Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 10, 2003, pp 143–152. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

A non-resonant multi-point boundary-value problem for a p-Laplacian type operator ^{∗}

Chaitan P. Gupta

Abstract

Let ϕ be an odd increasing homeomorphism from ℝ onto ℝ with ϕ(0) = 0, f : [0,1] × ℝ^{2} → ℝ be a function satisfying Caratheodory’s conditions and e(t) ∈ L^{1}[0, 1]. Let ξ_{i} ∈ (0, 1), a_{i} ∈ ℝ, i = 1, 2, …, m − 2, ∑_{i=1}^{m−2} a_{i} ≠ 1, 0 < ξ_{1} < ξ_{2} < ⋯ < ξ_{m−2} < 1 be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary-value problem

(ϕ(x′(t)))′ = f(t, x(t), x′(t)) + e(t), 0 < t < 1,

x(0) = 0, ϕ(x′(1)) = ∑_{i=1}^{m−2} a_{i}ϕ(x′(ξ_{i})).

This paper gives conditions for the existence of a solution for the above boundary-value problem using some new Poincaré type a priori estimates. In the case ϕ(t) ≡ t for t ∈ ℝ, this problem was studied earlier by Gupta, Trofimchuk in [2] and by Gupta, Ntouyas and Tsamatos in [1]. We give a priori estimates needed for this problem that are similar to a priori estimates obtained by Gupta, Trofimchuk in [2]. We then obtain existence theorems for the above multi-point boundary-value problem using the a priori estimates and Leray-Schauder continuation theorem.

1 Introduction

Let ϕ be an odd increasing homeomorphism from ℝ onto ℝ with ϕ(0) = 0, f : [0, 1]×ℝ^{2} → ℝ be a function satisfying Caratheodory’s conditions, e : [0, 1] ↦ ℝ be a function in L^{1}[0, 1], a_{i} ∈ ℝ, ξ_{i} ∈ (0, 1), i = 1, 2, …, m − 2, ∑_{i=1}^{m−2} a_{i} ≠ 1, 0 < ξ_{1} < ξ_{2} < ⋯ < ξ_{m−2} < 1 be given. We study the problem of existence of solutions for the m-point boundary-value problem

\begin{matrix}(ϕ(x′(t)))′ = f(t, x(t), x′(t)) + e(t), 0 < t < 1, \\ x(0) = 0, ϕ(x′(1)) = ∑_{i=1}^{m−2} a_{i}ϕ(x′(ξ_{i})). \end{matrix}  (1.1)

∗Mathematics Subject Classifications: 34B10, 34B15, 34G20.

Key words: Multi-point boundary-value problem, three-point boundary-value problem, p-Laplacian, Leray Schauder Continuation theorem, Caratheodory’s conditions.

\begin{matrix}c \\ ◯ \end{matrix}2003 Southwest Texas State University.

Published February 28, 2003.

144 A non-resonant multi-point boundary-value problem EJDE/Conf/10

This problem was studied earlier in the case ϕ(t) ≡ t for t ∈ ℝ, by Gupta, Trofimchuk in [2] and by Gupta, Ntouyas and Tsamatos in [1]. Gupta, Ntouyas and Tsamatos had studied the problem (1.1) when all of the a_{i} ∈ ℝ, i = 1, 2, …, m − 2, had the same sign by first studying the three-point boundaryvalue problem, for a given α ∈ ℝ, α ≠ 1, η ∈ (0, 1),

\begin{matrix}x′′(t) = f(t, x(t), x′(t)) + e(t), 0 < t < 1, \\ x(0) = 0, x′(1) = αx′(η), \end{matrix}  (1.2)

while Gupta, Trofimchuk in [2] studied the problem (1.1) when the a_{i} ∈ ℝ, i = 1, 2, …, m − 2, do not necessarily have the same sign.

We also study the three-point boundary-value problem analogue of (1.1), for a given α ∈ ℝ, α ≠ 1, η ∈ (0, 1),

\begin{matrix}(ϕ(x′(t)))′ = f(t, x(t), x′(t)) + e(t), 0 < t < 1, \\ x(0) = 0, ϕ(x′(1)) = αϕ(x′(η)). \end{matrix}  (1.3)

The purpose of this paper is to obtain conditions for the existence of a solution for the boundary-value problem (1.1), using new estimates and inequalities involving a function x(t), its derivative x′(t), the functions ϕ(x′(t)) and its derivative (ϕ(x′(t)))′. These results are motivated by the so called nonlocal boundary-value problem studied by Il’in and Moiseev in [5]. We may mention that the reason for studying the three-point boundary-value problem (1.3) is that in this case we obtain a better existence theorem using a priori estimates involving L_{2} norm.

We use the classical spaces C[0, 1], C^{k}[0, 1], L^{k}[0, 1], and L^{∞}[0, 1] of continuous, k-times continuously differentiable, measurable real-valued functions whose k-th power of the absolute value is Lebesgue integrable on [0, 1], or measurable functions that are essentially bounded on [0, 1]. We also use the Sobolev spaces W_{ϕ}^{2,k}(0, 1), k = 1, 2 defined by

W_{ϕ}^{2,k}(0, 1) = {x : [0, 1] → R : x, x′ abs. cont. on [0, 1], (ϕ(x′(t)))′ ∈ L^{k}[0, 1]}

with its usual norm. We denote the norm in L^{k}[0, 1] by ∥ · ∥_{k}, and the norm in L^{∞}[0, 1] by ∥ · ∥_{∞}.

2 A Priori Estimates

Let a_{i} ∈ ℝ, ξ_{i} ∈ (0, 1), i = 1, 2, …, m − 2, 0 < ξ_{1} < ξ_{2} < ⋯ < ξ_{m−2} < 1, with α = ∑_{i=1}^{m−2} a_{i} ≠ 1 be given. Let x(t) ∈ W_{ϕ}^{2,1}(0, 1) be such that x(0) = 0, ϕ(x′(1)) = ∑_{i=1}^{m−2} a_{i}ϕ(x′(ξ_{i})) be given. We are interested in obtaining a priori estimates of the form ∥ϕ(x′(t))∥_{∞} ≤ C∥(ϕ(x′(t)))′∥_{1}. The following theorem gives such an estimate. We recall that for a ∈ ℝ, a_{+} = max{a, 0}, a_{−} = max{−a, 0} so that a = a_{+} − a_{−} and |a| = a_{+} + a_{−}.

Theorem 2.1 Let a_{i} ∈ ℝ, ξ_{i} ∈ (0, 1), i = 1, 2, …, m − 2, 0 < ξ_{1} < ξ_{2} < ⋯ < ξ_{m−2} < 1, with α = ∑_{i=1}^{m−2} a_{i} ≠ 1 be given. Then for x(t) ∈ W_{ϕ}^{2,1}(0, 1) with x(0) = 0, ϕ(x′(1)) = ∑_{i=1}^{m−2} a_{i}ϕ(x′(ξ_{i})) we have

∥ϕ(x′(t))∥_{∞} ≤ \frac{1}{1 − τ} ∥(ϕ(x′(t)))′∥_{1}  (2.1)

where either τ = 0 or

τ = min \{\frac{∑_{i=1}^{m−2}(a_{i})_{+}}{∑_{i=1}^{m−2}(a_{i})_{−} + 1}, \frac{∑_{i=1}^{m−2}(a_{i})_{−} + 1}{∑_{i=1}^{m−2}(a_{i})_{+}}\}.

Proof We see that the assumption ϕ(x′(1)) = ∑_{i=1}^{m−2} a_{i}ϕ(x′(ξ_{i})) implies

ϕ(x′(1)) + ∑_{i=1}^{m−2}(a_{i})_{−}ϕ(x′(ξ_{i})) = ∑_{i=1}^{m−2}(a_{i})_{+}ϕ(x′(ξ_{i}))

and thus there exist λ_{1}, λ_{2} ∈ [0, 1] such that

(1 + ∑_{i=1}^{m−2}(a_{i})_{−})ϕ(x′(λ_{1})) = ∑_{i=1}^{m−2}(a_{i})_{+}ϕ(x′(λ_{2})).  (2.2)

If, now, either x′(λ_{1}) = 0 or x′(λ_{2}) = 0, so that either ϕ(x′(λ_{1})) = 0 or ϕ(x′(λ_{2})) = 0, then we clearly have

∥ϕ(x′(t))∥_{∞} ≤ ∥(ϕ(x′(t)))′∥_{1}.  (2.3)

Suppose, now, that x′(λ_{1}) ≠ 0 and x′(λ_{2}) ≠ 0, so that ϕ(x′(λ_{1})) ≠ 0 and ϕ(x′(λ_{2})) ≠ 0. It then follows easily from equation (2.2) that ϕ(x′(λ_{1})) ≠ ϕ(x′(λ_{2})), in view of the assumption α = ∑_{i=1}^{m−2} a_{i} ≠ 1. It then follows from equation (2.2), the estimate (2.3) and the equations

ϕ(x′(t)) = ϕ(x′(λ_{1})) + ∫_{λ_{1}}^{t}(ϕ(x′(s)))′ds,
ϕ(x′(t)) = ϕ(x′(λ_{2})) + ∫_{λ_{2}}^{t}(ϕ(x′(s)))′ds,

that

\begin{matrix}∥ϕ(x′(t))∥_{∞} ≤ \frac{1}{1 − τ} ∥(ϕ(x′(t)))′∥_{1} \\ with τ = min\{\frac{∑_{i=1}^{m−2}(a_{i})_{+}}{∑_{i=1}^{m−2}(a_{i})_{−} + 1}, \frac{∑_{i=1}^{m−2}(a_{i})_{−} + 1}{∑_{i=1}^{m−2}(a_{i})_{+}}\}. \end{matrix}

This completes the proof of the theorem. □

Remark We note that if a_{i} ≤ 0 for every i = 1, 2, …, m−2, then τ = 0. Also, if a_{i} ≥ 0 for every i = 1, 2, …, m − 2 so that α = ∑_{i=1}^{m−2} a_{i} = ∑_{i=1}^{m−2}(a_{i})_{+} ≥ 0 then τ = min{α, 1/α} ∈ [0, 1) since α ≠ 1, by assumption.

The following theorem gives a better estimate for the three-point boundaryvalue problem in the case of the L^{2} norm.

Theorem 2.2 Let α ∈ ℝ, α ≠ 1, η ∈ (0, 1) be given. Let x(t) ∈ W_{ϕ}^{2,2}(0, 1) be such that ϕ(x′(1)) = αϕ(x′(η)). Then

∥ϕ(x′(t))∥_{2} ≤ C(α, η)∥(ϕ(x′(t)))′∥_{2},  (2.4)

where

C(α, η) = \begin{cases}min\{\sqrt{F (α, η)}, 2/π\} & if α ≤ 0, \\ \sqrt{F (α, η)}, & if α > 0, \end{cases}

F (α, η) = \frac{1}{2(α − 1)^{2}} [α^{2}(1 − η)^{2} + (α^{2} − 2α)η^{2} + 1].

Proof If α ≤ 0, we note from ϕ(x′(1)) = αϕ(x′(η)) that there exists an ξ ∈ (η, 1) such that ϕ(x′(ξ)) = 0. It follows from the Wirtinger’s inequality [3, Theorem 256] that

∥ϕ(x′(t))∥_{2} ≤ \frac{2}{π} ∥(ϕ(x′(t)))′∥_{2}.  (2.5)

Next, we note, again, from ϕ(x′(1)) = αϕ(x′(η)) that for 0 < t < 1,

ϕ(x′(t)) = ∫_{0}^{t}(ϕ(x′(s)))′ds + \frac{α}{1 − α} ∫_{0}^{η}(ϕ(x′(s)))′ds − \frac{1}{1 − α} ∫_{0}^{1}(ϕ(x′(s)))′ds.

(2.6)

Accordingly, we have for t ∈ [0, η]

ϕ(x′(t)) = ∫_{0}^{t}(ϕ(x′(s)))′ds + \frac{α}{1 − α} ∫_{0}^{η}(ϕ(x′(s)))′ds − \frac{1}{1 − α} ∫_{0}^{1}(ϕ(x′(s)))′ds
= ∫_{0}^{t}(1 + \frac{α}{1 − α} − \frac{1}{1 − α})(ϕ(x′(s)))′ds

+ ∫_{t}^{η}(\frac{α}{1 − α} − \frac{1}{1 − α})(ϕ(x′(s)))′ds − \frac{1}{1 − α} ∫_{η}^{1}(ϕ(x′(s)))′ds

= − ∫_{t}^{η}(ϕ(x′(s)))′ds − \frac{1}{1 − α} ∫_{η}^{1}(ϕ(x′(s)))′ds,  (2.7)

and, for t ∈ [η, 1]

ϕ(x′(t)) = ∫_{0}^{t}(ϕ(x′(s)))′ds + \frac{α}{1 − α} ∫_{0}^{η}(ϕ(x′(s)))′ds − \frac{1}{1 − α} ∫_{0}^{1}(ϕ(x′(s)))′ds
= ∫_{0}^{η}(1 + \frac{α}{1 − α} − \frac{1}{1 − α})(ϕ(x′(s)))′ds + ∫_{η}^{t}(1 − \frac{1}{1 − α})(ϕ(x′(s)))′ds

−\frac{1}{1 − α} ∫_{t}^{1}(ϕ(x′(s)))′ds

= − ∫_{η}^{t} \frac{α}{1 − α}(ϕ(x′(s)))′ds − \frac{1}{1 − α} ∫_{t}^{1}(ϕ(x′(s)))′ds.  (2.8)

Let us, now, define a function K : [0, 1] × [0, 1] → ℝ by

K(t, s) = \begin{cases}−χ_{[t,η]}(s) − \frac{1}{1−α}χ_{[η,1]}(s), & for t ∈ [0, η], s ∈ [0, 1], \\ −\frac{α}{1−α}χ_{[η,t]}(s) − \frac{1}{1−α}χ_{[t,1]}(s), & for t ∈ [η, 1], s ∈ [0, 1]. \end{cases}  (2.9)

Now, we see from equations (2.7), (2.8) that

ϕ(x′(t)) = ∫_{0}^{1} K(t, s)(ϕ(x′(s)))′ds, for t ∈ [0, 1],  (2.10)

and

∥ϕ(x′(t))∥_{2}^{2} ≤ (∫_{0}^{1} ∫_{0}^{1}(K(t, s))^{2} ds dt)∥(ϕ(x′(s)))′∥_{2}^{2}.  (2.11)

Now, it is easy to see that

∫_{0}^{1} ∫_{0}^{1}(K(t, s))^{2} ds dt = \frac{1}{2(α − 1)^{2}} [α^{2}(1 − η)^{2} + (α^{2} − 2α)η^{2} + 1]. (2.12)

For α ≤ 0 the estimate (2.4) is now immediate from (2.5), (2.11), (2.12) and for α > 0, α ≠ 1, by (2.11), (2.12). This completes the proof of the theorem. □

Remark It is easy to see that C(−0.1, η) = 2/π, for all η ∈ (0, 1). Indeed, \sqrt{F (−0.1, η)} ≥ 0.648986183 and 2/π ≈ 0.6366197724. Also C(−2, 1/3) = \sqrt{11/54} and C(−2, 15/16) = 2/π, since \sqrt{F (−2, 15/16)} = \sqrt{1030}/48 > 2/π.

3 Existence Theorems

Definition A function f : [0, 1] × ℝ^{2} → ℝ satisfies Caratheodory’s conditions if

(i) For each (x,y) ∈ ℝ^{2}, the function t ∈ [0, 1] → f(t, x, y) ∈ ℝ is measurable on [0, 1]

(ii) for a.e. t ∈ [0, 1], the function (x, y) ∈ ℝ^{2} → f(t, x, y) ∈ ℝ is continuous on ℝ^{2} (iii) for each r > 0, there exists α_{r}(t) ∈ L^{1}[0, 1] such that |f(t, x, y)| ≤ α_{r}(t)

for a.e. t ∈ [0, 1] and all (x, y) ∈ ℝ^{2} with \sqrt{x^{2} + y^{2}} ≤ r.

Theorem 3.1 Let f : [0, 1] × ℝ^{2} → ℝ be a function satisfying Caratheodory’s conditions. Assume that there exist functions p(t), q(t), r(t) in L^{1}(0, 1) such that

|f(t, x_{1}, x_{2})| ≤ p(t)ϕ(|x_{1}|) + q(t)ϕ(|x_{2}|) + r(t)  (3.1)

for a.e. t ∈ [0, 1] and all (x_{1}, x_{2}) ∈ ℝ^{2}. Also let a_{i} ∈ ℝ, ξ_{i} ∈ (0, 1), i = 1, 2, …, m − 2, 0 < ξ_{1} < ξ_{2} < ⋯ < ξ_{m−2} < 1, with α = ∑_{i=1}^{m−2} a_{i} ≠ 1 be given. Then the boundary-value problem (1.1) has at least one solution in C^{1}[0, 1] provided

∥p(t)∥_{1} + ∥q(t)∥_{1} + τ < 1.  (3.2)

where τ is as defined in Theorem 2.1.

Proof It is easy to see that the boundary-value problem (1.1) is equivalent to the fixed point problem

x(t) = ∫_{0}^{t} ϕ^{−1}( ∫_{0}^{s}[f(τ, x(τ), x′(τ)) + e(τ)]dτ + A)ds,  (3.3)

where

A = ∑_{i=1}^{m−2}(\frac{a_{i}}{1 − ∑_{i=1}^{m−2} a_{i}}) ∫_{0}^{ξ_{i}}[f(τ, x(τ), x′(τ)) + e(τ)]dτ

−\frac{1}{1 − ∑_{i=1}^{m−2} a_{i}} ∫_{0}^{1}[f(τ, x(τ), x′(τ)) + e(τ)]dτ.

It is standard to check that the mapping

x(t) ∈ C^{1}[0, 1] ↦ ∫_{0}^{t} ϕ^{−1}(∫_{0}^{s}[f(τ, x(τ), x′(τ)) + e(τ)]dτ + A)ds ∈ C^{1}[0, 1],

is a compact mapping. We apply the Leray-Schauder Continuation theorem (see, e.g. [4]) to obtain the existence of a solution for (3.3) or equivalently to the boundary-value problem (1.1).

To do this, it suffices to verify that the set of all possible solutions of the family of equations

\begin{matrix}(ϕ(x′(t)))′ = λf(t, x(t), x′(t)) + λe(t), 0 < t < 1, \\ x(0) = 0, ϕ(x′(1)) = ∑_{i=1}^{m−2} a_{i}ϕ(x′(ξ_{i})), \end{matrix}  (3.4)

is, a priori, bounded in C^{1}[0, 1] by a constant independent of λ ∈ [0, 1]. We observe that if x ∈ W_{ϕ}^{2,1}(0, 1), with x(0) = 0, ϕ(x′(1)) = ∑_{i=1}^{m−2} a_{i}ϕ(x′(ξ_{i})) then x(t) = ∫_{0}^{t} x′(s)ds. Hence, |x(t)| ≤ ∥x′∥_{∞} for t ∈ [0, 1] and

∥ϕ(x′(t))∥_{∞} ≤ \frac{1}{1 − τ} ∥(ϕ(x′(t)))′∥_{1},

where τ is as defined in Theorem 2.1. Also, it is easy to see that ϕ(∥x′∥_{∞}) ≤ ∥ϕ(x′(t))∥_{∞}.

Let, now, x(t) be a solution of (3.4) for some λ ∈ [0, 1], so that x ∈ W_{ϕ}^{2,1}(0, 1) with x(0) = 0, ϕ(x′(1)) = ∑_{i=1}^{m−2} a_{i}ϕ(x′(ξ_{i})). We then get from the equation in (3.4) and Theorem 2.1 t hat

∥ϕ(x′(t))∥_{∞} ≤ \frac{λ}{1 − τ} ∥f(t, x(t), x′(t)) + e(t)∥_{1}

≤ \frac{1}{1 − τ} (∥p(t)ϕ(|x(t)|) + q(t)ϕ(|x′(t)|) + r(t)∥_{1} + ∥e(t)∥_{1})

≤ \frac{1}{1 − τ} (\{∥p(t)∥_{1} + ∥q(t)∥_{1}\}∥ϕ(|x′(t)|)∥_{∞} + ∥r(t)∥_{1} + ∥e(t)∥_{1})

≤ \frac{1}{1 − τ} (∥p(t)∥_{1} + ∥q(t)∥_{1})∥ϕ(x′(t))∥_{∞} + \frac{1}{1 − τ} (∥r(t)∥_{1} + ∥e(t)∥_{1}).

It follows from the assumption (3.2) that there is a constant c, independent of λ ∈ [0, 1], such that

∥x∥_{∞} ≤ ∥x′∥_{∞} ≤ c.

It is now immediate that the set of solutions of the family of equations (3.4) is, a priori, bounded in C^{1}[0, 1] by a constant, independent of λ ∈ [0, 1]. This completes the proof of the theorem. □

Remark Suppose that the the odd increasing homeomorphism ϕ in Theorem 3.1 is k-homogeneous, in the sense that ϕ(tx) = t^{k}ϕ(x) for t ≥ 0 and x ∈ ℝ. Then the existence condition 3.2 in Theorem 3.1 becomes

∥t^{k}p(t)∥_{1} + ∥q(t)∥_{1} + τ < 1.

Theorem 3.2 Let f : [0, 1] × ℝ^{2} → ℝ be a function satisfying Caratheodory’s conditions. Assume that there exist functions p(t), q(t), r(t) in L^{2}(0, 1) such that

|f(t, x_{1}, x_{2})| ≤ p(t)ϕ(|x_{1}|) + q(t)ϕ(|x_{2}|) + r(t)  (3.5)

for a.e. t ∈ [0, 1] and all (x_{1}, x_{2}) ∈ ℝ^{2}. Also let α ≠ 1, and η ∈ (0,1) be given. Then for any given e(t) in L^{2}(0, 1) the boundary-value problem (1.3) has at least one solution in C^{1}[0, 1] provided

\frac{1}{1 − τ} ∥p∥_{2} + C(α, η)∥q∥_{2} < 1,  (3.6)

where C(α, η) is as in Theorem 2.2.

Proof As in the proof of Theorem 3.1 it suffices to prove that the set of all possible solutions of the family of equations

\begin{matrix}(ϕ(x′(t)))′ = λf(t, x(t), x′(t)) + λe(t), 0 < t < 1, \\ x(0) = 0, ϕ(x′(1)) = αϕ(x′(η)), \end{matrix}  (3.7)

is, a priori, bounded in C^{1}[0,1] by a constant independent of λ ∈ [0,1]. For an x ∈ W_{ϕ}^{2,2}(0,1), with x(0) = 0, and ϕ(x′(1)) = αϕ(x′(η)) we use Theorem 2.1 and Theorem 2.2, above, to note that

∥ϕ(|x(t)|)∥_{2} ≤ ϕ(∥x∥_{∞}) ≤ ϕ(∥x′∥_{∞}) ≤ ∥ϕ(x′(t))∥_{∞}
≤ \frac{1}{1 − τ} ∥(ϕ(x′(t)))′∥_{1} ≤ \frac{1}{1 − τ} ∥(ϕ(x′(t)))′∥_{2}

and

∥ϕ(x′(t))∥_{2} ≤ C(α, η)∥(ϕ(x′(t)))′∥_{2}.  (3.8)

Now, for a solution x of the family of equations (3.7) for some λ ∈ [0,1] we have

∥(ϕ(x′(t)))′∥_{2} ≤ λ∥f(t, x(t), x′(t)) + e(t)∥_{2}

≤ ∥p(t)ϕ(|x(t)|) + q(t)ϕ(|x′(t)|) + r(t)∥_{2} + ∥e∥_{2} ≤ ∥p∥_{2}∥ϕ(|x(t)|)∥_{2} + ∥q∥_{2}∥ϕ(|x′(t)|)∥_{2} + ∥r∥_{2} + ∥e∥_{2}

≤ (\frac{1}{1 − τ} ∥p∥_{2} + C(α, η)∥q∥_{2})∥(ϕ(x′(t)))′∥_{2} + ∥r(t)∥_{2} + ∥e∥_{2},

in view of estimate (3.8), for a solution x of the family of equations (3.7) for some λ ∈ [0,1]. It then follows from (3.6) that there is a constant c independent of λ ∈ [0, 1] such that

∥(ϕ(x′(t)))′∥_{2} ≤ c,

for a solution x of the family of equations (3.7) for some λ ∈ [0,1]. Finally, we see, using Theorem 2.1 that

ϕ(∥x∥_{∞}) ≤ ϕ(∥x′∥_{∞}) ≤ \frac{1}{1 − τ} ∥(ϕ(x′(t)))′∥_{1} ≤ \frac{1}{1 − τ} ∥(ϕ(x′(t)))′∥_{2}

and accordingly, the set of solutions of the family of equations (3.7) is, a priori, bounded in C^{1}[0, 1] by a constant independent of λ ∈ [0, 1]. This completes the proof of Theorem 3.2. □

Example 3.3 Let α ≤ 0 and η ∈ (0, 1) be given and A ∈ ℝ. For e(t) ∈ L^{1}(0, 1), we consider the three point boundary-value problem

\begin{matrix}(ϕ(x′(t)))′ = t^{\frac{1}{2}} ϕ(|x(t)|) + Atϕ(|x′(t)|) + e(t), 0 < t < 1, \\ x(0) = 0, ϕ(x′(1)) = αϕ(x′(η)). \end{matrix}  (3.9)

We apply Theorem 3.1 to obtain a condition for the existence of a solution for the three-point boundary-value problem (3.9). Here p(t) = t^{1/2}, q(t) = At and τ = 0. Now, ∥p(t)∥_{1} = 2/3 and ∥q(t)∥_{1} = \frac{1}{2}|A|. Now, if

\frac{2}{3} + \frac{1}{2}|A| < 1,

or, equivalently |A| < 2/3, then Theorem 3.1 implies the existence of a solution for the three-point boundary-value problem (3.9).

Example 3.4 Let α = −2, η = \frac{1}{3} and A ∈ ℝ. For e(t) ∈ L^{2}(0, 1), we, next, consider the three point boundary-value problem

\begin{matrix}(ϕ(x′(t)))′ = t^{\frac{1}{4}} ϕ(|x(t)|) + At^{− \frac{1}{4}} ϕ(|x′(t)|) + e(t), 0 < t < 1, \\ x(0) = 0, ϕ(x′(1)) = αϕ(x′(η)). \end{matrix}  (3.10)

We apply Theorem 3.2 to obtain a condition for the existence of a solution for the three-point boundary-value problem (3.10). Here p(t) = t^{1/4}, q(t) = At^{−1/4}. Now, ∥p(t)∥_{2} = \sqrt{2/3} and ∥q(t)∥_{2} = \sqrt{2}|A|. Now the existence condition required to apply Theorem 3.2

\sqrt{\frac{2}{3}} + \sqrt{2}C(α, η)|A| < 1.  (3.11)

Since we have C(−2, \frac{1}{3}) = \sqrt{\frac{11}{54}}, we get from (3.11)

\sqrt{\frac{2}{3}} + \sqrt{\frac{22}{54}}|A| < 1.

Accordingly, we see from Theorem 3.2 that a solution for the three-point boundary-value problem (3.10) exists if |A| < \sqrt{54/22}(1 − \sqrt{2/3}) = .287 49.

Example 3.5 Let α = −2, η = 1/3 and A ∈ ℝ. For e(t) ∈ L^{2}(0, 1), we, next, consider the three point boundary-value problem

\begin{matrix}(ϕ(x′(t)))′ = t^{\frac{15}{32}} ϕ(|x(t)|) + Atϕ(|x′(t)|) + e(t), 0 < t < 1, \\ x(0) = 0, ϕ(x′(1)) = αϕ(x′(η)). \end{matrix}  (3.12)

We apply Theorem 3.2 to obtain a condition for the existence of a solution for the three-point boundary-value problem (3.12). Here p(t) = t^{15/32}, q(t) = At. Now, ∥p(t)∥_{2} = 4/\sqrt{31} and ∥q(t)∥_{2} = |A|/\sqrt{3}. Now the existence condition required to apply Theorem 3.2

\frac{4}{\sqrt{31}} + \frac{1}{\sqrt{3}}C(α, η)|A|) < 1.  (3.13)

Since, C(−2, 1/3) = \sqrt{11/54} and we get from (3.13)

\frac{4}{\sqrt{31}} + \sqrt{\frac{11}{162}}|A| < 1,

which implies

|A| < \sqrt{\frac{162}{11}}(1 − \frac{4}{\sqrt{31}}) = 1. 080 6.

Now, to apply Theorem 3.1 we see that

∥p(t)∥_{1} = ∫_{0}^{1} t^{\frac{15}{32}} dt = \frac{32}{47}

and ∥q(t)∥_{1} = \frac{1}{2}|A|. Accordingly, we see using Theorem 3.1 a solution for the three-point boundary-value problem (3.12) exists if

\frac{32}{47} + \frac{1}{2}|A| < 1,

or, equivalently, if

|A| < 2(1 − \frac{32}{47}) = \frac{30}{47} = 0.6383.

We thus see that Theorem 3.2 gives a better result than Theorem 3.1.

Remark Note that if we take for p > 1, the odd increasing homeomorphism ϕ : ℝ → ℝ defined by

ϕ(t) = |t|^{p−2}t for t ∈ ℝ,

then Theorems 3.1, 3.2 give existence theorems for the analogous three-point boundary-value problems for the one-dimensional analogue of the p-Laplacian. However, Theorems 3.1, 3.2 apply to more general differential operators than a p-Laplacian, since Theorems 3.1, 3.2 do not require the homeomorphism ϕ to be homogeneous as happens to be the case for the p-Laplacian.

References

[1] C. P. Gupta; S. K. Ntouyas and P. Ch. Tsamatos; Existence Results for m-Point Boundary Value Problems, Differential Equations Dynam. Systems, Vol. 2 (1994), 289-298.

[2] C. P. Gupta, S. I. Trofimchuk, Solvability of a multi-point boundary-value problem of Neumann type, Abstr. Anal. & Appl., 4, # 2, (1999) 71-81..

[3] G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, London/New York, 1967.

[4] J. Mawhin, Topological degree methods in nonlinear boundary value problems, in ”NSF-CBMS Regional Conference Series in Math.” No. 40, Amer. Math. Soc., Providence, RI, 1979.

[5] V.A. Il’in, and E. I. Moiseev, Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator, Differential Equations, Vol. 23, No. 8, (1988), pp. 979-987.

Chaitan P. Gupta Department of Mathematics, 084 University of Nevada, Reno Reno, NV 89557, USA email: gupta@unr.edu