GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 5, 1999, 415-420

TRIPLE POSITIVE SOLUTIONS FOR MULTIPOINT CONJUGATE BOUNDARY VALUE PROBLEMS

JOHN M. DAVIS, PAUL W. ELOE, AND JOHNNY HENDERSON

Abstract. For the nth order nonlinear differential equation y^{(n)}(t) = f(y(t)), t ∈ [0, 1], satisfying the multipoint conjugate boundary conditions, y^{(j)}(a_{i}) = 0, 1 ≤ i ≤ k, 0 ≤ j ≤ n_{i} − 1, 0 = a_{1} < a_{2} < ⋯ < a_{k} = 1, and ∑_{i=1}^{k} n_{i} = n, where f : ℝ → [0, ∞) is continuous, growth condtions are imposed on f which yield the existence of at least three solutions that belong to a cone.

1. Introduction

Let n ≥ 2 be an integer and k ∈ {2, 3, …, n}. Let 0 = a_{1} < a_{2} < ⋯ < a_{k} = 1 be k points, n_{i} ∈ {1, 2, …, n − 1}, 1 ≤ i ≤ k, be such that ∑_{i=1}^{k} n_{i} = n and define α_{i} = ∑_{j=i+1}^{k−1} n_{j} where 1 ≤ i ≤ k. We are concerned with the existence of multiple solutions for the nth order multipoint conjugate boundary value problem

y^{(n)}(t) = f(y(t)),  0 ≤ t ≤ 1,  (1.1)
y^{(j)}(a_{i}) = 0,  1 ≤ i ≤ k, 0 ≤ j ≤ n_{i} − 1,  (1.2)

where f : ℝ → [0, ∞) is continuous. We will impose growth conditions on f which ensure the existence of at least three solutions of (1.1), (1.2) that belong to a cone.

Recent attention has been directed toward obtaining multiple solutions for boundary value problems (BVPs) for ordinary differential equations (ODEs). While his methods are not the same as those applied here, Brykalov [1–3] has established the existence of multiple solutions for certain nonlinear BVPs for ODEs. Closer to this work, we refer the reader to the papers of Avery [4], Chyan and Davis [5], Chyan, Davis, and Yin [6], Davis and

415 Henderson [7,8], Henderson and Thompson [9], and Guo and Lakshmikantham [10]. This paper can be considered a complete generalization of [7] which deals with triple positive solutions for two point conjugate BVPs (i.e., when k = 2). Multipoint problems for higher order ODEs (specifically, conjugate problems) and the existence of multiple positive solutions have been studied by Eloe and Henderson in [11].

1991 Mathematics Subject Classification. Primary: 34B10; Secondary: 34B15.

Key words and phrases. Multiple positive solutions, multipoint conjugate boundary value problem, fixed points.

1072-947X/99/0900-0415$16.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

For the most part, each of the papers on the existence of triple positive solutions makes an application of the fixed point theorem by Leggett and Williams [12] which was developed using the fixed point index in ordered Banach spaces. Leggett and Williams [12] applied their fixed point theorem to prove the existence of three positive solutions for Hammerstein integral equations of the form y(x) = ∫_{Ω} G(x, s)g(s, y(s)) ds, Ω ⊂ ℝ^{N}, by making use of suitable inequalities imposed on G and g.

In Section 2, we provide some definitions and background results, and we state the Leggett–Williams Fixed Point Theorem. Then in Section 3, we impose growth conditions on f which allow us to apply the Leggett– Williams Fixed Point Theorem in obtaining three solutions of (1.1), (1.2) that lie in a cone.

2. Background and Definitions

Our main results will hinge on an application of the Legget–Williams Fixed Point Theorem which deals with fixed points of a cone preserving operator. For the convenience of the reader, we include here the necessary definitions from cone theory in Banach spaces.

Definition 2.1. Let B be a Banach space over ℝ. A nonempty, closed set P ⊂ B is said to be acone provided

(a) αu + βv ∈ P for all u, v ∈ P and all α, β ≥ 0, and

(b) u, −u ∈ P implies u = 0.

Definition 2.2. A Banach space B is called apartially ordered Banach space if there exists a partial ordering ⪯ on B satisfying

(a) u ⪯ v, for boldu, v ∈ B implies tu ⪯ tv, for all t ≥ 0, and

(b) u_{1} ⪯ v_{1} and u_{2} ⪯ v_{2}, for u_{1}, u_{2}, v_{1}, v_{2} ∈ B imply u_{1}+u_{2} ⪯ v_{1}+v_{2}.

Let P ⊂ B be a cone and define u ⪯ v if and only if v − u ∈ P. Then ⪯ is a partial ordering on B and we will say that ⪯ is the partial ordering induced by P. Moreover, B is a partially ordered Banach space with respect to ⪯.

We also state the following definitions for future reference.

Definition 2.3. The map α is anonnegative continuous concave functional on P provided α : P → [0, ∞) is continuous and

α(tx + (1 − t)y) ≥ tα(x) + (1 − t)α(y)

TRIPLE POSITIVE SOLUTIONS FOR MULTIPOINT CONJUGATE BVPS 417

for all x, y ∈ P and 0 ≤ t ≤ 1.

Definition 2.4. Let 0 < a < b be given and α be a nonnegative continuous concave functional on the cone P. Define the convex sets P_{r} and P(α, a, b) by

P_{r} = {y ∈ P | ∥y∥ < r} and P(α, a, b) = {y ∈ P | a ≤ α(y), ∥y∥ ≤ b}.

Next we state the Leggett–Williams Fixed Point Theorem. The proof can be found in Deimling’s text [13] and utilizes the fixed point index in ordered Banach spaces.

Theorem 2.1 (Leggett–Williams Fixed Point Theorem). Let A : \overline{P}_{c} → \overline{P}_{c} be a completely continuous operator and let α be a nonnegative continuous concave functional on P such that α(y) ≤ ∥y∥ for all y ∈ \overline{P}_{c}. Suppose there exist 0 < a < b < d ≤ c such that

(C1) {y ∈ P(α, b, d) | α(y) > b} ≠ ∅ and α(Ay) > b for y ∈ P(α, b, d), (C2) ∥Ay∥ < a for ∥y∥ ≤ a, and (C3) α(Ay) > b for y ∈ P(α, b, c) with ∥Ay∥ > d.

Then A has at least three fixed points y_{1}, y_{2}, and y_{3} such that ∥y_{1}∥ < a, b < α(y_{2}), and ∥y_{3}∥ > a with α(y_{3}) < b.

3. Triple Positive Solutions

In this section, we will impose growth conditions on f which allow us to apply Theorem 2.1 in regard to obtaining three solutions of (1.1), (1.2). We will apply Theorem 2.1 to a completely continuous operator whose kernel is the Green’s function, G(t, s), for

y^{(n)}(t) = 0,  (3.1)

satisfying the boundary conditions (1.2). It is fairly well known [14] that

(−1)^{α_{i}}G(t, s) > 0, for (t, s) ∈ (a_{i}, a_{i+1}) × (0, 1), 1 ≤ i ≤ k − 1. (3.2)

For s ∈ (0, 1), define

∥G(·, s)∥ = max_{t∈[0,1]} |G(t, s)|.  (3.3)

Eloe and Henderson [15] proved the following theorem which is a key estimate for our main result.

Theorem 3.1. Suppose y ∈ C^{(n)}[0, 1] is such that y^{(n)}(t) ≥ 0, t ∈ [0, 1], and that y satisfies the multipoint conjugate boundary conditions (1.2). Then, for each 1 ≤ i ≤ k − 1,

(−1)^{α_{i}}G(t, s) ≥ (a/4)^{m}∥G(·, s)∥,  t ∈ S_{i}, s ∈ (0, 1),  (3.4)

where S_{i} = [\frac{3a_{i}+a_{i+1}}{4} , \frac{a_{i}+3a_{i+1}}{4} ], a = min_{1≤i≤k−1}\{a_{i+1} − a_{i}\}, and m = max\{n −

n_{1}, n − n_{k}}.

Next, we define

K = ( max_{t∈[0,1]} ∫_{0}^{1} G(t, s) ds)^{−1} ,  (3.5)

L = ( min_{1≤i≤k−1} min_{t∈S_{i}} ∫_{S_{i}} G(t, s) ds)^{−1} .  (3.6)

Let B denote the Banach space C[0, 1] with the maximum norm ∥y∥ = max_{t∈[0,1]} |y(t)| and define the cone P ⊂ B by

P := {y ∈ B | (−1)^{α_{i}}y(t) ≥ 0, t ∈ [a_{i}, a_{i+1}] for 1 ≤ i ≤ k − 1} .

Let α : P → [0, ∞) be the nonnegative continuous concave functional

α(y) = min_{1≤i≤k−1} min_{t∈S_{i}} |y(t)|, for y ∈ P,

and let A : B → B be the operator

Ay(t) = ∫_{0}^{1} G(t, s)f(y(s)) ds.

We now present the main result of the paper.

Theorem 3.2. Let 0 < a < b < (\frac{4}{a})^{m} b ≤ c be such that f satisfies (i) f(w) < Ka, for 0 ≤ |w| ≤ a, (ii) f(w) ≥ Lb, for b ≤ |w| ≤ (\frac{4}{a})^{m} b, and (iii) f(w) ≤ Kc, for 0 ≤ |w| ≤ c.

Then, the boundary value problem (1.1), (1.2) has three positive solutions y_{1}, y_{2}, and y_{3} satisfying

∥y_{1}∥ < a, b < min_{1≤i≤k−1} min_{t∈S_{i}} |y_{2}(t)|, ∥y_{3}∥ > a with min_{1≤i≤k−1} min_{t∈S_{i}} |y_{3}(t)| < b.

Proof. We seek fixed points of A which satisfy the conclusion of the theorem. We observe first from the positivity of f and (3.2) that, for y ∈ P, (−1)^{α_{i}}Ay(t) ≥ 0 for t ∈ [a_{i}, a_{i+1}]. Thus, A : P → P.

We now show that the conditions of Theorem 2.1 are satisfied. Choose y ∈ \overline{P_{c}}. Then, ∥y∥ ≤ c and by assumption (iii), f(y(s)) ≤ Kc, s ∈ [0, 1]. Thus, from (3.5)

∥Ay∥ = max_{t∈[0,1]} ∫_{0}^{1} G(t, s)f(y(s)) ds ≤ max_{t∈[0,1]} ∫_{0}^{1} |G(t, s)|f(y(s)) ds

≤ max_{t∈[0,1]} ∫_{0}^{1} |G(t, s)|Kc ds = c.

Hence, A : \overline{P_{c}} → \overline{P_{c}}. In a similar way, if y ∈ \overline{P_{a}}, then assumption (i) yields f(y(s)) < Ka, s ∈ [0, 1], and it follows as above that A : \overline{P_{a}} → P_{a}. Consequently, condition (C2) of Theorem 2.1 is fulfilled.

To verify property (C1) of Theorem 2.1, we note that if we let

x(t) = \begin{cases}(−1)^{α_{i}}(4/a)^{m}b, & t ∈ S_{i}, 1 ≤ i ≤ k − 1, \\ \frac{x(a_{i+1})−x(a_{i})}{a_{i+1}−a_{i}}(t − a_{i}) + x(a_{i}), & t ∈ \overline{[0, 1] \ S_{i}}, 1 ≤ i ≤ k − 1, \end{cases}

then x(t) ∈ P(α, b, (\frac{4}{a})^{m} b). Moreover, α(x) = (\frac{4}{a})^{m} b > b. Hence

{y ∈ P(α, b, (4/a)^{m}b)|α(y)>b}≠∅.

Furthermore, if we choose y ∈P(α, b, (\frac{4}{a})^{m} b), then

α(y) = min_{1≤i≤k−1} min_{t∈S_{i}} |y(t)| ≥ b,

and so b ≤ |y(s)| ≤ (\frac{4}{a})^{m} b, s ∈ S_{i}, 1 ≤ i ≤ k − 1. Thus, for any y ∈ P(α, b, (\frac{4}{a})^{m} b), assumption (ii) yields f(y(s)) ≥ Lb, s ∈ S_{i}, 1 ≤ i ≤ k − 1, and by (3.6) we have

α(Ay) = min_{1≤i≤k−1} min_{t∈S_{i}} |Ay| =

= min_{1≤i≤k−1} min_{t∈S_{i}} ∫_{0}^{1}(−1)^{α_{i}}G(t, s)f(y(s)) ds >

> min_{1≤i≤k−1} min_{t∈S_{i}} ∫_{S_{i}}(−1)^{α_{i}}G(t, s)f(y(s)) ds ≥

≥ min_{1≤i≤k−1} min_{t∈S_{i}} ∫_{S_{i}}(−1)^{α_{i}}G(t, s)Lb ds = b.

Hence, condition (C1) of Theorem 2.1 is satisfied.

We finally exhibit that (C3) of Theorem 2.1 is satisfied. (In particular, we show, if y ∈ P(α, b, c) and ∥Ay∥ > (\frac{4}{a})^{m} b, then α(Ay) > b.) Thus we choose y ∈ P(α, b, c) such that ∥Ay∥ > (\frac{4}{a})^{m} b. Then, from (3.4),

α(Ay) = min_{1≤i≤k−1} min_{t∈S_{i}} ∫_{0}^{1}(−1)^{α_{i}}G(t, s)f(y(s)) ds ≥

≥ (\frac{a}{4})^{m} ∫_{0}^{1} ∥G(·, s)∥f(y(s)) ds ≥

≥ (\frac{a}{4})^{m} max_{t∈[0,1]} ∫_{0}^{1} |G(t, s)|f(y(s)) ds =

= (\frac{a}{4})^{m} ∥Ay∥ > b,

and (C3) of Theorem 2.1 is satisfied. Hence an application of Theorem 2.1 completes the proof.

References

1. S. A. Brykalov, Solvability of a nonlinear boundary value problem in a fixed set of functions. (Russian) Differentsial’nye Uravneniya 27(1991), 2027–2933.

2. S. A. Brykalov, A second-order nonlinear problem with two-point and integral boundary conditions. Georgian Math. J. 1(1994), No. 3, 243–249.

3. S. A. Brykalov, Solutions with a given maximum and minimum. (Russian) Differentsial’nye Uravneniya 29(1993), 938–942.

4. R. Avery, Existence of multiple positive solutions to a conjugate boundary value problem. MSR Hotline, 2(1998), No. 1.

5. C. J. Chyan and J. M. Davis, Existence of triple positive solutions for (n, p) and (p, n) boundary value problems. Preprint.

6. C. J. Chyan, J. M. Davis, and W. K. C. Yin, Existence of triple positive solutions for (k, n − k) right focal boundary value problems. Preprint.

7. J. M. Davis and J. Henderson, Triple positive solutions for (k, n − k) conjugate boundary value problems. Preprint.

8. J. M. Davis and J. Henderson, Triple positive symmetric solutions for a Lidstone boundary value problem. Preprint.

9. J. Henderson and H.B. Thompson, Existence of multiple solutions for some nth order boundary value problems. Preprint.

10. D. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones. Academic Press, San Diego, 1988.

11. P. W. Eloe and J. Henderson, Positive solutions and nonlinear multipoint conjugate eigenvalue problems. Electron. J. Differential Equations, (1997), No. 3, 1–11.

12. R. Leggett and L. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28(1979), 673–688.

13. K. Deimling, Nonlinear functional analysis. Springer-Verlag, New York, 1985.

14. W. Coppel, Disconjugacy. Lecture Notes in Mathematics 220, Springer-Verlag, Berlin and New York, 1971.

15. P. W. Eloe and J. Henderson, Inequalities for solutions of multipoint boundary value problems. Rocky Mountain J. Math. (in press).

(Received 14.12.1998)

Authors’ addresses:

John M. Davis, Johnny Henderson Department of Mathematics Auburn University Auburn, AL 36849 USA Paul W. Eloe Department of Mathematics University of Dayton Dayton, OH 45469 USA