A criterion for convergence of solutions of homogeneous delay linear differential equations

Josef Diblík
1999 Annales Polonici Mathematici  
The linear homogeneous differential equation with variable delayṡ is considered, where α j ∈ C(I, R + ), I = [t 0 , ∞), R + = (0, ∞), n j=1 α j (t) > 0 on I, τ j ∈ C(I, R + ), the functions t − τ j (t), j = 1, . . . , n, are increasing and the delays τ j are bounded. A criterion and some sufficient conditions for convergence of all solutions of this equation are proved. The related problem of nonconvergence is also discussed. Some comparisons to known results are given. 1991 Mathematics Subject
more » ... Classification: 34K15, 34K25. Key words and phrases: asymptotic convergence of solutions, linear homogeneous delay differential equation, topological principle of Ważewski (Rybakowski's approach). [115] 116 J. Diblík linear case the solution y(t * , ϕ)(t) is unique on its maximal existence interval D t * ,ϕ = [t * , ∞) ([10]). We say that a solution of (1.1) corresponding to the initial point t * is convergent or asymptotically convergent if it has a finite limit at ∞. Our aim in this paper is to formulate a criterion and some sufficient conditions for convergence of all solutions of (1.1). We also consider the related problem of nonconvergence of solutions of (1.1). Problems concerning asymptotic constancy of solutions, asymptotic convergence of solutions or existence of asymptotic equilibrium of various classes of retarded functional differential equations were investigated e.g. by O. Arino, I. Győri and M. Pituk [1], F. V. Atkinson and J. R. Haddock [2], R. Bellman and K. L. Cooke [3], I. Győri and M. Pituk [8, 9] and T. . Nonconvergence was considered e.g. by S. N. Zhang [18] and J. Diblík [7]. Some closely connected questions are discussed in the recent papers by J.Čermák [5, 6] (where, in some proofs, the fundamental results of F. Neuman [14, 15] concerning the transformation theory are used). In Section 2 some comparisons are given. Section 3 contains auxiliary lemmas and in the last Section 4 the proofs of the theorems are collected. In this paper the following is proved:
doi:10.4064/ap-72-2-115-130 fatcat:jrkqpxdznfbxhjdjejpbhge72u