solution of degree i. Then equation (1) has property (A). P r o o f . Let u(t) be a function of degree £, satisfying (1). We may suppose that u(t) is eventually positive. For the sake of contradiction we assume that £ G {1, 2 , . . . , n — 1} . Let 6A be a number associated with ix(t) by Lemma 1, Integrating (1) from t ( ^ ti) to oo we have oo Dn-iu(í) > / rn(s)F(s,u[g(s)]) ds. Repeating this procedure, we arrive at oo oo oo D£u(t) > ri+^se+i) / ... / rn(sn)F(sn,u[g(sn)]) dsn ... ds£+2dsi+1. t 5 £ + i S n - 1 (7) We multiply (7) by r^(t) and integrate over [t±:t] to obtain D/_! _[</(*)] >D/-i-(ť) t oo oo oo / rДs*) / ^ + 1 ( ^ + 1 ) "' rn(sn)F(sn,u[g(sn)]) d s n . . . d s ^ , í l S£ S£ + l s n - l where we also used the facts that g(t) > t and Di-iu(t) is an increasing function as £ > 1. If £ > 2, then we multiply (8) by r*_i [#(<)] (/(£) and integrate the 131 JOZEF DZURINA resulting inequality over [*i,£]. Continuing in this manner we obtain t si s£-2 s£-l OO D0u[g(t)] > Jr1[g(s1)]g'(s1)J . . . J rl.1[g(8e-1)]g'(st-i) J re{st) J ... t\ t\ _1 t\ s£ OO . . . / rn(sn)F(sn, u[g(sn)]) dsn ... dsx, t > U (9) Denote the right hand side of (9) by v(t) and define z(t) = r0 [g(t)]v(t). By repeated differentiation of z(t), one can verify that z(t) is a function of degree £ and, on the other hand, aez(t) + F(t,u[g(t)])=0. (10) Since _,[#(_)] > z(t), we obtain in view of (10) that z(t) is a solution of the differential inequality {aez(t) + F(t, z(t)) } sgnz(t) < 0 . But then [8; Corollary 1] of K u s a n o and N a i t o ensures that equation (Et) has also a solution of degree £, which contradicts the hypotheses. If £ = 1, then (8) implies oo oo D0u[g(t)] > / r i ( s i ) / r2(s2) / . . . / rn(sn)F(sn,u[g(sn)]) d s n . . . d , s i sl s2 sn-1 Again, denote the right hand side of the above inequality by v(t) and define z(t) = r0[g(t)]v(t). Proceeding similarly as above we can verify that equa-tion (Ei) has a solution of degree 1, which contradicts the hypotheses. The proof is complete. • Now, we apply our comparison principle to the linear form of equation (1), namely, to the advanced equation Lnu(t)+p(t)u[g(t)] = 0 , (11) where function p(t) is continuous and positive on [_o,oo). Let ik G {1, 2 , . . . , n — 1}, 1 ^ k ^ n — 1, and t , s E [t0, oo), we define /o = l , t Ik(t,s;rik,...,rh) = / rik(x) Ik-i(x, s\rik_x,... ,rh) dx . s 132 PROPERTY (A) OF ADVANCED FUNCTIONAL DIFFERENTIAL EQUATIONS For simplicity of notation, we put Ji(t, s) = r 0 [g(t)]li (t, s; r±(g)g',..., n(g)g') , Ki(t, s) = rn(t) Ii(t, s; r n _ i , . . . , r n _ ; ) . It is easy to see that Ji(t, s) = r 0 [g(t)]li (g(t),g(s);r1, ...,r{). We define i = 1, 2, . . . , n — 3 , t qi(t) = Гi+г(t) / Kn^i_2(s,ť)Ji-i(s,ť)p(s) ås ,  OO qn-i(t) = rn-2[g(t)]g'(t) / Jn-3(s,t)K0(s,t)p(s) ds, t where we suppose that both integrals are convergent. THEOREM 2. Suppose that (6) holds. Assume that the second order equations ^щz'(t)) +qi(t)z(t) = 0 (ЄІ) are oscillatory for i = 2 ,4 , . . . , n — 1 if n is odd and for i = 1,3,.. ., n — 1 if n is even. Then equation (11) has property (A). P r o o f . Let £ £ {1, 2 , . . . , n — 1} be fixed. By Theorem 1, equation (11) has not any solution of degree £ if the equation a£u(t) + p(t)u(t) = 0 has not any solution of degree £, which, by [7; Theorem B] and [10; Theorem 2], comes if equation (ei) is oscillatory. The proof is complete. • Let us consider another form of equation (11), namely, the advanced equation \r2 Let us denote \t)(Úť)U'{t)) ) +-#)«[-'(*)] =0. (-2) t Ri{t)= ГІ(S) ds, for i = 1,2. to 133 JOZEF DZURINA COROLLARY 1. Suppose that (6) holds. Equation (12) has property (A) if oo liminf R2(t) J(RX [g(s)} - Rx [g(t)])p(s) ds > ^ . (13) t P r o o f . By Theorem 2, equation (12) has property (A) if equation (e2) is oscillatory. By the well-known criterion of H i 11 e [3], condition liminf / r2(s) ds / q2(s) ds J > - , which is equivalent to (13), ensures oscillation of all solutions of (e2) . • E x a m p l e 1. Let us consider the advanced equation y'"(t)+p(t)y[g(t)] = 0. (14) By Corollary 1, equation (14) has property (A) if (6) holds and moreover if oo liminf t / (g(s) - g(t))p(s) ds > - . (15) t •OO J 4: E x a m p l e 2. The ordinary equation y,,,{t) + 3^yit)^°' t>0' (16) has a nonoscillatory solution y(t) == t1+1^, which is of degree 1, but the corresponding differential equation with advanced argument y///(*) + 3^Py ( A i ) = °' *>0' A > 1 ' (17) O / o in view of condition (15), has property (A) if A > . As a matter of fact we are able to relax the condition of monotonicity im-posed on the advanced argument in Theorems 1 and 2. Let us consider func-tional equation of the form (1) with larger advanced argument Q(t), where Q(t): [<o, oo) —» R is continuous. 134 PROPERTY (A) OF ADVANCED FUNCTIONAL DIFFERENTIAL EQUATIONS THEOREM 3 . Suppose that (5) and (6) hold and Q(t) > g(t). Let r0 be nondecreasing. Further assume that for z = l , 3 , . . . , n — 1 if n is even and for i = 2 , 4 , . . . , n — 1 if n is odd functional equation (Ei) has not any solution of degree i. Then the equation Lnu(t) + F(t,u[Q(t)])=0 (18) has property (A). P r o o f . By Theorem 1, equation (1) has property (A), and, by [8; Theo-rem 1], equation (18) has property (A). • Using similar arguments we can prove the following result. THEOREM 4. Suppose that (6) holds and Q(t) > g(t). Let r0 be nondecreas-ing. Further assume that equations (e^) are oscillatory for i = 1, 3 , . . . , n — 1 if n is even and for i = 2 , 4 , . . . , n — 1 if n is odd. Then the equation Lnu(t) + p(t)u[Q(t)\ = 0 has property (A). E x a m p l e 3. Let us consider the advanced equation \7^t){^(t)U'{t)) ) +P(')«[2* + cost] = 0. (19) Letting g(t) = 2t — 1 and applying Corollary 1 and Theorem 4, one gets that equation (19) has property (A) if (6) holds and (13) is satisfied with g(t) = 2t-l. Let 0 < i < n — 1. We denote Mi(t, s) = r0(t) Ii(t, s; n , . . . , n), M{(t) = M{(t, t0), Ki(t) = Ki(t,t0). We show that the conclusions of Theorems 1-4 can be strengthened as follows: THEOREM 5. Let (5) hold. Assume that equation (1) has property (A). Then every nonosdilatory solution u(t) of (1) satisfies lim 4 1 = 0 (20) t-oo r0(t) if and only if CO / / fn - l (* ) r n (0 |^ (*^0 l>( t ) ]^0[5(*) ] ) | dt = 00, for some c G K - {0} . The proof follows immediately from the following result, which can be found in [6]. 135 JOZEF DZURINA LEMMA 2. Let (5) hold. Let i, 0 < i < n — 1, be fixed. Equation (1) has a solution u(t) satisfying lim Diu(t) = a» G R - {0} , t—>oo /or some a$ i/ and only if oo / t f n - i - i ^ ^ ^ dt < oo . for some c G R - { 0 } . R e m a r k . The previous lemma provides effective sufficient and necessary condition for equation (1) not to have any solution u(t) of degree 0 satisfying lim Dou(t) — c / 0. On the other hand, for any nonoscillatory solution u(t) of t—KXD equation (1), condition (20) implies that lim D.iu(t) = 0 , for i = 0 ,1 , . . . , n - 1. (21) t—>oo Therefore, property (A) of equation (1) can be defined as above, or we can use the following definition. DEFINITION 1\ Equation (1) is said to have property (A) if for n even (1) is oscillatory and for n odd every nonoscillatory solution u(t) of (1) satisfies (21). R E F E R E N C E S [1] ERBE, L. : Oscillation criteria for second order nonlinear delay equations, Canad. Math . Bull. 16 (1973), 49-56. [2] ERBE, L. : Oscillation and asymptotic behavior of solutions of third order differential delay equations, SIAM J. Math . Anal. 7 (1976), 491-500. [3] HILLE, E . : Non-oscillation theorems, Trans. Amer. Math . Soc. 64 (1948), 234-258. [4] HARTMAN, P .—WINTNER, A. : Linear differential and difference equations with monotone solutions, Amer. J. Math . 75 (1953), 731-743. [5] KIGURADZE, I. T . : On the oscillation of solutions of the equation d m u/ dt171 + a(r)|u|nsignu = 0 (Russian), Mat. Sb. 65 (1964), 172-187. [6] KITAMURA, Y.—KUSANO, T . : Nonlinear oscillation of higher-order functional differ-ential equations with deviating arguments, J. Math . Anal. Appl. 77 (1980), 100-119. [7] KUSANO, T.—NAITO, M. : Oscillation criteria for general linear ordinary differential equations, Pacific J. Math . 92 (1981), 345-355. [8] KUSANO, T .—NAITO, M . : Comparison theorems for functional differential equations with deviating arguments, J. Math. Soc. Japan 3 (1981), 509-532. [9] MAHFOUD, W. E . : Comparison theorems for delay differential equations, Pacific J. Math. 83 (1979), 187-197. [10] TANAKA, K. : Asymptotic analysis of odd order ordinary differential equations, Hi-roshima Math. J. 10 (1980), 391-408. 136 PROPERTY (A) OF ADVANCED FUNCTIONAL DIFFERENTIAL EQUATIONS [11] T R E N C H , W . F . : Canonical forms and principal systems for general disconjugate equa-tions, Trans . Amer . Math . Soc. 189 (1974), 319-327 . Received May 13, 1992 Department of Mathematical Analysis Revised September 9, 1993 Šafárik University Jesennd 5 SK-041 54 Košice Slovakia 137 

Property (A) of advanced functional-differential equations

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Property (A) of advanced functional-differential equations

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Institute of Mathematics AS CR, v. v. i.