COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 5, Number 2, June 2006
Website: http://AIMsciences.org pp. 251–259
QUALITATIVE BEHAVIOR AND COMPUTATION OF MULTIPLE SOLUTIONS OF NONLINEAR BOUNDARY VALUE PROBLEMS
Grey Ballard, John Baxley and Nisrine Libbus Department of Mathematics, Wake Forest University P.O. Box 7388, Winston-Salem, NC 27109, USA Abstract. We consider nonlinear boundary value problems with multiple solutions. A method is proposed for the computation of such solutions which depends crucially on known a priori qualitative information about the behavior of the solutions. The method is a two-stage method where the second stage is a shooting method and initial values of the shooting parameters are found in a first stage which approximates the boundary value problem with a discrete approximation. Both nonsingular and singular problems are considered.
1. Introduction. The investigation of boundary value problems for ordinary differential equations which have multiple positive solutions has attracted much attention in recent years. Some of these efforts may be found in [1, 2, 4, 5, 6, 11, 13, 14]. Earlier work, which provided motivation for many of the these later papers, appears in [9, 16, 17]. Some of the results have been extended to semilinear elliptic partial differential equations, as for example in [10]. Here we consider the problem of computing the solutions of such boundary value problems. Our interest in such computational questions date back to a study of the paper by Taliaferro [18] and led to the earlier work [7, 8] which focused on computing the unique positive solution of certain singular boundary value problems. Our efforts on problems with multiple solutions were motivated by the very interesting work in the paper by Parter [17] and our first exposure to the paper by Henderson and Thompson [13]; the question of the computation of such multiple solutions was raised also in a stimulating discussion with John Davis in 2001. The current paper falls into two distinct parts. In the first part, we develop a procedure for the approximation of multiple solutions of nonsingular boundary value problems. The theoretical existence of such multiple solutions is guaranteed by the papers already referenced. Then we pass to the question of multiple solutions of singular boundary value problems. Here the theory has not yet been satisfactorily developed, although we have partial results which we are in the process of completing and we hope to make such results available soon. We shall exhibit an example of a singular problem with multiple solutions and describe a procedure for computing the solutions. The proposed procedure is a hybrid of the procedure we provide for nonsingular problems in the first part of this paper and the procedure we used in [7, 8] for singular problems with unique positive solutions. 2000 Mathematics Subject Classification. Primary: 34B15, 65L10; Secondary: 65L12, 34B16, 34B18, 34A45. Key words and phrases. Multiple solutions, qualitative behavior, shooting, iterative methods, nonlinear boundary value problems.
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The common thread running through these procedures is the use of a priori qualitative knowledge of the solutions to assist in their computation. As we will describe in more detail later, such qualitative information was used earlier on singular problems to allow computation on a smaller interval, not containing singular endpoints. For nonsingular problems with multiple solutions, qualitative knowledge is used to distinguish between the multiple solutions and aid in the computation of a desired solution. Both ideas will be used here for singular problems with multiple solutions. At their heart, all these methods involve some version of Newton’s method, so the procedures require initial knowledge to be sure the desired solution is the attractor for the sequence of iterates. 2. Nonsingular Problems with Multiple Solutions. The basic idea of our method will be made clear by use of the following example: y (6) = f (y),
(2.1)
where
0 ≤ y ≤ 1, C, C + (11530 − C)(y − 1), 1 < y < 2, f (y) = 11530, 2 ≤ y ≤ 17, (0 < C < 720, C constant) with the boundary conditions y(0) = y ′ (0) = y ′′′ (0) = y (4) (0) = 0, y ′ (1) = y ′′′ (1) = 0.
(2.2)
(2.3)
It is easy to check that our f (y) satisfies f (y) < 720a = 720, for 0 ≤ y ≤ a = 1,
f (y) ≥ 5760b = 11520, for 2 = b ≤ y ≤ 4b = 8, 0 ≤ f (y) ≤ 720c = 12240, for 0 ≤ y ≤ c = 17. The results contained in [3] (using the techniques illustrated in [2]) guarantee that this problem has at least three positive nondecreasing solutions y1 , y2 , y3 with y1 (1) < 1, y2 (1/2) < 2, 1 < y2 (1) < 8, y3 (1/2) > 2, y3 (1) < 17. We will use this information to help compute these solutions. (Note: the solution y1 can be easily found in closed form, but we will temporarily ignore that fact. We will return to it later.) We would like to solve the problem (2.1), (2.3) by shooting. So we consider the initial value problem consisting of (2.1) and the initial conditions y(0) = y ′ (0) = y ′′′ (0) = y (4) (0) = 0, y ′′ (0) = m y (5) (0) = s,
(2.4)
where m, s are shooting parameters. Each of the three solutions we seek should correspond to a pair (m, s) so that the solution y of the initial value problem satisfies y ′ (1) = y ′′′ (1) = 0. Denoting the solution of the initial value problem by y(t; m, s), we then seek to solve the equations y ′ (1; m, s) = y ′′′ (1; m, s) = 0 for the three pairs (m, s). It is natural to attempt some form of Newton’s method in two dimensions, but in order to be successful we need to have reasonably close starting pairs for each of the three solutions (m, s). Viewing the Newton iteration as a dynamical system,
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we at least need to have starting pairs in the basin of attraction of each of the three solutions. Thus, we next describe a method of obtaining seed values for (m, s) within such a basin of attraction. As in the approach taken in [2, 3], we write our boundary value problem (2.1), (2.3) in the form (L3 ◦ L2 ◦ L1 )y = f (y), (2.5) where L1 is defined by L1 (y) = y ′ with domain D1 = {y ∈ C 1 [0, 1] : y(0) = 0},
L2 is defined by L2 (y) = −y ′′ with domain
D2 = {y ∈ C 2 [0, 1] : y(0) = y(1) = 0},
and L3 is defined by L3 (y) = −y ′′′ with domain
D3 = {y ∈ C 3 [0, 1] : y(0) = y ′ (0) = y(1) = 0}.
We then discretize the problem by dividing the interval [0, 1] into n + 1 equal parts at the mesh points ti = i/(n + 1) and seek to approximate y(ti ), for i = 1, 2, · · · n. Letting yi be our approximation for y(ti ) and Y be the n-dimensional column vector with components yi , we then approximate the problem (2.5) by approximating Li with the discrete operator (matrix) h−i Ai where h = 1/(n + 1) and
A1 =
1 0 −1 1 0 −1 .. .. . . 0 0
−2 1 0 1 −2 1 1 −2 , A2 = 0 .. .. .. . . . 0 0 0 1 −3 1 0 0 ··· 0 3 −3 1 0 ··· 0 −1 3 −3 1 · · · 0 . A3 = 0 −1 3 −3 · · · 0 .. .. .. .. .. .. . . . . . . 0 0 0 0 · · · −3 0 ··· 0 ··· 1 ··· .. .. . . 0 ···
0 0 0
0 ··· 0 ··· 1 ··· .. .. . . 0 ···
0 0 0 .. . −2
,
Note that h−1 A1 is the usual approximation for the first derivative using a backward first order difference quotient, h−2 A2 is the usual approximation for the second derivative using a central divided second order difference quotient, and h−3 A3 approximates the third derivative using a backward difference quotient of the central divided difference quotient. Note also that (A1 (Y ))i = yi − yi−1 ,
for i = 1, 2, · · · , n if and only if y0 = 0,
(A2 (Y ))i = yi+1 − 2yi + yi−1 ,
for i = 1, 2, · · · , n if and only if y0 = yn+1 = 0, and, finally,
(A3 (Y ))i = −yi−2 + 3yi−1 − 3yi + yi+1 ,
for i = 1, 2, · · · , n if and only if y−1 = y0 = yn+1 = 0. We interpret y−1 = y0 as an approximation of the condition y ′ (0) = 0. Thus, the boundary conditions are “built in” automatically.
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Our discretized problem is then (h−3 A3 )(h−2 A2 )(h−1 A1 )Y = F (Y ), where F (Y ) is the n-vector whose ith component is f (yi ). We can re-write this problem in fixed point form as 2 −1 3 −1 (hA−1 1 )(h A2 )(h A3 )F (Y ) = Y.
(2.6)
Note that we can think of each of these operators on the left as performing an approximate integration. We view (2.6) as a discrete approximation for the continuous problem (2.5). We expect that this discrete problem will also have three postive solutions which approximate the three solutions of the continuous problem (2.5). We then solve (2.6) by iterating. This iteration is a discrete dynamical system which should have three fixed points. Even if we could find the exact fixed points for (2.6), they would be only approximations for the three solutions of (2.5) which are the goal of the computation. We used n = 31 so that h = 1/32 and our matrices Ai are 31 × 31. We did the iteration using MATLAB. We began with an initial seed vector Y0 which we believe approximates one of the desired solutions and iterated until |Yn+1 (15) − Yn (15)| + |Yn+1 (31) − Yn (31)| < 0.01,
i.e., the combined differences of two consecutive iterates in their 15th and 31st components differed by less than .01, calling this final iterate Z. Since we have a priori knowledge that the three solutions are nondecreasing and y1 (1) < 1, 2 < y3 (1/2) < y3 (1) < 17, to approximate y1 , we seeded with a monotone increasing vector whose maximum is 0.9 and to approximate y3 , we seeded with a monotone increasing vector whose maximum is 12.0. We got convergence with these seeds without problem, requiring generally from 3 to 6 iterates. However, approximating the middle solution y2 gives difficulty. It turns out (as one might expect) that the discrete approximation for y2 is a repeller for the iteration scheme. So we must somehow back into this solution. We took our final approximations Z1 and Z3 for y1 and y3 and chose Y = (Z1 + Z3 )/2 as a rough approximation for y2 . We then did one iteration step beginning with Y . If the next iterate was moving down toward Z1 , we replaced Z1 with Y , otherwise we replaced Z3 with Y . We then repeated this process until Z1 and Z3 differed by less than .01 in their last components. This process required generally around a dozen iterates. Next, we iterated Z1 (or Z3 ) once to get our approximation for the middle solution. The reason for this last iteration is that our final Z1 (or Z3 ) is a repeated average of the original Z1 and Z3 and, although close to the desired approximation for y2 does not have the right shape (i.e., curvature). One iteration re-shapes the vector without much change in value and gives a much better qualitative approximation Z2 for y2 . When doing the iterations, we computed and saved the values of 3 −1 −1 (h2 A−1 A1 Y, 2 )(h A3 )F (Y ) = h
which is an approximation for y ′ at the mesh points, as well as −2 (h3 A−1 A2 )(h−1 A1 )Y, 3 )F (Y ) = (h
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which is an approximation for y ′′′ at the mesh points. Using these approximations, we can approximate y ′ (h) − y ′ (0) y ′ (h) y ′′′ (h) − y ′′′ (0) y ′′′ (h) = , y (4) (0) ≈ = , h h h h y ′′′ (2h) − y ′′′ (h) y (4) (h) − y (4) (0) y (4) (h) ≈ , y (5) (0) ≈ . h h Since the goal of the iteration just described was to estimate these derivatives at t = 0, it should be clear why the shape, as well as the magnitude, of the vectors Yj , j = 1, 2, 3, is important. So now we have good seed values for the pair (m, s) and can return to the shooting method. We can go to any solver of initial value problems (we used RKF45) and shoot with these initial seeds, using a modified Newton’s method to adjust until satisfied. The iteration scheme we used to get these seed values for (m, s) could be viewed as a “predictor” and the shooting method beginning with these seeds as a “corrector.” Note that although the “middle” solution Y2 of the discrete problem (2.6) is a repellor for the iteration method used there, the corresponding pair (m2 , s2 ) which provide initial conditions for the solution y2 of the boundary value problem (2.1), (2.3) is an attractor for the Newton iteration. We conclude with some computational results in the case that C = 670. Using our iteration scheme for the discrete problem (2.6), and using the approximation Z1 for Y1 as described above as an approximation for y1 , we computed the approximations y ′′ (0) ≈
y1′ (1/32) ≈ .125. y1′′′ (1/32) ≈ −.211, y1′′′ (2/32) ≈ −.613, and calculated (5)
y1′′ (0) ≈ 4.0, y1 (0) ≈ −195.6.
Similarly, for the larger solution y3 , we found y3′ (1/32) ≈ 1.657,
y3′′′ (1/32) ≈ −1.919, y3′′′ (2/32) ≈ −5.74,
and calculated (5)
y3′′ (0) ≈ 53.02, y3 (0) ≈ −1947.58.
Finally, for the middle solution y2 , we found y2′ (1/32) ≈ .1814,
y2′′′ (1/32) ≈ −.254, y2′′′ (2/32) ≈ −.740,
and calculated (5)
y2′′ (0) ≈ 5.805, y2 (0) ≈ −237.44.
Using these values to seed our shooting method with RKF45, we found solutions y1 , y2 , y3 for which (5)
y1′′ (0) = 3.7222, y1 (0) = −223.3333, y1 (1) = .930555, (5)
y2′′ (0) = 6.8055, y2 (0) = −303.2909, y2 (1) = 1.8211, (5)
y3′′ (0) = 44.8698, y3 (0) = −1702.5216, y3 (1) = 11.7235.
We asked RKF45 to compute with an accuracy of 6 decimal places and stopped the iteration when two iterates agreed to 4 decimal places at t = 1. We noted earlier that the solution y1 can be found in closed form. It is simply y1 (t) =
67t2 4 (t − 2t3 + 2), 72
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GREY BALLARD, JOHN BAXLEY AND NISRINE LIBBUS
from which you can calculate y1 (1) = 67/72 = .930555,
y1′′ (0) = 4(67)/72 = 3.72222,
y (5) (0) = −240(67)/72 = −223.3333, a very pleasant agreement with our computational result! Note the several approximations which are used. For example, Z1 is an approximation for the solution Y1 of the discrete problem (2.6), Y1 is in turn an approximation for the corresponding solution y1 of the problem (2.5) and also of the boundary value problem (2.1), (2.3). So the seed values we use for the initial conditions (m1 , s1 ) which produce the solution y1 have these two sources of error, plus the error incurred by replacing slopes of tangent lines by slopes of secant lines. Even so, these values are sufficiently close to the true values to be within the basin of attraction of the true values with the Newton iteration. 3. Singular Problems with Unique Solutions. We review briefly our earlier approach to the computation of such unique positive solutions of singular problems; the approach was based on early results of Taliaferro [18], followed by work of Paul Waltman and others. For our purposes here, it is enough to review the results of Taliaferro. In his short but beautiful paper, he examined the boundary value problem φ(t) y ′′ = − λ , 0 < t < 1, y y(0) = 0, y(1) = 0, where φ(t) is a continuous positive function on (0, 1). Taliaferro proved that this problem has a unique positive solution if and only Z 1 t(1 − t)φ(t) dt < ∞, 0
and, moreover, Z
′
A = lim+ y (t) is finite if and only if t→0
B = lim− y ′ (t) is finite if and only if t→1
Z
1/2 0
φ(t)t−λ dt < ∞,
1
1/2
φ(t)(1 − t)−λ dt < ∞.
In the case that this first limit is finite and φ(t) ∼ ctα as t → 0+ , then α > −2 (in order for existence), λ < α + 1 (in order for a finite slope at t = 0), and Taliaferro proved that ctα−λ+2 y(t) = At − A−λ (1 + o(1)) (3.1) (α − λ + 1)(α − λ + 2) as t → 0+ , with a similar statement holding as t → 1− . Similarly, if the second limit is finite and φ(t) ∼ d(1 − t)β as t → 1− , then β > −2, λ < β + 1, and y(t) = B(1 − t) − B −λ (1 + o(1))
d(1 − t)β−λ+2 (β − λ + 1)(β − λ + 2)
(3.2)
as t → 1− , Although Taliaferro did not discuss any numerical efforts in the paper, it seemed apparent that he was motivated by computational goals with these qualitative results. Taliaferro also derived the asymptotic behavior of solutions at the endpoints
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of the interval in the case that the slope of the solution at one or both endpoint is infinite, but they will not be needed here. It is clear that the qualitative information contained in the asymptotic formulas (3.1), (3.2) should be useful in computation. Basically, the unknown values of A, the slope of the solution at t = 0 and B, the corresponding unknown slope of the solution at t = 1, were used as shooting parameters in [7, 8], where dependable initial value solvers (both RKF45 by Shampine and Watts and the Adams code of Shampine and Gordon) were used to solve examples of the form φ(t) , t0 ≤ t ≤ 1/2 yλ where 0 < t0 is small, using initial conditions at t0 specified by (3.1), with A as the shooting parameter. Similarly, the terminal value problem for 1/2 ≤ t ≤ t1 , (t1 < 1), was solved, using terminal conditions at t1 specified by an asymptotic formula (3.2), with B as a shooting parameter. The goal was to choose A and B so that the corresponding solutions y1 (t, A) and y2 (t, B) met nicely at t = 1/2: y ′′ = −
y1 (1/2, A) = y2 (1/2, B),
y1′ (1/2, A) = y2′ (1/2, B).
Then these two equations for A and B were solved using a modified Newton’s method to generate a sequence of pairs (Ak , Bk ) converging to the desired values of (A, B). This procedure used the qualitative information provided by Taliaferro to replace the interval [0, 1] with a slightly smaller interval [t0 , t1 ] so that the singular endpoints were not involved in the computation. Further details can be found in [7, 8]. 4. Singular Problems with Multiple Solutions. We will now report on some work which is far from finished. Our method will be illustrated on the example y ′′ = −f (t, y),
y(0) = y(1) = 0, where f (t, y) =
√
0 ≤ y ≤ 1,
t(1−t) √ , p y
2
2 (2 − y)t(1 − t) + 400(y − 1), 1 < y < 2, 40, 2 ≤ y ≤ 6, Theoretical work in progress on such problems shows that this example has at least three postive solutions. Initial computational work on such problems has also been done. We will illustrate the procedure on this example. It was chosen because one solution (the smallest) has a closed form : it is just y(t) = t(1 − t), so we have a simple way to assess the accuracy of our computation. The idea is to find the three solutions by the same “predictor-corrector” approach we took on the nonsingular example with three solutions. Our theoretical results indicate that we can expect three positive solutions y1 , t2 , y3 which are symmetric about t = 1/2, are concave down so that the maximum occurs at 1/2, and satisfy y1 (1/2) < 1, y2 (1/2) > 1, y2 (1/4) < 2, y3 (1/2) < 6, y3 (1/4) > 2. This qualitative information can be used as before with the iteration scheme. Now the differential equation is just L2 y = f (t, y),
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GREY BALLARD, JOHN BAXLEY AND NISRINE LIBBUS
and the discrete approximation is h−2 A2 Y = F (T, Y ), which we write in the fixed point form h2 A−1 2 F (T, Y ) = Y, where T is the n-vector whose kth component is tk = 1/k and the kth component of F (T, Y ) is just f (tk , yk ). As expected, the smallest solution Y1 and the largest solution Y3 are attractors for this iteration scheme while the middle solution Y2 is a repeller. Using an iterative procedure similar to that used on the nonsingular problem above, we found an approximating vector Y for the largest solution y3 to be Y
=
(0.5275, 1.0545, 1.5723, 2.0606, 2.5098, 2.9200, 3.2911, 3.6231, 3.9161, 4.1700, 4.3848, 4.5606, 4.6973, 4.7950, 4.8536, 4.8731, 4.8536, 4.7950, 4.6973, 4.5606, 4.3848, 4.1700, 3.9161, 3.6231, 3.2911, 2.9200, 2.5098, 2.0606, 1.5723, 1.0545, 0.5275)
The kth component of this vector is an approximation of y3 (tk ), where tk = k/32. Thus y3 (1/32) ≈ .5275 and so an approximation for y3′ (0) ≈ 32(.5275) = 16.88. Note this vector indicates the maximum of y3 to be approximately 4.873. Similarly, we find that for the smallest solution, y1 (1/32) ≈ .0303 (note that the exact solution t(1 − t) evaluates to .03027 at t = 1/32) so that y1′ (1/32) ≈ .97. Also, the maximum value of this vector is .2500 = 1/4, exactly the value of t(1 − t) at t = 1/2. For the middle solution, we obtain y2 (1/32) ≈ .0993, so that y2′ (1/32) ≈ 3.178, and a maximum value of 1.2746. Also we find from these approximations that y2 (1/4) ≈ .765 < 2 and y3 (1/4) ≈ 3.623 > 2, in agreement with the theoretical inequalities. We can also approximate derivatives at t = 1 in a similar way. Of course, in this example, the solutions are symmetric about t = 1/2, so approximations at t = 1 are easily obtained from the above approximations at t = 0. Each of the three solutions satisfy the asymptotic formulas of Taliaferro that we discussed earlier. So we can use them with the above estimates on A = yk′ (0) and B = yk′ (1) for each k = 1, 2, 3 as seeds for RKF45 and a modified Newton’s method as in our earlier discussion of singular problems with unique solutions to find the more accurate estimates y1′ (0) ≈ .9999993, y1 (1/2) ≈ .2499999988, y2′ (0) ≈ 3.234586, y2 (1/2) ≈ 1.29115169,
y3′ (0) ≈ 16.844603, y3 (1/2) ≈ 4.867396447. As before, the computation was performed on a smaller interval, in this case [t0 , t1 ] = [.001, .999], and we asked that RKF45 compute to 7 decimal places accurately and that the iteration continue until the error of meeting at t = 1/2 be less than 10−5 . In actual fact, the code reported that the error of meeting at the last iteration was less than 10−6 . Since we know the exact solution y1 , we can see we probably have accuracy to six decimal places. REFERENCES [1] J. Baxley, and P. Carroll, Nonlinear boundary value problems with multiple positive solutions, Proceedings of the 4th International Conference on Dynamical Systems and Differential
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[2]
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Equations, supplement to Discrete and Continuous Dynamical Systems, American Institute of Mathematical Sciences (2003), 83–90. J.V. Baxley, M.E. Cunningham and M.K. McKinnon, Higher order boundary value problems with multiple solutions: examples and techniques, Proceedings of the 5th International Conference on Dynamical Systems and Differential Equations, supplement to Discrete and Continuous Dynamical Systems, American Institute of Mathematical Sciences (2005), 84–90. J.V. Baxley and M.E. Cunningham, Higher order boundary value problems with multiple solutions, in preparation. J.V. Baxley and L.J. Haywood, Nonlinear boundary value problems with multiple solutions, Nonlinear Anal., 47 (2001), 1187–1198. J.V. Baxley and L.J. Haywood, Multiple positive solutions of nonlinear boundary value problems, Dynam. Contin. Discrete Impuls. Systems Ser. A, 10 (2003), 157–168. J.V. Baxley and C.R. Houmand, Nonlinear higher order boundary value problems with multiple positive solutions, J. Math. Anal. Applic., 286 (2003), 682–691. J.V. Baxley, Numerical methods for singular boundary value problems, in Proceedings of the Third International Colloquium in Numerical Analysis, Plovdiv, Bulgaria, (D. Bainov and V. Covachev, ed.), VSP BV, 1995, 5–24. J.V. Baxley and H.B. Thompson, Boundary behavior and computation of solutions of singular nonlinear boundary value problems, Comm. Appl. Anal., 4 (2000), 207–226. K.J. Brown, M.M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475–486. P. Dr´ abek and S. Robinson, Multiple positive solutions for elliptic boundary value problems, Rocky Mountain J., to appear. J.R. Graef, C. Qian and B. Yang, Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations, Proc. Amer. Math. Soc., 131 (2003), 577–585. D. Guo and V. Lakshmikantham, “Nonlinear Problems in Abstract Cones,” Academic Press, Boston, 1988 J. Henderson and H.B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc., 128 (2000), 2373–2379. J. Henderson and B. Thompson, Existence of multiple solutions for some n-th order boundary value problems, Comm. Appl. Nonlinear Anal., 7 (2000), 55–62. M.A. Krasnosel’ski˘i, “Positive Solutions of Operator Equations,” Noordhoff, Groningen, 1964. R. Leggett and L. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana U. Math. J., 28 (1979), 673–688. S.V. Parter, Solutions of a differential equation arising in chemincal reactor processes, SIAM J. Appl. Math., 26 (1974), 687–716. S. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal., 3 (1979), 897– 904.
Received March 2005; revised June 2005. E-mail address:
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