Stability and boundedness results for certain nonlinear vector differential equations of the fourth order

Abstract

We consider the equationin two cases: P ≡ 0 and P = 0. In the case P ≡ 0, the asymptotic stability of the zero solution X = 0 of the equation is investigated; in the case P = 0, the boundedness of all solutions of the equation is proved.

Some remarks on the stability and boundedness of solutions of certain differential equations of fourth-order

“…literature to derive suitable Lyapunov functions and hereby, in particular, many papers and books have been devoted to the study of stability and boundedness of solutions of certain second-, third-, fourth-, fi fth-and sixth order nonlinear differential equations (see, for example, Anderson [1], Barbasin [3], Cartwright [4], Chin ([5], [6]), Ezeilo ([8], [9]), Harrow ([10], [11]), Ku and Puri [13], Ku et al. [14], Ku ([15], [16]), Krasovskii [17], Leighton [18], Li [19], Marinosson [21], Miyagi and Taniguchi [22], Ogundare [23], Ponzo [24], Quian [25], Reissig et al [26], Schwartz and Yan [27], Shi-zong et al [28], Sinha ([29], [30]), Skidmore [31], Szegö [32], Tejumola ([33], [34]), Tiryaki and Tunç [35], Tunç ([36], [37], [38], [39], [40], [41], [42], [43]), Zubov [44], Wu and Xiong [45] and references quoted therein for some publications on these topics). So far, perhaps, the most effi cient tool for the study of the stability and boundedness of solutions of a given nonlinear differential equation is provided by Lyapunov theory.…”

Asymptotic Stability of Neutral Differential Systems with Variable Delay and Nonlinear Perturbations

“…Qualitative behaviour of solutions of differential equations with or without delay and/or randomness of various orders have received appreciable attention in recent years, see for instance the papers in [1][2][3][4][5][6][7][8][9]. These significant improvements in the study of differential equations are not unconnected to umpteen areas of applications in electrical networks containing lossless transmission lines [10][11][12], stability properties of electrical power systems, and macroeconomic models, the motion of nuclear reactors, feedback control loops involving sensors in integrated communication and control systems, energy or signal transmission, see [13][14][15][16][17][18].…”

A Stability Result for the Solutions of a Certain Fourth-Order Vector Differential Equation with Delay

“…Over the past years, many new results have been obtained on the stability for solutions of ordinary and functional differential equations of higher order without and with delay. For instance, we draw the attention of the interested reader to the book by Reissig et al [10] and the papers by Abou El-Ela et al [1,2,3], Adesina et al [4], Omeike [8,9], Sadek [11], Tunç [12,13,14] and the references cited therein. As far as we know, researches that discussed the stability of solutions to vector differential equations can briefly be summarized as follows:…”

“…First, in 2006 Tunç [13] gave sufficient conditions for the asymptotic stability of the trivial solution of equation:…”