QUASILINEARIZATION FOR SECOND ORDER SINGULAR BOUNDARY VALUE j?ROBLEMS
WITH SOLUTIONS IN WEIGHTED SPACES J. VASUNDHARA DEVI AND A. S. VATSALA
ABSTRACT. Inthis paper, we develop the method of quasilineariza- tion combined with the method of upper and lower solutions for singular second order boundary value problems in weighted spaces.
The sequences constructed converge uniformly and monotonically to the unique solution of the second singular order boundary value problem. Further we prove the rate of convergence is quadratic.
1. Introduction
The study of second order singular mixed boundary value problem has received much attention due to its application [3]. Also in [3] the motivation to look for existence of solutions to singular boundary value problems in Weighted Banach Spaces is presented. In this paper, we develop the method of generalized quasilinearization [1, 4] to singular boundary value problems in weighted spaces using upp~r and lower so- lutions. The method yields two monotone sequences which are solutions of linear singular boundary value problems in weighted spaces. Further, the sequences converge quadratically to the unique solution of the non- linear singular boundary value problem in weighted spaces. For this purpose, we have developed a comparison theorem [2] for second order singular boundary value problems. The results developed here paves way to study generalized quasilinearization method for singular problems in many different settings.
Received August30,1999.
2000 Mathematics Subject Classification: 34A45, 34B16.
Key words and phrases: quasilinearization method, singular boundary value prob- lems, weighted spaces, quadratic convergence.
2. Preliminaries
Consider the singular boundary value problem (SBVP) of 2nd order
{
- (tny')' = f(t, tn- 1y),
°
<t < 1(2.1) y(1) =Yb tim tn- 1y(t) =Yo with n >2
t-->O+
wheref :I x R ---+ R is continuous, 1= [0, 1].
DEFINITION 2.1. By a solution of the SBVP (2.1) we mean a function y E C2[(0, 1], R] with tn- 1yE C [[0, 1], R], andtny' E C [[0, 1], R] and (tn-1y)' E L1[0, 1] which satisfies the differential equation and the given boundary conditions.
Note: Whenever we write y(t) is a solution of the SBVP (2.1) we mean thattn- 1y(t) is a solution of SBVP (2.1). Similar notation is used for lower and upper solutions.
Throughout this paper, we work in the Banach space of functions E= {y E Cl[(0,1] ,R] Ixn- 1yl E C [[0, 1], R] and IxnY'1 E C [[0, 1], Rn with the norm
Iylo = Max { sup Ixn -1
y(x)
I'
sup IxnY'(X)I}.xE[O,lJ XE[O,l]
We now state the existence theorem for (2.1) from [3] which we need in our work.
THEOREM 2.1. Assume
(i) f :[0, 1] x R ---+ R is continuous.
(ii) Tbere exist an upper solution /3 and a lower solution a ofSBVP (2.1) witb t n- 1 a(t) :Stn- 1 /3(t) on (0, 1), a(1) :SY1 :S/3(1) and
limtn-1a(t) :SYo :S timtn-1/3(t).
t-->O+ t-->O+
Tben tbe SBVP (2.1) bas a solution y E E witb tn- 1a(t) :S tn- 1y(t) :S tn-1/3(t) for t E [0, 1], wbere
(2.2)
3. Main results
We first proceed to prove the comparison theorem which is an essential tool in our work.
THEOREM 3.1. Suppose that there exist y, z E E such that
{
- (tny,)' S; f(t, tn-Iy) (3.1)
- (tnz')' 2: f(t, tn-Iz) hold with
{
limtn-Iy(t) S; limtn-Iz(t)
(3.2) t-->OT t-->OT
andy(1) S; z(1).
Further, assume thatf :[0, 1J x R - t R is continuous and satisfies (3.3) f (t, tn-Ix) - f (t, tn-Iy) S; -Ltn-I(x - y),
whenever, x 2: y and L > O.
Then tn-1y(t) S; tn-Iz(t) on [0, 1J.
Proof. Suppose the conclusion fails, i.e. tn-Iy(t) > tn-Iz(t) at some t E (0, 1J. Then by continuity, we must have a local maximum at to E (0, 1). This implies that
(3.4) [tn-I(y - z)J'= 0 att =to and
(3.5) [tn-I(y - z)]"S; 0 att =to.
Now considering the relation (3.5), and differentiating it twice and sim- plifying using the relation (3.4), we obtain
o
> [tn-I(y - z)]"= ~(tny,)' _ ~(tnz')'
t t
which yields the inequality
-} (tny,)' 2: -}(tnz')' att =to E (0, 1J.
Using the relation (3.1) in the above inequality at t= to, we get f (to, t~-Iy) 2: - (t~y')' 2: - (t~z')' 2: f (to, t~-IZ)
which implies,
0:::; f (to, t~-ly) - f (to, t~-IZ) :::; _Lt~-l(y - z)< 0
a contradiction. Thus the proof is complete.
o
We next present the quasilinearization method which guarantees qua- dratic convergence of a sequence of functions to a solution9of the given SBVP.
THEOREM 3.2 Assume that
(i) ao, 130 E E w-e lower and upper solutions of the SBVP (2.1) such that tn-Iao(t) :::; f",-1/30(t) OIiI with
{
limtn-Iao(t) :::; Yo :::; timtn- l/3o(t)
(3.6) hO+ t-->O+
and ao(l):::;YI :::; /30(1);
(ii) f(t, tn-Iy) is convex (a)fuu exists, continuous and fuu (t,u) > 0 for everyt,u onn, wheren isgiven by
n= {(t,u);O<t:::; 1, andao:::; u:::; /3o};
(iii) fu (t, tn-ly) <0 onn.
Then there exists monotone sequences{tn-lak(t)} , {tn- l/3k(t)} with tn-Iao(t) :::; tn-Ial(t):::;···:::; tn-Ian(t) :::; tn- l/3n(t)
:::; ... :::; tn- l/3l(t) :::;tn- l/3o(t), t E [0, 1)
which converges uniformly and quadratically to the unique solution tn- l y(t) of the SBVP (2.1).
Proof. Integrating the assumption (ii) twice, we get
(3.7) j (t, tn-lX) 2:: f(t, tn-Iy)+fu(t, tn-Iy)(X - y)tn-\x 2:: y.
Consider the singular BVP (SBVP)
{
- (tna~)' =F (t, tn-lal; ao) (3.8)
with al(l) =Yl and lim tn-Ial(t) =Yo,
t-->O~
where F (t, tn-Ia!, ao) = f (t, tn-Iao)+fu (t, tn-Iao) (al - ao) tn-l.
Sinceao and/30 are lower and upper solutions of SBVP (2.1) with tn-Iao(t) :::; tn- l/3o(t),
using the relation (3.7), we get
- (tna~)' < f (t, tn-lao) F (t, tn-lao; ao) and
- (tn(3~)' > f (t, tn- l(3o)
> f (t, tn-lao) +fu (t, tn-lao) ((30 - ao)tn- l F (t, tn- l(3o; ao) .
The above two inequalities, along with relation (3.6) imply that ao and (30 are lower and upper solutions for the SBVP (3.8). Furthermore, ao and (30 satisfy the hypothesis of Theorem 2.1, hence we can conclude that there exists a function say al(t) E E such that al(t) is a solution of the SBVP (3.8) with
tn-lao(t) :::; tn-lal(t) :::; tn- l(3o(t), t E [0, 1] .
We now claim that this solution is unique. The proof is as follows. If possible, suppose thatal (t) is another solution of theSBVP (3.8). Then
{
- (tnZil' (t))' = F (t, tn-lal; ao) (3.9) with al(l) =Yl and limtn-lal(t) =Yo.
t-->o~
Now writing (3.8) and (3.9) as inequalities
{
- (tnal')' ~ F (t, tn-lal; ao)
(3.10) and
- (tnaD' ::; F (t, tn-lal; ao)
along with boundary conditions and observing that the function F sat- isfies the condition (3.3), we apply the comparison theorem and obtain the relationtn-lal(t) ~ tn-lal(t), t E [0, 1].
Now reversing the above inequalities in (3.10) and applying the com- parison theorem again, we get
tn-lal(t) ::; tn-lal(t), t E [0, 1].
The above inequalities together imply the uniqueness of the solution tn-lal(t) of the SBVP (3.8). We proceed next to show that there exists
a unique solution131 (t) for the SBVP
{
- (tn13~)' =G (t, tn- 1131; ao, (30) (3.11) with limtn- 1131(t) =Yo, 131(1) =Y1
t--->OT
such thattn-1ao(t) ::; tn- 1131(t) ::; tn-l13o(t);whereG (t, tn- 1131; ao, (30) =
f (t, (30)+fu (t, tn-1ao) (131 - (30) tn- 1.
The fact that ao, 130 are lower and upper solutions of SBVP (2.1), together with the relation (3.7) imply that
- (tna~)' < f (t, tn-1ao)
< f (t, tn-l13o) - fu (t, tn-1ao) (130 - ao) tn- 1 f (t, tn-l13o) +fu (t, tn-1aO) (aO - {30)tn- 1 - G (t, tn- 1aO; ao, (30)
and
(-tn13b)' 2: f (t, tn-l13o)
= G (t, tn-l13o; ao, (30) .
The above inequalities, along with relation (3.6) yield thatao, 130are also lower and upper solutions for the SBVP (3.11). Now using Theorem 2.1 we get the existence of a solution of SBVP (3.11). The proof of uniqueness is similar to the uniqueness proof of the solution of the SBVP (3.8). Thus we have {31 E E such that 131(t) is the unique solution of SBVP (3.11).
The proof oftn-1a1(t) ::; tn- 1{31(t) is as follows.
We know thattn-1ao ::; tn-l13o. Using it in relation (3.7), we get - (tnaD' F (t, tn-1a1; ao)
f (t, tn-1ao) +fu (t, tn-1ao) (a1 - ao) tn- 1
< f(t, tn-l13o) - fu(t, tn-1ao)(130 - ao)tn- 1 +fu(t, tn-1aO)(a1 - ao)tn- 1
- f(t, tn-l13o)+fu(t, tn-1aO)(a1 - (30)tn- 1 - G(t, tn-1a1; ao, (30).
SinceGsatisfies the relation (3.3), the above inequality together with the SBVP (3.11) and the boundary conditions satisfy the hypothesis of the comparison theorem. Hence, ,we havetn-1a1(t) ::; tn- 1{31(t), t E [0, 1].
Working in a similar fashion as above, we now proceed to show that for any k >1
tn-Io (t)k <_ tn-Iok+l(t) <_ tn- lfJk+1(.l (t) <_ tn- lfJk,(.l (t) whereOk(t), {3k(t) are known.
Consider the SBVP
{
_(tno~+1)' = F(t, tn-IOk+l, Ok)
(3.12) with lim tn- IOk+1(t) =Yo and Ok+1(1) = YI
t->O+
where
F(t, tn-Iok+l, Ok) ::::::: f(t, tn-IOk) +fu(t, tn-IOk)(Ok+1 - Ok)tn- l . Using the fact that tn-IOk_1 ::; tn-IOk in the relation (3.7) and sub- stituting in the equality below, we get
_(tno~)' F(t, tn-IOk, Ok-I)
f(t, tn-IOk_l)+fu(t, tn-Iak_l) (Ok - Ok_l)tn- l
< f(t, tn-IOk) F(t, tn-IOk, Ok)'
Since tn-Ioo ::; tn-IOk and fu is an increasing function in u, we have fu(t, tn-IOO) ::; fu (t, tn-IOk) .Using this in relation (3.7) and astn-IOk ::;
tn- l{3k ::;tn- l{3k_l, the equality below reduces to _(tn{3~)' G(t, tn- l{3k; 00, {3k-l)
f(t, tn- l{3k_l)+fu(t, tn-IoO)({3k - {3k_l)tn-1
> f(t, tn-IOk) +fu(t, tn-IOk)({3k_1 - Ok)tn- l - fu(t, tn- IOo)({3k_1 - {3k)tn- 1
> f(t, tn-IOk) +fu(t, tn-IOk)({3k - Ok)tn- l F(t, tn- l{3k; Ok)'
Thus we get the inequalities
-(tno')'k -< F(t tn-Io ., k, 0k) and
_(tn{3~)' ~ F(t, tn- l{3k; Ok) along with the boundary conditions.
Now discussing as earlier, we obtain that there is a unique solution (Yk+1(t) EE for the SBVP (3.12) such that
(3.13) tn-IO:k(t) :s;tn- IO:k+1(t) :s; tn-IfA(t), t E [0, 1].
Next consider theSBVP
{
_(tn(3~+I)' =G(t, tn- I(3k+I; 0:0, fA)
(3.14) with lim limtn- l(3k+l(t) = Yo and(3k+l(t) =YI'
t->OT
Again using the relation (3.7), fu is increasing inu, and the inequal- ities tn-Io:0 _< tn-Io: _kl_< tn-Io:k _< tn- l(3k _< tn- l(3 _kl_< tn- l(30 in the SBVP (3.14), we deduce as follows
_(tn(3~)' G(t, tn- l(3k; 0:0,(3k-l)
f(t, tn- l(3k_l)+fu(t, tn- IO:O)((3k - (3k_dtn- 1
> f(t, tn- l(3k) +fu(t, tn- l(3k)((3k_1 - (3k)tn- 1 - fu(t, tn- I
O:O)((3k_1 --(3k)tn- 1
> f(t, tn- l(3k)
G(t, tn- l(3k; 0:0, (3k).
Now considering
-(tno:'k)' F(t tn- l, O:k, O:k-l)
= f(t, tn-IO:k_l)+fu(t, tn-IO:k_d(O:k - O:k_l)tn- l
< f(t, tn- l(3k) +fu(t, tn-IO:k_l) x [-((3k - O:k_l)tn- l+(O:k - O:k_l)tn- l]
"- f(t, tn- l(3k) - fu(t, tn- IO:k_I)((3k - O:k)tn- l
< f(t, tn- l(3k) - fu(t, tn- IO:O)((3k - O:k)tn- l f(t, tn- l(3k) +fu(t, tn-IO:O)(O:k - (3k)tn- 1 G(t, tn-Io:k; 0:0, (3k).
Thus we get the inequalities
_(tn(3~)':2 G(t, tn- l(3k; 0:0, (3k)
_(tno:~)' :s; G(t, tn-Io:k; 0:0, (3k) together with the boundary conditions.
Now using the existence Theorem 2.1, we conclude that(3k+l(t) is the unique solution of (3.4) such that tn-IO:k(t) :s;tn- I(3k+I(t) :s; tn- l(3k(t).
Now consider the SBVPs (3.13) and (3.14). Working in a simi- lar fashion as in the proof of tn-1al(t) ~ tn- 1/31(t) we can show that tn-1ak+l(t) ~ tn- 1/3k+l(t).
Thus we have proved
(3.15)
These monotone sequences {tn-1ak+l(t)} and {tn- 1/3k+l(t)} are both bounded and hence they converge to tn-1p(t) and tn-1/,(t) point-wise respectively.
Next we show that {tn-1ak+l(t)} converges uniformly totn-1p(t). The proof for the convergence of the other sequence is similar.
Using the SBVP (3.12), the relation (3.15) and the integral equa- tion (2.2) it is easy to observe that { tn- 1ak+l (t) }, { (tn-l ak+1(t))'}, {tna~+1(t)},and {(tna~+1(t))'} are uniformly bounded sequences. Hence by Ascoli's theorem, we have that the sequence{tn-1ak+l(t)}, {tna~+l(t)}
have uniform convergent subsequences. Since {tn-1ak+l(t)} is a mono- tone sequence in E, it follows that the sequence is uniformly convergent in E. Thus {tn-l ak+1(t)} converges uniformly to a solution tn-1p(t) of the SBVP (2.1).
We now claim that this convergence is quadratic in nature. For proof, set
Then
- (tnp~+1(t))'
_ (tnp'(t))'+(tna~+l(t))'
f (t, tn-1p) - f (t, tn-1ak) - fu (t, tn-1ak) (ak+1 - ak)tn- 1 fu(t, tn-l~)(p - ak)tn- 1 - fu (t, tn-1ak) (ak+l - ad tn-\
< fu(t, tn-1p)(p - ak)tn- 1- fu(t, tn-1ak)(p - ak)tn- 1 +fu(t, tn-1ak)(p - ak+1)tn- 1
fuu(t, tn-l~) [(p - ak) (tn- 1)J2+fu(t, tn-1ak) (p - ak+l)tn- 1
< -MPk+l(t)tn- 1+N [Pk(t)tn- 1J2
< -MPk+l(t)tn- 1+N Ipkl~,
wheretn-l~ E (tn-1ak, tn-1p]. Thus we have the inequality (3.16) - (tnp~+1)' ::; -MPk+1(t)tn- 1+NIpkl~.
Clearly given M, N and Ipkl~, there exist z E R+ such that (3.17) 0=-Mztn- 1+N Ipkl~.
Now using the comparison theorem [2] for (3.16) and (3.17), we get Pk+l(t)tn- 1 ::; ztn-\ t E [0, 1].
(Le. )
which yields
N 2 Ip - ak+110 ::; M IPk10 .
This proves the quadratic convergence of the sequence {tn-1ak+1(t)}.
Similarly, we can prove the quadratic convergence of the sequence {tn-1
f3k+l (t) } to the solution of the SBVP (2.1). Further, it is noted that these sequences converge to the unique solution of (2.1) from hypothesis (iii) and the comparison Theorem 3.1. Thus the proof of the theorem is
complete. 0
References
[1] R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, American Elsevier, New York, 1965.
[2] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Academic Press, I (1969).
[3] R.Kannan, Donalo.Reagan, A Note on Singular Boundary Value Problems with Solutions on Weighted Spaces, Nonlinear Analysis, 37(1999), 791-796.
[4] V. Lakshmikantham and A. S. Vatsala, Generalized Quasilinearization for Non- linear Problems, Kluwer Academic Publishers, Dordrecht, 1998.
J. Vasundhara Devi
Department of Mathematics Gayathri Vidya Parishad College for P. G. Courses Vishakapatnam-530016 A. P. India
A. S. Vatsala
Department of Mathematics
University of Louisiana at Lafayette Lafayette, LA 70504
USA
E-mail: [email protected]
