Proc. Indian Acad. Sci. (Math. SCi.), Vol. 92, No. 1, September 1983, pp. 61-65. �9 Printed in India.

An approximate solvability scheme for a class of nonlinear equations

MOHAN JOSHI Mathematics Group, Birla Institute of Technology and Science, Pilani 333 031, India

MS received 5 January 1982; revised 6 June 1983

Abstract. An approximate solvability scheme for equations of the type u + K , ( u ) = w, in a

closed convex subset A of a Hilbert space X is given. Here, for each u ~ A, K. : X --* X is a bounded linear operator.

Keywords. Monotone operator; Leray-Schauder principle; Chandrasekhar's H-equation.

1. Introduction

The existence of solutions of equations of the type

u + K.(u) = w, (1)

where for each u ~ A, K.: X --* X is a bounded linear operator was earlier discussed [2]. A is a closed convex subset ofa Hilber t space X. Approximate solvability of such type of equations has not yet been dealt with. Our main purpose is to give an approximation scheme for (1), which is significant from the computational point of view. As an application of our abstract result, we discuss approximate solvability of nonlinear integral equations of the form

I' u(s) + u(s) K (s, t)u(t)dt = w(s). (2) o

These equations include Chandrasekhar's H-equation

f ~ s- ~F(t)u(t)dt=u(s). (3) 1 + u(s) s + t

Equation (3) plays an important role in the theory of radiative transfer in semi-infinite atmosphere [1], and hence the importance of our result needs no explanation.

2. Main result

Throughout this paper ' 4 ' denotes strong and ' ~ ' denotes weak convergence in a Hilbert space X . B[O, r] denotes the closed ball of radius 'r' in X. For u, w ~ X , (w, u) refers to inner product between u and w in X.

DEFINITION 2.1.

An operator T: X --, X is said to be monotone if

(Tu - To, u - o) >1 0 for all u, v in X.

61

62 Mohan Joshi

Let XI c X2 c . . . be a sequence of finite dimensional subspaces o f a Hilbert space X and P,: X --* X, a corresponding sequence of linear projections.

DEFINITION 2.2.

A tuple {{X.}, {P.}} is called an approximation scheme in the space X if P, is continuous for every n and for each x e X, P. x ~ x as n -~ oo.

In this section we shall give a constructive result concerning the existence of a solution of

u + K.(u) = w (4)

as a strong limit of solutions u. e X. of the 'approximate' equation

u,, + P,,K,,.(u,,) = w,,. (5)

DEFI~mON 2.3.

Equation (4) is said to be strongly (weakly) solvable if (5) has a solution u, e X, and there exists a subsequence of {u,} which converges to u strongly (weakly) and u is a solution of (4).

Here, for u e X, K , is a bounded linear operator on X. In the following K* denotes the conjugate of the bounded linear operator K.

THEOREM 2.1.

Let X be a Hilbert space with an approximation scheme { {X~}, { P~} } and A a closed convex subset of X. Assume that for each u e A, K. is a bounded linear monotone operator satisfying the following condition: (a) u~ --" u in A implies that K* �9 x - . .k(~p K*(v) for all v e X . Then (4) is approximately strongly solvable.

Proof: We first claim that K. : X ~ X is jointly weakly continuous. That is, u~ ~ u in A and vk --" v in X implies that K.t(vk) --" Ku(v). Consider (K~k(v~) -- Ku(v), x), x e X .

(K.,(vD - K.(v), x) = (K.,(v~)- K.(vD, x) + (K. (vD- K.(v), x)

= (v~, r* , (x ) - r * ( x ) ) + (r.(vD- r . ( O , x).

As k --, oo the first term in the right side of the above inequality tends to zero in view of assumption (a) and the second term tends to zero as K. is a continuous linear operator and hence also weakly continuous. This proves our claim.

We now consider the approximate equation

u. + P.K..(u.) = w..

Let T be the operator on X. defined by

Tu = u + P . K . ( u ) - w..

Define a closed and bounded set C as

C = {u e / l r~ X.: (w, u) .< (u, u) ~ (w, w)}.

Then T is a continuous operator on X, such that

(Tu, u) = (u, u) + (P.K.(u) , u) - (wu, u) I> (u, u) - (w, u) 1> 0, for u e C.

An approximate solvability scheme for a class o f nonlinear equations 63

Hence it follows by Leray-Schauder principle that Tu = 0 has a solution u. e C. That is

u. + P,,K,,.(u,,) = w,,.

{u.} is a bounded sequence in a Hilbert space and hence there exists a subsequence of it, which we again denote by {u~}, such that u. --- u. We claim that u is a solution of (4). As u. ~ u and w. ---, w it sufficies to show that P . K . . i u . ) ---" Ku(u) as n --* oo.

Consider

(P~K,.(u~) - K,(u) , x), x ~ X . (e.Ku,(un) - Ku(u), x)

= (P~K.,, (un) - P,K~ (u), x) + ( e . K ~ ( u ) - K~(u), x)

= ( K . . ( u . ) - g~(u) , x) + (g~. (u.) - g~(u), P , x - x) + (P~ g . ( u )

- K , , (u ) , x )

~< iK. , in.) - K , iu), x) + II g . , (u,) - g ,(u)II II e . x - x II + II P- Ku (u)

- K , iu)ll IIxll 0 as n ~ oo, since K.. (u.) --- K . (u)and P. x ~ x. This proves that (4)is approximately

weakly solvable. We now show that {u.} actually converges strongly to u. Consider

i l u , - u l l 2 = ( u , - u , u , - u ) = (w,, - w, u,, - u) + (K,,(u) - P,,K,,.(u,,), u,, - u)

= i w . - w , u , , - u ) + ( P , , K , , . ( u ) - P , , K , , . ( u , , ) , u , , - u )

- ( P , K , . ( u ) - P , K , ( u ) , u, - u) - ( P , K , ( u ) - K,(u), u, - u)

= ( w , - w . un - u) - ( K , . ( u , ) - K , . ( u ) , u~ - u)

- (g , , . (u , , ) - Ku , (u ) , u - P . ( u ) ) - ( r . . ( u ) - K ~ ( u ) , u,, - P,~u)

- (P , ,K , , (u ) -K , , (u ) , u , , - u )

<~ ( P , , w - w , u,, - u ) - ( K , , . ( u , , ) - K , , . i u ) , u - P , , u )

- ( r , . ( u ) - K . ( u ) , th - e . u ) - ( e , K . ( u ) - g . t u ) , u . - u)

by monotonicity of K , . The first, second and fourth term tend to 0 in view o f P , x ~ x for every x e X and the

third term tend to zero in view of (a) and uniform boundedness principle. This proves our theorem.

3. Application

As an application of our main theorem we obtain an approximate solvability result for nonlinear integral equations of the type

u (s) + u i s) S ~ K (s, t) u( t ) dt = w(s) (6)

in the space L2[0, 1]. We assume that K(s , t) is a Hilbert-Schmidt kernel. So eigenfunctions of K form a

complete or thonormal set in L2[0, 1]. Let el, e2 . . . . be eigen functions of K with eigenvalues 21, 22 . . . . . Define a sequence X, of finite dimensional subspaces of

64 Mohan Joshi

X = L2[O, 1] and linear projections P.: X -~ X. as follows:

X. = [el, e2 . . . . . e.], P.u = ~ O~ke k where u = k = l

~ , O~k ek . k = l

Then the approximate equation in the finite dimensional space X. is given by

u(s) + P.[u(s) ~ K (s, t)u(t)dt ] = P.w (7)

o o where u = Yk=~ ~kek. Taking inner product with e. we get (u,e.)+(P.[u(s)j~oK(s,t)u(t)dt],e n)

= (P.w, e.), which gives

O[. + ( (k~= l rtkek($) ) (j~_ l 2j~jej(s)), e.) = ft., (8)

1 where w = Z~= ~Pkek. Writing rj~. for ~oe~(s)ej(s)e.(s)ds in (8), we get an equivalent system of nonlinear equations

j=l k = l

Thus solvability of the approximate equation (7) is equivalent to the solvability of (9). One can now use the known techniques to solve the nonlinear system given by (9). ]For various reasons we skip the details regarding the computational aspect of (9). We have the following theorem giving the approximate solvability of (6).

THEOREM 3.1

Suppose that

(a) K(s, t) >1 0 a.e. on [0, 1] x [0, 1],

(b) ess sup ~ K2(s, t)dt < o0,

(c) w(s) >t 0 a.e. on [0, 1].

Then (6) is approximately strongly solvable in L2[0, 1].

Proof: Let A = {u e L2: u(s) >t 0 a.e. on [0, 1]} and let

K,,(v)(s) = v(s) ~1 o K(s, t)u(t)dt.

Then for each u cA, K, is a bounded linear operator on L2. Also, we have (K,(v), v) 1 1 = So(~oK(s, t)u(t)dt)v2(t)dt

>/0 foral l v~L2.

That is { K.} is monotone for each u ~ A. We now verify that the hypothesis (a) of Theorem 2.1 is satisfied.

K*(v) = v(s) ~ K(s, t)u(t)dt

= ~) Kv(s , t)u(t)dt,

An approximate solvability scheme for a class of nonlinear equations 65

where the new kernel Kv(s, t) = v(s)K(s, t). In view of assumption (b) of Theorem 3.1, it follows that K~ (s, t) is a Hilbert-Schmidt

kernel and hence the integral operator generated by it is completely continuous. That is u, ~ u implies that

~ Kv(s, t)u,(t)dt ~ ~ K~(s, t)u(t)dt,

which in turn implies that K*(v)--,K*(v). Thus the family {K,}, u ~ A of linear operators on X satisfies all conditions of Theorem 2.1 and hence (6) is approximately solvable.

As a corollary of this theorem we obtain an approximiate solvability result for Chandrasekhar's H-equation

l + u ( s ) ~ " s *(t)u(t)dt = u(s). (10) j o s + t

Here, the known function W(t) is assumed to be non-negative, bounded and measurable. Since equation (10) is not given in the standard form we first state a lemma which is useful in this direction and for the proof refer [I] .

LEMMA.

Suppose that Jn W(t)dt ~< 1/2 and that u ~ L 2 is a positive solution of the equation

u(s) I - 2 W(t)dt +u(s) W(t)u(t)dt = 1. (11) o s + t

Then ~n W(s)u(s)ds = 1 - ( 1 - 2 Sn W(s)ds) a/2 and u is also a solution of (10). Equation (11) is now in the standard form (6) with

K(~, t) = W(t) and w(s) -- - (12) c

Here c = [ 1 / ( 1 - 2 S n W(t)dt) U2] (without loss in generality we can assume that In W(t)dt < 1/2, one can tackle the case In W(t)dt = 1/2 as a limiting case of strict inequality). Thus K and w given by (12) satisfy all the requirements of Theorem 3.1 and hence we get the following solvability result for Chandrasekhar's equation (10).

THEOREM 3.2.

Let Jn W(t)dt ~< 1/2, then Chandrasekhar's H-equation is approximately strongly solvable.

References

[1] Chandrasekhar S 1960 Radiative transfer (Dover: New York) [2] Joshi M C and Srikanth P N 1978 Proc. Indian Acad. Sci. (Math. Sci,) A$7 169

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