QUASILINEARIZATION METHOD



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Journal of Applied Mathematics Stochastic Analysis 7, Number 4, 1994, 545-552. FURTHER GENERALIZATION OF GENERALIZED QUASILINEARIZATION METHOD V. LAKSHMIKANTHAM N. SHAHZAD 1 Florida Institute of Technology Program of Applied Mathematics Melbourne, FL 32901 USA (Received June, 1994; revised August, 1994) ABSTRACT The question whether it is possible to develop monotone sequences that converge to the solution quadratically when the function involved in the initial value problem admits a decomposition into a sum of two functions, is answered positively. This extends the method of generalized quasilinearization to a large class. Key words: Quasilinearization Method, Monotone Sequence, Quadratic Convergence. AMS (MOS) subject classifications: 34A34, 34A40. 1. Introduction It is well known [1, 2] that the method of quasilinearization offers an approach for obtaining approximate solutions to nonlinear differential equations. Consider the IVP x f(t,x), x(o) x o on J [0, T]. If f(t,x) is uniformly convex in x for all t E [0, T], then the method of quasilinearization provides a monotone increasing sequence of approximate solutions that converges quadratically to the unique solution. Moreover, the sequence provides lower bounds for the solution. Recently, the method of quasilinearization has received much attention after the publication of [9]. there has been a lot Since then, of activity in this area several interesting results have appeared (see, for example, [4, 5, 6, 7, 8, 10, 11]).. In this paper, we show that it is possible to develop monotone sequences that converge to the solution quadratically when f admits a decomposition into a sum of two functions F G with F + concave G + convex for some concave function for some convex function Theorem 2.1 extends Theorem 3.1 in [7] in the setup of [6]. However, we follow the direct approach discovered in [5] rather than the complicated multistage algorithmic method reported in [6]. We do not consider the corresponding results given in [7] that are generated by assuming various coupled upper lower solutions with suitable extra assumption to avoid monotony. 1permanent address: Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan. Printed in the U.S.A. @1994 by North Atlantic Science Publishing Company 545

546 V. LAKSHMIKANTHAM N. SHAHZAD Clearly, based on our results those of [7], it is not difficult to construct the proofs of other possible combinations. 2. Main Result Consider the initial value problem (IVP) x f(t,x), x(o) xo, t J [O,T] (2.1) where f E C[J x R, N]. Let a0, 30 E C [J, R] such that a 0 _</0 on J. We define a set f {(t,x):ao(t <_ x <_ o(t),t Theorem 2.1: Assume that A1) A) ao, o C [J,] such that a o <_ f(t, ao) o >- f(t,o) a o <_/o on J; f C[,],, f admits a decomposition f F +G where Fx, Gx, F z x, Gxx exist are continuous satisfying Fxx(t x) + Cxx(t, x) <_ 0 Gxx(t x) + Cxx(t, x) >_ 0 on f], where C[f,], Cx(t, x), Cx(t, x), Cxx(t, x), Cxx(t, x) exist, are continuous Cxx(t,x) < O, Cxx(t,x) > 0 on f. Then there exist monotone sequences {an(t)} solution of (2.1) the convergence is quadratic. Proof: In view of (A2) we see that {n(t)} which converge uniformly to the unique F(t,y) <_ H(t,x)- Hx(t,x)(x- y)- (t, y) G(t,x) >_ M(t,y) + Mx(t,y)(x- y)- (t, x) for x >_ y, x,y G f, where H(t,x) F(t,x) + (t,x) M(t,x) G(t,x) + (t,x). clear that whenever ao(t _< x 2 _< x _</o(t) (2.2) Also, it is Let al, fll be the solutions of linear IVPs f(t, xl) f(t, x2) <_ L(x I x2) for some L ) 0. (2.3) c i f(t,co)+ [Mx(t, a0) + Hx(t,O) Cx(t, Co)- Cx(t,/3o)](c1 -ao), O1(0 X0, (2.4) /31 f(t,/3o) + [Mx(t, ao) + Hx(t, 13o)- Cx(t, ao)-x(t, t3o)](131 /o), /1(0) X0 (2.5) where ao(0) _< x o _</3o(0). We shall prove that ao<_a 1 onj. To do this, let p-ao-casothat p(0)_<0. Then 0 1 <_ f(t, ao) If(t, ao) + {Mx(t, ao) + Hx(t, o) Cx(t, ao) Cx( t,/0)}(ctl Ct0)] [M x(t, ao) + H x(t, 0) x(t, c0) x(t, 0)]P"

Further Generalization of Generalized Quasilinearization Method 547 Since p(0) _< 0, we get Oo(t _< Ctl(t on J. Now Next we let p- ct I flo note that p(o) _< O. <_ f(t, Co) + [Mz(t, Co) + Hx(t, o) (t, Co) Cx(t,/0)](Ctl Ct0) f(t, fl0)" But since/3 o > Co, using (2.2) (A2) we have r(t, o) < H(t, flo) H(t, flo)(flo o) (t, o) G(t,/3o) >_ M(t, o) + M(t, Oo)(fl o Oo) (t, flo), or F(t, Co) <_ F(t,/3o) Hx(t, flo)(flo Co) + [O(t,/30) (t, Cto) G(t, o) >- G(t, Co) + Mx(t, Co)(fl o Co) [(t, flo) (t, Oo) ]. Now, by using the mean value theorem, (t, flo) (t, o) (t, )(flo o) 6(t, #o) 6(t, o) 6(t, 6)(#o o), where c o <, 5 </3o. Because Cx(t,x) is decreasing in x Cz(t,x) is increasing in x, it follows that (t, flo) (t, o) -< (t, o)(o o) (t, o) (t, o) -< (t, flo)(flo o). Thus we get _ F(t, Co) <_ F(t,/30) [Hx(t,/30) (t, ao)](flo Oo) G(t, /30) >_ G(t, Co) + [Mx(t, Co) Cx(t,/30)](/30 Co) which in turn yields P < [Mx(t, Co) + Hx(t, o) Cx( t, Co) Cx(t,/o)]P" Consequently, p(t) <_ 0 on J proving c /0 on Y.

548 V. LAKSHMIKANTHAM N. SHAHZAD As a result, we have s0(t Sl(t /0(t) on J. Similarly, we can show that ao(t <_/31(t _< flo(t) on J. We need to prove that al(t _</31(t on J so that it follows Since s 0 < S 1 < fl0, using (2.2) (2.4), we see that so(t _< Sl(t _< ill(t) <_ flo(t) on J. (2.6) s f(t, So) + [Mx(t So) + Hx(t,/30) (t, So) Cx(t, #0)](Sl S0) [F(t, So) + Hx(t, 3o)(S So) + [G(t, So) + Mx(t, S0)(S 1 S0) Cx(t, S0)(S 1 - S0) Cx(t,/0)(Sl S0) <_ if(t, So) + Hx(t, s1)(s I S0) [G(t, So) M(t, S0)(S 1 S0) Cx(t, S0)(Sl S0) Cx(t,/0)(Sl S0) <_ [F(t, Sl) (t, Sl) (t, S0) [a(t, Sl) (t, Sl) (t, S0) Cx(t, S0)(S 1 S0) Cx(t,/0)(Sl S0) <_ f(t, Sl) Cx(t, So)(S 1 s0) -t- Cx(t, flo)(sl So) )x(t, So)(S 1 s0) Cx(t, flo)(sl So) f(t, sl). Here we have used the mean value theorem the facts that Hz(t,x), x(t,x) are decreasing in x x(t,x) is increasing in x. Similarly, we can prove that /3] > f(t,31) therefore by Theorem 1.1.1 [3], it follows that sl(t </31(t on J which shows that (2.6) is valid. Assume now that for some k >1, s -< f(t, sk), k > f(t,k) sk(t _< k(t) on J, we shall show that where s k + /3 k + Sk(t) Sk + 1(t) k -t- (t) flk (t) on J, (2.7) are the solutions of linear IVP s Sk + f(t, sk) + [Mx(t sk) + Hx(t k) (t, sk) (t,/3k)](s k + 1 Sk) Sk + 1(0) X0, (2.8) k + f(t,) + [Mx(t, sk) + Hx(t, k) x( t, sk) Cx( t,/3k)](/3k + /k)

Further Generalization of Generalized Quasilinearization Method 549 Hence setting p- a k -a k + 1, it follows as before that p <_ [Mx(t ak) + Hz(t, ilk) Cz( t, ak) z(t, flk)]p on J p(0)- 0 which again implies p(t)<_ 0 on J. On the other h, letting p- ck+a-flk yields as before p <_ [ix(t, ck) + Ux(t, ilk) x(t, ck) Cx(t, /)]P" This proves that p(t) <_ O, since p(0) 0 therefore we have c k _< c k + -< k on J. In a similar manner, we can prove that k fl k -t-1 fl k on J. Now using (2.2), (2.8) the fact flk > ck +1 we get 0 q_ 1 f(t, ak) + [Mx(t, ak) + Hz(t, k) z(t, Ck) Cx(t, flk)](ck + Ok) [F(t, ak) + Hx(t, k)(ak + 1 Ck)] -+- [G(t, Ok) q- Mx(t, ak)(a k + 1 Ok)] Cx( t Ok)(Ok q- Ck Ok)- Cx( t flk)(ok + 1 Ok)] <- [F(t, ak) + Hx(t, ak + )(ak + 1 Ok)] -{- [G(t k) q- Mx(t, ak)(ak + Ok)] (t, Ok)(Oe k + Ok)- Cx( t k)(ok + Ok) [F(t ok q- 1) q- (t Ok q- 1)- (t Ok)] -+- [G(t k q- 1) + (t Ok q- 1) )x( ;t Ok)(Ok + Ok)- Cx( t k)(ok + Ok) f(t,ok + 1). Here we have used the mean value theorem the facts that Hx(t,x), Cx(t,x) are decreasing in x Cx(t,x) is increasing in x. Similar arguments yield k+l >f(t I hence Theorem 1.1 1 [3] shows that +1 + 1(t) -< flk + 1(t) on J which proves (2.7). Hence we obtain by induction 00 OZl Ct2 < OZn fin f12 fl fl O on d. Now using stard arguments, it is easy to show that the sequences {an(t)}, {fin(t)} converge uniformly monotonically to the unique solution x(t) of (2.1) on J. Finally, to prove quadratic convergence, we consider Pn + l( t)- x(t) n + l( t)-> O, qn + l( t)- n+l(t)-x(t) >_ O. Note that Pn + 1(0) 0 qn + 1(0)" We then have p +lx Ctn+ f(t, x) If(t, an) + {Mx(t, an) + Hx(t n) Cx( t an) Cx( t fln)](cn + (2.9)

_ 550 V. LAKSHMIKANTHAM N. SHAHZAD {H(t,x)- H(t, an) } + {M(t,x)- M(t, an) } {(t, x)- (t, an) } {(t, x) (t, an) } {Mx(t, an) + Hx(t, n) Cx( t an) Cx( t, n)}(an + an)] Hx(t, an)(x an) + Mx(t,x)(x an) Cx(t,x)(x an) _ {Mx(t an) + Hx(t, fin) Cx(t, an) Cx(t, fln)j(an + 1 X X an) Hzz(t, )(a n n)(x an) + Mxx(t, 5)(x an) 2 Cxx(t, 7)(x a.) 2 + Cxx(t, O)(n an)(x an) + {Mx(t, an) + Hx(t n) x(t, an) Cx(t, n)}pn + 1 {xx(t, O) Hxx(t,,)}Pn(qn + Pn) + {Mxx( t 5) Cxx(t, 7)}p2n + {M x(t, an) + Hx(t n) x(t, an) Cx(t, fin)} Pn + {xx(t, O)- Hxx(t, )}Pn(qn + Pn) + {Mxx( t, 5) Cxx(t, 7)}p2n -t- {ax(t, an) Fx(t, n)}pn -l- 1, wherean<(, 0<flnan<5, 7<x. Hence, we obtain where on Pn + 1 " (A + B + C)Pn(qn + Pn) + (A + C + D)p2n + (E + K)p n + 1 _< (A + B + C)(2P 2 n + q2n) + (A + C + D)p2n + (E + K)p n + 1 Qp2n + Rq2n + SPn + 1 I(t,x) < A, Ir(t,x) < B, I%x(t,x) < C, Ia(t,x) < D, la(t,x) <, IF(t,x) <K, Q-3A+2B+aC+D, R-A+B+C S-E+K. (2.10) Now Gronwall s inequality implies which yield the desired result Similarly, 0 _< Pn + 1(t) < / es(t S)[Qpn(S)+2 Rq2u(s)]ds on g 0 maxlx(t)- (t) < Qmxlx(t)-.(t)12/-Rmxl3.(tD-x(t)l 2 j +1 - q x n+l fln+l f(t,n)+[mx(t, an)+ Hx(t,n)-x(t, an)-x(t,n)](fln+ 1n)f(t, x)

Further Generalization of Generalized Quasilinearization Method 551 {H(t,n)- H(t,x)} + {M(t,n)- M(t,x)} + {(t, x)- (t, n) } + {(t, x) (t, fin)} + {Mz(t an) + Hx(t, n) Cx( t, an) Cx(t, fn)}(fln +,) <_ Hx(t,x)(n- x) + Mx(t, fln)( n x)- Cx(t,n)(n- X)x(t,x)(n X) + {Mx(t an) + Hx(t,/n) Cx( t, an) Cx( t,/n)}(/n + 1 x x <_ {xx(t, O) Hxx(t t)}(n x) 2 + {Mxx(t, 5) Cxx(t, ") )}(fin an)(n x) + {Gx(t, an) + Fx(t, n)}(n + 1 x) {xx(t, O) Hxz(t t)}q2n + {Mxx(t, 5) Czx(t, 7)}(qn + Pn)qn + {Gx(t, an) + rx(t, tn)}qn + 1 where a n <, O < fln a n < 5, 7 < x. Hence, we obtain q + <_ (A + B + C)q2n(A + C + D)q,(q n + p,) + (E + K)q n + 1 <_ (A + B + C)q2n + (A + C + D)(2qn + Pn) + (E + K)q n +1 2 -Q*q2n+R pn+sqn+ where the constants A, B, C, D, E, K S are as in (2.10) Q*-3A+B+3C+2D, R*-A+C+D. An application of Gronwall s inequality yields hence O <_ qn + l( t)-< es(t S) Q,q2n(s) + R pn(s) ds on J 0 maxj /n + l(t)- x(t) _< e-q*rnx.(t)- x(t) + R*maxj Ix(t) This completes the proof of the theorem. References [3] [4] [5] Bellman, R., Methods of Nonlinear Analysis Vol. II, Academic Press, New York, 1973. Bellman, R. Kalaba, R., Quasilinearization Nonlinear Boundary Value Problems, American Elsevier, New York, 1965. Ladde, G.S., Lakshmikantham, V., Vatsala, A.S., Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985. Lakshmikantham, V., An extension of the method of quasilinearization, (to appear in J.O.T.A.). Lakshmikantham, V., Further improvement to generalized quasilinearization method, (to appear in Nonlinear Analysis).

552 V. LAKSHMIKANTHAM N. SHAHZAD [6] [7] [8] [9] [10] [11] Lakshmikantham, V., Leela, S. McRae, F.A., Improved generalized quasilinearization method, (to appear in Nonlinear Analysis). Lakshmikantham, V., Leela, S. Sivasundaram, S., Extensions of the method of quasilinearization, (to appear in J.O.T.A.). Lakshmikantham, V. K6ksal, S., Another extension of the method of quasilinearization, (to appear in the Proceedings of DynamicSystems Applications). Lakshmikantham, V. Malek, S., Generalized quasilinearization, Nonlinear World 1 (1994), 59-63. Malek, S. Vatsala, A.S., Method of generalized quasilinearization for second order boundary value problem, Inequalities Applications 3 (1994), (to appear). McRae, F.A., Generalized quasilinearization of stochastic initial value problems, Stochastic Analysis Applications 13 (1994), (to appear).

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