On behaviours of functional Volterra integro-differential equations with multiple time lags

ABSTRACT In this paper, the authors consider a non-linear Volterra integro-differential equation (NVIDE) of first order with multiple constant time lags. They obtain new sufficient conditions on stability (S), boundedness (B), global asymptotic stability (GAS) of solutions, and in addition, every solution of the considered NVIDE belong to and The authors prove five new theorems on S, B, GAS, and properties of solutions. The technique of the proofs involves the construction of suitable Lyapunov functionals. The obtained conditions are nonlinear generalizations and extensions of those of Becker [Uniformly continuous L1 solutions of Volterra equations and global asymptotic stability. Cubo 11(3);2009:1–24], Graef et al. [Behavior of solutions of non-linear functional Voltera integro-differential equations with multiple delays. Dynam Syst Appl. 25(1–2);2016:39–46] and Tunç [A note on the qualitative behaviors of non-linear Volterra integro-differential equation. J Egyptian Math Soc. 24(2);2016:187–192; New stability and boundedness results to Volterra integro-differential equations with delay. J Egyptian Math Soc. 24(2);2016:210–213] and they improve some results can found in the literature. The results of this paper are new, and they have novelty and complete some results exist in the literature.

In particular, during the last four decades, numerous authors have obtained many interesting results on the qualitative properties of solutions of linear VIDEs and NVIDEs, such as stability (S), boundedness (B), global asymptotic stability (GAS), integrability, square integrability, oscillation, non-oscillation and etc. by means of the fixed point theory, the perturbation methods, the variations of parameters formulas, the Lyapunov's function or functional method, etc., (see ).
When we study the above-mentioned results on the S, B, GAS, integrability, square integrability, etc., of the solutions, it can be seen that during the investigations most of the mentioned results are proved by means of LFs in the literature. Indeed, this information shows the effectiveness and applicability of LFs during the investigations and applications. However, to the best of our information from the relevant literature, only in the proofs of a few results, the fixed point method or the perturbation theory or the variations of parameters formulas have been used. These cases can be verified and seen by studying the context of the mentioned papers and those found in their references. Here, we would not like to give the details of the applications of these methods.
Meanwhile, despite its long history, today the Lyapunov's second (direct) method still seems as a leading basic tool to reduce a complexed system into a relatively simpler system and makes available useful applications in the scientific topics just mentioned above. However, the Lyapunov characterizations for retarded NVIDEs with non-smooth functionals have still remained as an open problem in the related literature. Here, we try to provide an application of this fact for a NVIDE with multiple constant time lags. By this way, we would like to say that it is worth to investigate qualitative properties of solutions of NVIDEs with multiple constant time lags.
In this direction, we would like to summarize some related papers on the topic.
Becker [15] consider VIDE He commented some properties of solutions of VIDE (1) by the Lyapunov's functions. The author also gave examples to illustrate the results obtained. Later, Tunç [53] considered NVIDE In [53], sufficient conditions are introduced so that solutions of NVIDE (2) are stable, global asymptotic stable and so on, and it is also shown that the solutions have bounded derivatives by means of LFs.
Later, Tunç [54] studied the following NVIDE with a constant time lag Tunç [54] investigated the S, B and convergence of bounded solutions of NVIDE (3) when t 1 by constructing suitable LFs.
Finally, in 2016, Graef et al. [29] considered the following NVIDE with multiple constant time lags The authors constructed sufficient conditions so that solutions of NVIDE (4) are bounded, belong to L 1 [0, 1] and L 2 [0, 1]. The authors also discussed S and GAS of the zero solution of NVIDE (4).
In this paper, we consider the following NVIDE with constant multiple time lags where t ≥ 0, t − t i ≥ 0, t i is the fixed constant time lag, the functions a and g are continuous with x, x(t − t i )) and K i are real-valued and continuous functions on Hence, from NVIDE (5), we have Finally, if we compare VIDEs (1)-(4) with NVIDE (5), then the difference of NVIDE (5) and VIDEs (1)-(4) can be is seen, clearly. That is, NVIDE (5) is a new and different mathematical model. In addition, we obtain five new results on the qualitative properties of solutions NVIDE (5). These are the novelty and originality of this paper.
It should be noted that through the paper when we need x will represent x(t).
In view of assumption (A4), from (6) we see that From the calculation of the time derivative of the Lyapunov functional W, we have for all t ≥ 0. In view of assumption (A4), from (7) we reach that Integrating the former inequality from 0 to t and considering the inequality W(t) ≥ x 2 (t), we obtain Hence, we can obtain that every solution of NVIDE (5) is bounded for all t ≥ t 0 ≥ 0. Indeed, from (8) and where M(t 0 ) := 1 The obtained result shows that the zero solution of NVIDE (5) is stable. That is, for any given 1 . 0, let Thus, we can reach to the end of the proof of Theorem 1.
Proof. It is known from Theorem 1 that any solution x(t) of NVIDE (5) is bounded and for which (8) and (9) hold. If assumption (H1) holds, then from (7) we have for all t ≥ t 1 .
Integration of the last inequality from t 1 to t gives that: Hence, we see that x [ L 2 [0, 1).
If assumption (H2) holds, then from the definition of the functional W, we can get Again, we see that x [ L 2 [0, 1). This proves Theorem 2.

Assumption
The following assumption is needed for GAS of the zero solution of NVIDE (5).
(C1) There exists a positive constant K such that for all t ≥ t 1 .
Proof. From Theorem 2, we have that every solution of NVIDE (5) belongs to L 2 [0, 1). In addition, from NVIDE (5) and (9), it follows that Hence, we have showed that x ′ (t) is bounded. This result together with the fact x [ L 2 [0, 1) yields that x(t) 0 as t 1. That is, the zero solution of NVIDE (5) is GAS. So, we can conclude that the idea of Theorem 3 is true.

Assumption
(D1) There exist constants k 1 . 0 and b 1 , 0 ≤ b 1 , 1, such that Proof. Define the LF From (10), we find that W 1 (t, x(.)) ≥ |x| for all t ≥ t 0 − t. We know that for a continuously differentiable function h(t), |h(t)| has a right derivative, and this right derivative D r |h(t)| is given by Hence, from (10) the right derivative of W 1 is given by Then, we can conclude that Thus, it can be shown that Thus, the B of solutions and the S of the zero solution of NVIDE (5) can be seen as in Theorem 1.
To complete the rest of the proof, we re-revise the functional W 1 . Define Hence, by assumption (D1), we have Further, the right derivative of functional W b in (11) is calculated as From the assumptions of Theorem 4, we can conclude that By the assumptions of Theorem 4, it is now clear that In addition, the integration of (12) with consideration of W b (t)≥|x| yields that Therefore, the improper integral Proof. From Theorem 1, any solution of NVIDE (5) satisfies (7). To complete the proof of this theorem, we re-consider the Lyapunov functional W(t) = W(t, x(.)), which is used in the proof of Theorem 1. Obviously, we have Next, under the light of assumptions (A1)-(A5), the time derivative of the Lyapunov functional. W(t) = W(t, x(t)) can be re-arranged as |q i (t)|W(t).
By integrating this inequality from t 0 to t, we get W(t) ≤ W(t 0 ) + t t 0 n i=1 |q i (s)|W(s)ds.
Hence, an application of the Gronwall's inequality yields that Consequently, by the assumption |q i | [ L 1 [0, 1), one can arrive at the desirable result that every solution of NVIDE (5) is bounded.

Conclusion
We consider a functional NVIDE of first order with multiple constant time lags. The qualitative properties of the considered NVIDE such as S, B, GAS, L 1 [0, 1) and L 2 [0, 1) are investigated by defining some suitable LFs. The obtained results have a novelty and contribution to the literature, and they improve or generalize the results of Becker [15], Graef et al. [29] and Tunç [53,54] and those can be found in the relevant literature.

Disclosure statement
No potential conflict of interest was reported by the authors.