On the Unstable Solutions to Functional Vector Diﬀerential Equations of the Seventh Order

is paper studies the instability of the zero solution for a certain nonlinear functional vector diﬀerential equation of the seventh order with multiple deviating arguments. Under suﬃcient conditions, we prove a result on the instability of the zero solution. is work contributes and complements to previously known results in the literature.


Introduction
More than 100 years ago, the world famous mathematician Lyapunov established the Lyapunov direct method to study stability problems. From then on, the Lyapunov's direct method was also widely used to study the instability of solutions of ordinary differential equations and functional differential equations, see for example, Bereketoğlu [1], Sadek [2], Tejumola [3], Tunç [4][5][6][7], C. Tunç and E. Tunç [8], and the references therein. However, a review to date of the literature indicates that the instability of the solutions to the nonlinear functional vector differential equation of the seventh order has not been investigated. is paper is the �rst work on the subject. It is also known that the expressions of Lyapunov-Krasovskii functional are very complicated and hard to construct. In this paper, we de�ne a Lyapunov-Krasovskii functional [9] and base on the Krasovskii criteria to prove a new theorem on the topic for the nonlinear functional vector differential equation of the seventh order. In this paper, intend to make a contribution to the subject since the functional differential equations have an important place in various �elds of science and engineering.
Meanwhile, some respective contributions on the topic can be summarized as follows.
Let 0 be given, and let = ( in is de�ned by Let be an open subset of and consider the general autonomous delay differential systeṁ where ∶ → is continuous and maps closed and bounded sets into bounded sets. It follows from the conditions on that each initial value probleṁ has a unique solution de�ned on some interval [0 ), 0 < ∞. is solution will be denoted by ( )( ) so that e zero solution, = 0, oḟ= ( ) is stable if for each > 0 there exists = ( ) > 0 such that ‖ ‖ < implies that | ( )( )| < for all 0. e zero solution is said to be unstable if it is not stable.
Consider the linear constant coefficient differential equation of the seventh order: It is known from the qualitative behavior of solutions of linear differential equations that the zero solution of (13) is unstable if and only if, the associated auxiliary equation: ( ) ≡ 7 + 1 6 + 2 5 + 3 4 + 4 3 + 5 2 + 6 + 7 = 0 has at least one root with a positive real part. e existence of such a root naturally depends on (though not always all of) the coefficients 1 2 … 7 in (14). For example, if then it is clear from a consideration of the fact that the sum of the roots of (14) equals 1 and that at the least one root of (14) has a positive real part for arbitrary values of 2 , 3 , 4 , fact that the product of the roots (14) equals (− 7 ) will verify that at least one root of (14) have a positive real part if 1 = 0, 7 ≠ 0 (16) for arbitrary 2 , 3 , 4 , 5 and 6 . e condition 1 = 0 here in (16) is, however, super�uous when then for 0 = 7 < 0 and 0 if 0 is sufficiently large, thus showing that there is a positive real root of (14) subject to (17) and for arbitrary 1 , 2 , 3 , 4 , 5 and 6 . Moreover, a necessary and sufficient condition for (14) to has a purely imaginary root = ( real) is that the two equations then (14) cannot have any purely imaginary root whatever. It should be noted that there are no restrictions on the constants 2 , 4 , and 6 in (13).

Main Result
First, we give the following lemma.

Lemma 2. Let be a real symmetric -matrix and
where ′ and are constants, and are the eigenvalues of the matrix . en (Bellman [11] where is a real variable such that the integrals ∫ 0 − ∫ + ‖ 2 ‖ 2 are nonnegative, and are certain positive constants to be determined later in the proof.
e proof of the theorem is completed.