- Email: [email protected]
Generalized quasilinearization method for mixed boundary value problems
Applied Mathematics and Computation 133 (2002) 423–429 www.elsevier.com/locate/amc
Generalized quasilinearization method for mixed boundary value problems Bashir Ahmad a, Juan J. Nieto b, Naseer Shahzad a
c
c,*
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan b Departmento de Analisis Matematico, Facultad de Matematicas, Universidad de Santiago de Compostela, Spain Department of Mathematics, Faculty of Science, King Abdul Aziz University, P.O. Box 9028, Jeddah 21413, Saudi Arabia
Abstract A generalized quasilinearization method for a nonlinear mixed BVP is developed and a sequence of approximate solutions converging monotonically and quadratically to a solution of the given problem is presented. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Quasilinearization; Mixed BVP; Quadratic convergence
1. Introduction It is well known that the method of quasilinearization [1] provides an excellent tool for obtaining approximate solutions of nonlinear differential equations. This technique works fruitfully only for the problems involving convex/concave functions and gives the sequence of approximate solutions converging monotonically and quadratically to the solution. Recently, the convexity assumption was relaxed and the method was generalized and extended in various directions to make it applicable to a large class of problems [2,4–8]. Afterwards, Shahzad and Vatsala [12,13] and Shahzad and Sivasundaram [11] discussed the generalized quasilinearization method for second
*
Corresponding author. E-mail address: [email protected] (N. Shahzad).
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 2 4 3 - 0
424
B. Ahmad et al. / Appl. Math. Comput. 133 (2002) 423–429
order boundary value problems. More recently, Nieto [10] developed a generalized quasilinearization method for a nonlinear Dirichlet problem to obtain a sequence of approximate solutions that converges quadratically to the solution of the given problem. For a complete survey of the generalized quasilinearization method, see [9]. The aim of this paper is to consider a second order ordinary nonlinear differential equation with mixed boundary conditions and develop the method of quasilinearization for this problem without requiring the function involved to be convex/concave.
2. Preliminaries It is well known that the following mixed BVP w00 ðtÞ ¼ kwðtÞ; t 2 ½0; p; wð0Þ ¼ w0 ðpÞ ¼ 0; 2
has a nontrivial solution if and only if k ¼ ½ð2m 1Þ=2 ðm ¼ 1; 2; 3 . . .Þ. For 2 k 6¼ ½ð2m 1Þ=2 ðm ¼ 1; 2; 3 . . .Þ and ,ðtÞ 2 C½0; p, the unique solution of the problem w00 ðtÞ kwðtÞ ¼ ,ðtÞ;
t 2 ½0; p;
wð0Þ ¼ w0 ðpÞ ¼ 0; is wðtÞ ¼
Z
p
Gk ðt; tÞ,ðtÞ dt;
0
where hpffiffiffi i hpffiffiffi i 8 > cos k ðp tÞ sin kt ; > > < 1 0h6 t 6 t 6 ph i ðk > 0Þ pffiffiffi Gk ðt; tÞ ¼ pffiffiffi pffiffiffi i pffiffiffi > sin k t cos k ðp tÞ ; > k cos kp > : 06t6t6p t; 0 6 t 6 t 6 p G0 ðt; tÞ ¼ ðk ¼ 0Þ t; 0 6 t 6 t 6 p hpffiffiffiffiffiffiffi i hpffiffiffiffiffiffiffi i 8 > cosh k ðp tÞ sinh kt ; > > < 1 0 6 t 6 t 6 p hpffiffiffiffiffiffiffi i hpffiffiffiffiffiffiffi i ðk < 0Þ: pffiffiffiffiffiffiffi Gk ðt; tÞ ¼ pffiffiffiffiffiffiffi sinh k t cosh k ðp tÞ ; > k cosh kp > > : 06t6t6p
B. Ahmad et al. / Appl. Math. Comput. 133 (2002) 423–429
425
Let us consider the following nonlinear BVP x00 ðtÞ ¼ f ðt; xðtÞÞ;
t 2 ½0; p;
0
xð0Þ ¼ x ðpÞ ¼ 0;
ð2:1Þ
where f : ½0; p R ! R is a continuous real valued function. A function a 2 C 2 ½½0; p; R is a lower solution of (2.1) if a00 ðtÞ 6 f ðt; aðtÞÞ;
t 2 ½0; p;
að0Þ 6 0; a0 ðpÞ 6 0; and b 2 C 2 ½½0; p; R is an upper solution of (2.1) if b00 ðtÞ P f ðt; bðtÞÞ;
t 2 ½0; p;
0
bð0Þ P 0; b ðpÞ P 0: The following lemma plays a crucial role in the sequel and we sketch its proof for the sake of completeness. Lemma 2.1. Assume that a; b 2 C 2 ½½0; p; R are lower and upper solutions of (2.1), respectively, such that aðtÞ 6 bðtÞ for every t 2 ½0; p. Then there exists a solution xðtÞ of (2.1) such that aðtÞ 6 xðtÞ 6 bðtÞ for t 2 ½0; p. Proof. Let p : ½0; p R ! R be a mapping defined by pðt; xÞ ¼ maxfaðtÞ; minfx; bðtÞgg: Then, extend f ðt; xÞ to ½0; p R by setting F ðt; xÞ ¼ f ðt; pðt; xÞÞ: Observe that F ðt; xÞ is bounded. Let us consider the modified BVP x00 ðtÞ ¼ F ðt; xðtÞÞ; 0
xð0Þ ¼ x ðpÞ ¼ 0;
t 2 ½0; p;
ð2:2Þ
which is solvable, that is (2.2) has a solution x on ½0; p. Using the arguments similar to those of [3,10], it can be shown that a6x6b
on ½0; p:
Finally, since any solution of (2.2) is also a solution of (2.1), it follows that x is a solution of (2.1). This completes the proof.
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B. Ahmad et al. / Appl. Math. Comput. 133 (2002) 423–429
3. Main result Theorem 3.1. Assume that ðA1 Þ ao ; bo 2 C 2 ½½0; p; R are lower and upper solutions of (2.1), respectively, such that ao 6 bo on ½0; p, ðA2 Þ f 2 C½X; R is such that fx ðt; xÞ; fxx ðt; xÞ exist and are continuous for every ðt; xÞ 2 X, where X ¼ fðt; xÞ 2 ½0; p R : ao ðtÞ 6 x 6 bo ðtÞg; ðA3 Þ fx ðt; xÞ < 0 for every ðt; xÞ 2 X. Then there exists monotone nondecreasing sequence fan g which converges uniformly to a solution of (2.1) and the convergence is quadratic. Proof. Set Uðt; xÞ ¼ F ðt; xÞ f ðt; xÞ on ½0; p R; where F : ½0; p R ! R is such that F ðt; xÞ; Fx ðt; xÞ; Fxx ðt; xÞ are continuous on ½0; p R and Fxx ðt; xÞ P 0;
ðt; xÞ 2 ½0; p R:
ð3:1Þ
It further implies that F ðt; xÞ P F ðt; yÞ þ Fx ðt; yÞðx yÞ for x P y and therefore f ðt; xÞ P f ðt; yÞ þ Fx ðt; yÞðx yÞ ½Uðt; xÞ Uðt; yÞ:
ð3:2Þ
Now, consider the mixed BVP u00 ¼ gðt; u; ao Þ ¼ f ðt; ao Þ þ Fx ðt; ao Þðu ao Þ ½Uðt; uÞ Uðt; ao Þ; uð0Þ ¼ u0 ðpÞ ¼ 0: ð3:3Þ The inequality (3.2) together with ðA1 Þ imply a00o 6 f ðt; ao Þ ¼ gðt; ao ; ao Þ; b00o P f ðt; bo Þ P f ðt; ao Þ þ Fx ðt; ao Þðbo ao Þ ½Uðt; bo Þ Uðt; ao Þ ¼ gðt; bo ; ao Þ: In view of Lemma 2.1, there exists a solution a1 of (3.3) such that ao 6 a1 6 bo on ½0; p. Next, we consider the mixed BVP u00 ¼ gðt; u; a1 Þ; uð0Þ ¼ u0 ðpÞ ¼ 0:
ð3:4Þ
B. Ahmad et al. / Appl. Math. Comput. 133 (2002) 423–429
427
The inequality (3.2) yields a001 ¼ gðt; a1 ; ao Þ; ¼ f ðt; ao Þ þ Fx ðt; ao Þða1 ao Þ ½Uðt; a1 Þ Uðt; ao Þ 6 f ðt; a1 Þ ¼ gðt; a1 ; a1 Þ; b00o P f ðt; bo Þ P f ðt; a1 Þ þ Fx ðt; a1 Þðbo a1 Þ ½Uðt; bo Þ Uðt; a1 Þ ¼ gðt; bo ; a1 Þ; It follows again by Lemma 2.1 that there exists a solution a2 such that a1 6 a2 6 bo on ½0; p. Thus, ao 6 a1 6 a2 6 bo on ½0; p. Employing the same arguments successively, we conclude ao 6 a1 6 a2 . . . 6 an 6 b o
on ½0; p;
where the elements of the monotone sequence fan g are the solutions of the mixed BVP u00 ¼ gðt; u; an1 Þ ¼ f ðt; an1 Þ þ Fx ðt; an1 Þðu an1 Þ ½Uðt; uÞ Uðt; an1 Þ; uð0Þ ¼ u0 ðpÞ ¼ 0: The monotonicity of the sequence fan g ensures the existence of its (pointwise) limit x. Let us consider the linear mixed BVP u00 ¼ fn ðtÞ;
ð3:5Þ
uð0Þ ¼ u0 ðpÞ ¼ 0; where fn ðtÞ ¼ gðt; an ðtÞ; an1 ðtÞÞ;
t 2 ½0; p:
The continuity of g on X implies that the sequence ffn g is bounded in C½½0; p; R and so lim fn ðtÞ ¼ f ðt; xðtÞÞ;
n!1
t 2 ½0; p:
Here an ðtÞ ¼
Z
p
G0 ðt; sÞfn ðsÞ ds;
0
where an ðtÞ is a solution of (3.5). Thus fan g is bounded in C 2 ½½0; p; R and fan g " x uniformly on ½0; p. Consequently, Z p xðtÞ ¼ G0 ðt; sÞf ðs; xðsÞÞ ds; t 2 ½0; p: 0
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B. Ahmad et al. / Appl. Math. Comput. 133 (2002) 423–429
Hence x is a solution of (2.1). For quadratic convergence, we set pn ðtÞ ¼ xðtÞ an ðtÞ. Using the mean value theorem repeatedly, we obtain pn00 ðtÞ ¼ f ðt; xðtÞÞ gðt; an ðtÞ; an1 ðtÞÞ ¼ f ðt; xðtÞÞ f ðt; an1 ðtÞÞ Fx ðt; an1 ðtÞÞ½an ðtÞ an1 ðtÞ þ ½Uðt; an ðtÞÞ Uðt; an1 ðtÞÞ ¼ F ðt; xðtÞÞ F ðt; an1 ðtÞÞ Fx ðt; an1 ðtÞÞ½an ðtÞ an1 ðtÞ þ ½Uðt; an ðtÞÞ Uðt; xðtÞÞ ¼ Fx ðt; nÞ½xðtÞ an1 ðtÞ Fx ðt; an1 ðtÞÞ½an ðtÞ an1 ðtÞ þ ½Uðt; an ðtÞÞ Uðt; xðtÞÞ ¼ ðFx ðt; nÞ Fx ðt; an1 ðtÞÞÞ½xðtÞ an1 ðtÞ þ Fx ðt; an1 ðtÞÞ ½xðtÞ an ðtÞ þ ½Uðt; an ðtÞÞ Uðt; xðtÞÞ ¼ Fxx ðt; rÞ½n an1 ðtÞ½xðtÞ an1 ðtÞ þ Fx ðt; an1 ðtÞÞ½xðtÞ an ðtÞ Ux ðt; gÞ½xðtÞ an ðtÞ; where an1 ðtÞ 6 n 6 r 6 xðtÞ and an ðtÞ 6 g 6 xðtÞ: Take hn ðtÞ ¼ Fx ðt; an1 ðtÞÞ Ux ðt; gÞ; and 2 ðtÞ; kn ðtÞ ¼ Fxx ðt; rÞ½n an1 ðtÞ½xðtÞ an1 ðtÞ Mpn1
where 0 6 Fxx ðt; yÞ 6 M; ðt; yÞ 2 X. Clearly kn ðtÞ 6 0. Since Fx is nondecreasing and an1 ðtÞ 6 g, it follows by ðA3 Þ that there exists k < 0 and an integer N such that hn ðtÞ 6 k, t 2 ½0; p for n P N . Therefore, the error pn satisfies the mixed BVP 2 ðtÞ þ kn ðtÞ; pn00 ðtÞ kpn ðtÞ ¼ ½hn ðtÞ kpn ðtÞ þ Mpn1 0 pn ð0Þ ¼ pn ðpÞ ¼ 0:
Thus, we can write Z p
2 pn ðtÞ ¼ Gk ðt; sÞ ½hn ðsÞ kpn ðsÞ þ Mpn1 ðsÞ þ kn ðsÞ ds; 0
which gives pn ðtÞ 6 M
Z 0
p 2 Gk ðt; sÞpn1 ðsÞ ds;
n P N:
B. Ahmad et al. / Appl. Math. Comput. 133 (2002) 423–429
429
Hence, there exists a constant d > 0 such that 2
kpn k 6 dkpn1 k ;
n P N;
where kxk ¼ maxfjxðtÞj: t 2 ½0; pg.
References [1] R. Bellman, R. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, Elsevier, New York, 1965. [2] P.W. Eloe, Y. Zhang, A quadratic monotone iteration scheme for two-point boundary value problems for ordinary differential equations, Nonlinear Anal. 33 (1988) 443–453. [3] G.S. Ladde, V. Lakshmikantham, A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, MA, 1985. [4] V. Lakshmikantham, An extension of the method of quasilinearization, J. Optim. Theory Appl. 82 (1994) 315–321. [5] V. Lakshmikantham, Further improvement of generalized quasilinearization, Nonlinear Anal. 27 (1996) 223–227. [6] V. Lakshmikantham, S. Leela, F.A. McRae, Improved generalized quasilinearization method, Nonlinear Anal. 24 (1995) 1627–1637. [7] V. Lakshmikantham, N. Shahzad, Further generalization of generalized quasilinearization method, J. Appl. Math. Stochastics Anal. 7 (1994) 545–552. [8] V. Lakshmikantham, N. Shahzad, J.J. Nieto, Methods of generalized quasilinearization for periodic boundary value problems, Nonlinear Anal. 27 (1996) 143–151. [9] V. Lakshmikantham, A.S. Vatsala, Generalized Quasilinearization for Nonlinear Problems, Kluwer Academic Publishers, Boston, 1998. [10] J.J. Nieto, Generalized quasilinearization method for a second order ordinary differential equation with Dirichlet boundary conditions, Proc. Amer. Math. Soc. 125 (1997) 2599–2604. [11] N. Shahzad, S. Sivasundram, Extended quasilinearization method for boundary value problems, Nonlinear World 2 (1995) 311–319. [12] N. Shahzad, A.S. Vatsala, Extension of the method of generalized quasilinearization for second order boundary value problems, Appl. Anal. 58 (1995) 77–83. [13] N. Shahzad, A.S. Vatsala, Improved generalized quasilinearization method for second order boundary value problems, Dyn. Syst. Appl. 4 (1995) 79–85.
Generalized quasilinearization method for mixed boundary value problems Bashir Ahmad a, Juan J. Nieto b, Naseer Shahzad a
c
c,*
Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan b Departmento de Analisis Matematico, Facultad de Matematicas, Universidad de Santiago de Compostela, Spain Department of Mathematics, Faculty of Science, King Abdul Aziz University, P.O. Box 9028, Jeddah 21413, Saudi Arabia
Abstract A generalized quasilinearization method for a nonlinear mixed BVP is developed and a sequence of approximate solutions converging monotonically and quadratically to a solution of the given problem is presented. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Quasilinearization; Mixed BVP; Quadratic convergence
1. Introduction It is well known that the method of quasilinearization [1] provides an excellent tool for obtaining approximate solutions of nonlinear differential equations. This technique works fruitfully only for the problems involving convex/concave functions and gives the sequence of approximate solutions converging monotonically and quadratically to the solution. Recently, the convexity assumption was relaxed and the method was generalized and extended in various directions to make it applicable to a large class of problems [2,4–8]. Afterwards, Shahzad and Vatsala [12,13] and Shahzad and Sivasundaram [11] discussed the generalized quasilinearization method for second
*
Corresponding author. E-mail address: [email protected] (N. Shahzad).
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 2 4 3 - 0
424
B. Ahmad et al. / Appl. Math. Comput. 133 (2002) 423–429
order boundary value problems. More recently, Nieto [10] developed a generalized quasilinearization method for a nonlinear Dirichlet problem to obtain a sequence of approximate solutions that converges quadratically to the solution of the given problem. For a complete survey of the generalized quasilinearization method, see [9]. The aim of this paper is to consider a second order ordinary nonlinear differential equation with mixed boundary conditions and develop the method of quasilinearization for this problem without requiring the function involved to be convex/concave.
2. Preliminaries It is well known that the following mixed BVP w00 ðtÞ ¼ kwðtÞ; t 2 ½0; p; wð0Þ ¼ w0 ðpÞ ¼ 0; 2
has a nontrivial solution if and only if k ¼ ½ð2m 1Þ=2 ðm ¼ 1; 2; 3 . . .Þ. For 2 k 6¼ ½ð2m 1Þ=2 ðm ¼ 1; 2; 3 . . .Þ and ,ðtÞ 2 C½0; p, the unique solution of the problem w00 ðtÞ kwðtÞ ¼ ,ðtÞ;
t 2 ½0; p;
wð0Þ ¼ w0 ðpÞ ¼ 0; is wðtÞ ¼
Z
p
Gk ðt; tÞ,ðtÞ dt;
0
where hpffiffiffi i hpffiffiffi i 8 > cos k ðp tÞ sin kt ; > > < 1 0h6 t 6 t 6 ph i ðk > 0Þ pffiffiffi Gk ðt; tÞ ¼ pffiffiffi pffiffiffi i pffiffiffi > sin k t cos k ðp tÞ ; > k cos kp > : 06t6t6p t; 0 6 t 6 t 6 p G0 ðt; tÞ ¼ ðk ¼ 0Þ t; 0 6 t 6 t 6 p hpffiffiffiffiffiffiffi i hpffiffiffiffiffiffiffi i 8 > cosh k ðp tÞ sinh kt ; > > < 1 0 6 t 6 t 6 p hpffiffiffiffiffiffiffi i hpffiffiffiffiffiffiffi i ðk < 0Þ: pffiffiffiffiffiffiffi Gk ðt; tÞ ¼ pffiffiffiffiffiffiffi sinh k t cosh k ðp tÞ ; > k cosh kp > > : 06t6t6p
B. Ahmad et al. / Appl. Math. Comput. 133 (2002) 423–429
425
Let us consider the following nonlinear BVP x00 ðtÞ ¼ f ðt; xðtÞÞ;
t 2 ½0; p;
0
xð0Þ ¼ x ðpÞ ¼ 0;
ð2:1Þ
where f : ½0; p R ! R is a continuous real valued function. A function a 2 C 2 ½½0; p; R is a lower solution of (2.1) if a00 ðtÞ 6 f ðt; aðtÞÞ;
t 2 ½0; p;
að0Þ 6 0; a0 ðpÞ 6 0; and b 2 C 2 ½½0; p; R is an upper solution of (2.1) if b00 ðtÞ P f ðt; bðtÞÞ;
t 2 ½0; p;
0
bð0Þ P 0; b ðpÞ P 0: The following lemma plays a crucial role in the sequel and we sketch its proof for the sake of completeness. Lemma 2.1. Assume that a; b 2 C 2 ½½0; p; R are lower and upper solutions of (2.1), respectively, such that aðtÞ 6 bðtÞ for every t 2 ½0; p. Then there exists a solution xðtÞ of (2.1) such that aðtÞ 6 xðtÞ 6 bðtÞ for t 2 ½0; p. Proof. Let p : ½0; p R ! R be a mapping defined by pðt; xÞ ¼ maxfaðtÞ; minfx; bðtÞgg: Then, extend f ðt; xÞ to ½0; p R by setting F ðt; xÞ ¼ f ðt; pðt; xÞÞ: Observe that F ðt; xÞ is bounded. Let us consider the modified BVP x00 ðtÞ ¼ F ðt; xðtÞÞ; 0
xð0Þ ¼ x ðpÞ ¼ 0;
t 2 ½0; p;
ð2:2Þ
which is solvable, that is (2.2) has a solution x on ½0; p. Using the arguments similar to those of [3,10], it can be shown that a6x6b
on ½0; p:
Finally, since any solution of (2.2) is also a solution of (2.1), it follows that x is a solution of (2.1). This completes the proof.
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B. Ahmad et al. / Appl. Math. Comput. 133 (2002) 423–429
3. Main result Theorem 3.1. Assume that ðA1 Þ ao ; bo 2 C 2 ½½0; p; R are lower and upper solutions of (2.1), respectively, such that ao 6 bo on ½0; p, ðA2 Þ f 2 C½X; R is such that fx ðt; xÞ; fxx ðt; xÞ exist and are continuous for every ðt; xÞ 2 X, where X ¼ fðt; xÞ 2 ½0; p R : ao ðtÞ 6 x 6 bo ðtÞg; ðA3 Þ fx ðt; xÞ < 0 for every ðt; xÞ 2 X. Then there exists monotone nondecreasing sequence fan g which converges uniformly to a solution of (2.1) and the convergence is quadratic. Proof. Set Uðt; xÞ ¼ F ðt; xÞ f ðt; xÞ on ½0; p R; where F : ½0; p R ! R is such that F ðt; xÞ; Fx ðt; xÞ; Fxx ðt; xÞ are continuous on ½0; p R and Fxx ðt; xÞ P 0;
ðt; xÞ 2 ½0; p R:
ð3:1Þ
It further implies that F ðt; xÞ P F ðt; yÞ þ Fx ðt; yÞðx yÞ for x P y and therefore f ðt; xÞ P f ðt; yÞ þ Fx ðt; yÞðx yÞ ½Uðt; xÞ Uðt; yÞ:
ð3:2Þ
Now, consider the mixed BVP u00 ¼ gðt; u; ao Þ ¼ f ðt; ao Þ þ Fx ðt; ao Þðu ao Þ ½Uðt; uÞ Uðt; ao Þ; uð0Þ ¼ u0 ðpÞ ¼ 0: ð3:3Þ The inequality (3.2) together with ðA1 Þ imply a00o 6 f ðt; ao Þ ¼ gðt; ao ; ao Þ; b00o P f ðt; bo Þ P f ðt; ao Þ þ Fx ðt; ao Þðbo ao Þ ½Uðt; bo Þ Uðt; ao Þ ¼ gðt; bo ; ao Þ: In view of Lemma 2.1, there exists a solution a1 of (3.3) such that ao 6 a1 6 bo on ½0; p. Next, we consider the mixed BVP u00 ¼ gðt; u; a1 Þ; uð0Þ ¼ u0 ðpÞ ¼ 0:
ð3:4Þ
B. Ahmad et al. / Appl. Math. Comput. 133 (2002) 423–429
427
The inequality (3.2) yields a001 ¼ gðt; a1 ; ao Þ; ¼ f ðt; ao Þ þ Fx ðt; ao Þða1 ao Þ ½Uðt; a1 Þ Uðt; ao Þ 6 f ðt; a1 Þ ¼ gðt; a1 ; a1 Þ; b00o P f ðt; bo Þ P f ðt; a1 Þ þ Fx ðt; a1 Þðbo a1 Þ ½Uðt; bo Þ Uðt; a1 Þ ¼ gðt; bo ; a1 Þ; It follows again by Lemma 2.1 that there exists a solution a2 such that a1 6 a2 6 bo on ½0; p. Thus, ao 6 a1 6 a2 6 bo on ½0; p. Employing the same arguments successively, we conclude ao 6 a1 6 a2 . . . 6 an 6 b o
on ½0; p;
where the elements of the monotone sequence fan g are the solutions of the mixed BVP u00 ¼ gðt; u; an1 Þ ¼ f ðt; an1 Þ þ Fx ðt; an1 Þðu an1 Þ ½Uðt; uÞ Uðt; an1 Þ; uð0Þ ¼ u0 ðpÞ ¼ 0: The monotonicity of the sequence fan g ensures the existence of its (pointwise) limit x. Let us consider the linear mixed BVP u00 ¼ fn ðtÞ;
ð3:5Þ
uð0Þ ¼ u0 ðpÞ ¼ 0; where fn ðtÞ ¼ gðt; an ðtÞ; an1 ðtÞÞ;
t 2 ½0; p:
The continuity of g on X implies that the sequence ffn g is bounded in C½½0; p; R and so lim fn ðtÞ ¼ f ðt; xðtÞÞ;
n!1
t 2 ½0; p:
Here an ðtÞ ¼
Z
p
G0 ðt; sÞfn ðsÞ ds;
0
where an ðtÞ is a solution of (3.5). Thus fan g is bounded in C 2 ½½0; p; R and fan g " x uniformly on ½0; p. Consequently, Z p xðtÞ ¼ G0 ðt; sÞf ðs; xðsÞÞ ds; t 2 ½0; p: 0
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Hence x is a solution of (2.1). For quadratic convergence, we set pn ðtÞ ¼ xðtÞ an ðtÞ. Using the mean value theorem repeatedly, we obtain pn00 ðtÞ ¼ f ðt; xðtÞÞ gðt; an ðtÞ; an1 ðtÞÞ ¼ f ðt; xðtÞÞ f ðt; an1 ðtÞÞ Fx ðt; an1 ðtÞÞ½an ðtÞ an1 ðtÞ þ ½Uðt; an ðtÞÞ Uðt; an1 ðtÞÞ ¼ F ðt; xðtÞÞ F ðt; an1 ðtÞÞ Fx ðt; an1 ðtÞÞ½an ðtÞ an1 ðtÞ þ ½Uðt; an ðtÞÞ Uðt; xðtÞÞ ¼ Fx ðt; nÞ½xðtÞ an1 ðtÞ Fx ðt; an1 ðtÞÞ½an ðtÞ an1 ðtÞ þ ½Uðt; an ðtÞÞ Uðt; xðtÞÞ ¼ ðFx ðt; nÞ Fx ðt; an1 ðtÞÞÞ½xðtÞ an1 ðtÞ þ Fx ðt; an1 ðtÞÞ ½xðtÞ an ðtÞ þ ½Uðt; an ðtÞÞ Uðt; xðtÞÞ ¼ Fxx ðt; rÞ½n an1 ðtÞ½xðtÞ an1 ðtÞ þ Fx ðt; an1 ðtÞÞ½xðtÞ an ðtÞ Ux ðt; gÞ½xðtÞ an ðtÞ; where an1 ðtÞ 6 n 6 r 6 xðtÞ and an ðtÞ 6 g 6 xðtÞ: Take hn ðtÞ ¼ Fx ðt; an1 ðtÞÞ Ux ðt; gÞ; and 2 ðtÞ; kn ðtÞ ¼ Fxx ðt; rÞ½n an1 ðtÞ½xðtÞ an1 ðtÞ Mpn1
where 0 6 Fxx ðt; yÞ 6 M; ðt; yÞ 2 X. Clearly kn ðtÞ 6 0. Since Fx is nondecreasing and an1 ðtÞ 6 g, it follows by ðA3 Þ that there exists k < 0 and an integer N such that hn ðtÞ 6 k, t 2 ½0; p for n P N . Therefore, the error pn satisfies the mixed BVP 2 ðtÞ þ kn ðtÞ; pn00 ðtÞ kpn ðtÞ ¼ ½hn ðtÞ kpn ðtÞ þ Mpn1 0 pn ð0Þ ¼ pn ðpÞ ¼ 0:
Thus, we can write Z p
2 pn ðtÞ ¼ Gk ðt; sÞ ½hn ðsÞ kpn ðsÞ þ Mpn1 ðsÞ þ kn ðsÞ ds; 0
which gives pn ðtÞ 6 M
Z 0
p 2 Gk ðt; sÞpn1 ðsÞ ds;
n P N:
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Hence, there exists a constant d > 0 such that 2
kpn k 6 dkpn1 k ;
n P N;
where kxk ¼ maxfjxðtÞj: t 2 ½0; pg.
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