The main purpose of this paper is to investigate the existence of solutions of BVPs for a very general case in which both the system of ordinary differential equations and the boundary conditions are nonlinear. By employing the implicit function theorem, sufficient conditions for the existence of three-point boundary value problems are established.

1. Introduction

We consider existence of solutions at resonance to first-order three-point BVPs with nonlinear boundary conditions using results developed in [1, 2].

Consider

(1.1)

(1.2)

where M, N, and R are constant square matrices of order n, A(t) is an n × n matrix with continuous entries, E : [0,1] → ℝ is continuous, F : [0,1] × ℝⁿ × (−ɛ₀, ɛ₀) → ℝⁿ is a continuous function where ɛ₀ > 0, ℓ ∈ ℝⁿ, η ∈ (0,1), and g : ℝ³ⁿ → ℝⁿ is continuous.

Our existence theorem uses the implicit function theorem; see for example Nagle [3]. Nagle [3] extended the alternative method considered by Hale [4] for handling the periodic case of non-self-adjoint problems subject to homogeneous boundary conditions. These results extend the work of Feng and Webb [5] and Gupta [6] of three-point BVPs with linear boundary conditions for α = 1 and αη = 1 to nonlinear boundary conditions. Feng and Webb [5] studied the existence of solutions of the following BVPs (1.3) and (1.4):

(1.3)

(1.4)

where η ∈ (0,1), α ∈ ℝ, f : [0,1] × ℝ² → ×ℝ is a continuous function, and e : [0,1] → ℝ is a function in L¹[0,1]. Both of the problems are resonance cases under the assumption α = 1 for the problem (1.3), and αη = 1 for the problem (1.4). The problem for nonlinear boundary conditions for discrete systems has been studied by Rodriguez [7, 8]. Rodriguez [7] extended results of Halanay [9], who considered periodic boundary conditions and also extended those of Rodriguez [10] and Agarwal [11] who considered linear boundary conditions. To our knowledge there appears to be no research in the literature on multipoint BVPs for systems of first-order equations with nonlinear boundary conditions at resonance. The results of this paper fill this gap in the literature.

Our results are analogues for three-point boundary conditions of those periodic boundary conditions for perturbed systems of first-order equations at resonance considered by Coddington and Levinson [12] and Cronin [13, 14]. Moreover, our results extend the work of Urabe [15], Liu [16], and of Nagle [3], where he solved the two-point BVP using the Cesari-Hale alternative method.

2. Preliminaries

Now we state the following basic existence theorems for systems with a parameter and use them to formulate the existence results for problem (1.1) and (1.2).

Theorem 2.1 (see Coppel [17], Page 19.)

(i)
Let F(t, x, ɛ) be a continuous function of (t, x, ɛ) for all points (t, x) in an open set D and all values ɛ near .
(ii)
Let x(t, c, ɛ) be any noncontinuable solution of the differential equation
(2.1)
If is defined on the interval [0,1] and is unique, then x(t, c, ɛ) is defined on [0,1] for all (c, ɛ) sufficiently near and is a continuous function of its threefold arguments at any point .

Theorem 2.2 (see Coppel [17], Page 22.)

(i)
Let F(t, x, ɛ) be a continuous function of (t, x, ɛ) for all points (t, x) in a domain D and all values of the vector parameter ɛ near .
(ii)
Let be a solution of the differential equation
(2.2)
defined on a compact interval [0,1].
(iii)
Suppose that F has continuous partial derivatives F_x, F_ɛ at all points with t ∈ [0,1].

Then for all (c, ɛ) sufficiently near the differential equation

(2.3)

has a unique solution x(t, c, ɛ) over [0,1] that is close to the solution

of (ii). The continuous differentiability of F with respect to x and ɛ implies the additional property that the solution x(t, c, ɛ) is differentiable with respect to (t, c, ɛ) for (c, ɛ) near

We recall the following results of [2].

Lemma 2.3 (see [2].)Consider the system

(2.4)

where A(t) is an n × n matrix with continuous entries on the interval [0,1]. Let Y(t) be a fundamental matrix of (2.4). Then the solution of (2.4) which satisfies the initial condition

(2.5)

is x(t) = Y(t)Y⁻¹(0)c where c is a constant n-vector. Abbreviate Y(t)Y⁻¹(0) to Y₀(t). Thus x(t) = Y₀(t)c.

Lemma 2.4 (see [2].)Let Y(t) be a fundamental matrix of (2.4). Then any solution of (1.1) and (2.5) can be written as

(2.6)

The solution (1.1) satisfies the boundary conditions (1.2) if and only if

(2.7)

where ℒ = M + NY₀(η) + RY₀(1), 𝒩(c, α, η, ɛ) =

− g(c, x(η), x(1))),

, and x(t, c, ɛ) is the solution of (1.1) given x(0) = c.

Thus (2.7) is a system of n real equations in ɛ, c₁, …, c_n where c₁, …, c_n are the components of c. The system (2.7) is sometimes called the branching equations.

Next we suppose that ℒ is a singular matrix. This is sometimes called the resonance case or degenerate case. Now we consider the case rank ℒ = n − r, 0 < n − r < n. Let E_r denote the null space of ℒ, and let E_n−r denote the complement in ℝⁿ of E_r; that is,

(2.8)

Let x₁, …, x_n be a basis for ℝⁿ such that x₁, …, x_r is a basis for E_r and x_r+1, …, x_n a basis for E_n−r.

Let P_r be the matrix projection onto Ker ℒ = E_r, and P_n−r = I − P_r, where I is the identity matrix. Thus P_n−r is a projection onto the complementary space E_n−r of E_r, and

(2.9)

Without loss of generality, we may assume

(2.10)

We will identify P_rc with c^r = (c₁, …, c_r) and P_n−rc with c^n−r = (c_r+1, …, c_n) whenever it is convenient to do so.

Let H be a nonsingular n × n matrix satisfying

(2.11)

Matrix H can be computed easily. The nature of the solutions of the branching equations depends heavily on the rank of the matrix ℒ.

Lemma 2.5 (see [2].)The matrix ℒ has rank n − r if and only if the three-point BVP (2.4) and Mx(0) + Nx(η) + Rx(1) = 0 has exactly r linearly independent solutions.

Next we give a necessary and sufficient condition for the existence of solutions of x(t, c, ɛ) of three-point BVPs for ɛ > 0 such that the solution satisfies x(0) = c where c = c(ɛ) for suitable c(ɛ).

We need to solve (2.7) for c when ɛ is sufficiently small. The problem of finding solutions to (1.1) and (1.2) is reduced to that of solving the branching equations (2.7) for c as function of ɛ for |ɛ| < ɛ₀. So consider (2.7) which is equivalent to

(2.12)

Multiplying (2.7) by the matrix H and using (2.11), we have

(2.13)

where H𝒩((P_r + P_n−r)c, α, η, ɛ) =

− g(c, x(η), x(1))) and Hd =

Since the matrix H is nonsingular, solving (2.7) for c is equivalent to solving (2.13) for c. The following theorem due to Cronin [13, 14] gives a necessary condition for the existence of solutions to the BVP (1.1) and (1.2).

Theorem 2.6 (see [2].)A necessary condition that (2.13) can be solved for c, with |ɛ| < ɛ₀, for some ɛ₀ > 0 is P_rHd = 0.

If ℒ is a nonsingular matrix then the implicit function theorem is applicable to solve (2.7) uniquely for c as a function of ɛ in a neighborhood of the initial solution c (see Cronin [14]). The implicit function theorem may be stated as in Voxman and Goetschel [18, page 222].

Theorem 2.7 (the implicit function theorem). Let Ω ⊂ ℝⁿ × ℝ^m be an open set, and let F : Ω → ℝ^m be function of class C¹. Suppose (x₀, y₀) = 0. Assume that

(2.14)

where F = (F₁, …, F_m). Then there are open sets U ⊂ ℝⁿ and V ⊂ ℝ^m, with x₀ ∈ U and y₀ ∈ U, and a unique function f : U → V such that

(2.15)

for all x ∈ U with y₀ = f(x₀). Furthermore, f is of class C¹.

3. Main Results

In this section sufficient conditions are introduced for the existence of solutions to the BVP (1.1), (1.2). We recall the following Definition 1 of [2] to develop our main results.

Definition 3.1 (see [2].)Let E_r denote the null space of ℒ, and let E_n−r denote the complement in ℝⁿ of E_r. Let P_r be the matrix projection onto Ker ℒ = E_r, and P_n−r = I − P_r, where I is the identity matrix. Thus P_n−r is a projection onto the complementary space E_n−r of E_r. If E_n−r is properly contained in ℝⁿ, then E_r is an r-dimensional vector space where 0 < r < n. If c = (c₁, …, c_n), let P_rc = c^r and P_n−r = c^n−r, then define a continuous mapping Φ_ɛ : ℝ^r → ℝ^r, given by

(3.1)

where c^n−r(c^r, ɛ) = c^n−r is a differentiable function of c^r and ɛ. By abuse of notation we will identify P_rc and c^r when convenient and where the meaning is clear from the context so that in defining Φ_ɛ above from the context we interpreted P_rH𝒩 as (H𝒩₁, …, H𝒩_r). Similarly we will sometimes identify P_n−rc and c^n−r. Setting ɛ = 0, we have

(3.2)

where c^n−r(c^r, 0) = P_n−rHd; note that from the context c^n−r(c^r, 0) = P_n−rHd is interpreted as c^n−r(c^r, 0) = (Hd_r+1, …, Hd_n).

If E_r = ℝⁿ and P_r = I, then P_n−r = 0. Since P_n−r = 0, it follows that the matrix H is the identity matrix. Thus define a continuous mapping Φ_ɛ : ℝⁿ → ℝⁿ, given by Φ_ɛ(c) = 𝒩(c, α, η, ɛ). Setting ɛ = 0, we have Φ₀(c) = 𝒩(c, α, η, 0).

The following theorem is the main theorem of this paper and gives sufficient conditions for the existence of solutions of (1.1), (1.2) for |ɛ| < ɛ₀, for some ɛ₀ > 0. The existence theorem can be established using the implicit function theorem; see Theorem 2.7.

Theorem 3.2. If c = (c₁, …, c_n) ∈ ℝⁿ, let c^r = (c₁, …, c_r). Let the conditions (i), (ii), and (iii) of Theorem 2.2 hold, and let k₁ > 0, k > 0 and ɛ₀ > 0 be small enough so that (1.1) has a unique n-vector x(t, c, ɛ) defined on . Let , given by

(3.3)

where c^n−r(c^r, ɛ) = c^n−r is a differentiable function of c^r and ɛ, and

(3.4)

for

. If

and

(3.5)

for some

, then there is

, and δ > 0 such that (1.1), (1.2) has a unique solution x(t, c(ɛ), ɛ) for all

such that

and

Proof. The existence and uniqueness of a solution x(t, c, ɛ) for |ɛ| < ɛ₀ with x(0, c, ɛ) = c ∈ ℝⁿ follows directly from conditions (i), (ii), and (iii) of Theorem 2.2. Now

(3.6)

for some

, thus it follows from the implicit function theorem that there is

such that (3.3) has a unique solution (c₁, …, c_r) = (c₁(ɛ), …, c_r(ɛ)), with

, for all ɛ,

. From this it follows that x(t, c(ɛ), ɛ) is a unique solution of the BVP (1.1), (1.2) which satisfies the initial value x(0, c(ɛ), ɛ) = c(ɛ) and

and

, where

We now consider the BVP (1.1), (1.2) in the case r = n; that is, ℒ is the zero matrix, which is sometimes called the totally degenerate case.

Theorem 3.3 (compare with Theorem 3.8, page 69 of Cronin [14]). If r = n, a necessary condition in order that (2.7) has a solution for each ɛ with |ɛ| < ɛ₀ for some ɛ₀ > 0 is d = 0; that is,

(3.7)

Theorem 3.4. Let the conditions (i), (ii), and (iii) of Theorem 2.2 hold, and let k₁ > 0, k > 0 and ɛ₀ > 0 be small enough so that (1.1) has a unique solution x(t, c, ɛ) defined on . If r = n, d = 0, and

(3.8)

then there is

, and δ > 0 such that (1.1), (1.2) has a unique solution x(t, c(ɛ), ɛ) for all

such that

and

Proof. If r = n and d = 0, then P_n−r = 0. This implies P_r = I. Since P_n−r = 0, it follows that H = I, the identity matrix.

The existence and uniqueness of a solution x(t, c(ɛ), ɛ) for

with x(0, c, ɛ) = c ∈ ℝⁿ follows directly from conditions (i), (ii) and (iii) of Theorem 2.2. Now

(3.9)

(3.10)

for some

; thus it follows from the implicit function theorem that there is

such that (3.8) has a unique solution c = c(ɛ), with

, for all ɛ,

. From this it follows that x(t, c(ɛ), ɛ) is a unique solution of the BVP (1.1), (1.2) which satisfies the initial values x(0, c(ɛ), ɛ) = c(ɛ) ∈ ℝⁿ for all ɛ,

such that

and

4. Some Examples

To find c for ɛ small using Theorem 2.6, we need to compute Φ₀(c) from (3.3). We apply Theorem 3.2 to show the existence of solutions.

Example 4.1. α = 1, rank ℒ_α=1 = 1 < 2, ℓ_i ≡ 0 for i = 1,2.

Consider the BVP

(4.1)

where f ∈ C([0,1] × ℝ² × (−ɛ₀, ɛ₀); ℝ), e ∈ C[0,1], g ∈ C(ℝ⁶; ℝ²). Then the BVP (4.1) is equivalent to

(4.2)

(4.3)

where

(4.4)

By Lemma 2.4, we find ℒ:

(4.5)

The resonance happens if det (ℒ) = −1 + α = 0; that is the case where α = 1. For α = 1, rank ℒ_α=1 = 1; that is,

(4.6)

Let E₁ denote the null space of ℒ_α=1. Thus

is a basis for Ker (ℒ_α=1), and Ker (ℒ_α=1) = Span e₁. Let P₁ be the matrix projection onto Ker (ℒ_α=1).

. Set H =

so that Hℒ_α=1 = P₂. In system (4.2), (4.3) let

, and let g₂(c₁, c₂, x₁(1/2), x₂(1/2), x₁(1), x₂(1)) = 2x₁(1/2)/256π⁴. We need to show that P₁Hd = 0 which is a necessary condition in order to apply Theorem 2.6:

(4.7)

Since

, it follows that P₁Hd = 0. From the boundary condition (4.3), we have x₂(0) = c₂ = 0. Then, by the variation of constants formula, we obtain

(4.8)

Thus the BVP (4.2), (4.3) has a solution if α = 1, ɛ = 0; namely, x₁(t, c, 0) = c₁ + ((1 − cos 4πt)/16π²), x₂(t, c, 0) = sin 4πt/4π, x₁(0) = x₁(1/2) = x₁(1) = c₁, x₂(0) = x₂(1/2) = x₂(1) = 0. Setting ɛ = 0, thus

and g₂(c₁, x₁(1/2), x₁(1)) = 2c₁/256π⁴. Hence

(4.9)

If c₁ ≈ −3.5023 × 10⁻³ or c₁ ≈ 4.2938 × 10⁻³, then Φ₀(c₁) = 0 and

(4.10)

Hence by Theorem 3.2 there is , and δ > 0 such that the BVP (4.2), (4.3) has a unique solution x(t, c(ɛ), ɛ) which satisfies the initial values x(0, c(ɛ), ɛ) = c(ɛ) ∈ ℝ² for all such that c(0) = (c₁, 0) and |c(ɛ) − c(0)| < δ.

Example 4.2. Rank ℒ = 2 < 3.

Consider the BVP

(4.11)

(4.12)

where f_i ∈ C([0,1] × ℝ³ × (−ɛ₀, ɛ₀); ℝ), i = 1,2, 3, ℓ = (ℓ₁, ℓ₂, ℓ₃) ∈ ℝ³, g ∈ C(ℝ⁹; ℝ³),

(4.13)

By Lemma 2.4, the problem of solving (4.11), (4.12) is reduced to that of solving ℒc = ɛ𝒩(c, α, η, ɛ) + d for c provided solutions x(t, c, ɛ) of initial value problems exist on [0,1] for each (c, ɛ). Thus we find ℒ:

(4.14)

Since rank ℒ = 2, it follows that the matrix ℒ is singular. Let E₃ denote the null space of ℒ. Thus

is a basis for Ker (ℒ), and Ker (ℒ) = Span e₃. Let P₃ be the matrix projection onto Ker (ℒ).

. So

. Set

so that Hℒ = P₂.

(4.15)

Since d = 0, it follows that P₃Hd = 0. Thus a necessary condition of Theorem 2.6 holds. We also have P₂Hd = 0. To obtain Φ₀(c) we must first calculate x(t, c, 0); that is the solution of x^′ = A(t)x + e(t). By Lemma 2.3, and boundary condition (4.12), x^′ = A(t)x has a solution x(t) with x(0) = c = (c₁, c₂, c₃) ^T. We note that at ɛ = 0, P₂Hd = P₂c, where

and P₂Hd = 0. Hence c₁ = 0 and c₂ = 0. Thus

(4.16)

Thus the BVP (4.11), (4.12) has a solution if ɛ = 0; namely, x₁(t, c, 0) = x₂(t, c, 0) = 0 and x₃(t, c, 0) = c₃, and thus ℓ_i = 0, i = 1,2, x₃(π) = c₃ = −ℓ₃, and x₃(2π) = c₃:

(4.17)

where

(4.18)

Thus Φ_ɛ(c₃) = 𝒩₃(c³ ⊕ c²(c³, ɛ), α, η, ɛ), where c³ = P₃c = (0,0, c₃) and c² = P₂c = (c₁, c₂, 0). Setting ɛ = 0, we have Φ₀(c₃) = 𝒩₃(c³, α, η, 0), where c²(c³, 0) = P₂Hd = 0. Writing out the components and setting ɛ = 0, we obtain x₁(t, c, 0) = x₂(t, c, 0) = 0 and x₃(t, c, 0) = c₃. Hence

(4.19)

where x_i(π) = x_i(2π) = 0, i = 1,2, x₃(π) = c₃ = −ℓ₃, and x₃(2π) = c₃. Let

, and

. Hence

(4.20)

, then Φ₀(c₃) = 0 and

(4.21)

Hence by Theorem 3.2 there is

and δ > 0 such that the BVP (4.11), (4.12) has a unique solution x(t, c(ɛ), ɛ) which satisfies the initial values x(0, c(ɛ), ɛ) = c(ɛ) ∈ ℝ³ for all

such that c(0) = (0,0, c₃) and |c(ɛ) − c(0)| < δ.

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First-Order Three-Point Boundary Value Problems at Resonance Part III

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First-Order Three-Point Boundary Value Problems at Resonance Part III

Abstract

1. Introduction

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3. Main Results

4. Some Examples

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