Finite-Time Attractivity for Diagonally Dominant Systems with Off-Diagonal Delays

and Applied Analysis 3 where T is a given positive constant, aij : R → R, i, j 1, . . . , d, are continuous functions and τij > 0 for i, j 1, . . . , d with i / j. Define r : max { τij : i, j 1, . . . , d, i / j } . 2.2 Note that 2.1 is a special case of 1.2 . More precisely, the right hand side of 2.1 equals f t, xt , where f f1, . . . , fd : 0, T × C → R is defined as follows: fi ( t, φ ) : aii t φ 0 d ∑ j 1,j / i aij t φj (−τij ) . 2.3 Let S : 0, T × C → C denote the evolution operator of 2.1 . From 2.3 , we see that the function f is linear in the second argument. Therefore, the evolution operator S is also linear in the second argument. Our aim in this section is to provide a sufficient condition for the finite-time attractivity for the zero solution of 2.1 and thus for all solutions of 2.1 , see Remark 1.2. Before presenting the main result, we recall the notion of row diagonal dominance. We refer the reader to 14, Definition 7.10 for a discussion of this notion. System 2.1 is called row diagonally dominant if there exists a positive constant δ such that