Representation and stability of solutions of systems of functional differential equations with multiple delays

This paper is devoted to the study of systems of nonlinear functional differential equations with time-dependent coefficients and multiple variable increasing delays represented by functions gi(t) < t. The solution is found in terms of a piecewise-defined matrix function. Using our representation of the solution and Gronwall’s, Bihari’s and Pinto’s integral inequalities, asymptotic stability results are proved for some classes of nonlinear functional differential equations with multiple variable delays and linear parts given by pairwise permutable constant matrices. The derived theory is illustrated on nontrivial examples.

[1]  D. Khusainov,et al.  Boundary Value Problems for Delay Differential Systems , 2010 .

[2]  Josef Diblík,et al.  Representation of solutions of linear discrete systems with constant coefficients and pure delay , 2006 .

[3]  Josef Diblík,et al.  Fredholm’s boundary-value problems for differential systems with a single delay , 2010 .

[4]  J. Diblík,et al.  Representation of a solution of the Cauchy problem for an oscillating system with pure delay , 2008 .

[5]  D. Khusainov,et al.  Boundary-Value Problems for Weakly Nonlinear Delay Differential Systems , 2011 .

[6]  Elena Braverman,et al.  On exponential stability of linear differential equations with several delays , 2006 .

[7]  Jianhong Wu,et al.  Introduction to Functional Differential Equations , 2013 .

[8]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[9]  Milan Medved,et al.  Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts , 2011 .

[10]  I. Bihari.  A generalization of a lemma of bellman and its application to uniqueness problems of differential equations , 1956 .

[11]  Milan Medved,et al.  Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices , 2012 .

[12]  Tingxiu Wang.  Inequalities and stability for a linear scalar functional differential equation , 2004 .

[13]  L. Berezansky,et al.  On stability of some linear and nonlinear delay differential equations , 2006 .

[14]  Josef Diblík,et al.  Representation of solutions of discrete delayed system x(k+1)=Ax(k)+Bx(k−m)+f(k) with commutative matrices , 2006 .

[15]  Milan Medved,et al.  Sufficient conditions for the exponential stability of delay difference equations with linear parts defined by permutable matrices , 2012 .