THREE SYMMETRIC POSITIVE SOLUTIONS FOR LIDSTONE PROBLEMS BY A GENERALIZATION OF THE LEGGETT-WILLIAMS THEOREM

Electronic Journal of Differential Equations Vol. ( No. 4 pp. 1 15. ISSN: 17-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ejde.math.unt.edu (login: ftp THREE SYMMETRIC POSITIVE SOLUTIONS FOR LIDSTONE PROBLEMS BY A GENERALIZATION OF THE LEGGETT-WILLIAMS THEOREM RICHARD I. AVERY JOHN M. DAVIS & JOHNNY HENDERSON Abstract. We study the existence of solutions to the fourth order Lidstone boundary value problem y (4 (t f(y(t y (t y( y ( y (1 y(1. By imposing growth conditions on f and using a generalization of the multiple fixed point theorem by Leggett and Williams we show the existence of at least three symmetric positive solutions. We also prove analogous results for difference equations. 1. Introduction First we are concerned with the existence of multiple solutions for the fourth order Lidstone boundary value problem (BVP y (4 (t f(y(t y (t t 1 (1 y( y ( y (1 y(1 ( where f : R R [ is continuous. We will impose growth conditions on f which ensure the existence of at least three symmetric positive solutions of (1 (. There is much current attention focused on questions of positive solutions of boundary value problems for ordinary differential equations as well as for for finite difference equations; see [ 5 6 8 1 11 1 13 17 18 19 ] to name a few. Much of this interest is due to the applicability of certain fixed point theorems of Krasnosel skii [15] or Leggett and Williams [16] to obtain positive solutions or multiple positive solutions which lie in a cone. The recent book by Agarwal O Regan and Wong [1] gives a good overview for much of the work which has been done and the methods used. Mathematics Subject Classification. 34B18 34B7 39A1 39A99. Key words. Lidstone boundary value problem Green s function multiple solutions fixed points difference equation. c Southwest Texas State University and University of North Texas. Submitted December 6 1999. Published May 3. 1

R.I. AVERY J.M. DAVIS AND J. HENDERSON EJDE /4 In [3] Avery imposed conditions on g which yield at least three positive solutions to the second order conjugate BVP y (t+g(y(t t 1 (3 y( y(1 (4 using the Leggett-Williams Fixed Point Theorem. Henderson and Thompson [14] improved these results by using the symmetry of the associated Green s function and then Avery and Henderson [5] established similar results by applying the Five Functionals Fixed Point Theorem [4] (which is a generalization of the Leggett- Williams Fixed Point Theorem to obtain the existence of three positive solutions of certain BVPs. Davis Eloe and Henderson [8] imposed conditions on f to yield at least three positive solutions to the mth order Lidstone BVPs by applying the Leggett- Williams Fixed Point Theorem. Note that [8] is the only work which has allowed f to depend on higher order derivatives of y. This paper is in the same spirit as [5] and [8] since we apply the Five Functionals Fixed Point Theorem [4] and also allow f to depend on y. This derivative dependence generalizes [9] as well. In Section we provide some background results and state the Five Functionals Fixed Point Theorem. In Section 3 we impose growth conditions on f which allow us to apply this theorem in obtaining three symmetric positive solutions of (1 (. In Section 4 we prove discrete analogs of the results in Section 3.. Some Background Definitions and Results In this section we provide some background material from the theory of cones in Banach spaces in order that this paper be self-contained. We also state the Five Functionals Fixed Point Theorem for cone preserving operators. Definition 1. Let E be a Banach space over R. A nonempty closed set P Eis a cone provided (a αu + βv P for all u v P and all α β and (b u u P implies u. Definition. A Banach space E is a partially ordered Banach space if there exists a partial ordering on E satisfying (a u v foruv E implies tu tv for all t and (b u 1 v 1 and u v foru 1 u v 1 v E imply u 1 + u v 1 + v. Let P Ebeacone and define u v if and only if v u P. Then is a partial ordering on E and we will say that is the partial ordering induced by P. Moreover E is a partially ordered Banach space with respect to. We also state the following definitions for future reference. Definition 3. The map α is a nonnegative continuous concave functional on P provided α : P [ is continuous and α(tx +(1 ty tα(x+(1 tα(y for all x y Pand t 1. Similarly we say the map β is a nonnegative continuous convex functional on P provided β : P [ is continuous and β(tx +(1 ty tβ(x+(1 tβ(y for all x y P and t 1.

EJDE /4 THREE SYMMETRIC POSITIVE SOLUTIONS 3 Definition 4. An operator A is completely continuous if A is continuous and compact i.e. A maps bounded sets into precompact sets. Let γ β θ be nonnegative continuous convex functionals on P and α ψ be nonnegative continuous concave functionals on P. Then for nonnegative numbers h a b d andc we define the following convex sets: P (γc{x P γ(x<c} P(γαac{x P a α(xγ(x c} Q(γβdc{x P β(x d γ(x c} P (γθαabc{x P a α(x θ(x b γ(x c} Q(γβψhdc{x P h ψ(x β(x d γ(x c}. In obtaining multiple symmetric positive solutions of (1 ( the following socalled Five Functionals Fixed Point Theorem will be fundamental. Theorem 1. [4] Let P be a cone in a real Banach space E. Suppose α and ψ be nonnegative continuous concave functionals on P and γ β andθbe nonnegative continuous convex functionals on P such that for some positive numbers c and m α(x β(x and x mγ(x for all x P (γc. Suppose further that A : P (γc P(γc is completely continuous and there exist constants h d a b with <d<asuch that each of following is satisfied: (C1 {x P (γθαabc α(x >a} and α(ax >afor x P (γθαabc (C {x Q(γβψhdc β(x<d} and β(ax <dfor x Q(γβψhdc (C3 α(ax >aprovided x P (γαac with θ(ax >b (C4 β(ax <dprovided x Q(γβdc with ψ (Ax <h. Then A has at least three fixed points x 1 x x 3 P(γc such that β(x 1 <d a<α(x and d<β(x 3 with α(x 3 <a. 3. Three Symmetric Positive Solutions In this section we will impose growth conditions on f which allow us to apply Theorem 1 in regard to obtaining three symmetric positive solutions of (1 (. We will apply Theorem 1 in conjunction with a completely continuous operator whose kernel G(t s is the Green s function for v satisfying (4. In particular G(t s { t(1 s t s 1 s(1 t s t 1.

4 R.I. AVERY J.M. DAVIS AND J. HENDERSON EJDE /4 We will make use of various properties of G(t s namely r 1 r t t(1 t G(t s ds t 1 G(1/sds G(1/sds 1 1 r 1 r 1 t1 G(1/sds 1 <r (5 4r G(1/sds r 4 <r (6 16r G(t 1 sds + G(t 1 sds t 1 (t t 1 <t 1 <t <1/ (7 G(1/r 1 r 1 G(t r t <t 1/ (8 G(t 1 r r 1 G(t r t 1 t <t 1 <t 1/. (9 Let E C[ 1] be endowed with the imum norm v t 1 v(t and for <t 3 <1/ define the cone P Eby { P v E v(tv(1 t v(t v(tisconcaveforallt [ 1] } and v(t t 3 v. t [t 31 t 3] Finally we define the nonnegative continuous concave functionals α ψ and the nonnegative continuous convex functionals β θ γ on P by γ(v α(v ψ(v β(v v(t v(t 3 t [t 3] [1 t 31] t [ 1 r 1 1 r] v(tv(1/r t [ 1 r 1 1 r] v(tv(1/ v(t v(t 1 t [t 1t ] [1 t 1 t 1] θ(v v(t v(t t [t 1t ] [1 t 1 t 1] where t 1 t andrare nonnegative numbers such that We observe here that for each v P <t 1 <t < 1 and 1 r t. α(vv(t 1 v(1/ β(v (1 v v(1/ v(t 3 t 3 1 t 3 γ(v (11

EJDE /4 THREE SYMMETRIC POSITIVE SOLUTIONS 5 and also that y Pis a solution of (1 ( if and only if there exists a v Psuch that y(t G(t sv(s ds t 1 where v is of the form ( v(t G(t sf G(s τv(τdτ v(s ds t 1. In light of these preliaries we are now ready to present the main result of this section. Theorem. Suppose there exist <a<b< t t 1 b csuch that f satisfies each of the following growth conditions: ( ( (G1 f(zw < a (G f(zw [ ] b t t 1 b (G3 f(zw 8r r 4 b t 1(t t 1 c r t 3(1 t 3 for (zw [ ] [ c t for (zw c c 3(1 t 3 16t 3 t 3 ]. [ ] for all (zw a(r r a 3 8 [ a r a] [ ] bt 1 (t t 1 c(t 1 +t(1 t 4t 3 + bt(t t 1 t 1 Then the Lidstone BVP (1 ( has at least three symmetric positive solutions y 1 y y 3 suchthat (t c for i 13 and t [t y i 3] [1 t 31] t [t y 1 (t >b 1t ] [1 t 1 t 1] t [ 1 r 1 1 r ] y (t <a t [t y 3 1t ] [1 t 1 t 1] for some v 1 v v 3 P satisfying y i (t (t <b with t [ 1 r 1 1 r ] y 3 G(t sv i (s ds i 13. Proof. Define the completely continuous operator A by where Av(t Bv(s G(t sf(bv(sv(s ds G(s τv(τ dτ. (t >a We seek three fixed points v 1 v v 3 Pof A which satisfy the conclusion of the theorem. We note first that if v P then from the properties of G(t s Av(t Bv(t (Av (t f(b(v(tv(t t 1 Av(t 3 t 3 Av(1/ and Av(t Av(1 t t 1/. Consequently A : P P. Also for all v P by (1 we have α(v β(v and by (11 v 1 t 3 γ(v. If v P (γc then v 1 t 3 γ(v c t 3 which implies that for s [ 1] [ ] [ ] c c v(s and Bv(s. t 3 16t 3

6 R.I. AVERY J.M. DAVIS AND J. HENDERSON EJDE /4 By condition (G3 we have γ(av t [t 3] [1 t 31] Therefore A : P (γc P(γc. It is immediate that { ( v P G (t 3 sf(bv(sv(s ds ( c 1 G (t 3 s ds t 3 (1 t 3 c. { v Q b c t 1 γθαb t ( γβψ a r ac G(t sf(bv(sv(s ds } α(v >b } β(v<a and thus the first parts of (C1 and (C are satisfied. In order to show the second part of (C1 holds let v P (γθαb t t 1 b c. For each s [t 1 t ] [1 t 1 t 1 ] we have b v(s t t 1 b. Hence bt 1 (t t 1 Bv(s c(t 1 + t (1 t 4t 3 + bt (t t 1 t 1 and by condition (G and (7 we see α(av t [t 1t ] [1 t 1 t 1] t G (t 1 sf(bv(sv(s ds G(t sf(bv(sv(s ds t1 > G (t 1 sf(bv(sv(s ds + G (t 1 sf(bv(sv(s ds t 1 1 t ( b t ( b 1 t1 G (t 1 s ds + G (t 1 s ds t 1 (t t 1 t 1 t 1 (t t 1 1 t ( ( b t1 [(1 t 1 (1 t ] + t 1(t t 1 t 1 (t t 1 b. To verify the second part of (C let v Q(γβψ a r ac. This implies that for each s [ 1 r 1 ] 1 r wehave v(s [ ] a r a and Bv(s [ a(r r 3 a ]. 8

EJDE /4 THREE SYMMETRIC POSITIVE SOLUTIONS 7 Thus by condition (G1 and the calculations in (5 and (6 β(av t [ 1 r 1 1 r] r G(t sf(bv(sv(s ds G (1/sf(Bv(sv(s ds G(1/sf(Bv(sv(s ds + 1 r ( 8r c < r t 3 (1 t 3 + a. r 4 ( a c r t 3 (1 t 3 G(1/sf(Bv(sv(s ds ( r 4 To show (C3 holds suppose v P (γαbc withθ(av > t t 1 b. Using (9 we get α(av G(t sf(bv(sv(s ds t [t 1t ] [1 t 1 t 1] t 1 t G (t 1 sf(bv(sv(s ds G (t 1 s G(t s G(t sf(bv(sv(s ds G (t sf(bv(sv(s ds t 1 θ(ay t >b. Finally to show (C4 we take v Q(γβac withψ(av < a r. Using (8 we have β(av G(t sf(bv(sv(s ds t [ 1 r 1 1 r] G (1/sf(Bv(sv(s ds 8r G (1/s G (1/r s f(bv(sv(s ds G(1/r s r G (1/r s f(bv(sv(s ds r ψ(ay <a. Therefore the hypotheses of Theorem 1 are satisfied and there exist three positive solutions y 1 y andy 3 for the Lidstone BVP (1 (. Moreover these solutions are of the form y i (t G(t sv i (s ds i 13 for some v 1 v v 3 P.

8 R.I. AVERY J.M. DAVIS AND J. HENDERSON EJDE /4 Remark. We have chosen to perform the analysis when f is autonomous. However if f f(t w z and in addition for each (w z f(t w z is symmetric about t 1 then an analogous theorem would be valid with respect to the same cone P. 4. Discrete Analogs Motivated by the early multiple solutions results for difference equations [1 6] and specifically papers involving difference equations satisfying Lidstone boundary conditions [7] we want to extend Theorem to discrete problems. To this end we will again impose growth conditions on f which allow us to apply Theorem 1 and obtain three symmetric positive solutions of the discrete fourth order Lidstone BVP 4 y(t f(y(t y(t 1 a+ t b+ (1 y(a y(a y(b+y(b+4 (13 where f : R R [ is continuous. We will apply Theorem 1 in conjunction with a completely continuous operator whose kernel G 1 (t s is the Green s function for w(t 1 w(a +1w(b+3. In particular (t a 1(b +3 s a+1 t s b+3 G 1 (t s b+ a (s a 1(b +3 t a+1 s t b+3. b+ a We will write our solutions of (1 (13 in the form y(t b+1 sa+1 G (t sv(s where G (t s is the Green s function for w(t 1 w(a w(b+4 and v is a fixed point of a completely continuous operator with kernel G 1 (t s. In particular (t a(b +4 s a t s b+4 G (t s b+4 a (s a(b +4 t a s t b+4. b+4 a We will make use of various properties of G (t s andg 1 (t s namely b+3 sa+1 G (t s G 1 (t s (t a(b +4 t a t b+4 (t a 1(b +3 t a+1 t b+3

EJDE /4 THREE SYMMETRIC POSITIVE SOLUTIONS 9 3 sa+k 3 G(t m sc 1 (14 a+k 3 1 a+k G 1 (t m s+ sa+k 1 G 1 (t 1 s+ 3 sa+k 3 G (t 3 sc 4 a+k sa+k 1 G (t 1 s+ a+k 1 1 sa+1 a+k G (t m s+ sa+k 1 G (t m s+ G 1 (t m r a+ r b+ G 1 (t r a+ r b+ b+3 sb+5 k 3 G 1 (t m sc (15 1 s G 1 (t 1 sc 3 (16 1 s G (t 1 sc 5 b+3 k sa+k +1 1 G (t m s+ s G (t m sc 7 b+3 sb+5 k 1 G (t m sc 6 t m a 1 t a 1 a+ t t m (17 G 1 (t r G 1 (t r t a 1 t a 1 a+ t t t m (18 For completeness we have calculated and included the values of the above constants. C 1 (t m a 1(b +3 t m (t 3 a 1 ( C (t 3 a 1 ( C 3 (t 1 a[(t a ( +(b+4 t 1 ( (t 1 a 1 ( (b+3 t ( ] (b + a C 4 (t 3 a[(b +5 t 3 ( (t 3 a ( ] (b +4 a C 5 (t 1 a[(t +1 a ( +(b+5 t 1 ( (t 1 a ( (b+4 t ( ] (b +4 a C 6 (b+4 t m(t m a+(t 1 a ( (t a +1 ( C 7 (t a +1 ( (t 1 a ( Define b +4+a t m

1 R.I. AVERY J.M. DAVIS AND J. HENDERSON EJDE /4 and b +4 a M. Let E 1 {y y :[a b +4] R}be endowed with the imum norm y 1 y(t a t b+4 and for k M lett a+k and define the cone P 1 E 1 by y E 1 such that y(a + k y( for all k [M] y(t y(t 1 for all a +1 t b+3 P 1 (. and y(t t a y 1 t [a+k ] t m a Similarly let E {y y :[a+1b+3] R} be endowed with the imum norm v v(t a+1 t b+3 and define the cone P E by v E such that v(a +1+kv(b+3 k for all k [M 1] v(t v(t 1 for all a + t b+ P (. and v(t t a 1 v t [a+k ] t m a 1 Finally we define the nonnegative continuous concave functionals α ψ and the nonnegative continuous convex functionals β θ γ on P by γ(v ψ(v β(v α(v v(t v(t t [a+1a+k ] [ b+3] v(t v(t 3 t [a+k 3 3] v(t v(t m t [a+k 3 3] v(t v(t 1 t [a+k 1a+k ] [ 1] θ(v v(t v(t t [a+k 1a+k ] [ 1] where t 1 a + k 1 t + a + k and t 3 a + k 3 are nonnegative numbers such that We observe here that for each v P k 1 k k 3 [M]andk 1 k. α(vv(t 1 v(t m β(v (19 ( ( tm a 1 tm a 1 v v(t m v(t γ(v ( t a 1 t a 1 and also that y P 1 is a solution of (1 (13 if and only if there exists a v P such that y(t b+3 sa+1 G (t sv(s a t b+4

EJDE /4 THREE SYMMETRIC POSITIVE SOLUTIONS 11 where v is of the form v(t G 1 (t sf ( b+3 τ a+1 G (s τv(τv(s a+1 t b+3. In light of these preliaries we are now ready to present the main result of this section. ( Theorem 3. Suppose there exist < a < b < t a 1 t 1 a 1 b c such that f satisfies each of the following growth conditions: (Γ1 f(zw < a c C 1 C (t a 1(b+3 t C 1 for all (zw in [ a ] [ (t 3 a 1C 4 a (t m a(b +4 t m a ] (t 3 a 1 (t m a 1 t m a 1 a C 3 for all (zw in [ b C 5 b (t a 1C 7 + (t 1 a 1 (Γ f(zw < b (Γ3 f(zw c (t a 1(b+3 t [ c (t m a ( (b +4 t m (t a 1 c C 6 (t a 1(b +3 t for all (zw in ] ] [ b t ] a 1 t 1 a 1 b [ c (t m a 1 t a 1 Then the Lidstone BVP (1 (13 has at least three symmetric positive solutions y 1 y y 3 P 1 suchthat t [a+1a+k y i (t 1 c for i 13 ] [ b+3] t [a+k y 1 (t 1 >b 1a+k ] [ 1] t [a+k y (t 1 <a 3 3] and t [a+k y 3 (t 1 <b 1a+k ] [ 1] with t [a+k y 3 (t 1 >a 3 3] for some v 1 v v 3 P satisfying y i (t G (t sv i (s i 13. Proof. Define the completely continuous operator A by where Av(t G 1 (t sf(bv(sv(s Bv(s b+3 τ a+1 G (s τv(τ. ]. a+1 t b+3

1 R.I. AVERY J.M. DAVIS AND J. HENDERSON EJDE /4 We seek three fixed points v 1 v v 3 P of A which satisfy the conclusion of the theorem. We note first that if v P then from the properties of G 1 (t s and G (t s it follows that Av(t Bv(t and (Av(t 1 f(bv(tv(t a+ t b+ (Bv(t 1 v(t ( t a 1 t m a 1 a+ t b+ Av(t m and Av(a+k Av(for1 k M. Moreover Av(t Consequently A : P P. Also for all v P by (19 we have α(v β(v and by ( ( tm a 1 v γ(v. t a 1 If v P (γc then v ( tm a 1 γ(v c (t m a 1 t a 1 (t a 1 which implies that for s [a +1b+3] [ ] v(s c (t m a 1 and Bv(s t a 1 By condition (Γ3 we have γ(av t [a+1a+k ] [ b+3] ( c. [ c (t m a ( ] (b +4 t m. (t a 1 G 1 (t sf(bv(sv(s c t a 1(b +3 t Therefore A : P (γc P(γc. It is immediate that { v P (γθαb b (t a 1 b+ G 1 (t sf(bv(sv(s G 1 (t s } t 1 a 1 c : α(v >b { v Q(γβψ a (t 3 a 1 t m a 1 a c : β(v <a } and thus the first parts of (C1 and (C are satisfied. In order to show the second part of (C1 holds let v P (γθαb b (t a 1 t 1 a 1 c. For each s [a + k 1 a+k ] [ 1 ] we have b v(s b (t a 1 t 1 a 1. Hence b C 5 Bv(s b (t a 1C 7 c C 6 + (t 1 a 1 (t a 1(b +3 t

EJDE /4 THREE SYMMETRIC POSITIVE SOLUTIONS 13 and by condition (Γ and (16 we see α(av t [a+k 1a+k ] [ 1] a+k G 1 (t 1 sf(bv(sv(s G 1 (t sf(bv(sv(s 1 G 1 (t 1 sf(bv(sv(s + G 1 (t 1 sf(bv(sv(s sa+k 1 s ( ( b a+k 1 > G 1 (t 1 s+ G 1 (t 1 s C 3 sa+k 1 s b. To verify the second part of (C let v Q(γβψ a (t 3 a 1 t m a 1 a c. This implies that for each s [a + k 3 3 ] we have a (t 3 a 1 t m a 1 v(s a and a (t 3 a 1C 4 (t m a 1 Bv(s a (t m a(b +4 t m. Thus by condition (Γ1 and the calculations in (14 and (15 β(av t [a+k 3 3] a+k 3 1 G 1 (t m sf(bv(sv(s G 1 (t sf(bv(sv(s 3 G 1 (t m sf(bv(sv(s + G 1 (t m sf(bv(sv(s sa+k 3 + G 1 (t m sf(bv(sv(s sb+5 k 3 ( c < (t a 1(b +3 t a. ( a C + C 1 c C (t a 1(b +3 t C 1 C 1

14 R.I. AVERY J.M. DAVIS AND J. HENDERSON EJDE /4 To show (C3 holds suppose v P (γαb c withθ(av > b (t a 1 t. Using 1 a 1 (18 we get α(av t [a+k 1a+k ] [ 1] t 1 a 1 t a 1 G 1 (t 1 sf(bv(sv(s ( G1 (t 1 s G 1 (t sf(bv(sv(s G 1 (t s t 1 a 1 t a 1 θ(av >b. G 1 (t sf(bv(sv(s G 1 (t sf(bv(sv(s Finally to show (C4 we take v Q(γβa c withψ(av < a (t 3 a 1 t. Using m a 1 (17 we have β(av t [a+k 3 3] ( tm a 1 t 3 a 1 ( tm a 1 t 3 a 1 G 1 (t m sf(bv(sv(s G 1 (t sf(bv(sv(s ( G1 (t m s G 1 (t 3 sf(bv(sv(s G 1 (t 3 s b+ ψ(av G 1 (t 3 sf(bv(sv(s <a. The hypotheses of Theorem 1 are satisfied and as a result there exist three positive solutions y 1 y andy 3 P 1 for the Lidstone BVP (1 (13. Moreover these solutions are of the form for some v 1 v v 3 P. y i (t b+1 sa+1 G (t sv i (s i 13 Remark. We have chosen to perform the analysis when f is autonomous. However if f f(t w z and in addition for each (w z f(t w z is symmetric about t t m then an analogous theorem would be valid with respect to the same cone P.

EJDE /4 THREE SYMMETRIC POSITIVE SOLUTIONS 15 References [1] R.P. Agarwal D. O Regan and P.J.Y. Wong Positive Solutions of Differential Difference and Integral Equations Kluwer Academic Publishers Boston 1999. [] R.P. Agarwal and P.J.Y. Wong Double positive solutions of (n p boundary value problems for higher order difference equations Comput. Math. Appl. 3 (1996 1 1. [3] R.I. Avery Existence of multiple positive solutions to a conjugate boundary value problem Math. Sci. Res. Hot-Line (1998 1 6. [4] R.I. Avery A generalization of the Leggett-Williams fixed point theorem Math. Sci. Res. Hot-Line 3 (1999 9 14. [5] R.I. Avery and J. Henderson Three symmetric positive solutions for a second order boundary value problem Appl. Math. Lett. in press. [6] R.I. Avery and A.C. Peterson Multiple positive solutions of a discrete second order conjugate problem Panamer. Math. J. 8 (1998 1 1. [7] J.M. Davis P.W. Eloe and J. Henderson Comparison of eigenvalues for discrete Lidstone boundary value problems Dynam. Systems Appl. 8 (1999 381 387. [8] J.M. Davis P.W. Eloe and J. Henderson Triple positive solutions and dependence on higher order derivatives J. Math. Anal. Appl. 37 (1999 71 7. [9] J.M. Davis and J. Henderson Triple positive symmetric solutions for a Lidstone boundary value problem Differential Equations Dynam. Systems 7 (1999 31 33. [1] P.W. Eloe and J. Henderson Positive solutions for (n 1 1 conjugate boundary value problems Nonlinear Anal. 8 (1997 1669 168. [11] L.H. Erbe Boundary value problems for ordinary differential equations Rocky Mountain J. Math. 17 (1991 1 1. [1] L.H. Erbe S. Hu and H. Wang Multiple positive solutions of some boundary value problems J. Math. Anal. Appl. 184 (1994 64 648. [13] L.H. Erbe and H. Wang On the existence of positive solutions of ordinary differential equations Proc. Amer. Math. Soc. 1 (1994 743 748. [14] J. Henderson and H.B. Thompson Multiple symmetric positive solutions for a second order boundary value problem Proc.Amer.Math.Soc. in press. [15] M.A. Krasnosel skii Positive Solutions of Operator Equations Noordhoff Groningen 1964. [16] R.W. Leggett and L.R. Williams Multiple positive fixed points of nonlinear operators on ordered Banach spaces Indiana Univ. Math. J. 8 (1979 673 688. [17] F. Merdivenci Positive solutions for focal point problems for nth order difference equations Panamer. Math. J. 5 (1995 71 8. [18] F. Merdivenci Two positive solutions of a boundary value problem for difference equations J. Differ. Equations Appl. 1 (1995 6 7. [19] P.J.Y. Wong Triple positive solutions of conjugate boundary value problems Comput. Math. Appl. 36 (1998 19 35. [] P.J.Y. Wong and R.P. Agarwal Multiple solutions of difference and partial difference equations with Lidstone conditions Math. Comput. Modelling in press. Richard I. Avery College of Natural Sciences Dakota State University Madison SD 574 USA E-mail address: averyr@pluto.dsu.edu John M. Davis Department of Mathematics Baylor University Waco TX 76798 USA E-mail address: John M Davis@baylor.edu Johnny Henderson Department of Mathematics Auburn University Auburn AL 36849 USA E-mail address: hendej@mail.auburn.edu