CORRIGENDUM TO ON A CLASS OF DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH INFINITE DELAY

This paper serves as a corrigendum to the paper titled On a class of differential-algebraic equations with infinite delay appearing in EJQTDE no. 81, 2011. We present here a corrected version of Lemma 5.5 and Corollary 5.7.


Introduction
In Section 5 of [1] we investigated examples of applications of that paper's results to a particular class of implicit differential equations.For so doing we used a technical lemma from linear algebra that, unfortunately, turns out to be flawed.As briefly discussed below this affects only marginally our paper's results (just a corollary in Section 5 of [1]).
The simple example below shows that there is something wrong with Lemma 5.5 in [1].In the next section we provide an amended version of this result.
Example 1.1.Consider the matrices Clearly, ker C T = ker E T = span{( 0 1 )} for all t ∈ R. The matrices realize a singular value decomposition for E. Nevertheless which is not the form expected from Lemma 5.5 in [1].The problem, as it turns out, is that ker C = ker E.
Luckily, the impact of the wrong statement of [1, Lemma 5.5] on [1] is minor: all results and examples (besides Lemma 5.5, of course) remain correct, with the exception of Corollary 5.7 where it is necessary to assume the following further hypothesis: (A corrected statement of Corollary 5.7 of [1] can be found in the next section, Corollary 2.2.)

Corrected Lemma and its consequences
We present here a corrected version of Lemma 5.5 in [1].
Lemma 2.1.Let E ∈ R n×n and C ∈ C R, R n×n be respectively a matrix and a matrix-valued function such that Put r = rank E, and let P, Q ∈ R n×n be orthogonal matrices that realize a singular value decomposition for E. Then it follows that ) then C 12 (t) ≡ 0. Namely, in this case, Proof.Our proof is essentially a singular value decomposition (see, e.g., [2]) argument, based on a technical result from [3].
Observe that (2.1) imply rank E = rank C(t) = r > 0 for all t ∈ R. In fact, Since rank C(t) is constantly equal to r > 0, by inspection of the proof of Theorem 3.9 of [3, Chapter 3, §1] we get the existence of orthogonal matrixvalued functions U, V ∈ C R, R n×n and C r ∈ C(R, R r×r ) such that, for all t ∈ R, det C r (t) = 0 and ) be matrix-valued functions formed, respectively, by the first r and n − r columns of U and V .An argument involving Equation (2.5) shows that, for all t ∈ R, the space im C(t) is spanned by the columns of U r (t).Also, (2.5) imply that the columns of V 0 (t), t ∈ R, belong to ker C(t) for all t ∈ R. A dimensional argument shows that they constitute a basis ker C(t).Analogously, transposing (2.5), we see that the columns of V r (t) and U 0 (t) are bases of im C(t) T and ker C(t) T respectively. 1et now P r , Q r and P 0 , Q 0 be the matrices formed taking the first r and n−r columns of P and Q, respectively.Since P and Q realize a singular value decomposition of E, proceeding as above one can check that the columns of P r , Q r , P 0 and Q 0 span im E, im E T , ker E T , and ker E, respectively.
We claim that P T 0 U r (t) is constantly the null matrix in R (n−r)×r .To prove this, it is enough to show that for all t ∈ R, the columns of P 0 are orthogonal to those of U r (t).Let v and u(t), t ∈ R, be any column of P 0 and of U r (t), respectively.Since for all t ∈ R the columns of U r (t) are in im C(t), there is a vector w(t) ∈ R n with the property that u(t) = C(t)w(t), and T for all t ∈ R.This proves the claim.A similar argument shows that P T r U 0 (t) is identically zero as well.
where the maps C : R → R n×n and S : BU (−∞, 0], R n → R n are continuous, E is a (constant) n × n matrix, F is locally Lipschitz and S verifies condition (K) in [1].Suppose also that C and E satisfy (2.1) and (2.3), and that C is T -periodic.Let r > 0 be the rank of E and assume that there exists an orthogonal basis of R n ≃ R r × R n−r such that E has the form Assume also that, relatively to this decomposition of R n , ∂ 2 F 2 (ξ, η) is invertible for all x = (ξ, η) ∈ R r × R n−r .
Let Ω be an open subset of [0, +∞) ×C T (R n ) and suppose that deg(F , Ω∩ R n ) is well-defined and nonzero.Then, there exists a connected subset Γ of nontrivial T -periodic pairs for (2.7) whose closure in Ω is noncompact and meets the set (0, p) ∈ Ω : F (p) = 0 .This result follows as in [1] taking into account the modified version of the lemma.