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

Abstract

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].

Consider𝑥^′(𝑡)()()(𝑡)(0)𝜂(1)𝑥(0)𝜂(1),−𝐴𝑥=𝐻𝑡,𝑥,𝜀=𝜀𝐹𝑡,𝑥,𝜀+𝐸,0≤𝑡≤1,𝑀𝑥+𝑁𝑥+𝑅𝑥=ℓ+𝜀𝑔,𝑥,𝑥 (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]×ℝ^𝑛×(−𝜀₀,𝜀₀)→ℝ^𝑛 is a continuous function where 𝜀₀>0, ℓ∈ℝ^𝑛, 𝜂∈(0,1), and 𝑔∶ℝ^3𝑛→ℝ^𝑛 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):𝑦^′′=𝑓𝑡,𝑦,𝑦^′(𝑡)𝑦+𝑒,0≤𝑡≤1,_′(0)(1)𝜂,𝑦=0,𝑦=𝛼𝑦_′′=𝑓𝑡,𝑦,𝑦^′(𝑡)𝑦(0)(1)𝜂,+𝑒,0≤𝑡≤1,=0,𝑦=𝛼𝑦 (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,𝜀) be a continuous function of (𝑡,𝑥,𝜀) for all points (𝑡,𝑥) in an open set 𝐷 and all values 𝜀 near 𝜀.(ii)Let 𝑥(𝑡,𝑐,𝜀) be any noncontinuable solution of the differential equation 𝑥^′=𝐹(𝑡,𝑥,𝜀),with𝑥(0)=𝑐. (2.1) If 𝑥(𝑡,𝑐,𝜀) is defined on the interval [0,1] and is unique, then 𝑥(𝑡,𝑐,𝜀) is defined on [0,1] for all (𝑐,𝜀) sufficiently near (𝑐,𝜀) and is a continuous function of its threefold arguments at any point (𝑡,𝑐,𝜀).

Theorem 2.2 (see Coppel [17, Page 22]). (i) Let 𝐹(𝑡,𝑥,𝜀) be a continuous function of (𝑡,𝑥,𝜀) for all points (𝑡,𝑥) in a domain 𝐷 and all values of the vector parameter 𝜀 near 𝜀. (ii)Let 𝑥(𝑡,𝑐,𝜀) be a solution of the differential equation 𝑥^′=𝐹(𝑡,𝑥,𝜀),with𝑥(0)=𝑐 (2.2) defined on a compact interval [0,1]. (iii)Suppose that 𝐹 has continuous partial derivatives 𝐹_𝑥, 𝐹_𝜀 at all points (𝑡,𝑥(𝑡,𝑐,𝜀),𝜀) with 𝑡∈[0,1].
Then for all (𝑐,𝜀) sufficiently near (𝑐,𝜀) the differential equation 𝑥^′=𝐹(𝑡,𝑥,𝜀),with𝑥(0)=𝑐 (2.3) has a unique solution 𝑥(𝑡,𝑐,𝜀) over [0,1] that is close to the solution 𝑥(𝑡,𝑐,𝜀) of (ii). The continuous differentiability of 𝐹 with respect to 𝑥 and 𝜀 implies the additional property that the solution 𝑥(𝑡,𝑐,𝜀) is differentiable with respect to (𝑡,𝑐,𝜀) for (𝑐,𝜀) near (𝑐,𝜀).

We recall the following results of [2].

Lemma 2.3 (see [2]). Consider the system 𝑥^′=𝐴(𝑡)𝑥, (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 𝑥(0)=𝑐 (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 𝑥(𝑡,𝑐,𝜀)=𝑥(𝑡)=𝑌₀(𝑡)𝑐+^𝑡₀𝑌(𝑡)𝑌⁻¹(𝑠)𝐻(𝑠,𝑥(𝑠),𝜀)𝑑𝑠. (2.6) The solution (1.1) satisfies the boundary conditions (1.2) if and only if ℒ𝑐=𝜀𝒩𝑐,𝛼,𝜂,𝜀+𝑑, (2.7) where ℒ=𝑀+𝑁𝑌₀(𝜂)+𝑅𝑌₀(1), 𝒩(𝑐,𝛼,𝜂,𝜀) = −(∫^𝜂₀𝑁𝑌(𝜂)𝑌⁻¹(𝑠)𝐹(𝑠,𝑥(𝑠,𝑐,𝜀),𝜀)𝑑𝑠 + ∫¹₀𝑅𝑌(1)𝑌⁻¹(𝑠)𝐹(𝑠,𝑥(𝑠,𝑐,𝜀),𝜀)𝑑𝑠 − 𝑔(𝑐,𝑥(𝜂),𝑥(1))), 𝑑=−(∫^𝜂₀𝑁𝑌(𝜂)𝑌⁻¹(𝑠)𝐸(𝑠)𝑑𝑠 + ∫¹₀𝑅𝑌(1)𝑌⁻¹(𝑠)𝐸(𝑠)𝑑𝑠−ℓ), and 𝑥(𝑡,𝑐,𝜀) is the solution of (1.1) given 𝑥(0)=𝑐.

Thus (2.7) is a system of 𝑛 real equations in 𝜀,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 ℝ^𝑛 of 𝐸_𝑟; that is,ℝ^𝑛=𝐸_𝑛−𝑟⊕𝐸_𝑟(directsum). (2.8) Let 𝑥₁,…,𝑥_𝑛 be a basis for ℝ^𝑛 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, and𝑃²_𝑟=𝑃_𝑟,𝑃²_𝑛−𝑟=𝑃_𝑛−𝑟,𝑃_𝑛−𝑟𝑃_𝑟=𝑃_𝑟𝑃_𝑛−𝑟=0. (2.9) Without loss of generality, we may assume𝑃_𝑟𝑐=𝑐₁,…,𝑐_𝑟,0,…,0,𝑃_𝑛−𝑟𝑐=0,…,0,𝑐_𝑟+1,…,𝑐_𝑛. (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 satisfying𝐻ℒ=𝑃_𝑛−𝑟. (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,𝜀) of three-point BVPs for 𝜀>0 such that the solution satisfies 𝑥(0)=𝑐 where 𝑐=𝑐(𝜀) for suitable 𝑐(𝜀).

We need to solve (2.7) for 𝑐 when 𝜀 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 𝜀 for |𝜀|<𝜀₀. So consider (2.7) which is equivalent toℒ𝑃_𝑟+𝑃_𝑛−𝑟𝑐=𝜀𝒩𝑃_𝑟+𝑃_𝑛−𝑟𝑐,𝛼,𝜂,𝜀+𝑑. (2.12) Multiplying (2.7) by the matrix 𝐻 and using (2.11), we have𝑃_𝑛−𝑟𝑐=𝜀𝐻𝒩𝑃_𝑟+𝑃_𝑛−𝑟𝑐,𝛼,𝜂,𝜀+𝐻𝑑, (2.13) where 𝐻𝒩((𝑃_𝑟+𝑃_𝑛−𝑟)𝑐,𝛼,𝜂,𝜀) = −𝐻(∫^𝜂₀𝑁𝑌(𝜂)𝑌⁻¹(𝑠)𝐹(𝑠,𝑥(𝑠,𝑐,𝜀),𝜀)𝑑𝑠 + ∫¹₀𝑅𝑌(1)𝑌⁻¹(𝑠)𝐹(𝑠,𝑥(𝑠,𝑐,𝜀),𝜀)𝑑𝑠 − 𝑔(𝑐,𝑥(𝜂),𝑥(1))) and 𝐻𝑑 = −𝐻(∫^𝜂₀𝑁𝑌(𝜂)𝑌⁻¹(𝑠)𝐸(𝑠)𝑑𝑠 + ∫¹₀𝑅𝑌(1)𝑌⁻¹(𝑠)𝐸(𝑠)𝑑𝑠−ℓ).

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 |𝜀|<𝜀₀, for some 𝜀₀>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 𝜀 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⊂ℝ^𝑛×ℝ^𝑚 be an open set, and let 𝐹∶Ω→ℝ^𝑚 be function of class 𝐶¹. Suppose (𝑥₀,𝑦₀)=0. Assume that det𝜕𝐹₁𝜕𝑦₁⋯𝜕𝐹₁𝜕𝑦_𝑚⋯𝜕𝐹_𝑚𝜕𝑦₁⋯𝜕𝐹_𝑚𝜕𝑦_𝑚≠0evaluatedat𝑥₀,𝑦₀, (2.14) where 𝐹=(𝐹₁,…,𝐹_𝑚). Then there are open sets 𝑈⊂ℝ^𝑛 and 𝑉⊂ℝ^𝑚, with 𝑥₀∈U and 𝑦₀∈𝑈, and a unique function 𝑓∶𝑈→𝑉 such that 𝐹𝑥,𝑓(𝑥)=0 (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 ℝ^𝑛 of 𝐸_𝑟. Let 𝑃_𝑟 be the matrix projection onto Kerℒ=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 ℝ^𝑛, 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_𝜀∶ℝ^𝑟→ℝ^𝑟, given by Φ_𝜀𝑐₁,…,𝑐_𝑟=𝑃_𝑟𝐻𝒩𝑐^𝑟⊕𝑐^𝑛−𝑟𝑐^𝑟,𝜀,𝛼,𝜂,𝜀, (3.1) where 𝑐^𝑛−𝑟(𝑐^𝑟,𝜀)=𝑐^𝑛−𝑟 is a differentiable function of 𝑐^𝑟 and 𝜀. 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_𝜀 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 𝜀=0, we have Φ₀𝑐₁,…,𝑐_𝑟=𝑃_𝑟𝐻𝒩𝑐^𝑟⊕𝑃_𝑛−𝑟𝐻𝑑,𝛼,𝜂,0, (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=ℝ^𝑛 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_𝜀∶ℝ^𝑛→ℝ^𝑛, given by Φ_𝜀(𝑐)=𝒩(𝑐,𝛼,𝜂,𝜀). Setting 𝜀=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 |𝜀|<𝜀₀, for some 𝜀₀>0. The existence theorem can be established using the implicit function theorem; see Theorem 2.7.

Theorem 3.2. If 𝑐=(𝑐₁,…,𝑐_𝑛)∈ℝ^𝑛, 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 𝜀₀>0 be small enough so that (1.1) has a unique 𝑛-vector 𝑥(𝑡,𝑐,𝜀) defined on [0,1]×𝐵_𝑘1×[−𝜀₀,𝜀₀]. Let Φ_𝜀∶𝐵_𝑘⊆ℝ^𝑟→ℝ^𝑟, given by Φ_𝜀𝑐₁,…,𝑐_𝑟=𝑃_𝑟𝐻𝒩𝑐^𝑟⊕𝑐^𝑛−𝑟𝑐^𝑟,𝜀,𝛼,𝜂,𝜀, (3.3) where 𝑐^𝑛−𝑟(𝑐^𝑟,𝜀)=𝑐^𝑛−𝑟 is a differentiable function of 𝑐^𝑟 and 𝜀, and Φ₀𝑐₁,…,𝑐_𝑟=𝑃_𝑟𝐻𝒩𝑐^𝑟⊕𝑃_𝑛−𝑟𝐻𝑑,𝛼,𝜂,0 (3.4) for (𝑐^𝑟⊕𝑃_𝑛−𝑟𝐻𝑑)∈𝐵_𝑘×{𝑃_𝑛−𝑟𝐻𝑑}⊆𝐵_𝑘1. If Φ₀(𝑐₁,…,𝑐_𝑟)=0 and det𝜕Φ^𝑖₀𝑐₁,…,𝑐_𝑟𝜕𝑐_𝑗∣_{(𝑐1,…,𝑐𝑟)=(𝑐1,…,𝑐𝑟)}≠0, (3.5) for some (𝑐₁,…,𝑐_𝑟)∈𝐵_𝑘, then there is 𝜀, 0<𝜀≤𝜀₀, and 𝛿>0 such that (1.1), (1.2) has a unique solution 𝑥(𝑡,𝑐(𝜀),𝜀) for all |𝜀|<𝜀 such that 𝑐(0)=𝑐=(𝑐^𝑟⊕𝑃_𝑛−𝑟𝐻𝑑) and |𝑐(𝜀)−𝑐|<𝛿.

Proof. The existence and uniqueness of a solution 𝑥(𝑡,𝑐,𝜀) for |𝜀|<𝜀₀ with 𝑥(0,𝑐,𝜀)=𝑐∈ℝ^𝑛 follows directly from conditions (i), (ii), and (iii) of Theorem 2.2. Now Φ₀𝑐₁,…,𝑐_𝑟=𝑃_𝑟𝐻𝒩𝑐^𝑟⊕𝑃_𝑛−𝑟𝐻𝑑,𝛼,𝜂,0=0,det𝜕Φ^𝑖₀𝑐₁,…,𝑐_𝑟𝜕𝑐_𝑗∣_{(𝑐1,…,𝑐𝑟)=(𝑐1,…,𝑐𝑟)}≠0, (3.6) for some (𝑐₁,…,𝑐_𝑟)∈𝐵_𝑘, thus it follows from the implicit function theorem that there is 𝜀, 0<𝜀≤𝜀₀ such that (3.3) has a unique solution (𝑐₁,…,𝑐_𝑟)=(𝑐₁(𝜀),…,𝑐_𝑟(𝜀)), with |(𝑐₁(𝜀),…,𝑐_𝑟(𝜀))−(𝑐₁,…,𝑐_𝑟)|<𝛿, for all 𝜀, |𝜀|<𝜀. From this it follows that 𝑥(𝑡,𝑐(𝜀),𝜀) is a unique solution of the BVP (1.1), (1.2) which satisfies the initial value 𝑥(0,𝑐(𝜀),𝜀)=𝑐(𝜀) and 𝑐(0)=𝑐=(𝑐^𝑟⊕𝑃_𝑛−𝑟𝐻𝑑) and |𝑐(𝜀)−𝑐|<𝛿, where 𝑐(𝜀)=(𝑐^𝑟(𝜀)⊕𝑐^𝑛−𝑟(𝑐^𝑟(𝜀),𝜀)).

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 𝜀 with |𝜀|<𝜀₀ for some 𝜀₀>0 is 𝑑=0; that is, 𝑁𝑌𝜂^𝜂₀𝑌⁻¹(𝑠)𝐸(𝑠)𝑑𝑠+¹₀𝑅𝑌(1)𝑌⁻¹(𝑠)𝐸(𝑠)𝑑𝑠=ℓ. (3.7)

Theorem 3.4. Let the conditions (i), (ii), and (iii) of Theorem 2.2 hold, and let 𝑘₁>0, 𝑘>0 and 𝜀₀>0 be small enough so that (1.1) has a unique solution 𝑥(𝑡,𝑐,𝜀) defined on [0,1]×𝐵_𝑘1×[−𝜀₀,𝜀₀]. If 𝑟=𝑛, 𝑑=0, and Φ_𝜀(𝑐)=−^𝜂₀𝑁𝑌𝜂𝑌⁻¹(𝑠)𝐹(())−𝑠,𝑥𝑠,𝑐,𝜀,𝜀𝑑𝑠₁₀𝑅𝑌(1)𝑌⁻¹(𝑠)𝐹(𝑠,𝑥(𝑠,𝑐,𝜀),𝜀)𝑑𝑠+𝑔𝑐,𝑥𝜂,𝑥(1), (3.8) then there is 𝜀, 0<𝜀≤𝜀₀, and 𝛿>0 such that (1.1), (1.2) has a unique solution 𝑥(𝑡,𝑐(𝜀),𝜀) for all |𝜀|<𝜀 such that 𝑐(0)=𝑐 and |𝑐(𝜀)−𝑐|<𝛿.

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(𝜀),𝜀) for |𝜀|<𝜀<𝜀₀ with 𝑥(0,𝑐,𝜀)=𝑐∈ℝ^𝑛 follows directly from conditions (i), (ii) and (iii) of Theorem 2.2. NowΦ₀(𝑐)=−^𝜂₀𝑁𝑌𝜂𝑌⁻¹(𝑠)𝐹(())−𝑠,𝑥𝑠,𝑐,0,0𝑑𝑠₁₀𝑅𝑌(1)𝑌⁻¹(𝑠)𝐹(𝑠,𝑥(𝑠,𝑐,0),0)𝑑𝑠+𝑔𝑐,𝑥𝜂,𝑥(1). (3.9) If Φ₀(𝑐)=0,det𝜕Φ^𝑖₀(𝑐)𝜕𝑐_𝑗∣_𝑐=𝑐≠0, (3.10) for some 𝑐=(𝑐₁,…,𝑐_𝑛)∈𝐵_𝑘; thus it follows from the implicit function theorem that there is 𝜀, 0<𝜀≤𝜀₀ such that (3.8) has a unique solution 𝑐=𝑐(𝜀), with |𝑐−𝑐|<𝛿, for all 𝜀, |𝜀|<𝜀. From this it follows that 𝑥(𝑡,𝑐(𝜀),𝜀) is a unique solution of the BVP (1.1), (1.2) which satisfies the initial values 𝑥(0,𝑐(𝜀),𝜀)=𝑐(𝜀)∈ℝ^𝑛 for all 𝜀, |𝜀|<𝜀 such that 𝑐(0)=𝑐 and |𝑐(𝜀)−𝑐|<𝛿.

4. Some Examples

To find 𝑐 for 𝜀 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 𝑦^′′=𝜀𝑓𝑡,𝑦,𝑦^′(𝑡),(0),𝜀+𝑒𝑦′=𝜀𝑔₁𝑦(0),𝑦^′(0)1,𝑦2,𝑦^′12,𝑦(1),𝑦^′(1),𝑦(1)1−𝑦2=𝜀𝑔₂𝑦(0),𝑦^′(0)1,𝑦2,𝑦^′12,𝑦(1),𝑦^′(1), (4.1) where 𝑓∈𝐶([0,1]×ℝ²×(−𝜀₀,𝜀₀);ℝ), 𝑒∈𝐶[0,1], 𝑔∈𝐶(ℝ⁶;ℝ²). Then the BVP (4.1) is equivalent to 𝑥^′₁𝑥^′₂=01𝑥00𝑥¹₂0𝑓+𝜀𝑡,𝑥₁,𝑥₂+0𝑒(𝑡)𝑥,𝜀,0100¹(0)𝑥₂(0)+00𝑥−𝛼0¹12𝑥₂12+00𝑥10¹(1)𝑥₂(1)=𝜀𝑔₁𝑐₁,𝑐₂,𝑥₁12,𝑥₂12,𝑥₁(1),𝑥₂(1)𝜀𝑔₂𝑐₁,𝑐₂,𝑥₁12,𝑥₂12,𝑥₁(1),𝑥₂(1,) (4.2)(4.3) where ,𝐸(𝑡)=0𝑒(𝑡)()=0𝑓𝑀=0100,𝑁=00−𝛼0,𝑅=0010,𝐹𝑡,𝑥,𝜀𝑡,𝑥₁,𝑥₂,𝑔𝑐,𝜀₁,𝑐₂,𝑥₁𝜂,𝑥₂𝜂,𝑥₁(1),𝑥₂(1)=𝑔₁𝑐₁,𝑐₂,𝑥₁𝜂,𝑥₂𝜂,𝑥₁(1),𝑥₂(1)𝑔₂𝑐₁,𝑐₂,𝑥₁𝜂,𝑥₂𝜂,𝑥₁(1),𝑥₂(1),𝑌(𝑡)=𝑒^𝐴𝑡=1𝑡01,𝑌₀(𝑡)=𝑌(𝑡)𝑌⁻¹(0)=.1𝑡01 (4.4) By Lemma 2.4, we find ℒ: ℒ=𝑀+𝑁𝑌₀𝜂+𝑅𝑌₀(1)=++=.010000−𝛼01𝜂0100101101011−𝛼1−𝛼𝜂 (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, ℒ_𝛼=1=0101−𝜂. (4.6) Let 𝐸₁ denote the null space of ℒ_𝛼=1. Thus 𝑒₁=¹₀ is a basis for Ker(ℒ_𝛼=1), and Ker(ℒ_𝛼=1)=Span𝑒₁. Let 𝑃₁ be the matrix projection onto Ker(ℒ_𝛼=1). 𝑃₁=¹⁰₀₀. 𝑃₂=𝐼−𝑃₁=⁰⁰₀₁. Set 𝐻 = (1/(1−𝜂))^1−𝜂−1₀₁ so that 𝐻ℒ_𝛼=1=𝑃₂. In system (4.2), (4.3) let 𝑓(𝑡,𝑥₁,𝑥₂,𝜀)=𝑥₁𝑥²₂, 𝑒(𝑡)=cos4𝜋𝑡𝑔₁(𝑐₁,𝑐₂,𝑥₁(1/2),𝑥₂(1/2),𝑥₁(1),𝑥₂(1))=−𝑥²₁(1), 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: 𝑃₁1𝐻𝑑=210002−1001₀^1/212−𝑠cos4𝜋𝑠𝑑𝑠−¹₀()−1−𝑠cos4𝜋𝑠𝑑𝑠=2^01/212−𝑠cos4𝜋𝑠𝑑𝑠+¹₀()0.1−𝑠cos4𝜋𝑠𝑑𝑠 (4.7) Since ∫₀^1/2(1/2−𝑠)cos4𝜋𝑠𝑑𝑠=∫¹₀(1−𝑠)cos4𝜋𝑠𝑑𝑠=0, 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 𝑥()=𝑐𝑡,𝑐,01𝑡010¹+^𝑡₀1𝑡−𝑠001cos4𝜋𝑠𝑑𝑠. (4.8) Thus the BVP (4.2), (4.3) has a solution if 𝛼=1, 𝜀=0; namely, 𝑥₁(𝑡,𝑐,0)=𝑐₁+((1−cos4𝜋𝑡)/16𝜋²), 𝑥₂(𝑡,𝑐,0)=sin4𝜋𝑡/4𝜋, 𝑥₁(0)=𝑥₁(1/2)=𝑥₁(1)=𝑐₁, 𝑥₂(0)=𝑥₂(1/2)=𝑥₂(1)=0. Setting 𝜀=0, thus 𝑓(𝑡,𝑐₁+(1−cos4𝜋𝑡)/16𝜋²,sin4𝜋𝑡/4𝜋,0)=sin²4𝜋𝑡/16𝜋²(𝑐₁+(1−cos4𝜋𝑡)/16𝜋²),𝑔₁(𝑐₁,𝑥₁(1/2),𝑥₁(1))=−𝑐²₁ and 𝑔₂(𝑐₁,𝑥₁(1/2),𝑥₁(1))=2𝑐₁/256𝜋⁴. Hence Φ₀𝑐₁=₀^1/2𝑓𝑠,𝑐₁+1−cos4𝜋𝑠16𝜋²,sin4𝜋𝑠4𝜋,0𝑑𝑠+¹_1/22(1−𝑠)𝑓𝑠,𝑐₁+1−cos4𝜋𝑠16𝜋²,sin4𝜋𝑠4𝜋,0𝑑𝑠+𝑔₁𝑐₁,𝑥₁12,𝑥₁(1)−2𝑔₂𝑐₁,𝑥₁12,𝑥₁(1)=−𝑐²₁+𝑐₁128𝜋²+32048𝜋⁴. (4.9) If 𝑐₁≈−3.5023×10⁻³ or 𝑐₁≈4.2938×10⁻³, then Φ₀(𝑐₁)=0 and 𝜕Φ₀𝑐₁𝜕𝑐₁∣_{(𝑐1≈−3.5023×10−3)}≠0,𝜕Φ₀𝑐₁𝜕𝑐₁∣_{(𝑐1≈4.2938×10−3)}≠0. (4.10)
Hence by Theorem 3.2 there is 𝜀,  0<𝜀≤𝜀₀ and 𝛿>0 such that the BVP (4.2), (4.3) has a unique solution 𝑥(𝑡,𝑐(𝜀),𝜀) which satisfies the initial values 𝑥(0,𝑐(𝜀),𝜀)=𝑐(𝜀)∈ℝ² for all |𝜀|<𝜀 such that 𝑐(0)=(𝑐₁,0) and |𝑐(𝜀)−𝑐(0)|<𝛿.

Example 4.2. Rank ℒ=2<3.
Consider the BVP 𝑥^′₁𝑥^′₂𝑥^′₃=010𝑥−100000¹𝑥₂𝑥₃𝑓+𝜀¹𝑡,𝑥₁,𝑥₂,𝑥₃𝑓,𝜀₂𝑡,𝑥₁,𝑥₂,𝑥₃𝑓,𝜀₃𝑡,𝑥₁,𝑥₂,𝑥₃𝑥,𝜀,100010000¹(0)𝑥₂(0)𝑥₃(0)+−100𝑥0−1000−1¹(𝜋)𝑥₂(𝜋)𝑥₃(𝜋)+000𝑥1−10000¹()𝑥2𝜋₂()𝑥2𝜋₃()=ℓ2𝜋₁+𝜀𝑔₁𝑐₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃()ℓ2𝜋₂+𝜀𝑔₂c₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃()ℓ2𝜋₃+𝜀𝑔₃𝑐₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃(),2𝜋 (4.11)(4.12) where 𝑓_𝑖∈𝐶([0,1]×ℝ³×(−𝜀₀,𝜀₀);ℝ), 𝑖=1,2,3, ℓ=(ℓ₁,ℓ₂,ℓ₃)∈ℝ³, 𝑔∈𝐶(ℝ⁹;ℝ³), 𝑒(𝑡),(𝑡)=≡0,𝐴=010−100000,𝑀=100010000,𝑁=−1000−1000−1𝑅=0001−10000,𝑌cos𝑡sin𝑡0−sin𝑡cos𝑡0001,𝑌⁻¹(𝑡)=,𝑌cos𝑡−sin𝑡0sin𝑡cos𝑡0001₀(𝜋)=−1000−10001,𝑌₀()=,𝐹2𝜋100010001𝑡,𝑥₁,𝑥₂,𝑥₃=𝑓,𝜀¹𝑡,𝑥₁,x₂,𝑥₃𝑓,𝜀₂𝑡,𝑥₁,𝑥₂,𝑥₃𝑓,𝜀₃𝑡,𝑥₁,𝑥₂,𝑥₃,𝑔𝑐,𝜀₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃(=𝑔2𝜋)₁𝑐₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃()𝑔2𝜋₂𝑐₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃()𝑔2𝜋₃𝑐₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃(.2𝜋) (4.13) By Lemma 2.4, the problem of solving (4.11), (4.12) is reduced to that of solving ℒ𝑐=𝜀𝒩(𝑐,𝛼,𝜂,𝜀)+𝑑 for 𝑐 provided solutions 𝑥(𝑡,𝑐,𝜀) of initial value problems exist on [0,1] for each (𝑐,𝜀). Thus we find ℒ: ℒ=𝑀+𝑁𝑌₀(𝜋)+𝑅𝑌₀()=.2𝜋200110000 (4.14) Since rank ℒ=2, it follows that the matrix ℒ is singular. Let 𝐸₃ denote the null space of ℒ. Thus 𝑒₃=⁰⁰₁ is a basis for Ker(ℒ), and Ker(ℒ)=Span𝑒₃. Let 𝑃₃ be the matrix projection onto Ker(ℒ). 𝑃₃=⁰⁰⁰⁰⁰⁰₀₀₁. So 𝑃₂=𝐼−𝑃₃=¹⁰⁰⁰¹⁰₀₀₀. Set 𝐻=^1/200_−1/210001 so that 𝐻ℒ=𝑃₂. 𝒩𝑐,𝛼,𝜂,𝜀=−^𝜋₀𝑁𝑌(𝜋)𝑌⁻¹(𝑠)𝑓₁𝑠,𝑥₁(𝑠,𝑐,𝜀),𝑥₂(s,𝑐,𝜀),𝑥₃()𝑓𝑠,𝑐,𝜀,𝜀₂𝑠,𝑥₁(𝑠,𝑐,𝜀),𝑥₂(𝑠,𝑐,𝜀),𝑥₃()𝑓𝑠,𝑐,𝜀,𝜀₃𝑠,𝑥₁(𝑠,𝑐,𝜀),𝑥₂(𝑠,𝑐,𝜀),𝑥₃()−𝑠,𝑐,𝜀,𝜀𝑑𝑠_02𝜋𝑅𝑌(2𝜋)𝑌⁻¹(𝑠)𝑓₁𝑠,𝑥₁(𝑠,𝑐,𝜀),𝑥₂(𝑠,𝑐,𝜀),𝑥₃()𝑓𝑠,𝑐,𝜀,𝜀₂𝑠,𝑥₁(𝑠,𝑐,𝜀),𝑥₂(𝑠,𝑐,𝜀),𝑥₃()𝑓𝑠,𝑐,𝜀,𝜀₃𝑠,𝑥₁(𝑠,𝑐,𝜀),𝑥₂(𝑠,𝑐,𝜀),𝑥₃()+𝑔𝑠,𝑐,𝜀,𝜀𝑑𝑠₁𝑐₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃()𝑔2𝜋₂𝑐₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃()𝑔2𝜋₃𝑐₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃()=𝒩2𝜋₁𝒩𝑐,𝛼,𝜂,𝜀₂𝒩𝑐,𝛼,𝜂,𝜀₃.𝑐,𝛼,𝜂,𝜀 (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 𝜀=0, 𝑃₂𝐻𝑑=𝑃₂𝑐, where 𝑃₂𝑐=¹⁰⁰⁰¹⁰₀₀₀^𝑐1𝑐2_𝑐3 and 𝑃₂𝐻𝑑=0. Hence 𝑐₁=0 and 𝑐₂=0. Thus 𝑥()=00𝑐𝑡,𝑐,0cos𝑡sin𝑡0−sin𝑡cos𝑡0001₃. (4.16) Thus the BVP (4.11), (4.12) has a solution if 𝜀=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: 𝑃₃=1𝐻𝒩𝑐,𝛼,𝜂,𝜀0000000012−1002𝒩10001¹𝒩𝑐,𝛼,𝜂,𝜀₂𝒩𝑐,𝛼,𝜂,𝜀₃=00𝒩𝑐,𝛼,𝜂,𝜀₃,𝑐,𝛼,𝜂,𝜀 (4.17) where 𝒩₃𝑐,𝛼,𝜂,𝜀=^𝜋₀𝑓₃𝑠,𝑥₁(𝑠,𝑐,𝜀),𝑥₂(𝑠,𝑐,𝜀),𝑥₃(𝑠,𝑐,𝜀),𝜀𝑑𝑠+𝑔₃𝑐₁,𝑐₂,𝑐₃,𝑥₁(𝜋),𝑥₂(𝜋),𝑥₃(𝜋),𝑥₁(2𝜋),𝑥₂(2𝜋),𝑥₃(2𝜋). (4.18) Thus Φ_𝜀(𝑐₃)=𝒩₃(𝑐³⊕𝑐²(𝑐³,𝜀),𝛼,𝜂,𝜀), where 𝑐³=𝑃₃𝑐=(0,0,𝑐₃) and 𝑐²=𝑃₂𝑐=(𝑐₁,𝑐₂,0). Setting 𝜀=0, we have Φ₀(𝑐₃)=𝒩₃(𝑐³,𝛼,𝜂,0), where 𝑐²(𝑐³,0)=𝑃₂𝐻𝑑=0. Writing out the components and setting 𝜀=0, we obtain 𝑥₁(𝑡,𝑐,0)=𝑥₂(𝑡,𝑐,0)=0 and 𝑥₃(𝑡,𝑐,0)=𝑐₃. Hence Φ₀𝑐₃=^𝜋₀𝑓₃𝑠,𝑥₁𝑠,0,0,𝑐₃,0,𝑥₂𝑠,0,0,𝑐₃,0,𝑥₃𝑠,0,0,𝑐₃,0,0𝑑𝑠+𝑔₃𝑐₃,𝑥₃(𝜋),𝑥₃(2𝜋), (4.19) where 𝑥_𝑖(𝜋)=𝑥_𝑖(2𝜋)=0, 𝑖=1,2, 𝑥₃(𝜋)=𝑐₃=−ℓ₃, and 𝑥₃(2𝜋)=𝑐₃. Let 𝑓₃(𝑡,𝑥₁(𝑡,0,0,𝑐₃,0),𝑥₂(𝑡,0,0,𝑐₃,0),𝑥₃(𝑡,0,0,𝑐₃,0),0)=−𝑐²₃sin𝑡, and 𝑔₃(𝑐₃,𝑥₃(𝜋),𝑥₃(2𝜋))=𝑐⁴₃. Hence Φ₀𝑐₃=−^𝜋₀𝑐²₃sin𝑠𝑑𝑠+𝑐⁴₃=𝑐²₃cos𝑡∣^𝜋₀+𝑐⁴₃. (4.20) If 𝑐₃=±√2, then Φ₀(𝑐₃)=0 and 𝜕Φ₀𝑐₃𝜕𝑐₃∣_{(𝑐3√=±2)}≠0. (4.21) Hence by Theorem 3.2 there is 𝜀, 0<𝜀≤𝜀₀ and 𝛿>0 such that the BVP (4.11), (4.12) has a unique solution 𝑥(𝑡,𝑐(𝜀),𝜀) which satisfies the initial values 𝑥(0,𝑐(𝜀),𝜀)=𝑐(𝜀)∈ℝ³ for all |𝜀|<𝜀 such that 𝑐(0)=(0,0,𝑐₃) and |𝑐(𝜀)−𝑐(0)|<𝛿.