1. Introduction

The Lyapunov's direct method containing an energy like function has come to be a powerful implement in the qualitative study of ordinary differential equations (ODEs). Over the last five decades years, this technique has also been applied to functional differential equations (FDEs) and (IDEs) by many researchers. It is well known that while Lyapunov functions are made use in the investigation of (ODEs), more generally (LFs) are used in studying (FDEs) and (IDEs) (see, for example, Burton (^{6}) (^{7}). As (VIDEs) can also be treated as (FDEs), (LFs) have been constructed or defined exclusively for various (VIDEs) by Adivar and Raffoul (^{1}), Becker (^{2},(^{3}),(^{4}), Burton (^{5}),(^{6}),(^{7}), Burton et al. (^{8}), Burton and Mahfoud (^{9}),(^{10}), Corduneanu (^{11}), Diamandescu (^{12}), Eloe et al. (^{13}), Furumochi and Matsuoka (^{14}), Graef and Tunç (^{15}), Graef et al. (^{16}), Grimmer and Seifert (^{17}), Gripenberg et al. (^{18}), Hara et al. (^{19}), Islam and Raffoul (^{20}), Jordan (^{21}), Levin (^{22}), Mahfoud (^{23}), Miller (^{24}), Raffoul (^{25}),(^{26}),(^{27}), Raffoul and Ren ((^{28})), Raffoul and Unal (^{29}), Rama Mohana Rao and Raghavendra (^{30}), Rama Mohana Rao and Srinivas (^{31}), Staffans (^{32}), Tunç (^{33}),(^{34}),(^{35}), Tunç and Ayhan (^{38}), Vanualailai and Nakagiri (^{39}), Wang (^{40}), Wazwaz (^{41}), Zhang (^{42}), Da Zhang (^{43}) and many relative papers in their references in order to study the stability (S), instability (I), uniform stability (US), exponential stability (ES), boundedness (B), etc. properties that equations.

As differentiated from this line, in 2007, Raffoul (^{26}) presents some basic theorems which provide a way of constructing (LFs) for the vector (VIDE)

Using (LFs), the author obtains sufficient conditions guaranteeing the (EAS) of the trivial solution to system

where *x* is an *n-* dimensional vector, the function *G* is continuous function for *t* ∈
0, ∞) and x ∈ ℜ^{
n
} with *G*(*t*,0)=0. A typical example of (1.2) is (VIDE) (1.1) . The results of Raffoul (^{26}) are applied to (VIDE) (1.1) with _{
ʄ(x)=xn
} , *n* is positive and rational. Two examples are discussed in (^{26}).

Then, Raffoul (^{26}) applies the theory obtained for equation (1.2) to the real (VIDE)

The author (^{26}) describes specific conditions guaranteeing the trivial solution of (VIDE) (1.3) is (UEAS).

In addition, more recently, Raffoul and Ren (^{29}) considered the n-dimensional stochastic differential equation (SDE) of the form

for *t* ∈ ℜ^{+}, *t* ≥ 0. The authors provide general theorems about boundedness or bounded in probability of solutions to the former nonlinear delay stochastic differential equation. In (^{29}), the analysis is based on the successful construction of suitable (LFs), and the authors also offer several good examples as application of their theorems.

In this paper, instead of (VIDE) (1.3) without time-lag and the (SDE) in (^{29}), we pay attention to the below nonlinear (VIDE) with constant time-lag

where *x(t)= φ2(t)* for 0≤ *t*≤ t_{0}, t-τ ≥ 0, τ is fixed constant time-lag, x ∈ ℜ, α(t):
0,∞ )→ (0,∞ ), *ϕ*: ℜ → ℜ, *ψ* ,*h*:
0,∞ ) × ℜ ×ℜ → ℜ are continuous functions with *ϕ*(0)= *ψ*(*t*,0,0)=h(*t*,0,0)=0 and *C(t,s,x(s))* is a continuous function for 0≤ *s*≤ *t*<∞ and *x(s)* with *C*(*t,s*,0)=0.

Let

Then, it may obtained from (VIDE) (1.4) that

in which and throughout the paper, when we need *x* represents *x(t)*.

In (^{(26)}), Raffoul also considers the following real (VIDEs)

The author (^{26}) gives sufficient conditions under which the trivial solutions of (VIDE) (1.6) and (VIDE) (1.7) are (UEAS). It should be noted that (VIDE) (1.6) and (VIDE) (1.7) are equivalent when _{
δ =k1
} . However, the results given in (^{26}) with respect to that (VIDEs) are different since different (LFs) are used therein.

In this paper, instead of (VIDE) (1.6) and (VIDE) (1.7), we consider the nonlinear (VIDEs) with constant time-lag

respectively, in which *x(t)*= φ_{4}
*(t)*, 0≤ *t*≤ _{
t0
} , *t*-τ ≥ 0, τ is fixed constant time-lag, *x* ∈ ℜ, γ(*t*):
0,∞) → (0,∞) and *h, f, p, q*: ℜ →ℜ are continuous functions with *h(0)=f(0)=p(0)=q(0)=0*, and the function *K*(t,s) is continuous for 0≤ *s*≤ *t*<∞.

Let

Then, we have from (VIDE) (1.8) and (VIDE) (1.9) that

respectively.

We investigate here the (UEAS) of the trivial solutions of (VIDEs) (1.4), (1.8) and (1.9) via new suitable (LFs).

It is notable form the mentioned information that Raffoul (^{26}) considers certain non-linear (VIDEs) without time-lag. In spite of this case, here, we consider certain new non-linear (VIDEs) with time-lag. In fact, when we choose *ϕ*(*x*)=*x*, *ψ* (*s,x(s),x(s*-*τ*))= *ʄ*(*s,x(s)*), *g(t,x(t))*= *h(t,x(t),x(t-τ ))* and take zero instead of the term *t-τ*, then our equation, equation (VIDE) (1.4) reduces to (VIDE) (1.3) discussed by Raffoul (^{26}). In addition, if take zero instead of the term *t-τ*, *h(x)=x*, *p*
(*x*)= *x*
and *ʄ(x)=x*, *p*
= *x*
, then (VIDEs)(1.8) and (1.9) reduce to (VIDEs) (1.6) and (1.7), respectively. That is, (VIDEs) (1.4), (1.8) and (1.9) include and extend (VIDEs) (1.3), ({1.6) and (1.7) discussed by Raffoul (^{26}). Moreover, our results improve the related results of (^{26}) from the cases of without delay to the more general cases with time-lag.

Finally, if we set the coefficient of the stochastic term in the (SDE) investigated in (^{29}) equal to zero, then we have the following delay (VIDE) of the form

Hence, it is clear that the former delay (VIDE) includes the delay (VIDEs) (1.4), (1.6) and (1.7). However, the results established in (^{2}) and that we are going to establish here, Theorems 2.1-2.3, are different fromeach other. Because the results proved in (^{29}), their assumptions and the (LFs) constructed therein are different form that we are going to give and prove in this paper. By the way we would like to state that the results in (^{29}) and those we are going to give here complement to each other.

All the mentioned information shows that the present paper has novel and original results and makes contribution to the literature.

We now consider the differential system given by (FDE) (1.2). Let _{
t0
} ≥ 0. Then, for each continuous function *φ*:
0,_{
t0
}
→ ℜ^{n}, there exists in any case one function _{
x(t)=x(t,t0,ϕ)
} , x(.) ∈ *C*
_{
t0,I
}
, appeasing (FDE) (1.2) for 0≤ _{
t0
} ≤ *t*≤ *I* such that *x*(_{
t, t0
} , *φ* )= *φ* for 0≤ *t*≤ _{
t0
} .

**Definition 1.1.** The trivial solution of (FDE) (1.2) is (EAS) if for a positive constant λ any solution _{
x(t,t0,φ )
} of (FDE) (1.2) fulfills

for all *t* ≥ _{
t0
} , where *K*(|*φ*|,_{
t0
} ) is a positive constant depends on *φ* and _{
t0
} , and *φ* is a stated initial function, which is bounded and continuous.

If the constant *K* in (1.12) is independent of _{
t0
} , then it is said that the trivial solution of (FDE) (1.2) is (UEAS).

The following three basic theorems are needed for the (S) results of this paper.

**Theorem 1.1.** (Raffoul (^{26})) Assume that *D* ⊂ ℜ^{n} contains the origin and there exists a *(LF) V : D*→
0,∞ ) such that the below assumptions hold for all *(t,x)* ∈
0,∞ )× *D*:

*(A1)* λ_{1} ||*x*||^{
ρ
} ≤ *V*(*t,x*(.)),

*(A2)* ⩒(*t,x*) ≤ λ_{3}
*V*(*t,x*(.)) + L *exp* ( - δt),

where λ_{1} >0, λ_{3} >0, ρ > 0, δ > 0, *L* ≥ 0 are real constants and 0< ϵ *min* {λ_{3},δ}.

Then the trivial solution of (FDE) (1.2) is (UEAS).

**Proof.** See Raffoul (^{26}).

**Theorem 1.2.** (Raffoul (^{26})) Assume that *D* ⊂ ℜ^{n} contains the origin and there exists a continuously differentiable (*LF*) *V*:
0,∞ )× *D*→
0,∞ ) such that the below assumptions hold for all (*t,x*) ∈
0,∞)× D:

*(B1)* λ_{1} ||*x*||^{ρ} ≤ *V*(*t,x*(.)) ≤ λ_{2}
*W*
_{2} (|*x*|) + λ_{2} ʃ^{t}
_{0}
*ϕ*
_{1} (*t,s*)*W*
_{3}(|*x*(*s*)|)*ds*,

(*B2*) ⩒(t,x(.))≤ -λ_{3}
*W*
_{4} (|*x*|) - λ_{3} ʃ^{t}
_{0}
*ϕ*
_{2} (*t,s*)*W*
_{5}(|*x*(*s*)|)*ds* + *L**
*exp* ( *- δ t*),

where λ_{1}> 0, λ_{2}> 0, λ_{3}> 0, ρ>0, δ > 0 and *L** > 0 are real constants, and *ϕ*
_{1}(*t,s*) and *ϕ*
_{2}(*t,s*) are scalar valued and continuous functions for 0≤ *s*≤ *t*<∞. If the below assumption

*(B3)*
*W*
_{2}(|*x*|) - *W*
_{4} (|*x*|) + ʃ^{t}
_{0}
*ϕ*
_{1}(*t,s*)*W*
_{3} |*x*(*s*)| - _{
ϕ2
} (*t,s*)*W*
_{3}|*x*(*s*)|
*ds* ≤ *L*
_{1}
*exp* ( - *δ t*)

holds for some positive constants _{
L1
} and *B* with ʃ^{t}
_{0}
*ϕ*
_{1}(t,s) *ds* ≤ *B*, then the trivial solution of (FDE) (1.2) is (UEAS).

**Proof.** See Raffoul (^{26}).

**Theorem 1.3.** (Raffoul (^{26})) Assume that *D* ⊂ ℜ^{n} contains the origin and there exists a continuously differentiable (*LF*) *V*:
0,∞ )× *D* →
0,∞ ) such that the following assumptions hold for all (*t,x*) ∈
0,∞ )× *D*:

*(C1)* λ_{1}(*t*)||*x*||^{ρ} ≤ *V*(*t,x*(.)) ≤ λ_{2}(*t*)*W*
_{2} (|*x*|) + λ_{2}(*t*) ʃ^{t}
_{0}
*ϕ*
_{1}(*t,s*)*W*
_{3}(||*x*(*s*)||)*ds*,

*(C2)* ⩒(*t,x*(.)) ≤- λ_{3}(*t*)*W*
_{4} (||*x*||) - λ_{3}(*t*) ʃ^{t}
_{0}
*ϕ*
_{2} (*t,s*)*W*
_{5} (||*x*(*s*)||)*ds* +
exp ( - *δt*),

where λ_{1}(*t*), λ_{2}(*t*), and λ_{3}(*t*) are some positive and continuous functions, with λ_{1}(*t*), is non-decreasing, ρ > 0, δ > 0 and
> 0 are some real constants, and *ϕ*
_{1}(*t,s*) ≥ 0 and *ϕ*
_{2}(*t,s*) ≥ 0 are scalar valued and continuous functions for 0≤ s ≤ t <∞. If the assumption

*(C3)*
*W*
_{2} (||*x*||) - *W*
_{4} (||*x*||) + ʃ^{t}
_{0}
*ϕ*
_{1}(*t,s*)*W*
_{3} |*x(s)*| - *ϕ*
_{2}(*t,s*)*W*
_{3} |*x*(*s*)|
*ds* ≤ *L*
_{2}
*exp* ( - *δt*)

holds for some positive constants _{
L2
} , *B* and *N* with ʃ^{t}
_{0}
*ϕ*
_{1}(*t,s*) *ds* ≤ *B* and λ_{2}(*t*)≤ *N*, then the trivial solution of (FDE) (1.2) is (UEAS).

**Proof.** See Raffoul (^{26}).

2. Exponential stability

**A. Assumptions**

Let _{
δ0, δ1, ϵ, β, τ, K0
} and *K* be positive constants. Assume that the following assumptions are true:

*(D1) ϕ*
_{1}(*x*) ≥ *δ*
_{1} for *x* ∈ ℜ,

*(D2)* |*ψ* (*t,x,x*(*t - τ* ))| ≤ σ (t) min (|*x*|,|*x*(*t - τ* )|),

|*h*(*t,x,x*(*t - τ* ))| ≤ *ϴ(t)*min (|*x*|
,| *x*(*t - τ* )|
), where σ (*t*) and *ϴ* (*t*) are positive, bounded and continuous functions,

*(D3)*|*C*(*t,s,x(s)*)| ≥ *K*
_{0}
|*C*(*u+τ,s,x(s)*)| *du* with 0≤ *s*≤ *t*≤ *u*<∞,

*(D4)* |*C*(*t + τ ,s,x(s)*)| ≥ *K*
_{0}
|*C*(*u+τ,s,x(s)*)| du, and

*(D5)* α (t)δ_{1}-
-*βσ*(*t*)
|*C*(*u+τ,t,x*(*t*))| du ≥ *K* with _{
K0
} ≥
>0.

**Theorem 2.1.** If assumptions (*D1*)-(*D5*) are true, then the trivial solution of (*VIDE*) (1.4) is (*UEAS*).

**Proof.** Describe a (*LF*) *V*
_{1} = *V*
_{1}(*t*)= *V*
_{1}(*t,x*(*t*)) by

where *β* >0, *β* ∈ ℜ, which is determined later in the proof.

Positive definiteness of the (*LF*) _{
V1
} is clear. That is,

Differentiating the (*LF*) _{
V1
} with respect to *t*, we obtain from (2.1) that

Then, it follows from (2.2), the hypotheses of Theorem 2.1 and the reality 2|*mn*| ≤ m^{2}+n^{2} that

If _{
ϴ2
} (*t*) ≤ exp(-*δt*) for a positive constant *δ*, λ_{1} =1, *ρ* =1 and *L* = 2^{-1}, then we can show the verification of all the assumptions of Theorem 1.1. That is, the trivial solution of (VIDE) (1.4) is (UEAS). This finishes the proof of Theorem 2.1.

**B. Assumptions**

Let α_{1}, *δ* and τ be positive constants such that the following assumptions are correct:

*(E1)*
*h*(0) = *p*(0) = 0, *h*
_{1}(*x*)≥1, *x* ∈ ℜ, |*p*(*x*)|≤ α_{1}
^{1/2}|*x*|, 0 < α_{1} < 1,

*(E2)* 2*γ*(*t*) - exp ( - *δt*)
|*K*(*t,s*)|*ds* -
exp ( -*δ* (*u+ τ* ))|*K*(*u + τ ,t*)|*du* ≥1,

where *γ(t)* is a positive, bounded and continuous function,

*(E3)* (- 1 +
α_{1})exp (- *δt*)| *K*(*t,s*)| ≥
exp (- *δ*(*u + τ* )|*K*(*u + τ ,s*)| *du with* 0≤ *s*≤ *t*≤ *u*<∞ .

**Theorem 2.2.** If assumptions (*E1*)-(*E3*) are true, then the null solution of (*VIDE*) (1.8) is (*UEAS*).

**Proof.** We describe a _{
(LF) V2 =V2(t)=V2(t,x(t))
} by

where λ >0, λ ∈ ℜ, and we determine that constant later.

Obviously, it follows that the _{
(LF) V2
} is positive definite, since

Differentiating the (*LF*) _{
V2
} , it follows from (*LF*) (2.3) and (*VIDE*) (1.8) that

By the disparity 2|*mn*| ≤ *m*
^{2}+*n*
^{2} and the assumption *g*
_{1}(*x*)≥ 1, we reach that

Let *a*=
and *b*= 3. By using the Young's inequality,

and assumption (E1), we have from (2.4) that

Setting former inequality into (2.4), we have

By assumption (E2), we have

Then, it follows from (2.3) and (2.5), respectively, that

and

Hence, we can conclude that assumptions (B1) and (B2) of Theorem 1.2 hold.

Then, in view of the assumption

we get

Therefore, the last estimate shows that assumption (B3) of Theorem 1.2 holds for _{
L1
} =0. That is, the trivial solution of (VIDE) (1.8) is (UEAS). Thus, the desirable result is available for the proof of Theorem 2.2.

**C. Assumptions**

Let *β* and _{
k2
} be positive constants. Assume that the below assumptions hold:

*(F1)* ⨍(0)= *q*(0)=0, ⨍_{1}(*x*)≥ 1, (*x ≠* 0), |*q*(*x*)| ≤ *β*
|*x*|, 0< *β* < 1,

*(F2)*
_{
k2
} +2γ(t)-
|*K*(*t,s*)| *ds*-
|*K*(*u+τ,t*)| *du*≥ 1 with 0≤ *s*≤ *t*≤ *u*<∞, where *γ*(*t*) is a positive, bounded and continuous function,

*(F3)*
|*K*(*t,s*)| *ds*<∞ and
|*K*(*u+τ ,s*)| duds<∞ with 0≤ *s*≤ *t*≤ *u*<∞ and (1 -
*β*) exp (-*δt*) |*K*(*t,s*)| ≥
exp (-*δ*(*u+τ* )) |*K*(*u+τ ,s*)| *du*.

**Theorem 2.3.** Let _{
δ ,k1, k2
} be positive constants such that 1<*δ* =_{
k1+k2
} . If assumptions *(F1)*-*(F3)* are correct, then the null solution of *(VIDE)*
*(1.9)* is *(UEAS)*.

**Proof.** We define a *(LF)*
*V*
_{3}=*V*
_{3}(*t*)=*V*
_{3}(*t,x(t)*) by

where *μ* >0, *μ* ∈ ℜ, and it is determined later.

It can be seen the positive definiteness of the (*LF*) *V*
_{3}, since

Differentiating the (*LF*) *V*
_{3} with respect to *t*, along solutions of (*VIDE*) (1.9), we obtain from (*LF*) (2.6) and (*VIDE*) (1.9) that

In view of the inequality 2|*mn*| ≤ _{
m2
} +_{
n2
} and the assumption ⨍_{1}(*x*)≥ 1, we have

Let *a*=
and *b*=3. By using the Young's inequality,

and assumption (F1), we arrive at

Placing the former inequality into (2.7), we obtain

Let *μ* =1 and
|*K*(*t,s*)| *ds*. Then, using assumption (F2) of Theorem 2.3, we conclude that

Then, it follows from (2.6) and the last estimate that

Hence, we can conclude that assumptions (B1) and (B2) of Theorem 1.3 hold.

Then, in view of the assumption

we get

Therefore, the last estimate shows that assumption *(C2)* of Theorem 1.3 holds for
=0. That is, the trivial solution of *(VIDE)*
*(1.9)* is *(UEAS).* This brings to an end the proof of Theorem 2.3.

**Remark 2.1.** By Theorems 2.1, 2.2 and 2.3, we improve and extend the stability results of Raffoul (^{26}) obtained for *(VIDEs)* without time-lag to their more general and time-lag forms. In addition, Theorems 2.1, 2.2 and 2.3 complement to the results in the references like Adivar and Raffoul (^{(1)}), Becker (^{2}),(^{3}),(^{4}), Burton (^{5}),(^{6}),(^{7}), Burton et al. (^{8}), Burton and Mahfoud (^{9}),(^{10}), Eloe et al. (^{13}), Furumochi and Matsuoka (^{14}), Graef and Tunç (^{15}), Graef et al. (^{16}), Grimmer and Seifert (^{17}), Gripenberg et al. (^{18}), Hara et al. (^{19}), Islam and Raffoul (^{20}), Jordan (^{21}), Levin (^{22}), Mahfoud (^{23}), Miller (^{24}), Raffoul (^{25}),(^{26}),(^{27}),(^{28}), Raffoul and Ren (^{29}), Raffoul and Unal (^{29}), Rama Mohana Rao and Raghavendra (^{30}), Rama Mohana Rao and Srinivas (^{31}), Staffans (^{32}), Tunç (^{33}),(^{34}),(^{35}) Tunç and Ayhan (^{38}), Vanualailai and Nakagiri (^{39}), Wang (^{40}), Wazwaz (^{41}), Zhang (^{42}), Da Zhang (^{43}).

By this way, we also mean that the non-linear (*VIDEs*) thought about and the appointed assumptions here are separate from that in the sources mentioned above. It is noticeable that investigators working on the qualitative properties of (*VIDEs*) or (*IDEs*) may derive advantages from that results obtained here. These cases show the newness and contribution of the current paper.