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Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices. (English) Zbl 1244.34096
The authors derive a representation of a solution of nonlinear delay differential equations using a so-called multidelayed matrix exponential, which generalizes the results of {\it D. Y. Khusainov} and {\it G. V. Shuklin} [Stud. Univ. Žilina, Math. Ser. 17, No. 1, 101--108 (2003; Zbl 1064.34042)] for autonomous linear delay systems with one delay defined by permutable matrices. In addition, sufficient conditions for exponential stability of the trivial solution of nonlinear delay differential equations with several delays for different types of nonlinearities are presented. Finally, an application of one of the stability criteria to a biological model is provided.

##### MSC:
 34K20 Stability theory of functional-differential equations
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##### References:
 [1] Khusainov, D. Ya.; Shuklin, G. V.: Linear autonomous time-delay system with permutation matrices solving, Studies of the university of žilina 17, 101-108 (2003) · Zbl 1064.34042 [2] Boichuk, A.; Diblík, J.; Khusainov, D.; Ružičková, M.: Boundary-value problems for weakly nonlinear delay differential systems, Abstract and applied analysis, 19 (2011) · Zbl 1222.34075 · doi:10.1155/2011/631412 [3] Boichuk, A.; Diblík, J.; Khusainov, D.; Ružičková, M.: Fredholm’s boundary-value problems for differential systems with a single delay, Nonlinear analysis 72, 2251-2258 (2010) · Zbl 1190.34073 · doi:10.1016/j.na.2009.10.025 [4] Khusainov, D. Ya.; Diblík, J.; Ružičková, M.; Lukáčová, J.: Representation of a solution of the Cauchy problem for an oscillating system with pure delay, Nonlinear oscillations 11, No. 2, 276-285 (2008) · Zbl 1276.34055 [5] Hale, J.: Theory of functional differential equations, (1977) · Zbl 0352.34001 [6] Medved’, M.; Pospíšil, M.; Škripková, L.: Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts defined by permutable matrices, Nonlinear analysis: theory, methods & applications 74, 3903-3911 (2011) · Zbl 1254.34104 [7] Bihari, I. A.: A generalization of a lemma of Bellman and its application to uniqueness problem of differential equation, Acta Mathematica Academy of science hungarica 7, 81-94 (1956) · Zbl 0070.08201 · doi:10.1007/BF02022967 [8] Pinto, M.: Integral inequalities of bihari-type and applications, Funkcialaj ekvacioj 33, 387-403 (1990) · Zbl 0717.45004