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

Authors: Mesliza Mohamed; Bevan Thompson; Muhammad Sufian Jusoh
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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)x′-A(t)x=H(t,x,ɛ)=ɛF(t,x,ɛ)+E(t),0≤t≤1,(1.2)Mx(0)+Nx(η)+Rx(1)=l+ɛg(x(0),x(η),x(1)),
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)y′′=f(t,y,y′)+e(t),0≤t≤1,y′(0)=0,y(1)=αy(η),(1.4)y′′=f(t,y,y′)+e(t),0≤t≤1,y(0)=0,y(1)=αy(η),
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)
LetF(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 ɛ¯.(ii)
Letx(t,c,ɛ) be any noncontinuable solution of the differential equation
(2.1)x′=F(t,x,ɛ),withx(0)=c.
If x(t,c¯,ɛ¯) 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 (c¯,ɛ¯) and is a continuous function of its threefold arguments at any point (t,c¯,ɛ¯).Theorem 2.2 (see Coppel [17, Page 22]).
(i)
LetF(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 ɛ¯.(ii)
Letx(t,c¯,ɛ¯) be a solution of the differential equation
(2.2)x′=F(t,x,ɛ¯),withx(0)=c¯
defined on a compact interval [0,1].(iii)
Suppose thatF has continuous partial derivatives Fx, Fɛ at all points (t,x(t,c¯,ɛ¯),ɛ¯) with t∈[0,1].
Then for all(c,ɛ) sufficiently near (c¯,ɛ¯) the differential equation
(2.3)x′=F(t,x,ɛ),withx(0)=c
has a unique solution x(t,c,ɛ) over [0,1] that is close to the solution x(t,c¯,ɛ¯) 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 (c¯,ɛ¯).We recall the following results of [2].Lemma 2.3 (see [2]).
Consider the system(2.4)x′=A(t)x,
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)x(0)=c
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]).
LetY(t) be a fundamental matrix of (2.4). Then any solution of (1.1) and (2.5) can be written as
(2.6)x(t,c,ɛ)=x(t)=Y0(t)c+∫0tY(t)Y-1(s)H(s,x(s),ɛ)ds.
The solution (1.1) satisfies the boundary conditions (1.2) if and only if
(2.7)Lc=ɛN(c,α,η,ɛ)+d,
where ℒ=M+NY0(η)+RY0(1), 𝒩(c,α,η,ɛ) = -(∫0ηNY(η)Y-1(s)F(s,x(s,c,ɛ),ɛ)ds + ∫01RY(1)Y-1(s)F(s,x(s,c,ɛ),ɛ)ds − g(c,x(η),x(1))), d=-(∫0ηNY(η)Y-1(s)E(s)ds + ∫01RY(1)Y-1(s)E(s)ds-ℓ), 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,…,cn where c1,…,cn are the components of c. The system (2.7) is sometimes called the branching equations.Next we suppose thatℒ 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 Er denote the null space of ℒ, and let En-r denote the complement in ℝn of Er; that is,(2.8)Rn=En-r⊕Er(directsum).
Let x1,…,xn be a basis for ℝn such that x1,…,xr is a basis for Er and xr+1,…,xn a basis for En-r.LetPr be the matrix projection onto Ker ℒ=Er, and Pn-r=I-Pr, where I is the identity matrix. Thus Pn-r is a projection onto the complementary space En-r of Er, and(2.9)Pr2=Pr,Pn-r2=Pn-r,Pn-rPr=PrPn-r=0.
Without loss of generality, we may assume(2.10)Prc=(c1,…,cr,0,…,0),Pn-rc=(0,…,0,cr+1,…,cn).
We will identify Prc with cr=(c1,…,cr) and Pn-rc with cn-r=(cr+1,…,cn) whenever it is convenient to do so.LetH be a nonsingular n×n matrix satisfying(2.11)HL=Pn-r.
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 matrixℒ 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 ofx(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)L(Pr+Pn-r)c=ɛN((Pr+Pn-r)c,α,η,ɛ)+d.
Multiplying (2.7) by the matrix H and using (2.11), we have(2.13)Pn-rc=ɛHN((Pr+Pn-r)c,α,η,ɛ)+Hd,
where H𝒩((Pr+Pn-r)c,α,η,ɛ) = -H(∫0ηNY(η)Y-1(s)F(s,x(s,c,ɛ),ɛ)ds + ∫01RY(1)Y-1(s)F(s,x(s,c,ɛ),ɛ)ds − g(c,x(η),x(1))) and Hd = -H(∫0ηNY(η)Y-1(s)E(s)ds + ∫01RY(1)Y-1(s)E(s)ds-ℓ).Since the matrixH 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 PrHd=0.Ifℒ 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).
LetΩ⊂ℝn×ℝm be an open set, and let F:Ω→ℝm be function of class C1. Suppose (x0,y0)=0. Assume that
(2.14)det⁡(∂F1∂y1⋯∂F1∂ym⋯∂Fm∂y1⋯∂Fm∂ym)≠0evaluatedat(x0,y0),
where F=(F1,…,Fm). Then there are open sets U⊂ℝn and V⊂ℝm, with x0∈U and y0∈U, and a unique function f:U→V such that
(2.15)F(x,f(x))=0
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]).
LetEr denote the null space of ℒ, and let En-r denote the complement in ℝn of Er. Let Pr be the matrix projection onto Kerℒ=Er, and Pn-r=I-Pr, where I is the identity matrix. Thus Pn-r is a projection onto the complementary space En-r of Er. If En-r is properly contained in ℝn, then Er is an r-dimensional vector space where 0<r<n. If c=(c1,…,cn), let Prc=cr and Pn-r=cn-r, then define a continuous mapping Φɛ:ℝr→ℝr, given by
(3.1)Φɛ(c1,…,cr)=PrHN(cr⊕cn-r(cr,ɛ),α,η,ɛ),
where cn-r(cr,ɛ)=cn-r is a differentiable function of cr and ɛ. By abuse of notation we will identify Prc and cr when convenient and where the meaning is clear from the context so that in defining Φɛ above from the context we interpreted PrH𝒩 as (H𝒩1,…,H𝒩r). Similarly we will sometimes identify Pn-rc and cn-r. Setting ɛ=0, we have
(3.2)Φ0(c1,…,cr)=PrHN(cr⊕Pn-rHd,α,η,0),
where cn-r(cr,0)=Pn-rHd; note that from the context cn-r(cr,0)=Pn-rHd is interpreted as cn-r(cr,0)=(Hdr+1,…,Hdn).
IfEr=ℝn and Pr=I, then Pn-r=0. Since Pn-r=0, it follows that the matrix H is the identity matrix. Thus define a continuous mapping Φɛ:ℝn→ℝn, given by Φɛ(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.
Ifc=(c1,…,cn)∈ℝn, let cr=(c1,…,cr). 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 [0,1]×B¯k1×[-ɛ0,ɛ0]. Let Φɛ:B¯k⊆ℝr→ℝr, given by
(3.3)Φɛ(c1,…,cr)=PrHN(cr⊕cn-r(cr,ɛ),α,η,ɛ),
where cn-r(cr,ɛ)=cn-r is a differentiable function of cr and ɛ, and
(3.4)Φ0(c1,…,cr)=PrHN(cr⊕Pn-rHd,α,η,0)
for (cr⊕Pn-rHd)∈B¯k×{Pn-rHd}⊆B¯k1. If Φ0(c1¯,…,cr¯)=0 and
(3.5)det⁡∂Φ0i(c1,…,cr)∂cj∣(c1,…,cr)=(c1¯,…,cr¯)≠0,
for some (c1¯,…,cr¯)∈B¯k, then there is ɛ¯, 0<ɛ¯≤ɛ0, and δ>0 such that (1.1), (1.2) has a unique solution x(t,c(ɛ),ɛ) for all |ɛ|<ɛ¯ such that c(0)=c¯=(c¯r⊕Pn-rHd) and |c(ɛ)-c¯|<δ.Proof.
The existence and uniqueness of a solutionx(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)Φ0(c1¯,…,cr¯)=PrHN(c¯r⊕Pn-rHd,α,η,0)=0,det⁡∂Φ0i(c1,…,cr)∂cj∣(c1,…,cr)=(c1¯,…,cr¯)≠0,
for some (c1¯,…,cr¯)∈B¯k, thus it follows from the implicit function theorem that there is ɛ¯, 0<ɛ¯≤ɛ0 such that (3.3) has a unique solution (c1,…,cr)=(c1(ɛ),…,cr(ɛ)), with |(c1(ɛ),…,cr(ɛ))-(c1¯,…,cr¯)|<δ, for all ɛ, |ɛ|<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 c(0)=c¯=(cr¯⊕Pn-rHd) and |c(ɛ)-c¯|<δ, where c(ɛ)=(c¯r(ɛ)⊕cn-r(c¯r(ɛ),ɛ)).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]).
Ifr=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)∫0ηNY(η)Y-1(s)E(s)ds+∫01RY(1)Y-1(s)E(s)ds=l.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 [0,1]×B¯k1×[-ɛ0,ɛ0]. If r=n, d=0, and
(3.8)Φɛ(c)=-∫0ηNY(η)Y-1(s)F(s,x(s,c,ɛ),ɛ)ds-∫01RY(1)Y-1(s)F(s,x(s,c,ɛ),ɛ)ds+g(c,x(η),x(1)),
then there is ɛ¯, 0<ɛ¯≤ɛ0, and δ>0 such that (1.1), (1.2) has a unique solution x(t,c(ɛ),ɛ) for all |ɛ|<ɛ¯ such that c(0)=c¯ and |c(ɛ)-c¯|<δ.Proof.
Ifr=n and d=0, then Pn-r=0. This implies Pr=I. Since Pn-r=0, it follows that H=I, the identity matrix.The existence and uniqueness of a solutionx(t,c(ɛ),ɛ) for |ɛ|<ɛ¯<ɛ0 with x(0,c,ɛ)=c∈ℝn follows directly from conditions (i), (ii) and (iii) of Theorem 2.2. Now(3.9)Φ0(c)=-∫0ηNY(η)Y-1(s)F(s,x(s,c,0),0)ds-∫01RY(1)Y-1(s)F(s,x(s,c,0),0)ds+g(c,x(η),x(1)).
If Φ0(c¯)=0,(3.10)det⁡∂Φ0i(c)∂cj∣c=c¯≠0,
for some c¯=(c1¯,…,cn¯)∈B¯k; thus it follows from the implicit function theorem that there is ɛ¯, 0<ɛ¯≤ɛ0 such that (3.8) has a unique solution c=c(ɛ), with |c-c¯|<δ, for all ɛ, |ɛ|<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, |ɛ|<ɛ¯ such that c(0)=c¯ and |c(ɛ)-c¯|<δ.


## 4. Some Examples

To findc 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 ℒα=1=1<2, ℓi≡0 for i=1,2.
Consider the BVP(4.1)y′′=ɛf(t,y,y′,ɛ)+e(t),y′(0)=ɛg1(y(0),y′(0),y(12),y′(12),y(1),y′(1)),y(1)-y(12)=ɛg2(y(0),y′(0),y(12),y′(12),y(1),y′(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)(x1′x2′)=(0100)(x1x2)+ɛ(0f(t,x1,x2,ɛ))+(0e(t)),(4.3)(0100)(x1(0)x2(0))+(00-α0)(x1(12)x2(12))+(0010)(x1(1)x2(1))=(ɛg1(c1,c2,x1(12),x2(12),x1(1),x2(1))ɛg2(c1,c2,x1(12),x2(12),x1(1),x2(1))),
where
(4.4)M=(0100),N=(00-α0),R=(0010),E(t)=(0e(t)),F(t,x,ɛ)=(0f(t,x1,x2,ɛ)),g(c1,c2,x1(η),x2(η),x1(1),x2(1))=(g1(c1,c2,x1(η),x2(η),x1(1),x2(1))g2(c1,c2,x1(η),x2(η),x1(1),x2(1))),Y(t)=eAt=(1t01),Y0(t)=Y(t)Y-1(0)=(1t01).
By Lemma 2.4, we find ℒ:
(4.5)L=M+NY0(η)+RY0(1)=(0100)+(00-α0)(1η01)+(0010)(1101)=(011-α1-αη).
The resonance happens if det(ℒ)=-1+α=0; that is the case where α=1. For α=1, rank ℒα=1=1; that is,
(4.6)Lα=1=(0101-η).
Let E1 denote the null space of ℒα=1. Thus e1=(10) is a basis for Ker(ℒα=1), and Ker(ℒα=1)=Spane1. Let P1 be the matrix projection onto Ker(ℒα=1). P1=(1000). P2=I-P1=(0001). Set H = (1/(1-η))(1-η-101) so that Hℒα=1=P2. In system (4.2), (4.3) let f(t,x1,x2,ɛ)=x1x22, e(t)=cos4πtg1(c1,c2,x1(1/2),x2(1/2),x1(1),x2(1))=-x12(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:
(4.7)P1Hd=2(1000)(12-101)(0∫01/2(12-s)cos⁡⁡4πsds-∫01(1-s)cos⁡⁡4πsds)=2(-∫01/2(12-s)cos⁡⁡4πsds+∫01(1-s)cos⁡⁡4πsds0).
Since ∫01/2(1/2-s)cos4πsds=∫01(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
(4.8)x(t,c,0)=(1t01)(c10)+∫0t(1t-s01)(0cos⁡4πs)ds.
Thus the BVP (4.2), (4.3) has a solution if α=1, ɛ=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 ɛ=0, thus f(t,c1+(1-cos4πt)/16π2,sin4πt/4π,0)=sin24πt/16π2(c1+(1-cos4πt)/16π2),g1(c1,x1(1/2),x1(1))=-c12 and g2(c1,x1(1/2),x1(1))=2c1/256π4. Hence
(4.9)Φ0(c1)=∫01/2f(s,c1+1-cos⁡4πs16π2,sin⁡4πs4π,0)ds+∫1/21{2(1-s)f(s,c1+1-cos⁡4πs16π2,sin⁡4πs4π,0)}ds+g1(c1,x1(12),x1(1))-2g2(c1,x1(12),x1(1))=-c12+c1128π2+32048π4.
If c1≈-3.5023×10-3 or c1≈4.2938×10-3, then Φ0(c1)=0 and
(4.10)∂Φ0(c1)∂c1∣(c1≈-3.5023×10-3)≠0,∂Φ0(c1)∂c1∣(c1≈4.2938×10-3)≠0.
Hence by Theorem3.2 there is ɛ¯,  0<ɛ¯≤ɛ0 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 |ɛ|<ɛ¯ such that c(0)=(c1,0) and |c(ɛ)-c(0)|<δ.Example 4.2.
Rankℒ=2<3.
Consider the BVP(4.11)(x1′x2′x3′)=(010-100000)(x1x2x3)+ɛ(f1(t,x1,x2,x3,ɛ)f2(t,x1,x2,x3,ɛ)f3(t,x1,x2,x3,ɛ)),(4.12)(100010000)(x1(0)x2(0)x3(0))+(-1000-1000-1)(x1(π)x2(π)x3(π))+(0001-10000)(x1(2π)x2(2π)x3(2π))=(l1+ɛg1(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π))l2+ɛg2(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π))l3+ɛg3(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π))),
where fi∈C([0,1]×ℝ3×(-ɛ0,ɛ0);ℝ), i=1,2,3, ℓ=(ℓ1,ℓ2,ℓ3)∈ℝ3, g∈C(ℝ9;ℝ3),
(4.13)e(t)≡0,A=(010-100000),M=(100010000),N=(-1000-1000-1),R=(0001-10000),Y(t)=(cos⁡tsin⁡t0-sin⁡tcos⁡t0001),Y-1(t)=(cos⁡t-sin⁡t0sin⁡tcos⁡t0001),Y0(π)=(-1000-10001),Y0(2π)=(100010001),F(t,x1,x2,x3,ɛ)=(f1(t,x1,x2,x3,ɛ)f2(t,x1,x2,x3,ɛ)f3(t,x1,x2,x3,ɛ)),g(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π))=(g1(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π))g2(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π))g3(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π))).
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)L=M+NY0(π)+RY0(2π)=(200110000).
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. 							(4.15)N(c,α,η,ɛ)=-∫0πNY(π)Y-1(s)(f1(s,x1(s,c,ɛ),x2(s,c,ɛ),x3(s,c,ɛ),ɛ)f2(s,x1(s,c,ɛ),x2(s,c,ɛ),x3(s,c,ɛ),ɛ)f3(s,x1(s,c,ɛ),x2(s,c,ɛ),x3(s,c,ɛ),ɛ))ds-∫02πRY(2π)Y-1(s)(f1(s,x1(s,c,ɛ),x2(s,c,ɛ),x3(s,c,ɛ),ɛ)f2(s,x1(s,c,ɛ),x2(s,c,ɛ),x3(s,c,ɛ),ɛ)f3(s,x1(s,c,ɛ),x2(s,c,ɛ),x3(s,c,ɛ),ɛ))ds+(g1(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π))g2(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π))g3(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π)))=(N1(c,α,η,ɛ)N2(c,α,η,ɛ)N3(c,α,η,ɛ)).
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 P2c=(100010000)(c1c2c3) and P2Hd=0. Hence c1=0 and c2=0. Thus
(4.16)x(t,c,0)=(cos⁡⁡tsin⁡⁡t0-sin⁡⁡tcos⁡⁡t0001)(00c3).
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 ℓi=0, i=1,2, x3(π)=c3=-ℓ3, and x3(2π)=c3:
(4.17)P3HN(c,α,η,ɛ)=(000000001)(1200-1210001)(N1(c,α,η,ɛ)N2(c,α,η,ɛ)N3(c,α,η,ɛ))=(00N3(c,α,η,ɛ)),
where
(4.18)N3(c,α,η,ɛ)=∫0πf3(s,x1(s,c,ɛ),x2(s,c,ɛ),x3(s,c,ɛ),ɛ)ds+g3(c1,c2,c3,x1(π),x2(π),x3(π),x1(2π),x2(2π),x3(2π)).
Thus Φɛ(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)Φ0(c3)=∫0πf3(s,x1(s,0,0,c3,0),x2(s,0,0,c3,0),x3(s,0,0,c3,0),0)ds+g3(c3,x3(π),x3(2π)),
where xi(π)=xi(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)=-c32sint, and g3(c3,x3(π),x3(2π))=c34. Hence
(4.20)Φ0(c3)=-∫0π{c32sin⁡⁡s}ds+c34=c32cos⁡⁡t∣0π+c34.
If c3=±2, then Φ0(c3)=0 and
(4.21)∂Φ0(c3)∂c3∣(c3=±2)≠0.
Hence by Theorem 3.2 there is ɛ¯, 0<ɛ¯≤ɛ0 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 |ɛ|<ɛ¯ such that c(0)=(0,0,c3) and |c(ɛ)-c(0)|<δ.



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*Source: 357651-2012-04-08.xml*