1. Introduction

Qualitative behaviour of solutions of various second order differential equations with and without delay have been extensively discussed in the literature and are still receiving attention of authors because of their practical applications. Readers are referred to the books of Burton et al. ^{5}^{)(}^{6}^{)(}^{8} Hale ^{11}, Kolmanovskii and Myshkis ^{13}, ^{15}, Lakshmikantham et. al. ^{16}, Yoshizawa ^{30}^{)(}^{31}^{)(}^{32}, which contain general results on the subject matter and the papers of Ademola ^{1}, Ademola et. al. ^{2}^{)(}^{3}, Alaba and Ogundare ^{4}, Domoshnitsky;^{9}, Grioryan ^{10}, Jin and Zengrong ^{12}, Kroopnick^{14},Ogundare et. al. ^{17}, Ogundare and Afuwape ^{18}, Ogundare and Okecha ^{19}, Tunç ^{20}-^{24}, Wang and Zhu ^{26}, Xu ^{27}, Yeniçerioğlu ^{28},^{29} and the references cited therein.

In ^{18} the authors discussed boundedness and stability properties of solutions of

where *f, g* and **p** are continuous functions in their respective arguments *t, x* and *x’*. In ^{20} the author discussed boundedness of solutions to

where *c, b, q* and *f* are continuous functions defined on **R**
^{+} x **R**
^{2}, **R**, **R**
^{+} and **R**
^{+} R + respectively. Recently, in ^{4} the authors studied the second order non autonomous damped and forced nonlinear ordinary differential equation of the form

where the functions a, b, c, f, g and p depend only on the arguments displayed explicitly.

Finally in ^{1} the author considered stability, boundedness and existence of unique periodic solutions to the following second order ordinary differential equation

where
and p are continuous functions in their respective arguments on R, R^{+} x R^{2}, R^{+}, R and R^{+} x R^{2} respectively.

Unfortunately, the problem of uniform asymptotic stability (when the function p = 0), uniform boundedness, uniform ultimate boundedness, existence and uniqueness of periodic solutions to second order nonlinear delay differential equation (1.1), where the nonlinear functions g, h and p contain variable deviating arguments and the second ordered derivative contains a variable coefficient, is yet to be investigated. The purpose of this paper therefore is to fill this gap. We will consider

where , g, h, p and are continuous functions in their respective arguments, (i.e. with and . The primes indicate differentiation with respect to the independent variable t. If , then equation is equivalent to system of first order delay differential equations

where
is a constant to be determined later and the derivatives
and
exist and continuous for all x and t. The work is motivated by the recent works in ^{1},^{3},^{4},^{20}. The obtained results are new, in fact according to our observation from relevant literature, this is the first paper on second order delay differential equations where the highest ordered derivatives contains variable coefficient. In Section 2 we discussed the basic mathematical tools that will be used in the sequel. In Section 3 the main results are stated and proved while examples are given in Section 4.

2. Preliminary Results

Consider the following general nonlinear non-autonomous delay differential equation

Lemma 2.1. (See ^{32} pp 206).Suppose that
and
is periodic in t of period
, and consequently for any
there exists an
such that
implies
. Suppose that a continuous Lyapunov functional
exists, defined on
is the set of
such that with
(𝐻 may be large) and that
satisfies the following conditions:

(i) , where a(r) and b(r) are continuous, increasing and positive for

(ii) , where c(r) is continuous and positive for .

Suppose that there exists an , such that

where is a constant which is determined in the following way: By the condition on there exist and such that and is defined by . Under the above conditions, there exists a periodic solution of (2.1) of period /. In particular, the relation (2.2) can always be satisfied if h is sufficiently small.

Lemma 2.2. (See ^{32} pp 188).Suppose that
is defined and continuous on
and that there exists a continuous Lyapunov functional
defined on
which satisfy the following conditions:

(iii) for the associated system

we have , where for or , we understand that the condition is satisfied in the case V’ can be defined.

Then, for given initial value , there exists a unique solution of (2.1).

Lemma 2.3. (See ^{32} pp 190). Suppose that a continuous Lyapunov functional
exists, defined on
which satisfies the following conditions:

(i) , where a(r) and b(r) are continuous, increasing and positive,

(ii) , where c(r) is continuous and positive for ,

then the zero solution of system (2.1) is uniformly asymptotically stable

Lemma 2.4. (See ^{5}; pp 317).Let
be continuous and locally Lipschitz in
. If

(ii) , for some where Wi(i = 0,1,2,3,4) are wedges.

Then X_{
t
} of system 2.1 is uniformly bounded and uniformly ultimately bounded for bound M.

3. Main Results

We will start with the following notations, let
and
. Let (x_{
t
} , y_{
t
} ) be any solution of the system (1.2), the continuously differentiable functional used in this investigation is V = V(t, x_{
t
} , y_{
t
} ) defined as

where the function is defined in Section 1, are positive constants and the value of will be determined later. Next, we state the main results as follows.

Theorem 3.1. Further to the basic assumptions on the functions and p, suppose that and M are positive constants such that

then the solution (x_{
t
} , y_{
t
} ) of system (1.2) is uniformly bounded and uniformly ultimately bounded.

Remark 1. We observed the following:

(i) If and , equation (1.1) reduces to linear constant coefficients differential equation

and conditions (i) to (v) of Theorem 3.1 specializes to the corresponding Routh-Hurwitz criteria and

(ii) When
and
, equation (1.1) reduces to a special case discussed in ^{20}, thus Theorem 3.1 includes and improves the results in ^{20}.

(iii) Whenever
and
, equation (1.1) reduces to that discussed in ^{4}, thus Theorem 3.1 improves the boundedness results in ^{(}^{4}.

(iv) When
and
equation (1.1) becomes that considered in ^{18}, hence our results extend the results in ^{(}^{18}.

(v) If
and
, equation (1.1) coincides with (1.1) in ^{(}^{1}, thus the results in ^{(}^{1} are contained in this work.

In what follows we will state and prove a result that would be useful in the proof of Theorem 3.1 and the subsequent results.

Lemma 3.2. Under the hypotheses of Theorem 3.1 there exist positive constants and such that

for all and y. Furthermore, there exist positive constants and such that

Proof. Let (x_{
t
} , y_{
t
} ) be any solution of system (1.2), from equation (3.1) for
, we have V(t, 0,0) = 0 for all
. Also the functional V defined in (3.1) can be recast in the form

In view of condition (i) of Theorem 3.1, , and the fact that

for all and y, there exists a positive constant such that

In addition, from inequality (3.6) we find that

Moreover, since for all x and the fact that the inequality holds, there exist positive constants and such that

and

Inequalities (3.6) and (3.10) established the inequality (3.3) with
and
equivalent to D_{
0
} , D_{
1
} and D_{
2
} respectively.

Next, the differentiation of the functional V defined by equation (3.1) with respect to the independent variable t along the solution path of system (1.2) after simplification is

Where

Now from conditions (ii) and (iii) of Theorem 3.1, we find that

for all
and y. Using conditions (i), (ii) and (iii) of Theorem 3.1 in W_{
2
} we have

for all and y.Employing estimates

inequality (3.12) becomes

for all $
and y. Moreover, from hypothesis (i) and let
it follows from W_{
3
} that

for all
and y. Using inequality (3.13) and
for all x in W_{
3
} , we obtain

for all and y. Finally, for all x, h’ for all x and the inequality imply that

for all
and y. Inserting estimate W_{
j
} (j = 1,…,4) in equation (3.11), we obtain

for all and y. Furthermore, and choose

there exist positive constants and such that

and

Inequality (3.16) satisfies the inequality (3.4) with
and
equivalent to D_{
3
} and D_{
4
} respectively. This completes the proof of Lemma 3.2. Next, we will give the prove of Theorem 3.1, using some of the estimates of Lemma 3.2.

Proof of Theorem 3.1 Let (x_{
t
} , y_{
t
} ) be any solution of system (1.2). From inequalities (3.10) and (3.10) hypothesis (i) of Lemma 2.4 holds. Furthermore, using assumption (v) of Theorem 3.1 in estimate (3.16), noting that
, there exist positive constants
and
such that

where
with
choosing sufficiently small and
Inequality (3.18) satisfies condition (ii) of Lemma 2.4. Thus by Lemma 2.4 the solution (x_{
t
} , y_{
t
} ) of system (1.2) is uniformly bounded and uniformly ultimately bounded. This completes the proof of Theorem 3.1.

Next, if the forcing term
of equation (1.1) is replaced by a function p_{
1
} (t) where p_{
1
} (t) is defined on R^{+}, we obtain a special case of equation (1.1) as

Equation (3.19) is equivalent to system of first order equations

(3.20)

where the functions g, h and are defined in Section 1. We obtain the following result.

Theorem 3.3. If assumptions (i) to (iv) of Theorem 3.1 hold and assumption (v) is replaced by boundedness of the function p_{
1
} (t), then the solution (x_{
t
} , y_{
t
} ) of system (3.20) is uniformly bounded and uniformly ultimately bounded.

Proof. Let (x_{
t
} , y_{
t
} ) be any solution of system (3.20), the remaining part of the prove is similar to the proof of Theorem 3.1 hence it is omitted. This completes the proof of Theorem 3.3.

Furthermore, if the function
of equation (1.1) is replaced by p_{
2
} (t,x,x’) defined on R^{+} x R^{2} we have the following equation

Equation (3.21) can be written as system of first order differential equations

and we have following result.

Theorem 3.4. If the forcing term p^{
2
} of system (3.22) is bounded and assumptions (i) to (iv) of Theorem 3.1 hold, then the solution (x_{
t
} , y_{
t
} ) of system (3.22) is uniformly bounded and uniformly ultimately bounded.

Proof. Let (x_{
t
} , y_{
t
} ) be any solution of system (3.22), the remaining part of the prove is similar to the proof of Theorem 3.1 hence it is omitted. This completes the proof of Theorem 3.4.

Next, if of equation 1.1 is zero we have the following special case

Equation (3.23) as system of first order differential equations are as follow

and we have following result.

Theorem 3.5. If assumptions (i) to (iv) of Theorem 3.1 hold, then the trivial solution of system (3.24) is uniformly asymptotically stable.

Proof. Let (x_{
t
} , y_{
t
} ) be any solution of system (3.24), from equation (3.1) we have V(t, 0,0) = 0 for all
. Moreover, the inequalities (3.6) and (3.10) satisfy condition (i) of Lemma 2.3. Also if
= 0, estimate (3.16) becomes

for all and y where is defined in (3.17). Inequality (3.25) establish assumption (ii) of Lemma 2.3, hence by Lemma 2.3 the trivial solution of system (3.24) is uniformly asymptotically stable. This completes the proof of Theorem 3.5.

Next, we will state and proofs existence and uniqueness results of the solutions of system (1.2).

Theorem 3.6. If assumptions of Theorem 3.1 hold, then there exists a periodic solution of system 1.2 of period 𝜔.

Proof. Let (x_{
t
} , y_{
t
} ) be any solution of system (1.2), from inequalities (3.6), (3.10) and estimate (3.9), assumption (i) of Lemma 2.1 hold. Moreover, using hypothesis (v) of Theorem 3.6 and inequality
in estimate (3.16) there exist positive constants
and
such that

where and Inequality (3.26) satisfies assumption (ii) of Lemma 2.1, thus by Lemma 2.1 the periodic solution of system (1.2) exists and is of period This completes the proof of Theorem 3.6.

Theorem 3.7. If assumptions of Theorem 3.1 are satisfied, then there exists a unique solution of system (1.2).

Proof. Let (x_{
t
} , y_{
t
} ) be any solution of system (1.2), in view of estimates (3.6), (3.7), (3.9) and (3.26), assumptions of Lemma 2.2 hold, thus by Lemma 2.2 solution of system (1.2) is unique. This completes the proof of Theorem 3.7.

4. Examples

Example 4.1. Consider the second order delay differential equation

Equation (4.1) is equivalent to

Comparing systems (1.2) and (4.2) we find that:

(i) The function

Since

for all x, it follows that

for all x. Furthermore,

for all x. The paths of and its derivative, , are shown in Figure 1

(ii) The function

Since the fraction

(iii) The function

or

Since

for all x it follows that

Furthermore, the derivative of the function h with respect to x is

Noting that

for all x, it follows that

and

for all x. The behaviour of the functions h(x)/x, H(x) and h’(x) are shown in Figure 4

(iv) Using estimates (i) to (iii) of Example 4.1 with inequality (3.2) and equation (3.15) become

respectively.

(v) The function

It is not difficult to show that

From items (i) to (v) of Example 4.1, the assumption of Theorem 3.1, Theorem 3.6 and Theorem 3.7 hold, thus by Theorem 3.1, Theorem 3.6 and Theorem 3.7 the solution (x_{
t
} , y_{
t
} ) of system (4.2).

(i) is uniformly bounded and uniformly ultimately bounded;

(ii) possess a periodic solution of period 𝜔; and

(iii) is unique.

Also, if in system (4.2), items (i) to (iv) of Example 4.1 are equivalent to hypotheses (i) to (iv) of Theorem 3.5, then by Theorem 3.5 the trivial solution of system 4.2 is uniformly asymptotically stable.

Example 4.2.Consider also, the second order delay differential equation

Equation (4.3) in its equivalent form is

Comparing system (1.2) with system (4.4), we find that

(i) the function

Let

It is not difficult to show that

for all x. Moreover,

for all x. The behaviour of the functions and are shown in Figure 3

(ii) the function

from where we obtain

(iii) the function

This can be recast in the form

where

It is not difficult to show that

for all x. It follows that

for all x. Alternatively,

It follows that

for all x. The behaviour of the functions and are shown in Figure 3

(iv) From items (i) - (iii) of Example 4.2, choose inequality 3.2 and equation (3.15) become

respectively.

(v) Finally, the function

where we obtain

From items (i) to (v) of Example , assumptions of Theorem 3.1, Theorem 3.6 and Theorem 3.7 hold, thus by Theorem 3.1, Theorem 3.6 and Theorem 3.7 the solution (x_{
t
} , y_{
t
} ) of system (4.4)

(i) is uniformly bounded and uniformly ultimately bounded;

(ii) possess a periodic solution of period ; and

(iii) is unique.

Also, if in system (4.4), items (i) to (iv) of Example 4.2 satisfy the assumptions of Theorem 3.5, then by Theorem 3.5 the trivial solution of system 4.4 is uniformly asymptotically stable.