On the stability and behavior of solutions in mixed differential equations with delays and advances.

In this study, some new results are obtained on asymptotic behavior and stability analysis of the mixed type differential equation x′t=∫−10xt−r1θdvθ+∫−10xt+r2θdηθ,where xt∈ℝ, r1θ and r2θ are real nonnegative continuous functions on −1,0, and vθ and ηθ are real valued functions of bounded variation on −1,0. These results were obtained by using an appropriate real root of the characteristic equation. Moreover, by the use of two appropriate distinct real roots of the corresponding characteristic equation, a new result on the behavior of solutions is established. Five examples are also given to illustrate our results. We also presented the application of the obtained results in the special case of constant coefficients and have given three different cases in one example. The results obtained in this article show that real roots play an important role. [ABSTRACT FROM AUTHOR]