Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 18, 1-12; http://www.math.u-szeged.hu/ejqtde/

Positive Solutions for Systems of nth Order Three-point Nonlocal Boundary Value Problems

J. Henderson^{1} and S. K. Ntouyas^{2}

1 Department of Mathematics, Baylor University Waco, Texas 76798-7328 USA e-mail: Johnny Henderson@baylor.edu

2 Department of Mathematics, University of Ioannina 451 10 Ioannina, Greece email: sntouyas@uoi.gr

Abstract

Intervals of the parameter λ are determined for which there exist positive solutions for the system of nonlinear differential equations, u^{(n)} + λa(t)f(v) = 0, v^{(n)} + λb(t)g(u) = 0, for 0 < t < 1, and satisfying three-point nonlocal boundary conditions, u(0) = 0, u′(0) = 0, …, u^{(n−2)}(0) = 0, u(1) = αu(η), v(0) = 0, v′(0) = 0, …, v^{(n−2)}(0) = 0, v(1) = αv(η). A Guo-Krasnosel’skii fixed point theorem is applied.

Key words and phrases: Three-point nonlocal boundary value problem, system of differential equations, eigenvalue problem. AMS (MOS) Subject Classifications: 34B18, 34A34

1 Introduction

We are concerned with determining intervals of the parameter λ (eigenvalues) for which there exist positive solutions for the system of differential equations,

\begin{matrix}u^{(n)} + λa(t)f(v) = 0, 0 < t < 1, \\ v^{(n)} + λb(t)g(u) = 0, 0 < t < 1, \end{matrix}  (1)

satisfying the three-point nonlocal boundary conditions,

\begin{matrix}u(0) = 0, u′(0) = 0, …, u^{(n−2)}(0) = 0, u(1) = αu(η), \\ v(0) = 0, v′(0) = 0, …, v^{(n−2)}(0) = 0, v(1) = αv(η), \end{matrix}  (2)

where 0 < η < 1, 0 < αη^{n−1} < 1 and (A) f, g ∈ C([0, ∞), [0, ∞)),

(B) a, b ∈ C([0, 1], [0, ∞)), and each does not vanish identically on any subinterval,

(C) All of f_{0} := lim_{x→0^{+}} \frac{f(x)}{x}, g_{0} := lim_{x→0^{+}} \frac{g(x)}{x}, f_{∞} := lim_{x→∞} \frac{f(x)}{x} and g_{∞} := lim_{x→∞} \frac{g(x)}{x} exist as real numbers.

There is currently a great deal of interest in positive solutions for several types of boundary value problems. While some of the interest has focused on theoretical questions [5, 9, 13, 26], an equal amount of interest has been devoted to applications for which only positive solutions have meaning [1, 8, 17, 18]. While most of the above studies have dealt with scalar problems, some recent work has addressed questions of positive solutions for systems of boundary value problems [3, 12, 14, 15, 16, 19, 22, 25, 27, 30]. In addition, some studies have been directed toward positive solutions for nonlocal boundary value problems; see, for example, [4, 6, 10, 17, 18, 19, 21, 22, 20, 24, 26, 28, 29, 30].

Additional attention has been directed toward extensions to higher order problems, such as in [2, 4, 7, 8, 11, 23, 29]. Recently Benchohra et al. [3] and Henderson and Ntouyas [12] studied the existence of positive solutions of systems of nonlinear eigenvalue problems. Here we extend these results to eigenvalue problems for systems of higher order three-point nonlocal boundary value problems.

The main tool in this paper is an application of the Guo-Krasnosel’skii fixed point theorem for operators leaving a Banach space cone invariant [9]. A Green’s function plays a fundamental role in defining an appropriate operator on a suitable cone.

2 Some preliminaries

In this section, we state some preliminary lemmas and the well-known Guo-Krasnosel’skii fixed point theorem.

Lemma 2.1 [4] Let 0 < η < 1, 0 < αη^{n−1} < 1; then for any u ∈ C[0, 1] the following boundary value problem

u^{(n)}(t) = 0, 0 < t < 1  (3)
u(0) = 0, u′(0) = 0, …, u^{(n−2)}(0) = 0, u(1) = αu(η),  (4)

has a unique solution

u(t) = ∫_{0}^{1} k(t, s)u^{(n)}(s)ds

where k(t, s) : [0, 1] × [0, 1] → ℝ^{+} is defined by

k(t, s) = \{ \begin{matrix}\frac{a(η,s)t^{n−1}}{(n−1)!} , & 0 ≤ t ≤ s ≤ 1, \\ \frac{a(η,s)t^{n−1}+(t−s)^{n−1}}{(n−1)!} , & 0 ≤ s ≤ t ≤ 1, \end{matrix}  (5)

and

a(η, s) = \{ \begin{matrix}−\frac{(1−s)^{n−1}}{1−αη^{n−1}} , & η ≤ s, \\ −\frac{(1−s)^{n−1}−(η−s)^{n−1}}{1−αη^{n−1}} , & s ≤ η. \end{matrix}

Lemma 2.2 [4] Let 0 < α^{n−1} < 1. Let u satisfy u^{(n)}(t) ≤ 0, 0 < t < 1, with the nonlocal conditions (2). Then

inf_{t∈[η,1]} u(t) ≥ γ∥u∥,

where γ = min \{αη^{n−1}, \frac{α(1−η)}{1−αη}, η^{n−1}\} .

Define θ(s) = max_{t∈[0,1]} |k(t, s)|. From Lemma 1.2 in [4], we know that

|k(t, s)| ≥ γθ(s), t ∈ [η, 1], s ∈ [0, 1].  (6)

By simple calculation we have (see [11])

θ(s) = max_{t∈[0,1]} |k(t, s)| ≤ \frac{(1 − s)^{n−1}}{(1 − αη^{n−1})(n − 1)!}, s ∈ (0, 1).  (7)

We note that a pair (u(t), v(t)) is a solution of eigenvalue problem (1), (2) if, and only if,

u(t) = −λ ∫_{0}^{1} k(t, s)a(s)f (−λ ∫_{0}^{1} k(s, r)b(r)g(u(r))dr) ds, 0 ≤ t ≤ 1, (8)

where

v(t) = −λ ∫_{0}^{1} k(t, s)b(s)g(u(s))ds, 0 ≤ t ≤ 1.

Values of λ for which there are positive solutions (positive with respect to a cone) of (1), (2) will be determined via applications of the following fixed point theorem.

Theorem 2.1 Let B be a Banach space, and let P ⊂ B be a cone in B. Assume Ω_{1} and Ω_{2} are open subsets of B with 0 ∈ Ω_{1} ⊂ \overline{Ω}_{1} ⊂ Ω_{2}, and let

T : P ∩ (\overline{Ω}_{2} \ Ω_{1}) → P

be a completely continuous operator such that, either

(i) ||T u|| ≤ ||u||, u ∈ P ∩ ∂Ω_{1}, and ||T u|| ≥ ||u||, u ∈ P ∩ ∂Ω_{2}, or

(ii) ||T u|| ≥ ||u||, u ∈ P ∩ ∂Ω_{1}, and ||T u|| ≤ ||u||, u ∈ P ∩ ∂Ω_{2}.

Then T has a fixed point in P ∩ (\overline{Ω}_{2} \ Ω_{1}).

3 Positive solutions in a cone

In this section, we apply Theorem 2.1 to obtain solutions in a cone (that is, positive solutions) of (1), (2). For our construction, let B = C[0, 1] with supremum norm, ∥ · ∥, and define a cone P ⊂ B by

P = {x ∈ B | x(t) ≥ 0 on [0, 1], and min_{t∈[η, 1]} x(t) ≥ γ∥x∥} .

For our first result, define positive numbers L_{1} and L_{2} by

L_{1} := max {[γ^{2} ∫_{η}^{1} θ(r)a(r)f_{∞}dr]^{−1} , [γ^{2} ∫_{η}^{1} θ(r)a(r)g_{∞}dr]^{−1}} ,

and

L_{2} := min {[∫_{0}^{1} θ(r)a(r)f_{0}dr]^{−1} , [∫_{0}^{1} θ(r)b(r)g_{0}dr]^{−1}} .

Theorem 3.1 Assume conditions (A), (B) and (C) are satisfied. Then, for each λ satisfying

L_{1} < λ < L_{2},  (9)

there exists a pair (u, v) satisfying (1), (2) such that u(x) > 0 and v(x) > 0 on (0, 1).

Proof. Let λ be as in (9). And let ϵ > 0 be chosen such that

max {[γ^{2} ∫_{η}^{1} θ(r)a(r)(f_{∞} − ϵ)dr]^{−1} , [γ^{2} ∫_{η}^{1} θ(r)a(r)(g_{∞} − ϵ)dr]^{−1}} ≤ λ

and

λ ≤ min {[∫_{0}^{1} θ(r)a(r)(f_{0} + ϵ)dr]^{−1} , [∫_{0}^{1} θ(r)b(r)(g_{0} + ϵ)dr]^{−1}} .

Define an integral operator T : P → B by

T u(t) := −λ ∫_{0}^{1} k(t, s)a(s)f (−λ ∫_{0}^{1} k(s, r)b(r)g(u(r))dr) ds, u ∈ P.  (10)

We seek suitable fixed points of T in the cone P.

By Lemma 2.2, T P ⊂ P. In addition, standard arguments show that T is completely continuous.

Now, from the definitions of f_{0} and g_{0}, there exists an H_{1} > 0 such that

f(x) ≤ (f_{0} + ϵ)x and g(x) ≤ (g_{0} + ϵ)x, 0 < x ≤ H_{1}.

Let u ∈ P with ∥u∥ = H_{1}. We first have from (7) and choice of ϵ,

−λ ∫_{0}^{1} k(s, r)b(r)g(u(r))dr ≤ λ ∫_{0}^{1} θ(r)b(r)g(u(r))dr
≤ λ ∫_{0}^{1} θ(r)b(r)(g_{0} + ϵ)u(r)dr
≤ λ ∫_{0}^{1} θ(r)b(r)dr(g_{0} + ϵ)∥u∥
≤ ∥u∥
= H_{1}.

As a consequence, we next have from (7), and choice of ϵ,

T u(t) = −λ ∫_{0}^{1} k(t, s)a(s)f (−λ ∫_{0}^{1} k(s, r)b(r)g(u(r))dr) ds
≤ λ ∫_{0}^{1} θ(s)a(s)f (−λ ∫_{0}^{1} k(s, r)b(r)g(u(r))dr) ds
≤ λ ∫_{0}^{1} θ(s)a(s)(f_{0} + ϵ) [−λ ∫_{0}^{1} k(s, r)b(r)g(u(r))dr] ds
≤ λ ∫_{0}^{1} θ(s)a(s)(f_{0} + ϵ)H_{1}ds
≤ H_{1}
= ∥u∥.

So, ∥T u∥ ≤ ∥u∥. If we set

Ω_{1} = {x ∈ B | ∥x∥ < H_{1}},

then

∥T u∥ ≤ ∥u∥, for u ∈ P ∩ ∂Ω_{1}.  (11)

Next, from the definitions of f_{∞} and g_{∞}, there exists \overline{H}_{2} > 0 such that

f(x) ≥ (f_{∞} − ϵ)x and g(x) ≥ (g_{∞} − ϵ)x, x ≥ \overline{H}_{2}.

Let

H_{2} = max \{2H_{1}, \frac{\overline{H}_{2}}{γ} \} .

Let u ∈ P and ∥u∥ = H_{2}. Then,

min_{t∈[η,1]} u(t) ≥ γ∥u∥ ≥ \overline{H}_{2}.

Consequently, from (8) and choice of ϵ,

−λ ∫_{0}^{1} k(s, r)b(r)g(u(r))dr ≥ λγ ∫_{η}^{1} θ(r)b(r)g(u(r))dr
≥ λγ ∫_{η}^{1} θ(r)b(r)g(u(r))dr
≥ λγ ∫_{η}^{1} θ(r)b(r)(g_{∞} − ϵ)u(r)dr
≥ λγ ∫_{η}^{1} θ(r)b(r)(g_{∞} − ϵ)drγ∥u∥
≥ ∥u∥
= H_{2}.

And so, we have from (8) and choice of ϵ,

T u(η) ≥ λγ ∫_{η}^{1} θ(s)a(s)f (−λ ∫_{η}^{1} k(s, r)b(r)g(u(r))dr) ds
≥ λγ ∫_{η}^{1} θ(s)a(s)(f_{∞} − ϵ) [−λ ∫_{η}^{1} k(s, r)b(r)g(u(r))dr] ds
≥ λγ ∫_{η}^{1} θ(s)a(s)(f_{∞} − ϵ)H_{2}ds

≥ λγ^{2} ∫_{η}^{1} θ(s)a(s)(f_{∞} − ϵ)H_{2}ds
≥ H_{2}
= ∥u∥.

Hence, ∥T u∥ ≥ ∥u∥. So, if we set

Ω_{2} = {x ∈ B | ∥x∥ < H_{2}},

then

∥T u∥ ≥ ∥u∥, for u ∈ P ∩ ∂Ω_{2}.  (12)

Applying Theorem 2.1 to (11) and (12), we obtain that T has a fixed point u ∈ P ∩ (\overline{Ω}_{2} \ Ω_{1}). As such, and with v defined by

v(t) = −λ ∫_{0}^{1} k(t, s)b(s)g(u(s))ds,

the pair (u, v) is a desired solution of (1), (2) for the given λ. The proof is complete.□

Prior to our next result, we introduce another hypothesis. (D) g(0) = 0 and f is an increasing function. We now define positive numbers L_{3} and L_{4} by

L_{3} := max {[γ^{2} ∫_{η}^{1} θ(r)a(r)f_{0}dr]^{−1} , [γ^{2} ∫_{η}^{1} θ(r)a(r)g_{0}dr]^{−1}} ,

and

L_{4} := min {[∫_{0}^{1} θ(r)a(r)f_{∞}dr]^{−1} , [∫_{0}^{1} θ(r)b(r)g_{∞}dr]^{−1}} .

Theorem 3.2 Assume conditions (A)–(D) are satisfied. Then, for each λ satisfying

L_{3} < λ < L_{4},  (13)

there exists a pair (u, v) satisfying (1), (2) such that u(x) > 0 and v(x) > 0 on (0, 1). Proof. Let λ be as in (13). And let ϵ > 0 be chosen such that

max {[γ^{2} ∫_{η}^{1} θ(r)a(r)(f_{0} − ϵ)dr]^{−1} , [γ^{2} ∫_{η}^{1} θ(r)a(r)(g_{0} − ϵ)dr]^{−1}} ≤ λ

and

λ ≤ min {[∫_{0}^{1} θ(r)a(r)(f_{∞} + ϵ)dr]^{−1} , [∫_{0}^{1} θ(r)b(r)(g_{∞} + ϵ)dr]^{−1}} .

Let T be the cone preserving, completely continuous operator that was defined by (10).

From the definitions of f_{0} and g_{0}, there exists H_{1} > 0 such that

f(x) ≥ (f_{0} − ϵ)x and g(x) ≥ (g_{0} − ϵ)x, 0 < x ≤ H_{1}.

Now g(0) = 0 and so there exists 0 < H_{2} < H_{1} such that

λg(x) ≤ \frac{H_{1}}{∫_{0}^{1} θ(r)b(r)dr}, 0 ≤ x ≤ H_{2}.

Choose u ∈ P with ∥u∥ = H_{2}. Then

−λ ∫_{0}^{1} k(s, r)b(r)g(u(r))dr ≤ λ ∫_{0}^{1} θ(r)b(r)g(u(r))dr
≤ λ ∫_{0}^{1} θ(r)b(r)g(u(r))dr
≤ \frac{∫_{0}^{1} θ(r)b(r)H_{1}dr}{∫_{0}^{1} θ(r)b(s)ds}
≤ H_{1}.

Then, by (8) and (D)

T u(η) ≥ λγ ∫_{η}^{1} θ(s)a(s)f (λγ ∫_{η}^{1} θ(r)b(r)g(u(r))dr) ds
≥ λγ ∫_{η}^{1} θ(s)a(s)(f_{0} − ϵ)λγ ∫_{η}^{1} θ(r)b(r)g(u(r))drds
≥ λγ ∫_{η}^{1} θ(s)a(s)(f_{0} − ϵ)λγ^{2} ∫_{η}^{1} θ(r)b(r)(g_{0} − ϵ)∥u∥drds
≥ λγ ∫_{η}^{1} θ(s)a(s)(f_{0} − ϵ)∥u∥ds
≥ λγ^{2} ∫_{η}^{1} θ(s)a(s)(f_{0} − ϵ)∥u∥ds
≥ ∥u∥.

So, ∥T u∥ ≥ ∥u∥. If we put

Ω_{1} = {x ∈ B | ∥x∥ < H_{2}},

then

∥T u∥ ≥ ∥u∥, for u ∈ P ∩ ∂Ω_{1}.  (14)

Next, by definitions of f_{∞} and g_{∞}, there exists \overline{H}_{1} such that

f(x) ≤ (f_{∞} + ϵ)x and g(x) ≤ (g_{∞} + ϵ)x, x ≥ \overline{H}_{1}.

There are two cases, (a) g is bounded, and (b) g is unbounded.

For case (a), suppose N > 0 is such that g(x) ≤ N for all 0 < x < ∞. Then, for u ∈ P

−λ ∫_{0}^{1} k(s, r)b(r)g(u(r))dr ≤ Nλ ∫_{0}^{1} θ(r)b(r)dr.

Let

M = max {f(x) | 0 ≤ x ≤ Nλ ∫_{0}^{1} θ(r)b(r)dr} ,

and let

H_{3} > max {2H_{2}, Mλ ∫_{0}^{1} θ(s)a(s)ds} .

Then, for u ∈ P with ∥u∥ = H_{3},

T u(t) ≤ λ ∫_{0}^{1} θ(s)a(s)Mds
≤ H_{3}
= ∥u∥,

so that ∥T u∥ ≤ ∥u∥. If

Ω_{2} = {x ∈ B | ∥x∥ < H_{3}},

then

∥T u∥ ≤ ∥u∥, for u ∈ P ∩ ∂Ω_{2}.  (15)

For case (b), there exists H_{3} > max\{2H_{2}, \overline{H}_{1}\} such that g(x) ≤ g(H_{3}), for 0 < x ≤ H_{3}. Similarly, there exists H_{4} > max {H_{3}, λ ∫_{0}^{1} θ(r)b(r)g(H_{3})dr)} such that f(x) ≤ f(H_{4}), for 0 < x ≤ H_{4}. Choosing u ∈ P with ∥u∥ = H_{4}, we have by (D) that

T u(t) ≤ λ ∫_{0}^{1} θ(s)a(s)f (λ ∫_{0}^{1} θ(r)b(r)g(H_{3})dr) ds

≤ λ ∫_{0}^{1} θ(s)a(s)f(H_{4})ds
≤ λ ∫_{0}^{1} θ(s)a(s)ds(f_{∞} + ϵ)H_{4}
≤ H_{4}
= ∥u∥,

and so ∥T u∥ ≤ ∥u∥. For this case, if we let

Ω_{2} = {x ∈ B | ∥x∥ < H_{4}},

then

∥T u∥ ≤ ∥u∥, for u ∈ P ∩ ∂Ω_{2}.  (16)

In either of the cases, application of part (ii) of Theorem 2.1 yields a fixed point u of T belonging to P ∩ (\overline{Ω}_{2} \ Ω_{1}), which in turn yields a pair (u, v) satisfying (1), (2) for the chosen value of λ. The proof is complete. □

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(Received June 12, 2007)