First-order Three-Point BVPs at Resonance (II)

This paper deals with existence of solutions to three-point BVPs in perturbed systems of ﬁrst-order ordinary diﬀerential equations at resonance. An existence theorem is established by using the Theorem of Borsuk and some examples are given to illustrate it. A result for computing the local degree of polynomials whose terms of highest order have no common real linear factors is also presented.


Introduction
In this paper, we consider 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 continuous, F : [0, 1] × R n × (−ε 0 , ε 0 ) → R n is a continuous function and ε ∈ R such that | ε |< ε 0 , and η ∈ (0, 1). 1 Corresponding author EJQTDE, 2011 No. 68, p. 1 The work is motivated by Cronin [6,7] who considered the problem of finding periodic solutions of perturbed systems.We adapt her approach to study three-point BVPs with linear boundary conditions using the methods and results of Cronin [6,7].The three-point BVP (1), ( 2) is called resonant or degenerate in the case that the rank of matrix L = n − r , 0 < n − r < n, that is the matrix L = M + NY 0 (η) + RY 0 (1) is singular where M, N and R are the constant n × n matrices given in (1), and Y (t) is a fundamental matrix of linear system x ′ = A(t)x and Y 0 (t) = Y (t)Y −1 (0).In studying the resonant case, we will use a finite-dimensional version of the Lyapunov Schmidt procedure (see [7]).
Recently, Mohamed et al. [30] established the existence of solutions at resonance for the following nonlinear boundary conditions where M, N and R are constant square matrices of order n, A(t) is an n × n matrix with continuous entries, In this paper, we make use of the Theorem of Borsuk to show the existence of solutions of the BVP (1), (2) under suitable assumptions on the coefficients.We obtain the existence of solutions of three-point BVPs at resonance for general BVPs.We also present a result for computing the degree of ψ 0 (c) = (ψ 1 0 (c 1 , c 2 ), ψ 2 0 (c 1 , c 2 )) at (0, 0) where the ψ 0 (c 1 , c 2 ) are polynomials whose terms of highest order have no common real linear factors; see Cronin [7] p. 296-297.This result is for homogeneous polynomials in two variables which need not be odd functions while Borsuk's Theorem holds for continuous odd functions in any dimensions.
These results generalize the degenerate case of periodic BVPs considered by Cronin [6,7], and also the degenerate case of three-point BVP [13,30].

Preliminaries
Lemma 2.1.Consider the system where A(t) is an n × n matrix with continuous entries on the interval [0,1].Let Y (t) be a fundamental matrix of (5).Then the solution of (5) which satisfies the initial condition Lemma 2.2.[30] Let Y (t) be a fundamental matrix of (5).Then any solution of ( 1) and ( 6) can be written as The solution (1) satisfies the boundary conditions (2) if and only if EJQTDE, 2011 No. 68, p. 3 where and x(t, c, ε) is the solution of (1) given x(0) = c.
nents of c.The system (8) 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 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 P 2 r = P r , P 2 n−r = P n−r and P n−r P r = P r P n−r = 0.
Without loss of generality, we may assume We will identify P r c with c r = (c 1 , • • • , c r ) and it is convenient to do so.
Let H be a nonsingular n × n matrix satisfying Matrix H can be computed easily (see Cronin [7]).The nature of the solutions of the branching equations depends heavily on the rank of the matrix L.
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 (8) for c when ε is sufficiently small.The problem of finding solutions to ( 1) and ( 2) is reduced to that of solving the branching equations ( 8) for c as function of (8) which is equivalent to Multiplying ( 8) by the matrix H and using ( 11), we have where ) and Since the matrix H is nonsingular, solving (8) for c is equivalent to solving (12) for c.
The following theorem due to Cronin [6,7] gives a necessary condition for the existence of solutions to the BVP ( 1) and (2).
Theorem 2.4.A necessary condition that ( 12) can be solved for c, with | ε | < ε 0 , for some where c n−r (c r , ε) = c n−r is a differentiable function of c r and ε, P r HN is interpreted as . Similarly we will sometimes identify P n−r c and c n−r .Setting ε = 0, we have where c n−r (c r , 0) = P n−r Hd; note that from the context c n−r (c r , 0) = P n−r Hd is interpreted it follows that the matrix H is the identity matrix.Thus define a continuous mapping

Main Results
Now we state the well known Theorem of Borsuk (see, for example, Piccinini, Stampacchia and Vidossich [31] p. 211).
Theorem 3.1.Let B k ⊆ R n be a bounded open set that is symmetrical with respect to the origin (that is B k = −B k ) and contains the origin.If Φ 0 : Bk → R n is continuous and antipodal ) is an odd number (and thus nonzero).
Next we introduce the computation of the topological degree of a mapping in Euclidean 2-space defined by homogeneous polynomials.The methods and notations described below EJQTDE, 2011 No. 68, p. 6 come from Cronin [7,8].Let where C 1 , C 2 are constants.(We include the possibility that some a i = ∞ or some b j = ∞; equivalently, that the factor y −a i x is equal to −x or the factor y −b j x is equal to −x).The topological degree is resolved by examining the changes of sign of Φ 1 0 (c 1 , c 2 ) and Φ 2 0 (c 1 , c 2 ) as c 1 , c 2 varies over the boundary of the ball B k with centre at the origin and arbitrary radius when computing the topological degree of (Φ 1 0 , Φ 2 0 ).We may omit the following factors since none of them affect the degree of (Φ 1 0 , Φ 2 0 ) on B k at 0.
where a i and b j have complex conjugates in Φ 1 0 , respectively, Φ 2 0 .

Factors
which appear with even exponents where a i and b j are real.
3. Factors (c 1 − a i c 2 ) and (c 1 − a i+1 c 2 ), if there exists a pair a i , a i+1 (i < i + 1) such that no b j lies between them (i.e., there is no b j such that a i < b j < a i+1 ).
Similarly for pairs b j , b j+1 .
If there are no remaining factors in Φ 1 0 or Φ 2 0 , then the topological degree is zero.We now state the second main theorem in this paper (see Cronin [7] p. [38][39][40]. Theorem 3.2.If we assume that the terms of highest degree of Φ 1 0 (c 1 , c 2 ) and Φ 2 0 (c 1 , c 2 ) are homogenous polynomials with no common real linear factors after reduction using the EJQTDE, 2011 No. 68, p. 7 conditions 1, 2, 3, and 4 above, then for some integer p ≤ min{m, n}.In the first case the degree is p, while in the second case the degree is −p.Hence for B k , a ball with centre at the origin and sufficiently large radius.Then for sufficiently Hence there is a solution x(t, c, ε) of the BVP (1), (2) Remark 3.3.In this paper, we find that an arbitrarily small change in A(t) will affect the structure of the set of solutions, and the value of the local degree will depend on how the function f (t, y, y ′ , ε) is changed.

Applications and Examples
In this section, we apply our results from the previous section and we start by considering the degenerate case for α = √ 2 in the interval [0, 2π] with rank L (α= √ 2) = 1 < 2. Thus, we consider where EJQTDE, 2011 No. 68, p. 8 Then we study the totally degenerate case, rank L = 0 for general boundary conditions and give an example where Borsuk's Theorem or Theorem 3.2 applies.We consider where We will use the following facts in solving the examples.if and only if both n and m are even.Rank L (α= √ 2) = 1 < 2, α = √ 2 and y ′ (0) = 0.
Since Φ 0 (c 1 ) is odd, the local degree is odd and therefore nonzero.Then for sufficiently large Next we apply Borsuk's Theorem in Example 1, and then Theorem 3. Rank L = 0.

Definition 2 . 5 .
[30] 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 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 EJQTDE, 2011 No. 68, p. 5

4 .
Factors (c 1 − a r c 2 ) and (c 1 − a s c 2 ), if a r and a s are the smallest and largest of the array of numbers a 1 , • • • , a n , b 1 , • • • , b m .Similarly factors (c 1 − b r c 2 ) and (c 1 − b s c 2 ), if b r and b s are the smallest and largest of the array of numbers

1 0
sin n 2πs cos m 2πs ds = 0(20) if and only if both n and m are even.