Uniform-ultimate boundedness of solutions to vector lienard equation with delay

Boundedness of Vector Linéard Equation with Multiple Variable Delays

“…Moreover, the Jacobian matrices corresponding to H i (X), G(X) are defined by As for the contributions of this article, we would like to summarize some of them briefly in the following sentences. It can be observed that the ODEs of second order without and with delay of Adeyanju [13] and Adeyanju and Tunç [14], respectively, are particular cases of our Equation (3). The ODEs of Adeyanju [13] and Adeyanju and Tunç [14] are without delay and with a constant delay, respectively.…”

“…It can be observed that the ODEs of second order without and with delay of Adeyanju [13] and Adeyanju and Tunç [14], respectively, are particular cases of our Equation (3). The ODEs of Adeyanju [13] and Adeyanju and Tunç [14] are without delay and with a constant delay, respectively. However, our ODE (3) has two variable time delays.…”

“…As for the motivation of this article, recently, Adeyanju [13] and Adeyanju and Tunç [14] considered the following n-dimensional ODEs of second order without delay and with the constant delay τ, respectively:…”

“…Adeyanju [13] discussed boundedness, uniformly boundedness and some other behaviors of solutions of the first ODE via the Lyapunov second method. Next, Adeyanju and Tunç [14] constructed suitable new sufficient conditions that allow for the uniform ultimate boundedness of solutions of the second ODE with constant delay via the LKF method as a basic tool. On the other hand, we would like to outline some related works as follows: the stability and boundedness of solutions of vector differential equations by Lyapunov's second method (see [13,[15][16][17][18][19][20]), qualitative behavior of ordinary and functional differential equations (see [2,21]), stability, boundedness and periodicity of solutions of delay differential equations of second order (see [22][23][24]) and existence, uniqueness, stability, etc., of ordinary and functional differential equations (see [12]); for some other results, see [1,14,25,26].…”

“…Motivated by the works of Adeyanju [13], Adeyanju and Tunç [14] and those mentioned above, we consider the following system of ODEs of second order with two multiple variable time delays:…”