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On a class of weakly regular singular two point boundary value problems—I

On a class of weakly regular singular two point boundary value problems—I
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  NnnlimwAna(,w~~, Thmy, Merhodr&Applications, Vol.Zl,No. l,pp. l-12,1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/96 15.00 + 0.00 0362-546X(95)00006-2 ON A CLASS OF WEAKLY REGULAR SINGULAR TWO POINT BOUNDARY VALUE PROBLEMS-I? R.K. PANDEY Department of Mathematics & Astronomy, Lucknow University, Lucknow 226 007, India (Received 8 August 1994; received in revised form 29 October 1994; received for publication 10 February 1995) Key words and phrases: Monotone iterative method, eigenfunction expansion. 1. INTRODUCTION Consider a class of singular two point boundary value problems - (p(x)y’)’ =p(x)f(x, y), O<xsb (1) y(O) =A, y(b) = B, (2) where A and B are finite constants. Using finite differences and splines second order and fourth order numerical methods have been developed by several authors ([ll-[7], [81 and [9]>. As far as existence-uniqueness is concerned, Ciarlet, Natterer and Varga [4] have suggested that if p(x) satisfies 6) p(x) > 0 in (0, b); (ii) p(x) E Ci(O, b); and (iii) l/p(x) E L,[O, b]. Then the above problems (1) and (2) can be reduced to a regular one by change of variable z(x) = / k/d 5). 0 The function p(x) =xa, (Y E [O,l), x E (0, l] does satisfy all the assumptions and so far p(x) = x a, (Y E [O, 1) we can reduce the singular boundary value problems (1) and (2) to a regular problem, and hence from the result of [lo] it follows that the singular problems (1) and (2) have unique solutions on [0, l] provided there exists a Lipschitz constant L satisfying If(x, y) -f(x,w>l < Lly - WI and L < {n- (1 -c&. (3) In the present paper we apply a monotone iterative method directly to the singular problems (1) and (2) to establish the existence-uniqueness of the problem and for p(x) =x~, cx E 10, 11, x E (0, 11 has a unique solution provided L <k,(a), (4) where k, is the first positive zero of J iJ?zj,2. Now as (Y approaches one, i.e. as the order of the singularity increases the upper bound for the constant L in (3) approaches zero while in (4) ,it is always a positive constant. In fact the positive constant k, 2 {r (1 - a)}’ for all tDedicated to Professor R. K S. Rathore.  R. K. PANDEY ‘I. When Singular Problem is Solved @irectly 2. When Singular Ptwbism is Reduced to Regular Problem Alpha Fig. 1 cy E (O,l> which can be seen from Fig. 1. Thus it is justified to consider the singular problem directly. In this work an existence theorem (theorem 4) has been established for the boundary value problems (1) and (2) in a region The function U,(X) satisfies MU, rp(x)f( x, u,), u,(O) = 0, u,(b) 2 B, where the operator A4 is defined by My = -(p(x)y’)’ and the function U,(X) satisfies the reverse differential inequalities. It is assumed that the function p(x) is analytic at x = 0 and there is a constant L, such that L,(y - w) ~f(x, y> -f( x,w) in the region D. The proof is accomplished by use of a sequence {u,} for which Mu,+1 -kpWUn+l =F(x,u,), u,+,(O) = 0, u,+,(b) =B, where F(x, y) =p(x)f(x, y> - kp(x)y. U n d er certain conditions on f(x, y> and the constant k, the sequence {u,(x)} with initial iterate u,(x) is a monotone nonincreasing sequence which converges uniformly to a solution u(x) of (1) and (2) in D. The analogous sequence {u,(x)}  On a class of weakly regular singular two point boundary value problems 3 with initial iterate u,(x) is a monotone nondecreasing sequence converging to a solution u(x) of (1) and (2) in the region D. Any other solution z(x) satisfies u(x) I z(x) I u(x). Under an additional condition on the constant L,, the solution is unique (theorem 5). 2. EXISTENCEANDUNIQUENESS OFTHE CORRESPONDING LINEAR BOUNDARYVALUEPROBLEM In this section we discuss existence-uniqueness of the linear boundary value problem My - kp(x)y =fCx>, 0 <x I b, y(O) = 0, y(b) = B, since the boundary conditions y(O) =A, y(b) = B can be reduced to y(O) = 0, y(b) = B without adding any more singularity to the differential equation. The function p(x) satisfies A-l: 6) p(x)> 0 on (0, b); (ii) p(x) E C1 (0, b), and for some r > b; (iii) xp’ (x)/p(x) is analytic in {z:lzl < r} with Taylor expansion xp’ (x)/p(x) = 6, + b,x + . . . , b, E K4 1). Under certain conditions on k, we have established that Green’s function for the boundary value problem exists and is unique (lemma 3 and corollary 2). Nonnegativity of the Green’s function is also shown (corollary 1). This property of the Green’s function works as a basic tool for establishing existence-uniqueness of the nonlinear problem in the Section 2. Preliminaries. Following the notations of [ll], let 4(x> = 4(x, h) and 8(x) = 6(x, h) be the solutions of y” + (A - q(x))y = 0, O<x<b, (5) q(x) is a continuous function on (0, b], such that db) = 0, 4’(b) = -1; e(b) = 1, e’(b) = 0 and a solution 0(x) + Z+(x) of (5) which satisfies a real boundary condition at x = a, 0 < a < b, as follows {e(a) + Z+(a)} cos p + {0’(a) + l+‘(a)} sin p = 0 (6) where p is real. Let m(h) be the limit of I(h) as a approaches to zero and we assume that the complex valued function m(h) is a single-valued analytic function whose singularities are simple poles (A,}~=0 on the real axis and corresponding residues are {r,}I=,. Then the results similar to the theorem 2.7 (i), (ii) and theorem 2.17 of [ll] can be obtained as stated in [ll, page 231. They are as follows: THEOREM 1. Let f(x) be the integral of an absolutely continuous function and let q(x)f(x) -f”(x) E &CO, b), f(b) = 0, and lim W(p(x, A), f(x)> = 0, x-0  4 R.K.PANDEY where rp(x, A) is an L, (0, b) solution of (1) for every nonreal A and W(cp, f) wronskian of cp and f. Then m f(x) = c c&&(x), Osxsb, n=O the series being absolutely and uniformly convergent on [O, b]. THEOREM 2. Let f(x) E L,(O, b). Then / b {f(x)}‘dx = f c,2. 0 n=O THEOREM 3. Let f(x) be in L,(O, b) and Wx, A) be the solution of y” + (A - q(x))y =fCx), 0 <x I b, satisfying y(b) = 0. Then for h not equal to any of the A, n=O where the series is absolutely convergent. Now consider the singular Sturm-Liouville problem Mz = hp(x)z, O<xsb (7) z(0) = 0, z(b) = 0. It is easy to see that all the eigenvalues of the Sturm-Liouville problem are real, positive and simple and corresponding eigenfunctions are orthogonal with respect to the inner product (f, g> = /” p(x)f(x)g(x) dx. 0 Next we will verify that the function m(A) corresponding to the differential equation (4) satisfying z(b) = 0 is a single-valued analytic function having singularities on the real axis. By taking y(x) = J(p(x)) z(x), the differential equation (7) can be reduced to (5) for q(x) = p2/4 + p’/2, /3(x) =p’(x)/p(x), p(x) E C2 (0, b) and p(x) > 0 on (0, bl. Define the functions 4(x, A) and 0(x, A) as $4x, A) = dy,(x)y,(b) -y&)y&b)l, 0(x, A) = d[y,(x)y;(b) -y2(x)y;Cb)l, where d = l/W(y,, y2), a nonzero constant and yi(x), y*(x) are two linearly-independent solutions of (5) with q(x) = p2/4 + /3’/2. Using the Frobenius series method it is easy to see that the solutions yi(x) and y,(x) are given by y,(x) = &?i7 (a, + a2x2 + . . . ), a, 2 0  On a class of weakly regular singular two point boundary value problems and y*(x) = &m X1-bo f c,xm ) c,fO, m=O 1 where the coefficients a,‘s and c,‘s depend on the integral power of A. Let 0(x) + I+(x) be a general solution satisfying the boundary condition (6) at x = a, then l( *) = _ e(a) cot p + e’(a) 44a> cot p + @(a> . (8) Now we will show that as a + 0, the limit m(h) of I( A) is a single-valued analytic function having simple poles on the real axis. As x + 0 we can approximate y, and y2 as follows yl(x) =u,vlpx> + O(X(*+bo/2)), y*(x) = COX(l-~~~~ + 0(x(2-b0’2)), y;(x) = (p’(x)/2\lpx))u, + 0(x” +bo’2’) y;(x) = (p’(x)/243iqc,x”-~O’ + )/&J (1 - b&,x-bo + 0(x(‘-b9. The numerator 0(a) cot (Y + 8’ (a) in (8) can be written as 8 (u)cot CY 8 (a) = d[{y, (a) cot a + y; my; (b) - {y,b)cot a + y; my; (b)l = d[ (a,m cota + (p’ (4’2~) u,)y;(b) - ( q)u(‘-b$@iY cot a + (p’(a)/2~jc,u”-b~) +fiu> (1 - b,)c,u-bo)y; (b) + O(u(2-b~‘2)(cotrzI) + o(u’*-b./2’)]. Let ( c,u(l-bO)Vlp(a)cotcu + (p’(u)/2~)c,~“-~of+ Jp(a)(l - b,)cOu-bo} =c(u,~Cota+ (p’(u)/2&qu,), where c is a constant, then cota = [((l - b,)u-bv, + (p’(u)/2p(u))~‘-~0c, -c~p’~u>/2p~u>~u,~/~u~c - C@+b9]. It is easy to see (i) for c 0, m and as a + 0 cot a = (1 - b,)u-bv,/uoc - (p’(u)/2p(u)) = 0(1/u); and
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