Dynamic Systems and Applications 15 (2006) 21-34

EXISTENCE OF SOLUTIONS FOR NONLINEAR MULTI-POINT PROBLEMS ON TIME SCALES

DOUGLAS R. ANDERSON

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

Dedicated to John Baxley on the occasion of his retirement from Wake Forest University.

ABSTRACT. Consider the nonlinear second-order Sturm-Liouville-type multipoint boundary value problem

(py^{∇})^{Δ} (t) − q(t)y(t) + h(t)f(t, y(t)) = 0, t_{1} < t < t_{n},

αy(t_{1}) − βp(t_{1})y^{∇}(t_{1}) = ∑_{i=2}^{n−1} a_{i}y(t_{i}), γy(t_{n}) + δp(t_{n})y^{∇}(t_{n}) = ∑_{i=2}^{n−1} b_{i}y(t_{i}).

Conditions for the existence of at least one, two, or three positive solutions on time scales are discussed, employing various fixed point theorems on cones.

AMS (MOS) Subject Classification. 34B15, 39A10.

1. INTRODUCTION

We are interested in the second-order multipoint time scale eigenvalue problem

(py^{∇})^{Δ} (t) − q(t)y(t) + h(t)f(t, y(t)) = 0, t_{1} < t < t_{n}, (1.1)

αy(t_{1}) − βp(t_{1})y^{∇}(t_{1}) = ∑_{i=2}^{n−1} a_{i}y(t_{i}), γy(t_{n}) + δp(t_{n})y^{∇}(t_{n}) = ∑_{i=2}^{n−1} b_{i}y(t_{i}), (1.2)

where

(1.3)  p, q : [t_{1}, t_{n}] → (0, ∞), p ∈ C^{Δ}[t_{1}, t_{n}), q ∈ C[t_{1}, t_{n}];

the points t_{i} ∈ 𝕋_{κ}^{κ} for i ∈ {1, 2, …, n} with t_{1} < t_{2} < ⋯ < t_{n};

(1.4) α, β, γ, δ ∈ [0, ∞), αγ + αδ + βγ > 0, a_{i}, b_{i} ∈ [0, ∞), i ∈ {2, …, n − 1};

the continuous function f : [0, ∞) → [0, ∞) is such that the following exist:

f_{0} := lim_{y→0^{+}} \frac{f(·, y)}{y}, f_{∞} := lim_{y→∞} \frac{f(·, y)}{y};

Received August 23, 2004                1056-2176 $15.00 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

22 ANDERSON

and the right-dense continuous function h : [t_{1}, t_{n}] → [0, ∞) satisfies some suitable conditions to be developed. Problem (1.1), (1.2) is an extension of the three-point problems discussed in Sun and Li [1, 2], Anderson [3], Atici and Guseinov [4], Kaufmann [5], Kaufmann and Raffoul [6]. Other related three-point problems on time scales include Anderson, Avery, and Henderson [7], Peterson, Raffoul, and Tisdell [8], and a singular case in DaCunha, Davis, and Singh [9]. Some of the work on multipoint time-scale problems includes Anderson [10, 11] and Kong and Kong [12]. See also Ma [13] and Ma and Thompson [14] for related results when 𝕋 = ℝ. Other work on the existence of solutions to dynamic equations on time scales (measure chains) includes Chyan and Henderson [15], Erbe and Peterson [16, 17], and Henderson [18]. For more general information concerning dynamic equations on time scales, introduced by Aulbach and Hilger [19] and Hilger [20], see the excellent text by Bohner and Peterson [21] and their edited text [22].

2. FOUNDATIONAL LEMMAS

We first construct the Green’s function for the second-order boundary value problem

(py^{∇})^{Δ} (t) − q(t)y(t) + u(t) = 0, t_{1} < t < t_{n}, (2.1) αy(t_{1}) − βp(t_{1})y^{∇}(t_{1}) = 0, γy(t_{n}) + δp(t_{n})y^{∇}(t_{n}) = 0, (2.2)

where α, β, γ, δ are real numbers such that |α| + |β| ≠ 0, |γ| + |δ| ≠ 0. The techniques here are similar to those found in [4].

Denote by ϕ and ψ the solutions of the corresponding homogeneous equation

(2.3)  (py^{∇})^{Δ} (t) − q(t)y(t) = 0, t ∈ [t_{1}, t_{n}),

under the initial conditions

(2.4)  ψ(t_{1}) = β, p(t_{1})ψ^{∇}(t_{1}) = α,
(2.5)  ϕ(t_{n}) = δ, p(t_{n})ϕ^{∇}(t_{n}) = −γ,

so that ψ and ϕ satisfy the first and second boundary conditions in (2.2), respectively. Set

(2.6)  d = −W_{t}(ψ, ϕ) = p(t)ψ^{∇}(t)ϕ(t) − ψ(t)p(t)ϕ^{∇}(t).

Since the Wronskian of any two solutions is independent of t, evaluating at t = t_{1}, t = t_{n}, and using the boundary conditions (2.4), (2.5) yields

d = αϕ(t_{1}) − βp(t_{1})ϕ^{∇}(t_{1}) = γψ(t_{n}) + δp(t_{n})ψ^{∇}(t_{n}).

In addition d ≠ 0 if and only if the homogeneous equation (2.3) has only the trivial solution satisfying the boundary conditions (2.2). For the proof of the following lemma, see Theorem 4.2 and Remark 4.1 in [4].

Lemma 2.1. Assume (1.3) and (1.4) hold. If d ≠ 0, then the nonhomogeneous boundary value problem (2.1), (2.2), has a unique solution y for which the formula

y(t) = ∫_{t_{1}}^{t_{n}} G(t, s)u(s)Δs, t ∈ [ρ(t_{1}), t_{n}]

holds, where the function G(t, s) is given by

(2.7)  G(t, s) = \frac{1}{d} \begin{cases}ψ(t)ϕ(s), & ρ(t_{1}) ≤ t ≤ s ≤ t_{n} \\ ψ(s)ϕ(t), & ρ(t_{1}) ≤ s ≤ t ≤ t_{n}, \end{cases}

and G(t, s) is the Green’s function of the boundary value problem (2.1), (2.2).

Lemma 2.2. Assume (1.3) and (1.4) hold. Then the functions ψ and ϕ satisfy

ψ(t) ≥ 0, t ∈ [ρ(t_{1}), t_{n}], ψ(t) > 0, t ∈ (ρ(t_{1}), t_{n}], p(t)ψ^{∇}(t) ≥ 0, t ∈ [ρ(t_{1}), t_{n}],

ϕ(t) ≥ 0, t ∈ [ρ(t_{1}), t_{n}], ϕ(t) > 0, t ∈ [ρ(t_{1}), t_{n}), p(t)ϕ^{∇}(t) ≤ 0, t ∈ [ρ(t_{1}), t_{n}].

Proof. The proof is very similar to the proof of Lemma 5.1 in [4] and is omitted.

Set

(2.8)  D := |\begin{matrix}− ∑_{i=2}^{n−1} a_{i}ψ(t_{i}) & d − ∑_{i=2}^{n−1} a_{i}ϕ(t_{i}) \\ d − ∑_{i=2}^{n−1} b_{i}ψ(t_{i}) & − ∑_{i=2}^{n−1} b_{i}ϕ(t_{i}) \end{matrix}| .

Lemma 2.3. Assume (1.3) and (1.4) hold. If D ≠ 0 and u ∈ C_{rd}[t_{1}, t_{n}], then the nonhomogeneous dynamic equation (2.1) with boundary conditions (1.2) has a unique solution y for which the formula

(2.9)  y(t) = ∫_{t_{1}}^{t_{n}} G(t, s)u(s)Δs + A(u)ψ(t) + B(u)ϕ(t), t ∈ [ρ(t_{1}), t_{n}]

holds, where the function G(t, s) is the Green’s function (2.7) of the boundary value problem (2.1), (2.2), and the functionals A and B are defined by

(2.10)  A(u) := \frac{1}{D} |\begin{matrix}∑_{i=2}^{n−1} a_{i} ∫_{t_{1}}^{t_{n}} G(t_{i}, s)u(s)Δs & d − ∑_{i=2}^{n−1} a_{i}ϕ(t_{i}) \\ ∑_{i=2}^{n−1} b_{i} ∫_{t_{1}}^{t_{n}} G(t_{i}, s)u(s)Δs & − ∑_{i=2}^{n−1} b_{i}ϕ(t_{i}) \end{matrix}| ,

(2.11)  B(u) := \frac{1}{D} |\begin{matrix}− ∑_{i=2}^{n−1} a_{i}ψ(t_{i}) & ∑_{i=2}^{n−1} a_{i} ∫_{t_{1}}^{t_{n}} G(t_{i}, s)u(s)Δs \\ d − ∑_{i=2}^{n−1} b_{i}ψ(t_{i}) & ∑_{i=2}^{n−1} b_{i} ∫_{t_{1}}^{t_{n}} G(t_{i}, s)u(s)Δs \end{matrix}| .

Proof. It can be verified that for a solution y of the nonhomogeneous equation (2.1) under the nonhomogeneous boundary conditions (1.2), the formula (2.9) holds, where G(t, s) is given by (2.7). We thus show that the function y given in (2.9) is a solution of (2.1) with conditions (1.2) only if A and B are given by (2.10) and (2.11), respectively. If y as in (2.9) is a solution of (2.1), (1.2), then

y(t) = \frac{1}{d} ∫_{t_{1}}^{t} ϕ(t)ψ(s)u(s)Δs + \frac{1}{d} ∫_{t}^{t_{n}} ψ(t)ϕ(s)u(s)Δs + Aψ(t) + Bϕ(t)

for some constants A and B. Taking the nabla derivative and multiplying by p yields

py^{∇} = \frac{pϕ^{∇}}{d} ∫_{t_{1}}^{t} ψ(s)u(s)Δs + \frac{pψ^{∇}}{d} ∫_{t}^{t_{n}} ϕ(s)u(s)Δs + Apψ^{∇} + Bpϕ^{∇};

the delta derivative of this expression is

(py^{∇})^{Δ} = (\frac{pϕ^{∇}}{d} )^{Δ} ∫_{t_{1}}^{σ(t)} ψ(s)u(s)Δs + \frac{pϕ^{∇}}{d} ψ(t)u(t) + A (pψ^{∇})^{Δ} + B (pϕ^{∇})^{Δ}

+ (\frac{pψ^{∇}}{d} )^{Δ} ∫_{σ(t)}^{t_{n}} ϕ(s)u(s)Δs − \frac{pψ^{∇}}{d} ϕ(t)u(t).

Using Theorem 1.75 from [21], and the fact that ψ and ϕ are solutions to (2.3), we obtain

(py^{∇})^{Δ} (t) = \frac{q(t)}{d} ∫_{t_{1}}^{t} ϕ(t)ψ(s)u(s)Δs + \frac{q(t)}{d}ϕ(t)μ_{σ}(t)ψ(t)u(t) + \frac{u(t)}{d}p(t)ϕ^{∇}(t)ψ(t)

+\frac{q(t)}{d} ∫_{t}^{t_{n}} ψ(t)ϕ(s)u(s)Δs − \frac{q(t)}{d}ψ(t)μ_{σ}(t)ϕ(t)u(t)

−\frac{u(t)}{d}p(t)ψ^{∇}(t)ϕ(t) + q(t)(Aψ(t) + bϕ(t)).

Recall that d is in terms of the Wronskian of ψ and ϕ in (2.6); it follows that

(py^{∇})^{Δ} (t) = q(t)y(t) − u(t).

Now

y(t_{1}) = \frac{ψ(t_{1})}{d} ∫_{t_{1}}^{t_{n}} ϕ(s)u(s)Δs + Aψ(t_{1}) + Bϕ(t_{1})

and

p(t_{1})y^{∇}(t_{1}) = \frac{p(t_{1})ψ^{∇}(t_{1})}{d} ∫_{t_{1}}^{t_{n}} ϕ(s)u(s)Δs + Ap(t_{1})ψ^{∇}(t_{1}) + Bp(t_{1})ϕ^{∇}(t_{1});

multiply the first line by α and the second by −β, and use (1.2) and (2.4) to see that (2.12)

B [αϕ(t_{1}) − βp(t_{1})ϕ^{∇}(t_{1})] = ∑_{i=2}^{n−1} a_{i} (∫_{t_{1}}^{t_{n}} G(t_{i}, s)u(s)Δs + Aψ(t_{i}) + Bϕ(t_{i})) .

At the other end,

y(t_{n}) = \frac{ϕ(t_{n})}{d} ∫_{t_{1}}^{t_{n}} ψ(s)u(s)Δs + Aψ(t_{n}) + Bϕ(t_{n})

and

p(t_{n})y^{∇}(t_{n}) = \frac{p(t_{n})ϕ^{∇}(t_{n})}{d} ∫_{t_{1}}^{t_{n}} ψ(s)u(s)Δs + Ap(t_{n})ψ^{∇}(t_{n}) + Bp(t_{n})ϕ^{∇}(t_{n});

consequently (2.13)

A [γψ(t_{n}) + δp(t_{n})ψ^{∇}(t_{n})] = ∑_{i=2}^{n−1} b_{i} (∫_{t_{1}}^{t_{n}} G(t_{i}, s)u(s)Δs + Aψ(t_{i}) + Bϕ(t_{i})) .

Combining (2.12) and (2.13) and using (2.6), we arrive at the system of equations

− A ∑_{i=2}^{n−1} a_{i}ψ(t_{i}) + B [αϕ(t_{1}) − βp(t_{1})ϕ^{∇}(t_{1}) − ∑_{i=2}^{n−1} a_{i}ϕ(t_{i})]

= ∑_{i=2}^{n−1} a_{i} ∫_{t_{1}}^{t_{n}} G(t_{i}, s)u(s)Δs

A [γψ(t_{n}) + δp(t_{n})ψ^{∇}(t_{n}) − ∑_{i=2}^{n−1} b_{i}ψ(t_{i})] − B ∑_{i=2}^{n−1} b_{i}ϕ(t_{i})

= ∑_{i=2}^{n−1} b_{i} ∫_{t_{1}}^{t_{n}} G(t_{i}, s)u(s)Δs.

Again using (2.6) at both t_{1} and t_{n}, we verify (2.10) and (2.11) hold.

Lemma 2.4. Let (1.3) and (1.4) hold, and assume

(2.14)  D < 0, d − ∑_{i=2}^{n−1} a_{i}ϕ(t_{i}) > 0, d − ∑_{i=2}^{n−1} b_{i}ψ(t_{i}) > 0

for D and d given in (2.8) and (2.6), respectively. If u ∈ C_{rd}[t_{1}, t_{n}] with u ≥ 0, the unique solution y as in (2.9) of the problem (2.1), (1.2) satisfies y(t) ≥ 0 for t ∈ [t_{1}, t_{n}].

Proof. From the previous lemmas and assumptions, we know that the Green’s function (2.7) satisfies G(t, s) ≥ 0 on [ρ(t_{1}), t_{n}] × [ρ(t_{1}), t_{n}]. Hypotheses (1.3), (1.4), and (2.14) applied to (2.10) and (2.11) imply that A(u), B(u) ≥ 0.

Suppose (2.14) does not hold. For example, let n = 3, p(t) ≡ 1 = α = γ, q(t) ≡ 0 = β = δ = a_{2}, and t_{1} = 0. Then (2.1), (1.2) becomes

y^{∇Δ}(t) + u(t) = 0, t_{1} < t < t_{3}, y(t_{1}) = 0, y(t_{3}) = b_{2}y(t_{2}).

Note that ψ(t) = t, d = t_{3}, and D = t_{3}(b_{2}t_{2} − t_{3}). If D > 0, then b_{2}t_{2} > t_{3}, and there is no positive solution; see Lemma 5 in [3].

Lemma 2.5. Let (1.3), (1.4), and (2.14) hold, and fix

ξ_{1}, ξ_{2} ∈ 𝕋_{κ}^{κ}, ρ(t_{1}) < ξ_{1} < ξ_{2} < t_{n}.

If u ∈ C_{rd}[t_{1}, t_{n}] with u ≥ 0, the unique solution y as in (2.9) of the time scale boundary value problem (2.1), (1.2) satisfies

min_{t∈[ξ_{1},ξ_{2}]} y(t) ≥ Γ∥y∥, ∥y∥ := max_{t∈[ρ(t_{1}),t_{n}]} y(t),

where

(2.15)  Γ := min \{\frac{ϕ(ξ_{2})}{ϕ(ρ(t_{1}))}, \frac{ψ(ξ_{1})}{ψ(t_{n})}\} ∈ (0, 1).

26 ANDERSON

Proof. From (1.3), (2.7), and Lemma 2.2,

(2.16)  0 ≤ G(t, s) ≤ G(s, s),  t ∈ [ρ(t_{1}), t_{n}],

so that

y(t) ≤ ∫_{t_{1}}^{t_{n}} G(s, s)u(s)Δs + A(u)ψ(t_{n}) + B(u)ϕ(ρ(t_{1})), ∀t ∈ [ρ(t_{1}), t_{n}].

For t ∈ [ξ_{1}, ξ_{2}], the Green’s function (2.7) satisfies

(2.17) \frac{G(t, s)}{G(s, s)} = \begin{cases}\frac{ϕ(t)}{ϕ(s)} : & ρ(t_{1}) ≤ s ≤ t ≤ t_{n} \\ \frac{ψ(t)}{ψ(s)} : & ρ(t_{1}) ≤ t ≤ s ≤ t_{n} \end{cases} ≥ \begin{cases}\frac{ϕ(ξ_{2})}{ϕ(ρ(t_{1}))} : & ρ(t_{1}) ≤ s ≤ t ≤ t_{n} \\ \frac{ψ(ξ_{1})}{ψ(t_{n})} : & ρ(t_{1}) ≤ t ≤ s ≤ t_{n} \end{cases} ≥ Γ,

for Γ as in (2.15), and

y(t) = ∫_{t_{1}}^{t_{n}} \frac{G(t, s)}{G(s, s)}G(s, s)u(s)Δs + A(u)ψ(t) + B(u)ϕ(t)
≥ ∫_{t_{1}}^{t_{n}} ΓG(s, s)u(s)Δs + A(u)ψ(ξ_{1}) + B(u)ϕ(ξ_{2})
≥ Γ (∫_{t_{1}}^{t_{n}} G(s, s)u(s)Δs + A(u)ψ(t_{n}) + B(u)ϕ(ρ(t_{1}))) ≥ Γ∥y∥.

3. RESULTS ON CONES

Assume the right-dense continuous function h satisfies (3.1)

h : [t_{1}, t_{n}] → [0, ∞) and there exists t_{∗} ∈ (σ(t_{1}), ρ(t_{n})) such that h(t_{∗}) > 0.

Then there exist ξ_{1}, ξ_{2} as in Lemma 2.5 such that

ξ_{1} < t_{∗} < ξ_{2}, ∫_{ξ_{1}}^{ξ_{2}} G(t, s)h(s)Δs > 0, t ∈ (ρ(t_{1}), t_{n}).

In the following, let Γ be the constant defined in (2.15) with respect to such constants ξ_{1}, ξ_{2}. Let τ ∈ [ξ_{1}, ξ_{2}] be determined by

(3.2)  ∫_{ξ_{1}}^{ξ_{2}} G(τ, s)h(s)Δs = min_{t∈[ξ_{1},ξ_{2}]} ∫_{ξ_{1}}^{ξ_{2}} G(t, s)h(s)Δs > 0.

For G(t, s) in (2.7) and A, B as in (2.10), (2.11), respectively, define the constant

(3.3)  K := ∫_{t_{1}}^{t_{n}} G(s, s)h(s)Δs + A(h)ψ(t_{n}) + B(h)ϕ(ρ(t_{1})).

Let S denote the Banach space C[ρ(t_{1}), t_{n}] with the norm ∥y∥ = sup_{t∈[ρ(t_{1}),t_{n}]} |y(t)|. Define the cone P ⊂ S by

(3.4)  P = {y ∈ S : y(t) ≥ 0 on [ρ(t_{1}), t_{n}] and y(t) ≥ Γ∥y∥ on [ξ_{1}, ξ_{2}]},

where Γ is given in (2.15). Since y is a solution of (1.1), (1.2) if and only if

y(t) = ∫_{t_{1}}^{t_{n}} G(t, s)h(s)f(s, y(s))Δs+A(hf(·, y))ψ(t)+B(hf(·, y))ϕ(t), t ∈ [ρ(t_{1}), t_{n}],

define, for y ∈ P, the operator T : P → S by

(3.5) (T y)(t) := ∫_{t_{1}}^{t_{n}} G(t, s)h(s)f(s, y(s))Δs + A(hf(·, y))ψ(t) + B(hf(·, y))ϕ(t).

If y ∈ P, then by (2.16) we have

(T y)(t) = ∫_{t_{1}}^{t_{n}} G(t, s)h(s)f(s, y(s))Δs + A(hf(·, y))ψ(t) + B(hf(·, y))ϕ(t)

≤ ∫_{t_{1}}^{t_{n}} G(s, s)h(s)f(s, y(s))Δs + A(hf(·, y))ψ(t_{n}) + B(hf(·, y))ϕ(ρ(t_{1})),

so that for t ∈ [ξ_{1}, ξ_{2}],

(T y)(t) = ∫_{t_{1}}^{t_{n}} G(t, s)h(s)f(s, y(s))Δs + A(hf(·, y))ψ(t) + B(hf(·, y))ϕ(t)

≥ ∫_{t_{1}}^{t_{n}} \frac{G(t, s)}{G(s, s)}G(s, s)h(s)f(s, y(s))Δs + A(hf(·, y))ψ(ξ_{1}) + B(hf(·, y))ϕ(ξ_{2})

≥ Γ (∫_{t_{1}}^{t_{n}} G(s, s)h(s)f(s, y(s))Δs + A(hf(·, y))ψ(t_{n}) + B(hf(·, y))ϕ(ρ(t_{1})))

≥ Γ∥T y∥.

Therefore T : P → P. Moreover, T is completely continuous by a typical application of the Ascoli-Arzela Theorem.

Lemma 3.1. [23, 24] Let P be a cone in a Banach space S, and let B be an open, bounded subset of S with B_{P} := B ∩ P ≠ ∅ and \overline{B}_{P} ≠ P. Assume that T : \overline{B}_{P} → P is a compact map such that y ≠ T y for y ∈ ∂B_{P}. The following results hold.

(i) If ∥T y∥ ≤ ∥y∥ for y ∈ ∂B_{P}, then i_{P} (T, B_{P}) = 1. (ii) If there exists an η ∈ P \{0} such that y ≠ T y + λη for all y ∈ ∂B_{P} and all

λ > 0, then i_{P} (T, B_{P}) = 0. (iii) Let U be open in P such that \overline{U}_{P} ⊂ B_{P}. If i_{P} (T, B_{P}) = 1 and i_{P} (T, U_{P}) = 0,

then T has a fixed point in B_{P} \\overline{U}_{P}; the same is true if i_{P} (T, B_{P}) = 0 and i_{P} (T, U_{P}) = 1.

For the cone P given above in (3.4) and any positive real number r, define the convex set

P_{r} := {y ∈ P : ∥y∥ < r},

and, for Γ in (2.15), the set

Ω_{r} := {y ∈ P : min_{t∈[ξ_{1},ξ_{2}]} y(t) < Γr} .

28 ANDERSON

Lemma 3.2. [23] The set Ω_{r} has the following properties.

(i) Ω_{r} is open relative to P. (ii) P_{Γr} ⊂ Ω_{r} ⊂ P_{r}. (iii) y ∈ ∂Ω_{r} if and only if min_{t∈[ξ_{1},ξ_{2}]} y(t) = Γr. (iv) If y ∈ ∂Ω_{r}, then Γr ≤ y(t) ≤ r for t ∈ [ξ_{1}, ξ_{2}].

As in [2], we introduce the following notation for further reference. Let

f_{Γr}^{r} := min \{ min_{t∈[ξ_{1},ξ_{2}]} \frac{f(t, y)}{r} : y ∈ [Γr, r]\} ,

f_{0}^{r} := max \{ max_{t∈[ρ(t_{1}),t_{n}]} \frac{f(t, y)}{r} : y ∈ [0, r]\} ,

f^{a} := lim sup_{y→a} max_{t∈[ρ(t_{1}),t_{n}]} \frac{f(t, y)}{y},

f_{a} := lim inf_{y→a} min_{t∈[ξ_{1},ξ_{2}]} \frac{f(t, y)}{y} (a := 0^{+}, ∞).

In the next two lemmas, we sanction conditions on f guaranteeing that i_{P} (T, P_{r}) = 1 or i_{P} (T, Ω_{r}) = 0.

Lemma 3.3. Let K be as in (3.3). If the conditions

f_{0}^{r} ≤ 1/K and y ≠ T y for y ∈ ∂P_{r}

hold, then i_{P} (T, P_{r}) = 1.

Proof. From (2.10) and (2.11),

|A(hf(·, y))| ≤ A(h)∥f(·, y)∥, |B(hf(·, y))| ≤ B(h)∥f(·, y)∥.

Let y ∈ ∂P_{r}; using (2.16) we have

(T y)(t) = ∫_{t_{1}}^{t_{n}} G(t, s)h(s)f(s, y(s))Δs + A(hf(·, y))ψ(t) + B(hf(·, y))ϕ(t)
≤ ∥f(·, y)∥ (∫_{t_{1}}^{t_{n}} G(s, s)h(s)Δs + A(h)ψ(t_{n}) + B(h)ϕ(ρ(t_{1})))
≤ (r/K)K = r = ∥y∥.

It follows that for y ∈ ∂P_{r}, ∥T y∥ ≤ ∥y∥. By Lemma 3.1 (i), i_{P} (T, P_{r}) = 1.

Lemma 3.4. Let τ be as in (3.2), and let

(3.6)  M^{−1} := ∫_{ξ_{1}}^{ξ_{2}} G(τ, s)h(s)Δs.

If the conditions

f_{Γr}^{r} ≥ MΓ and y ≠ T y for y ∈ ∂Ω_{r}

hold, then i_{P} (T, Ω_{r}) = 0.

Proof. Let η(t) ≡ 1 for t ∈ [ρ(t_{1}), t_{n}], so that η ∈ ∂P_{1}. Suppose there exist y_{∗} ∈ ∂Ω_{r} and λ_{∗} > 0 such that y_{∗} = T y_{∗} + λ_{∗}η. Then for t ∈ [ξ_{1}, ξ_{2}],

y_{∗}(t) = (T y_{∗})(t) + λ_{∗}η(t) ≥ ∫_{ξ_{1}}^{ξ_{2}} G(t, s)h(s)f(s, y(s))Δs + λ_{∗}
≥ MΓr ∫_{ξ_{1}}^{ξ_{2}} G(τ, s)h(s)Δs + λ_{∗} = Γr + λ_{∗}.

But this implies that Γr ≥ Γr + λ_{∗}, a contradiction. Consequently, y_{∗} ≠ T y_{∗} + λ_{∗}η for y_{∗} ∈ ∂Ω_{r} and λ_{∗} > 0, so by Lemma 3.1 (ii) we have i_{P} (T, Ω_{r}) = 0.

4. ONE OR TWO SOLUTIONS

In what follows, we seek fixed points in the cone P of T given in (3.5) using the lemmas established in the previous sections; these fixed points will be positive solutions to the multi-point boundary value problem (1.1), (1.2).

Theorem 4.1. Let Γ, K, and M be as given in (2.15), (3.3), and (3.6), respectively. Assume that there exist constants c_{1}, c_{2}, c_{3} ∈ ℝ with 0 < c_{1} < Γc_{2} and c_{2} < c_{3} such that

(H1)  f_{0}^{c_{1}}, f_{0}^{c_{3}} ≤ 1/K, f_{Γc_{2}}^{c_{2}} ≥ MΓ, and y ≠ T y for y ∈ ∂Ω_{c_{2}},

or that there exist constants c_{1}, c_{2}, c_{3} ∈ ℝ with 0 < c_{1} < c_{2} < Γc_{3} such that

(H2)  f_{Γc_{1}}^{c_{1}}, f_{Γc_{3}}^{c_{3}} ≥ MΓ, f_{0}^{c_{2}} ≤ 1/K, and y ≠ T y for y ∈ ∂P_{c_{2}}.

Then (1.1), (1.2) has two positive solutions in P as in (3.4). Additionally, if in (H1) the condition f_{0}^{c_{1}} ≤ 1/K is replaced by f_{0}^{c_{1}} < 1/K, then (1.1), (1.2) has a third positive solution in P_{c_{1}}.

Proof. Assume (H2) holds; the case for (H1) is similar and is omitted. We show that either T has a fixed point in ∂Ω_{c_{1}} or in P_{c_{2}}\\overline{Ω}_{c_{1}}. From Lemma 3.4, if y ≠ T y for y ∈ ∂Ω_{c_{1}} ∪ ∂Ω_{c_{3}}, then i_{P} (T, Ω_{c_{1}}) = 0 and i_{P} (T, Ω_{c_{3}}) = 0. Since f_{0}^{c_{2}} ≤ 1/K and y ≠ T y for y ∈ ∂P_{c_{2}}, Lemma 3.3 implies that i_{P} (T, P_{c_{2}}) = 1. By Lemma 3.2 (ii), Ω_{c_{1}} ⊂ P_{c_{1}} ⊂ P_{c_{2}}. From Lemma 3.1 (iii), T has a fixed point in P_{c_{2}}\\overline{Ω}_{c_{1}}. In the same way P_{c_{2}} ⊂ P_{Γc_{3}} ⊂ Ω_{c_{3}} and T has a fixed point in Ω_{c_{3}}\\overline{P}_{c_{2}}.

Corollary 4.2. Suppose there exists a constant c > 0 such that either

(H1′)  0 ≤ f^{0}, f^{∞} < 1/K, f_{Γc}^{c} ≥ MΓ, and y ≠ T y for y ∈ ∂Ω_{c}

or

(H2′)  M < f_{0}, f_{∞} ≤ ∞, f_{0}^{c} ≤ 1/K, and y ≠ T y for y ∈ ∂P_{c}

holds. Then (1.1), (1.2) has two positive solutions in P.

Proof. Since (H1′) implies (H1) and (H2′) implies (H2), the result follows.

The proofs of the following two results are similar to those given above and are omitted.

Theorem 4.3. Assume that there exist constants c_{1}, c_{2} ∈ ℝ with 0 < c_{1} < Γc_{2} such that

(H3)  f_{0}^{c_{1}} ≤ 1/K and f_{Γc_{2}}^{c_{2}} ≥ MΓ,

or that there exist constants c_{1}, c_{2} ∈ ℝ with 0 < c_{1} < c_{2} such that

(H4)  f_{Γc_{1}}^{c_{1}} ≥ MΓ and f_{0}^{c_{2}} ≤ 1/K.

Then (1.1), (1.2) has a positive solution.

Corollary 4.4. Suppose either

(H3′)  0 ≤ f^{0} < 1/K and M < f_{∞} ≤ ∞

or

(H4′)  0 ≤ f^{∞} < 1/K and M < f_{0} ≤ ∞

holds. Then (1.1), (1.2) has a positive solution.

5. THREE SOLUTIONS

To prove the existence of at least three positive solutions to (1.1), (1.2), we will use the Leggett-Williams fixed point theorem [24, 25]:

Theorem 5.1. Let P be a cone in the real Banach space S, T : \overline{P_{c}} → \overline{P_{c}} be completely continuous and ϖ be a nonnegative continuous concave functional on P with ϖ(y) ≤ ∥y∥ for all y ∈ \overline{P_{c}}. Suppose there exist constants 0 < a < b < ℓ ≤ c such that the following conditions hold:

(i) {y ∈ P (ϖ, b, ℓ) : ϖ(y) > b} ≠ ∅ and ϖ(T y) > b for all y ∈ P (ϖ, b, ℓ); (ii) ∥T y∥ < a for ∥y∥ ≤ a; (iii) ϖ(T y) > b for y ∈ P (ϖ, b, c) with ∥T y∥ > ℓ.

Then T has at least three fixed points y_{1}, y_{2}, and y_{3} in \overline{P_{c}} satisfying:

∥y_{1}∥ < a, ϖ(y_{2}) > b, a < ∥y_{3}∥ with ϖ(y_{3}) < b.

Define the continuous concave functional ϖ : P → [0, ∞) by ϖ(y) := min_{t∈[ξ_{1},ξ_{2}]} y(t). Moreover, we take

P_{c} = {y ∈ P : ∥y∥ < c}, P (ϖ, a, b) := {y ∈ P : a ≤ ϖ(y), ∥y∥ ≤ b}.

Theorem 5.2. Take ξ_{1}, ξ_{2} as in Lemma 2.5, Γ as in (2.15), K as in (3.3), and M as in (3.6). Assume there exist constants 0 < a < b such that the following conditions hold:

MULTI-POINT PROBLEMS ON TIME SCALES 31

(i) f(t, y) ≥ bM for all t ∈ [ξ_{1}, ξ_{2}], y ∈ [b, b/Γ]; (ii) f(t, y) < a/K for all t ∈ [ρ(t_{1}), t_{n}], y ∈ [0, a]; (iii) one of the following is satisfied:

(A) lim_{y→∞} max_{t∈[ρ(t_{1}),t_{n}]} \frac{f(t, y)}{y} < \frac{1}{K};

(B) there exists a constant c > a/Γ such that f(t, y) ≤ c/K for t ∈ [ρ(t_{1}), t_{n}] and y ∈ [0, c].

Then (1.1), (1.2) has at least three positive solutions y_{1}, y_{2}, and y_{3} in \overline{P_{c}} satisfying:

∥y_{1}∥ < a, min_{t∈[ξ_{1},ξ_{2}]} y_{2}(t) > b, a < ∥y_{3}∥ with min_{t∈[ξ_{1},ξ_{2}]} y_{3}(t) < b.

Proof. We seek fixed points in the cone P, given in (3.4), of T as in (3.5) using Theorem 5.1 above. Throughout the proof we set ℓ := b/Γ. The technique here is similar to that used in a third-order discrete case in [26].

Claim 1: If condition (A) holds, then there exists a number c such that c > ℓ and T : \overline{P}_{c} → P_{c}. If

lim sup_{y→∞} max_{t∈[ρ(t_{1}),t_{n}]} \frac{f(t, y)}{y} < \frac{1}{K} ,

then there exists a θ > 0 and ϵ < \frac{1}{K} such that if y > θ, then max_{t∈[ρ(t_{1}),t_{n}]} \frac{f(t,y)}{y} < ϵ. For λ := max{f(t, y) : y ∈ [0, θ], t ∈ [ρ(t_{1}), t_{n}]}, we have f(t, y) ≤ ϵy + λ for all y ≥ 0, for all t ∈ [ρ(t_{1}), t_{n}]. Pick any

c > max \{ℓ, \frac{λ}{1/K − ϵ}\} .

Then y ∈ \overline{P}_{c} implies that

∥T y∥ = max_{t∈[ρ(t_{1}),t_{n}]} ∫_{t_{1}}^{t_{n}} G(t, s)h(s)f(s, y(s))Δs + A(hf(·, y))ψ(t) + B(hf(·, y))ϕ(t)
≤ (ϵ∥y∥ + λ) ( max_{t∈[ρ(t_{1}),t_{n}]} ∫_{t_{1}}^{t_{n}} G(t, s)h(s)Δs + A(h)ψ(t) + B(h)ϕ(t))
≤ (ϵ∥y∥ + λ)K < ϵcK + c(1 − ϵK) = c.

Thus, T : \overline{P}_{c} → P_{c} and Claim 1 has been shown.

Claim 2: If there exists a positive number r such that y ∈ [0, r] implies f(t, y) < r/K for t ∈ [ρ(t_{1}), t_{n}], then T : \overline{P}_{r} → P_{r}. If y ∈ \overline{P}_{r}, then

∥T y∥ = max_{t∈[ρ(t_{1}),t_{n}]} ∫_{t_{1}}^{t_{n}} G(t, s)h(s)f(s, y(s))Δs + A(hf(·, y))ψ(t) + B(hf(·, y))ϕ(t)
≤ (r/K) ( max_{t∈[ρ(t_{1}),t_{n}]} ∫_{t_{1}}^{t_{n}} G(t, s)h(s)Δs + A(h)ψ(t) + B(h)ϕ(t)) ≤ r.

As a result, T : \overline{P}_{r} → P_{r} and Claim 2 has been shown. We have thus seen in the two previous claims that if either condition (A) or condition (B) holds, then there exists a number c such that c > ℓ and T : \overline{P}_{c} → P_{c}. Note from Claim 2 with r = a we get, using hypothesis (ii), that T : \overline{P}_{a} → P_{a}.

Claim 3: {y ∈ P (ϖ, b, ℓ) : ϖ(y) > b} ≠ ∅ and ϖ(T y) > b for all y ∈ P (ϖ, b, ℓ).

It is easy to see that y(t) ≡ (ℓ + b)/2 ∈ {y ∈ P (ϖ, b, ℓ) : ϖ(y) > b} ≠ ∅. Next, let y ∈ P (ϖ, b, ℓ), τ as in (3.2), and M as in (3.6). Using (i),

ϖ(T y) = min_{t∈[ξ_{1},ξ_{2}]} ∫_{t_{1}}^{t_{n}} G(t, s)h(s)f(s, y(s))Δs + A(hf(·, y))ψ(t) + B(hf(·, y))ϕ(t)
> min_{t∈[ξ_{1},ξ_{2}]} ∫_{ξ_{1}}^{ξ_{2}} G(t, s)h(s)f(s, y(s))Δs ≥ bM/M = b,

by the definition of ℓ. Consequently, y ∈ P (ϖ, b, ℓ) yields ϖ(T y) > b.

Claim 4: If y ∈ P (ϖ, b, c) and ∥T y∥ > ℓ, then ϖ(T y) > b.

Let Γ be as in (2.15) and recall that T : P → P. If y ∈ P (ϖ, b, c) and ∥T y∥ > ℓ, then

ϖ(T y) = min_{t∈[ξ_{1},ξ_{2}]}(T y)(t) ≥ Γ∥T y∥ > Γℓ = b.

Since all of the hypotheses of the Leggett-Williams existence theorem are satisfied, (1.1), (1.2) has at least three positive solutions.

Corollary 5.3. Suppose that there exist constants

0 < α_{1} < β_{1} < ℓ_{1} < α_{2} < β_{2} < ℓ_{2} < α_{3} < ⋯ < α_{m}, m ∈ ℕ,

where ℓ_{i} := β_{i}/Γ, such that the following conditions hold:

(i′) f(t, y) ≥ β_{i}M for all t ∈ [ξ_{1}, ξ_{2}], y ∈ [β_{i}, ℓ_{i}]; (ii′) f(t, y) < α_{i}/K for all t ∈ [ρ(t_{1}), t_{n}], y ∈ [0, α_{i}].

Then (1.1), (1.2) has at least 2m − 1 positive solutions.

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