ne. J. Ma. Mah. Vo1. {1978)1-1 BEHAVOR OF SECOND ORDER NONLNEAR DFFERENTAL EQUATONS RNA LNG Deparmen of Mahemaics California Sae Universiy Los Angeles, California 93 (Received November 9, 1977 and in revised form March 31, 1978) ABSTRACT. Qualiaive behavior of second order nonlinear differenial equaions wih variable coefficiens is sudied. includes properies such as posllvly, number of zeroes, oscillaory behavior, boundedness and monoonicly of he soluions. i. NTRODUCTON. Second order nonlinear differenial equaions of he form y() + p()y() + q()yn() where n is an ineger >_, occur in many physical problems, such as he massspring sysems and saellie (see Ames [i], Mclachlan [] and Sruble [3]) and nuclear energy disribuion (see Canosa and Cole [4, 5]). n his work, qualiaive behavior of real-valued soluions of (i.i) is sudied. Wih cerain condiions on he coefficiens p() and q(), and n, properies such as poslivly, number of zeroes, boundedness and monoonlcly
RNA LNG are obained. would be assumed ha he coefficiens and heir derivaives are coninuous real-valued funcions on he inerval of ineres. The work is divided ino four pars; he firs par deals wih he case of p() < and q() <, he second par wih he case of p() < and q() > O, he hird par wih p() > and q() < and he fourh par wih p() > and q() > O. Papers in he pas, Skldmore [6], Abramovlch [7], Rankln [8], and Grimmer and Paula [9] have sudied behavior of second order linear differenial equaions. Nonllnear differenial equaions have been invesigaed in Chen [i] and Chen, Yeh and Yu [11], he former is on oscillaory behavior of bounded solulons and he laer on asympoic behavior of soluions. The resuls here are of a differen naure and are independen of heirs.. CASE OF p() < AND q() <. THEOREM.1. f (i) p() <, >, () q() < O, > and (3) n is odd, hen eiher y() > O, > or y() <, >. The graph is concave upward for y >, and concave downward for y <. PROOF. Equaion (i.i) can be wrien in he form + (p() + q()yn-l)y O. (.1) Le z() be real-valued and saisfy he linear equaion + p()z. (.) By a heorem in Harman [1, p. 346-347], z() has no zero. Since n i is even and q() <, we have p() + q()y n-1 < p() and by Surm Firs Comparison Theorem, z() has a leas a zero on (,=), if y has a zero on (,), a conradicion. The concaviy of he graph follows from (.i) wrien in he form
NONLNEAR DFFERENTAL EQUATONS 3 y n -p()y- q()y The case of n being even is considered in he following heorem. THEOREM.. f () p() <, >, () q() <, > and (3) n is even, hen y() < O, for all >. PROOF. Since q() < and n is even, we have + p()y + q()y n < + p()y, herefore < + p()y and by Bellman and Kalaba [13, p. 67], y() < for all >. CASE O p(c) < AND q(=) > O. THEOREM 3.1. f () p() <, >, () q() >, > and (3) n is even, hen y() >, for all >. PROOF. METHOD i. f in (.i), we le y() -z(), hen he equaion becomes -E- p()z + q()(-l)nzn--, since n is even,. + p()z q()z n and by Theorem., z() < for all >. Therefore y() > for all >. METHOD. Since q() > and n is even, we have y + p()y < y + p()y + q()y herefore + p()y < and by Bellman and Kalaba [13, p. 67], y() > for all >.
4 RNA LNG 4. CASE OF p() > AND q() <. f n is odd, he following wo heorems on he number of zeroes of he soluion can be obained. THEOREM 4.1. f (i) p() >, a < < b, () q() <, a < < b and (3) n is odd, hen a necessary condiion for y o have wo zeroes on (a, b] is ha a b p()d > 4 PROOF. Equaion (i.i) can be wrien in he form + (p() + q()yn-l)y. Le z() be real-valued and saisfy he linear equaion Since q() < and (n- i) is even, + p()z. p() + q()y n- < p() and by Hrman [1], z() has a leas wo zeroes on (a, b) if y() has wo zeroes on (a, b]. By Lyapunov Theorem in Harman [1, p. 346], a necessary condiion for z() o have wo zeroes on [a, b] is ha b p()d > 4 THEOREM 4.. f (i) p() >, < < T, () q() <, < < T, (3) n is odd, (4) y() has N zeroes on (,T], hen T N < (T p()d) + i.
NONLNEAR DFFTAL EQUATONS 5 PROOF. As in he proof of Theorem 4.1, i can be shown ha if z() has M zeroes on (,T), hen N <_ M. Bu by Harman [1, p. 346-347 ], T M < (T p()d + i and he conclusion follows. n he nex wo heorems, n is assumed o be even. THEOREM 4.3. f () p() >, >_, () q() <, >_ and (3) n is even, hen y() <, for >. PROOF. Since q() < and n is even, + p()y + q()y n < + p()y, herefore, < ; + p()y and by Bellman and Kalaba [13, p. 67], y() <_, for all >. f in addiion o he hypoheses of Theorem 4.3, we assume ha () >, hen y is negaive and monoonic increasing. THEOREM 4.4. f (i) () >, () p() >, >, (3) q() <, > and (4) n is even, hen y is mononic increasing. PROOF. negraion of (i.i) from o leads o () ()+ p(s)yds + q(s)yn ds. Since p > and y < by Theorem 4.3, q < and n is even, () () " r() () + p(s)yds + q(s)yn ds,
6 RNA LNG herefore, and so y is monoonic increasing. 5. CASZ p(=) > AD q() >. () () > O, () > (o) > o For p() >, q() > and n odd, he following heorems on he oscillaory behavior and boundedness of he soluions can be obained. THEOREM 5.1. (i) f p() >, >, () q() >, >, (3) n is odd and (4) z() is a real-valued soluion o + p() z, hen y() oscillaes more rapidly han z(), for >. PROOF. Equalon (.i) can be wrien in he form + (p() + q()yn-1)y O. Le z() be a real-valued soluion o E + p()z which has been widely discussed. Since q() > and (n i) is even, p() < p() + q()y n-1 and he conclusion follows from comparison heorems in Harman and Sanchez [1, 14]. THEOREM 5.. f (i) p() >, () >, >_, () q() >, () >, >, (3) n is odd and (4) y has successive exrema a l,, <, hen ly() < ly(l). (The ampliudes of oscillaions do no grow.) PROOF. The proof is by conradicion, so assume ha ly(l) < [Y() Muliplicaion of (i.i) by leads o + p() + q()yn.
NONLNEAR DFFERENTAL EQUATONS 7 negraing (5.1) from o, we ge P() d + q()yn d. Since and p() a ;g (p()y() P(i)y(i) ()y 1 d) q()yn d i yn+ 1 yn+ [ yn+ n + i (q() () q() () () dr), (5.) becomes P()Y() P(l)Y(l) + yn+ n + 1 q() () )yn+l n + q(l (l) ()y d + n +1 ()yn+1 d < y()(p( ) P(l)) + n+l n + i y ()(q() q(l)) herefore (yn+ yn+l () P(l)(y() y(l)) < n + i q(l) (l) <, since q(l) > and (n + i) is even, which is a conradicion since he lef hand side is posiive. THEOREM 5.3. f (i) p() > i, () <, >, () q() >, () < and (3) n is odd, hen y() is bounded for all.
8 RNA LNG PROOF. Muliplicaion of (1.1) by and inegraion of he resuling equaion from o lead o #() + p() y() where C is a consan, herefore (s)y ds + yn+l (s)yn+lds C, n+ q() () n+l yn- y()(p() + q() ()) (s)y ds Since p > i, q >, n is odd, < and <, i follows ha and so y is bounded. y() < C, for all THEOREM 5.4. f () p() >, () >, >, () q() >, () <, >, and (3) n is odd, hen y() is bounded for all. PROOF. As in Theorem 5.3, equaion (i.) is muliplied by and he resuling equaion inegraed from o, giving, () + P()y() + n--- )yn+ q( () n+l Since q >, (n + i) is even and <_, p()y() < C + (s)y ds ds =C+ (s)y ds. so p()y() _< [C[ + P(s)Y (S)p(s) ds
NONLNEAR DFFERENTAL EQUATONS 9 and by Gronwall s inequaliy, Harman [1, p. 4 herefore p()y() _< C exp C p() p() p(s)(s) ds y() < p for all. and q" n he nex heorem, (i.i) is assumed o have consan coefficiens P THEOREM 5.5. f (i) P >, () q > and (3) n is odd, hen y is oscillaory. PROOF. The equaion is 9 + poy + q yn. By linearizaion in McLachlan [1, p. 16], 9+ay=, where where A is a consan. i a (P A sin e + q An sin n e) sine de, Since P >, q > and n is odd, he inegrand in is posiive, so a >. i a --w (P sin e + q A n- s inn+! e)de
i RNA LNG By Theorem 5.4, y is bounded. The fac ha ld=m and y is bounded imply ha y is oscillaory, see Harman [1, p. 354]. n he nex wo heorems, n is assumed o be even. THEOREM 5.6. f (i) p() >, >, () q() >, > and (3) n is even, hen y() >, for >. PROOF. The equaion is + p()y + q()y n. Le y() -z(), hen -. p()z + (-l)nq() z n. Since n is even, + p()z q()z n and by Theorem 4.3, z <. Therefore y() > for >. f in addiion o he hypoheses of Theorem 5.6, we assume ha () <, hen y is posiive and monoonic decreasing. THEORE.7. Tf () ()!, () p() >, >_, (3) q() >, and (4) n is even, hen y is monoonic decreasing. PROOF. negraion of (i.) from o leads o ()- ()+ f p(s)y ds + q(s)n ds " Since p > and y > by Theorem 5.6, q > and n is even,
NONLNEAR DFFERENTAL EQUATONS 11 () () < O, so () < ;(o) < o and y is monoonic decreasing. REFERENCES i. Ames, W. F. Nonlinear. Ordln.ary Differenial Equal.o.ns in Tra.nsp.or Processes., Academic Press, New York and London, 196 8.. McLachlan, N. W. Ordinary Non-Linear Differenial Equaion.s. in Engin.ee.ring and Physlcal Sciences, nd ed., Oxford Universiy Press, London and New York, 1958. 3. Sruble, R. A. Nonlinear Differenial Equaions, McGraw-Hill Company, New York and London, 196. 4. Canosa, J. and J. Cole. Asympoic Behavior of Cerain Nonlinear Boundary- Value Problems, J. Mah. Phys. 9 (1968) 1915-191. 5. Canosa, J. Expansion Mehod for Nonlinear Boundary-Value Problems, J. Mah. Phys. 8 (1967) 18-183. 6. Skidmore, A. On he Differenlal Equaion y" + p(x)y f(x), J. Mah. Anal. App1. 43 (1973) 46-55. 7. Abramovich, S. On he Behavior of he Soluions of y" + p(x)y f(xi, J. Mah. Anal. Appl. 5 (1975) 465-47. 8. Rankin, S. M. Oscillaion Theorems for Second-Order Nonhomogeneous Linear Differenial Equaions, J. Mah. Anal. Appl. 53 (1976) 55-553. 9. Grimmer, R. C. and W. T. Paula. Nonoscillaory Soluions of Forced Second- Order Linear Equaions, J. Mah. Anal. Appl. 56 (1976) 45-459. i. Chen, L. S. Some Oscillaion Theorem for Differenial Equaions wih Funcional Argumens, J. Mah. Anal. Appl. 58 (1977) 83-87. ii. Chen, M. P., C. C. Yeh and C. S. Yu. Asympoic Behavior of Nonoscillaory Soluions of Nonlinear Differenial Equaions wih Rearded Argumens, J. Mah. Anal. Appl. 59 (1977) 11-15. 1. Harman, P. Ordinary Differenial Equaions, John Wiley and Sons, New York and London, 1964.
1 RNA LNG 13. Bellman, R. E. and R. E. Kalaba. uasilinearizalon and Nonlinear Boundary- Value Problems, American Elsevler Publishing Company, New York, 1965. 14. Sanchez, D. A. Ordinary Differenial Equaions and Sabiliy Theory, W. H. Freeman and Company, San Francisco and London, 1968. KEF WORDS AND PHRASES. Nonlin differenial equaion, osry and ympoie behavior of solons, boundedn and onooniy of olons. AMS(MOS) SUBJECT CLASSFCATONS (197). 34C1, 34C15, 34D5.