Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 5, 1-6; http://www.math.u-szeged.hu/ejqtde/

EXISTENCE OF A POSITIVE SOLUTION TO A RIGHT FOCAL BOUNDARY VALUE PROBLEM

RICHARD I. AVERY, JOHNNY HENDERSON AND DOUGLAS R. ANDERSON

Abstract. In this paper we apply the recent extension of the Leggett-Williams Fixed Point Theorem which requires neither of the functional boundaries to be invariant to the second order right focal boundary value problem. We demonstrate a technique that can be used to deal with a singularity and provide a non-trivial example.

1. Introduction

The recent topological proof and extension of the Leggett-Williams fixed point theorem [3] does not require either of the functional boundaries to be invariant with respect to a functional wedge and its proof uses topological methods instead of axiomatic index theory. Functional fixed point theorems (including [2, 4, 5, 6, 8]) can be traced back to Leggett and Williams [7] when they presented criteria which guaranteed the existence of a fixed point for a completely continuous map that did not require the operator to be invariant with regard to the concave functional boundary of a functional wedge. Avery, Henderson, and O’Regan [1], in a dual of the Leggett-Williams fixed point theorem, gave conditions which guaranteed the existence of a fixed point for a completely continuous map that did not require the operator to be invariant relative to the concave functional boundary of a functional wedge. We will demonstrate a technique to take advantage of the added flexibility of the new fixed point theorem for a right focal boundary value problem.

2. Preliminaries

In this section we will state the definitions that are used in the remainder of the paper.

Definition 1. Let E be a real Banach space. A nonempty closed convex set P ⊂ E is called a cone if it satisfies the following two conditions:

(i) x ∈ P, λ ≥ 0 implies λx ∈ P; (ii) x ∈ P, −x ∈ P implies x = 0.

Every cone P ⊂ E induces an ordering in E given by

x ≤ y if and only if y − x ∈ P.

Definition 2. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

EJQTDE, 2010 No. 5, p. 1 Definition 3. A map α is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E if α : P → [0, ∞) is continuous and

2000 Mathematics Subject Classification. 34B10.

Key words and phrases. Fixed-point theorems, positive solutions, singularities, cone.

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

for all x, y ∈ P and t ∈ [0, 1]. Similarly we say the map β is a nonnegative continuous convex functional on a cone P of a real Banach space E if β : P → [0, ∞) is continuous and

β(tx + (1 − t)y) ≤ tβ(x) + (1 − t)β(y)

for all x, y ∈ P and t ∈ [0, 1].

Let α and ψ be non-negative continuous concave functionals on P and δ and β be non-negative continuous convex functionals on P; then, for non-negative real numbers a, b, c and d, we define the following sets:

(1)  A := A(α, β, a, d) = {x ∈ P : a ≤ α(x) and β(x) ≤ d},
(2)  B := B(α, δ, β, a, b, d) = {x ∈ A : δ(x) ≤ b},

and

(3)  C := C(α, ψ, β, a, c, d) = {x ∈ A : c ≤ ψ(x)}.

We say that A is a functional wedge with concave functional boundary defined by the concave functional α and convex functional boundary defined by the convex functional β. We say that an operator T : A → P is invariant with respect to the concave functional boundary, if a ≤ α(T x) for all x ∈ A, and that T is invariant with respect to the convex functional boundary, if β(T x) ≤ d for all x ∈ A. Note that A is a convex set. The following theorem is an extension of the original Leggett-Williams fixed point theorem [7].

Theorem 4. [Extension of Leggett-Williams] Suppose P is a cone in a real Banach space E, α and ψ are non-negative continuous concave functionals on P, δ and β are non-negative continuous convex functionals on P, and for non-negative real numbers a, b, c and d the sets A, B and C are as defined in (1), (2) and (3). Furthermore, suppose that A is a bounded subset of P, that T : A → P is completely continuous and that the following conditions hold:

(A1) {x ∈ A : c < ψ(x) and δ(x) < b} ≠ ∅ and {x ∈ P : α(x) < a and d < β(x)} = ∅; (A2) α(T x) ≥ a for all x ∈ B; (A3) α(T x) ≥ a for all x ∈ A with δ(T x) > b; (A4) β(T x) ≤ d for all x ∈ C; and, (A5) β(T x) ≤ d for all x ∈ A with ψ(T x) < c.

Then T has a fixed point x^{∗} ∈ A.

3. Right Focal Boundary Value Problem

In this section we will illustrate the key techniques for verifying the existence of a positive solution for a boundary value problem using the newly developed extension of the Leggett-Williams fixed point theorem, applying the properties of a Green’s function, bounding the

EJQTDE, 2010 No. 5, p. 2 nonlinearity by constants over some intervals, and using concavity to deal with a singularity. Consider the second order nonlinear focal boundary value problem

x′′(t) + f(x(t)) = 0, t ∈ (0, 1), (4)

x(0) = 0 = x′(1), (5)

where f : ℝ → [0, ∞) is continuous. If x is a fixed point of the operator T defined by

T x(t) := ∫_{0}^{1} G(t, s)f(x(s))ds,

where

G(t, s) = \begin{cases}t & : t ≤ s, \\ s & : s ≤ t, \end{cases}

is the Green’s function for the operator L defined by

Lx(t) := −x′′,

with right-focal boundary conditions

x(0) = 0 = x′(1),

then it is well known that x is a solution of the boundary value problem (4), (5). Throughout this section of the paper we will use the facts that G(t, s) is nonnegative, and for each fixed s ∈ [0, 1], the Green’s function is nondecreasing in t.

Define the cone P ⊂ E = C[0, 1] by

P := {x ∈ E : x is nonnegative, nondecreasing, and concave} .

For fixed ν, τ, μ ∈ [0, 1] and x ∈ P, define the concave functionals α and ψ on P by

α(x) := min_{t∈[τ,1]} x(t) = x(τ), ψ(x) := min_{t∈[μ,1]} x(t) = x(μ),

and the convex functionals δ and β on P by

δ(x) := max_{t∈[0,ν]} x(t) = x(ν), β(x) := max_{t∈[0,1]} x(t) = x(1).

In the following theorem, we demonstrate how to apply the Extension of the Leggett-Williams Fixed Point Theorem (Theorem 4), to prove the existence of at least one positive solution to (4), (5).

Theorem 5. If τ, ν, μ ∈ (0, 1] are fixed with τ ≤ μ < ν ≤ 1, d and m are positive real numbers with 0 < m ≤ dμ and f : [0, ∞) → [0, ∞) is a continuous function such that

(a) f(w) ≥ \frac{d}{ν−τ} for w ∈ [τd, νd],

(b) f(w) is decreasing for w ∈ [0, m] with f(m) ≥ f(w) for w ∈ [m, d], and

(c) ∫_{0}^{μ} s f (\frac{ms}{μ}) ds ≤ \frac{2d−f(m)(1−μ^{2})}{2},

then the operator T has at least one positive solution x^{∗} ∈ A(α, β, τd, d).

EJQTDE, 2010 No. 5, p. 3 Proof. Let a = τd, b = νd = \frac{aν}{τ}, and c = dμ. Let x ∈ A(α, β, a, d) then if t ∈ (0, 1), by the properties of the Green’s function (T x)′′(t) = −f(x(t)) and T x(0) = 0 = (T x)′(1), thus

T : A(α, β, a, d) → P.

We will also take advantage of the following property of the Green’s function. For any y, w ∈ [0, 1] with y ≤ w we have

(6)  min_{s∈[0,1]} \frac{G(y, s)}{G(w, s)} ≥ \frac{y}{w}.

By the Arzela-Ascoli Theorem it is a standard exercise to show that T is a completely continuous operator using the properties of G and f, and by the definition of β, we have that A is a bounded subset of the cone P. Also, if x ∈ P and β(x) > d, then by the properties of the cone P,

α(x) = x(τ) ≥ (\frac{τ}{1}) x(1) = τβ(x) > τd = a.

Therefore,

{x ∈ P : α(x) < a and d < β(x)} = ∅.

For any K ∈ (\frac{2d}{2−μ}, \frac{2d}{2−ν} ) the function x_{K} defined by

x_{K}(t) ≡ ∫_{0}^{1} KG(t, s)ds = \frac{Kt(2 − t)}{2} ∈ A,

since

α(x_{K}) = x_{K}(τ) = \frac{Kτ(2 − τ)}{2} > \frac{dτ(2 − τ)}{2 − μ} ≥ dτ = a,

β(x_{K}) = x_{K}(1) = \frac{K}{2} < \frac{d}{2 − ν} ≤ d,

and x_{K} has the properties that

ψ(x_{K}) = x_{K}(μ) = \frac{Kμ(2 − μ)}{2} > (\frac{2d}{2 − μ}) (\frac{μ(2 − μ)}{2}) = dμ = c

and

δ(x_{K}) = x_{K}(ν) = \frac{Kν(2 − ν)}{2} < (\frac{2d}{2 − ν} ) (\frac{ν(2 − ν)}{2}) = dν = b.

Hence

{x ∈ A : c < ψ(x) and δ(x) < b} ≠ ∅.

Claim 1: α(T x) ≥ a for all x ∈ B.

Let x ∈ B. Thus by condition (a),

α(T x) = ∫_{0}^{1} G(τ, s) f(x(s)) ds ≥ (\frac{a}{τ(ν − τ)}) ∫_{τ}^{ν} G(τ, s) ds
= (\frac{a}{τ(ν − τ)}) (τ(ν − τ)) = a.

Claim 2: α(T x) ≥ a, for all x ∈ A with δ(T x) > b. Let x ∈ A with δ(T x) > b. Thus by the properties of G (6),

α(T x) = ∫_{0}^{1} G(τ, s) f(x(s)) ds ≥ (\frac{τ}{ν} ) ∫_{0}^{1} G(ν, s) f(x(s)) ds
= (\frac{τ}{ν} ) δ(T x) > (\frac{τ}{ν} ) (dν) = a.

Claim 3: β(T x) ≤ d, for all x ∈ C. Let x ∈ C, thus by the concavity of x, for s ∈ [0, μ] we have

x(s) ≥ \frac{cs}{μ} ≥ \frac{ms}{μ}.

Hence by properties (b) and (c),

β(T x) = ∫_{0}^{1} G(1, s) f(x(s)) ds = ∫_{0}^{1} s f(x(s)) ds
= ∫_{0}^{μ} s f(x(s)) ds + ∫_{μ}^{1} s f(x(s)) ds
≤ ∫_{0}^{μ} s f (\frac{ms}{μ}) ds + f(m) ∫_{μ}^{1} s ds
≤ \frac{2d − f(m)(1 − μ^{2})}{2} + \frac{f(m)(1 − μ^{2})}{2} = d.

Claim 4: β(T x) ≤ d, for all x ∈ A with ψ(T x) < c. Let x ∈ A with ψ(T x) < c. Thus by the properties of G (6),

β(T x) = ∫_{0}^{1} G(1, s) f(x(s)) ds ≤ (\frac{1}{μ}) ∫_{0}^{1} G(μ, s) f(x(s)) ds
= (\frac{1}{μ}) T x(μ) = (\frac{1}{μ}) ψ(T x) ≤ (\frac{1}{μ}) c = d.

Therefore, the hypotheses of Theorem 4 have been satisfied; thus the operator T has at least one positive solution x^{∗} ∈ A(α, β, a, d). □ We note that because of the concavity of solutions, the proof of Theorem 5 remains valid for certain singular nonlinearities as presented in this example. Example: Let

d = \frac{5}{4}, τ = \frac{1}{16}, μ = \frac{3}{4}, and ν = \frac{15}{16}.

EJQTDE, 2010 No. 5, p. 5 Then the boundary value problem

x′′ + \frac{1}{\sqrt{x}} + \sqrt{x} = 0,

with right-focal boundary conditions

x(0) = 0 = x′(1),

has at least one positive solution x^{∗} which can be verified by the above theorem, with

5/64 ≤ x^{∗}(1/16) and x^{∗}(1) ≤ 5/4.

References

[1] R. I. Avery, J. Henderson and D. O’Regan, Dual of the compression-expansion fixed point theorems, Fixed Point Theory and Applications, 2007 (2007), Article ID 90715, 11 pages.

[2] D. R. Anderson and R. I. Avery, Fixed point theorem of cone expansion and compression of functional type, J. Difference Equations Appl., 8(2002), 1073–1083.

[3] D. R. Anderson, R. I. Avery and J. Henderson, A topological proof and extension of the Leggett-Williams fixed point theorem, Communications on Applied Nonlinear Analysis, 16(2009), 39–44.

[4] D. Guo, A new fixed point theorem, Acta Math. Sinica, 24(1981), 444–450.

[5] D. Guo, Some fixed point theorems on cone maps, Kexeu Tongbao, 29(1984), 575–578.

[6] M. A. Krasnosel’skii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, The Netherlands, 1964.

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

[8] J. Sun and G. Zhang, A generalization of the cone expansion and compression fixed point theorem and applications, Nonlin. Anal., 67(2007), 579-586.

[9] E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer-Verlag, New York, 1986.

(Received November 23, 2009)

College of Arts and Sciences, Dakota State University, Madison, South Dakota 57042 USA

  E-mail address: rich.avery@dsu.edu

Department of Mathematics, Baylor University, Waco, Texas 76798 USA

  E-mail address: Johnny Henderson@baylor.edu

Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562 USA

  E-mail address: andersod@cord.edu

EJQTDE, 2010 No. 5, p. 6