On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations
UDC 517.9 In this paper, two new Lyapunov functionals are defined. We apply these functionals to get sufficient conditions guaranteeing the asymptotic stability, uniform stability, and boundedness of solutions of certain nonlinear Volterra integro-differential equations of the first order. The resul...
Gespeichert in:
| Datum: | 2020 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2020
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6037 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512239176384512 |
|---|---|
| author | Tunç, C. Mohammed , S. A. Tunç, C. Mohammed , S. A. |
| author_facet | Tunç, C. Mohammed , S. A. Tunç, C. Mohammed , S. A. |
| author_sort | Tunç, C. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:49:28Z |
| description | UDC 517.9
In this paper, two new Lyapunov functionals are defined. We apply these functionals to get sufficient conditions guaranteeing the asymptotic stability, uniform stability, and boundedness of solutions of certain nonlinear Volterra integro-differential equations of the first order. The results obtained are improvements and extensions of known results that can be found in literature. We also suggest examples to show the applicability of our results and for the sake of illustrations. Using MATLAB-Simulink, in particular cases we clearly show the behavior of orbits of Volterra integro-differential equations under consideration. |
| doi_str_mv | 10.37863/umzh.v72i12.6037 |
| first_indexed | 2026-03-24T03:25:37Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i12.6037
UDC 517.9
C. Tunç (Van Yüzüncü Yil Univ., Turkey),
S. A. Mohammed (College Basic Education, Univ. Duhok, Iraq)
ON ASYMPTOTICALLY STABILITY, UNIFORMLY STABILITY
AND BOUNDEDNESS OF SOLUTIONS OF NONLINEAR VOLTERRA
INTEGRO-DIFFERENTIAL EQUATIONS*
ПРО АСИМПТОТИЧНУ СТIЙКIСТЬ, РIВНОМIРНУ СТIЙКIСТЬ
ТА ОБМЕЖЕНIСТЬ РОЗВ’ЯЗКIВ НЕЛIНIЙНИХ
IНТЕГРО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ ВОЛЬТЕРРА
In this paper, two new Lyapunov functionals are defined. We apply these functionals to get sufficient conditions guaranteeing
the asymptotic stability, uniform stability, and boundedness of solutions of certain nonlinear Volterra integro-differential
equations of the first order. The results obtained are improvements and extensions of known results that can be found
in literature. We also suggest examples to show the applicability of our results and for the sake of illustrations. Using
MATLAB-Simulink, in particular cases we clearly show the behavior of orbits of Volterra integro-differential equations
under consideration.
Наведено означення двох нових функцiоналiв Ляпунова. Цi функцiонали використано для отримання достатнiх
умов, що гарантують асимптотичну стiйкiсть, рiвномiрну стiйкiсть та обмеженiсть розв’язкiв нелiнiйних iнтегро-
диференцiальних рiвнянь Вольтерра першого порядку. Отриманi результати удосконалюють та розширюють вiдомi
результати, що вже були опублiкованi. Наведено приклади застосування отриманих результатiв. За допомогою
MATLAB-Simulink в окремих випадках показано поведiнку орбiт розглянутих iнтегро-диференцiальних рiвнянь
Вольтерра.
1. Introduction. Volterra integral and integro-differential equations, integral equations and integro-
differential equations have many applications in sciences and engineering (see Burton [2], Rah-
man [7], Wazwaz [20] and the cited references therein). Due to these facts, in the last years,
stability, asymptotic stability, uniform stability, boundedness, exponentially stability, etc., of linear
and nonlinear Volterra integro-differential equations, Volterra integral equations, integral equations
and integro-differential equations have been discussed by many researches. In particular, as a brief
information, the reader can referee to the articles of Becker [1], Furumochi and Matsuoka [3], Graef
et al. [4], Mahfoud [5], Raffoul [6], Rama Mohana Rao and Srinivas [8], Tunç [9 – 13], Tunç and
Mohammed [14], Tunç and Tunç [15 – 17], Wang [18, 19] and the works mentioned in that sources
for the former scientific results that can be found in the literature on the diverse qualitative behaviors
of various of Volterra integro-differential equations, Volterra integral equations, integral equations
and integro-differential equations. For some very recent interesting works on the various qualitative
properties of solutions of certain nonlinear Volterra integro-differential equations without and with
delay, we would also like to mention the papers of Tunç [21, 22], Tunç and Tunç [23 – 25].
As a distinguished information from this line, the following article is notable.
In 2000, Wang [19] considers the Volterra integro-differential equation
* This research was supported by Van Yüzüncü Yıl University (grant FAP-2016-5550).
c\bigcirc C. TUNÇ, S. A. MOHAMMED, 2020
1708 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
ON ASYMPTOTICALLY STABILITY, UNIFORMLY STABILITY AND BOUNDEDNESS . . . 1709
dx
dt
= A(t)x(t) +
t\int
0
C(t, s)x(s)ds, (1.1)
in which t is non-negative and real variable, x \in \Re n, n \geq 1, A(.) and C(.) are (n \times n)-matrices,
which are continuous for 0 \leq t < \infty and 0 \leq s \leq t < \infty , respectively.
Wang [19] proves three theorems related to the stability, uniform stability and asymptotic stability
of solutions of Volterra integro-differential equation (1.1). The author gives an example verifying the
established assumptions. The results obtained in Wang [19] are variants of the results that can be
found throughout Burton [2].
In this article, motivated by the results of Wang [19], we first take into consideration the nonlinear
Volterra integro-differential equation
dx
dt
= - A(t)x+
t\int
0
C(t, s)g(s, x(s))ds+ h(t, x), (1.2)
where t is non-negative and real variable, x \in \Re n, A(.) and C(.) have the same properties as in
the Volterra integro-differential equation (1.1), g : \Re + \times \Re n \rightarrow \Re n and h : \Re + \times \Re n \rightarrow \Re n are
continuous functions with \Re + = [0,\infty ), and g(s, 0) = 0.
We will discuss the stability, uniform stability of trivial solution and boundedness of solutions of
Volterra integro-differential equation (1.2) by help of appropriate Lyapunov functionals for the cases
of h(.) \equiv 0 and h(.) \not = 0, respectively.
It follows that the Volterra integro-differential equation (1.1) discussed in Wang [19] is a linear
equation. However, the Volterra integro-differential equation (1.2), to be discussed here, is a non-
linear Volterra integro-differential equation. This case is a clear improvement and contribution from
the linear Volterra integro-differential equation (1.1) to the nonlinear Volterra integro-differential
equation (1.2). This fact is the first originality of this article.
Furthermore, Volterra integro-differential equation (1.2) includes and extends the Volterra integro-
differential equation (1.1) discussed in Wang [19]. Next, Wang [19] discusses the stability and uni-
form stability of the solution x(t) \equiv 0 of Volterra integro-differential equation (1.1) when h(.) \equiv 0.
However, in addition to these two problems, we will establish the hypotheses to get the boundedness
solutions of the Volterra integro-differential equation (1.2), when h(.) \not = 0. This fact is the second
originality of this article and its contribution to the literature.
Moreover, if choose g(.) = x(s) and h(.) = 0, then Volterra integro-differential equation (1.2)
reduces to Volterra integro-differential equation (1.1) studied by Wang [19]. This information clearly
shows the other contribution of this article to the subject.
In addition, we define here two new Lyapunov functionals to proceed the proofs of main results.
It is well-known that the definition or construction of suitable Lyapunov functionals remains as an
unsolved problem in the related literature by this time. Succeeding this fact for the general problems
considered such as in this paper is the third original property of this article.
Furthermore, two specific examples are given to make clear the correctness and applicability of
our hypotheses to be given. In particular cases, using MATLAB-Simulink, it is clearly shown the
behaviors of the orbits of the Volterra integro-differential equations considered. According to our
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1710 C. TUNÇ, S. A. MOHAMMED
observations, this fact is not appeared in the papers or books in the references of this article and the
others. This fact is another originality and contribution of this article to the topic. Indeed, we would
like to state by all of these information the originality and novelty properties of the results of this
paper.
Throughout this article, when we need x will represent x(t).
Let x \in \Re n, \| x\| =
\Bigl( \sum n
i=1
x2i
\Bigr) 1/2
, A be an (n\times n)-matrix, \| A\| = \mathrm{m}\mathrm{a}\mathrm{x}1\leq j\leq n
\Bigl( \sum n
i=1
a2ij
\Bigr) 1/2
and P be an (n\times n)-positive definite constant matrix.
We represent the characteristic roots of the symmetric matrix
1
2
(AT (t)P + PA(t))
by \lambda j(t, P,A), j = 1, 2, . . . , n.
We suppose that
\lambda m(t, P,A) \equiv \mathrm{m}\mathrm{i}\mathrm{n}
1\leq j\leq n
\lambda j(t, P,A)
and
\lambda M (t, P,A) \equiv \mathrm{m}\mathrm{a}\mathrm{x}
1\leq j\leq n
\lambda j(t, P,A),
which are minimum and maximum eigenvalues of the symmetric matrix
1
2
\bigl(
AT (t)P + PA(t)
\bigr)
,
respectively.
Lemma 1.1 [19]. If x \in \Re n, then
\lambda m(t, P,A)\| x\| 2 \leq xT
\biggl[
1
2
\bigl(
AT (.)P + PA(.)
\bigr) \biggr]
x \leq \lambda M (t, P,A)\| x\| 2
. (1.3)
The proof of the inequality (1.3) can be easily done. We omit the details.
The following lemma is also well-known for the linear algebra.
Lemma 1.2. Let P be a real symmetric (n\times n)-matrix and
a\prime \geq \lambda i(P ) \geq a > 0, i = 1, 2, . . . , n,
where a\prime and a are constants. Then
a\prime \langle x, x\rangle \geq \langle x, Px\rangle \geq a \langle x, x\rangle
and
a\prime
2 \langle x, x\rangle \geq \langle Px, Px\rangle \geq a2 \langle x, x\rangle .
2. Asymptotic stability. Let h(.) \equiv 0.
A. Hypotheses: Suppose the following hypotheses hold:
(A1) Let P be an (n\times n)-matrix with constant elements, which is positive definite and symmetric
such that
\| P\| \leq K2, where K2 > 0, K2 \in \Re ;
(A2) K3\| x\| \leq \| g(t, x)\| \leq K4\| x\| , where K3 and K4 are some positive real constants;
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
ON ASYMPTOTICALLY STABILITY, UNIFORMLY STABILITY AND BOUNDEDNESS . . . 1711
(A3) \theta (t) = 2\lambda m(t, P,A) - K2K4
\int t
0
\| C(t, s)\| ds - K2K
- 2
3 K3
4
\int \infty
t
\bigm\| \bigm\| C(u, t)
\bigm\| \bigm\| du \geq K1 such
that
\infty \int
0
\bigm\| \bigm\| C(t, s)
\bigm\| \bigm\| ds < \infty and
\infty \int
t
\bigm\| \bigm\| C(u, t)
\bigm\| \bigm\| du < \infty ,
where K1 > 0, K1 \in \Re .
Theorem 2.1. If hypotheses (A1) – (A3) hold, then the zero solution of Volterra integro-differential
equation (1.2) is asymptotic stable.
Proof. We define a Lyapunov functional V0 = V0(t, x(.)) by
V0 = xTPx+ \gamma 1
t\int
0
\infty \int
t
\bigm\| \bigm\| C(u, s)
\bigm\| \bigm\| du\bigm\| \bigm\| g(s, x(s))\bigm\| \bigm\| 2ds, (2.1)
where \gamma 1 > 0, \gamma 1 \in \Re , and we determine this constant later in the proof.
The positive definiteness of the Lyapunov functional V0 and the existence of the estimate
xTPx \leq V0
are clear.
Differentiating the Lyapunov functional V0 with respect to t, it can be obtained from (2.1) and
Volterra integro-differential equation (1.2) that
d
dt
V0 = - xT
\bigl\{
AT (t)P + PA(t)
\bigr\}
x+
t\int
0
gT (s, x(s))CT (t, s)ds\times Px+
+xTP \times
t\int
0
C(t, s)g(s, x(s))ds+ \gamma 1\| g(t, x)\| 2
\infty \int
t
\bigm\| \bigm\| C(u, t)
\bigm\| \bigm\| du -
- \gamma 1
t\int
0
\bigm\| \bigm\| C(t, s)
\bigm\| \bigm\| \bigm\| \bigm\| g(s, x(s))\bigm\| \bigm\| 2ds. (2.2)
Let \gamma 1 =
K2K4
K2
3
. Applying hypotheses (A1) – (A3) and an elementary inequality, we can get from (2.2)
that
d
dt
V0 \leq - \theta (t)\| x\| 2. (2.3)
Hence
d
dt
V0 \leq - K1\| x\| 2
by (2.3) and hypothesis (A3). The proof of the asymptotic stability of the zero solution of equa-
tion (1.2) is completed.
Theorem 2.1 is proved.
Remark 2.1. We can also conclude that the sufficient hypotheses (A1) – (A3) guarantee the stabi-
lity of the zero solution of Volterra integro-differential equation (1.2).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1712 C. TUNÇ, S. A. MOHAMMED
3. Bounded solutions. Let h(.) \not = 0.
B. Hypothesis: Let the following hypothesis holds:
(B1) \| h(.)\| \leq \Omega (t), where \Omega is a non-negative and continuous function such that \Omega \in L1(0,\infty ),
that is,
\infty \int
0
\Omega (s)ds < \infty .
Theorem 3.1. If hypotheses (A1) – (A3) and (B1) are satisfied, then all solutions of Volterra
integro-differential equation (1.2) are bounded.
Proof. We keep in the mind the Lyapunov functional V0 = V0(t, x(.)) given by (2.1). Hence, in
view of h(t, x) \not = 0 and the time derivative of functional V0, an application of hypotheses (A1) – (A3)
and (B1) makes enable that
V \prime
0 \leq hT (t, x)Px+ xTPh(t, x) \leq K2\Omega (.) + \Omega (.)V0(t, x(.)).
By evaluating the integral of the previous inequality between 0 and t, we arrive at
V0(t, x(t)) \leq V0(0, x(0)) +K2
t\int
0
\Omega (s)ds+
t\int
0
\Omega (s)V0(s, x(s))ds.
By an application of the Gronwall’s inequality, we have
xTPx \leq V0(t, x(t)) \leq K5 \mathrm{e}\mathrm{x}\mathrm{p}
\left( \infty \int
0
\Omega (s)ds
\right) ,
where K5 = V0(0, x(0)) +K2
\int \infty
0
\Omega (s)ds, K5 > 0, so that
xTPx \leq K2
5 .
Hence, we can conclude that all solutions of Volterra integro-differential equation (1.2) are bounded.
Theorem 3.1 is proved.
4. Uniform stability. In this paper, finally, we consider the nonlinear Volterra integro-differential
equation
dx
dt
= - A(t)x+
t\int
0
C(t, s, x(s))ds, (4.1)
with C(t, s, 0) = 0, t \geq 0, x \in \Re n, A(.) is an (n\times n)-matrix function and C(.) is an (n\times 1)-vector
function, which are continuous for the arguments displayed clearly.
It is clear that Volterra integro-differential equation (4.1) is a nonlinear generalization of Volterra
integro-differential equation (1.1) discussed by Wang [19]. Here, we take into consideration the
uniform stability of the zero solution of this Volterra integro-differential equation.
C. Hypotheses: We assume that following hypotheses are true:
(C1)
\bigm\| \bigm\| C(t, s, x)
\bigm\| \bigm\| \leq K(t, s)\| x\| , where K(t, s) is a continuous scalar function for t \geq s \geq 0;
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
ON ASYMPTOTICALLY STABILITY, UNIFORMLY STABILITY AND BOUNDEDNESS . . . 1713
(C2)
\int t
0
\int \infty
t
\| K(u, s)\| duds < LK - 1
2 , (K2, L \in \Re ,K2, L > 0), where K(u, s) is a continuous
scalar function for the arguments shown;
(C3) \rho (t) = 2\lambda m(t, P,A) - K2
\int t
0
\bigm\| \bigm\| K(t, s)
\bigm\| \bigm\| ds - K2
\int \infty
t
\bigm\| \bigm\| K(u, t)
\bigm\| \bigm\| du \geq K6 with
\infty \int
0
\bigm\| \bigm\| K(t, s)
\bigm\| \bigm\| ds < \infty ,
\infty \int
0
\bigm\| \bigm\| K(u, t)
\bigm\| \bigm\| du < \infty ,
where K6 is a given positive constant and K(u, t) and K(t, s) are continuous scalar functions for
the arguments shown.
Theorem 4.1. If hypotheses (A1) and (C1) – (C3) are satisfied, then the zero solution of Volterra
integro-differential equation (4.1) is uniformly stable.
Proof. Let us define a Lyapunov functional V = V (t, x(.)) by
V (t, x(.)) = xTPx+ \beta
t\int
0
\infty \int
t
\| K(u, s)\| du \| x(s)\| 2 ds, (4.2)
where \beta > 0, \beta \in \Re , and we specific this constant in the proof.
The positive definiteness of the Lyapunov functional V is clear.
Differentiating the Lyapunov functional V with respect to t, we obtain from (4.2) and Volterra
integro-differential equation (4.1) that
d
dt
V = - xT
\bigl\{
AT (t)P + PA(t)
\bigr\}
x+
t\int
0
CT
\bigl(
t, s, x(s)
\bigr)
ds\times Px+
+xTP \times
t\int
0
C(t, s, x(s))ds+ \beta
\infty \int
t
\bigm\| \bigm\| K(u, t)
\bigm\| \bigm\| du\bigm\| \bigm\| x\bigm\| \bigm\| 2 - \beta
t\int
0
\bigm\| \bigm\| K(t, s)
\bigm\| \bigm\| \| x(s)\| 2 ds. (4.3)
Let \beta = K2. Take into consideration equality (4.3), make an application of hypotheses (A1),
(C1) – (C3) and use an elementary inequality, we can obtain
d
dt
V \leq - \rho (t)\| x\| 2. (4.4)
Hence, we have from (4.4) and hypothesis (C3) that
d
dt
V \leq - K6\| x\| 2. (4.5)
Let us now show that the uniform stability of zero solution of Volterra integro-differential equa-
tion (4.1).
Since the matrix P is real, symmetric and positive definite, then it is clear that
a\prime \| x\| 2 \geq xTPx \geq a
\bigm\| \bigm\| x\bigm\| \bigm\| 2, (4.6)
where a\prime > 0 and a > 0, which are some constants.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1714 C. TUNÇ, S. A. MOHAMMED
Let x \in \Re n and | x| be any norm. In addition, let C denote the Banach space of continuous
functions \phi : [t0 - \tau , t0] \rightarrow \Re n with
\| \phi \| t0 := \mathrm{s}\mathrm{u}\mathrm{p}
t0 - \tau \leq t\leq t0
\bigm| \bigm| \phi (t)\bigm| \bigm| .
We assume that x(t) = x(t, t0, \phi ) is a solution of equation (4.1) on [t0 - \tau ,\infty ) such that
x(t) = \phi (t) on [t0 - \tau , t0], t0 \geq 0, where \phi is the initial function with \phi \in C[t0 - \tau , t0].
Hence, we can write form (4.5), (4.6), (C1) and Lemma 1.2 that
a \| x(t, t0, \phi )\| 2 \leq \langle \phi (t0), P\varphi (t0)\rangle \leq V (t, x(.)) \leq V (t0, \phi (.)) \leq
\leq \phi T (t0)P\phi (t0) + \beta
t0\int
0
\infty \int
t0
\| K(u, s)\| du \| \phi (s)\| 2 ds \leq
\leq a\prime \| \phi (t0)\| 2 + \beta L \| \phi \| 2 .
Due to the discussion made, it follows that for each \varepsilon > 0, we can choose a positive constant
like \delta =
\biggl(
a
a\prime + \beta L
\biggr) 2 \varepsilon
4
such that for any solution of equation (4.1), the inequality \| \phi (t)\| < \delta ,
t \in [t0 - \tau , t0], implies that
a
\bigm\| \bigm\| x(t, t0, \phi )\bigm\| \bigm\| 2 \leq a\prime \delta 2 + \beta L\delta 2 \leq a\varepsilon 2
16
,
that is,
\| x(t, t0, \phi )\| \leq \varepsilon
4
< \varepsilon , t \geq t0.
It is clear that the constant \delta does not depend on the constant t0. Therefore, we can say that the
solution x(t) \equiv 0 of Volterra integro-differential equation (4.1) is uniformly sable. Thus, we can
conclude the desired result.
Remark 4.1. We can also conclude that the sufficient hypotheses (A1) and (C1) – (C3) guarantee
the stability of the zero solution of Volterra integro-differential equation (4.1).
Example 4.1. In particular case, we consider the nonlinear Volterra integro-differential equation
of the form \Biggl(
x\prime 1
x\prime 2
\Biggr)
=
\left( 1 - 3
2
\mathrm{e}\mathrm{x}\mathrm{p}(t)
1
2
\mathrm{e}\mathrm{x}\mathrm{p}(3t)
- \mathrm{e}\mathrm{x}\mathrm{p}(3t) - 2 \mathrm{e}\mathrm{x}\mathrm{p}(t)
\right) \Biggl( x1
x2
\Biggr)
+
+
t\int
0
\left( 1
4
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- 2t+ s - x21(s)
\bigr)
0
0
1
4
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- 2t+ s - x22(s)
\bigr)
\right) \Biggl( x1(s)
x2(s)
\Biggr)
ds,
where t \geq 0 and
\biggl(
x1
x2
\biggr)
\in \Re 2.
If we compare the former Volterra integro-differential equation with Volterra integro-differential
equation (4.1), then it follows that
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
ON ASYMPTOTICALLY STABILITY, UNIFORMLY STABILITY AND BOUNDEDNESS . . . 1715
A(t) =
\left( - 1 +
3
2
\mathrm{e}\mathrm{x}\mathrm{p}(t) - 1
2
\mathrm{e}\mathrm{x}\mathrm{p}(3t)
\mathrm{e}\mathrm{x}\mathrm{p}(3t) 2 \mathrm{e}\mathrm{x}\mathrm{p}(t)
\right) ,
C
\bigl(
t, s, x(s)
\bigr)
=
\left( 1
4
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- 2t+ s - x21(s)
\bigr)
0
0
1
4
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- 2t+ s - x22(s)
\bigr)
\right) \Biggl( x1(s)
x2(s)
\Biggr)
,
\bigm\| \bigm\| C(t, s, x(s))
\bigm\| \bigm\| \leq
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\left( 1
4
\mathrm{e}\mathrm{x}\mathrm{p}( - t+ s) 0
0
1
4
\mathrm{e}\mathrm{x}\mathrm{p}( - t+ s)
\right)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\Biggl(
x1(s)
x2(s)
\Biggr) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| =
\bigm\| \bigm\| K(t, s)
\bigm\| \bigm\| \| x\| ,
\bigm\| \bigm\| K(t, s)
\bigm\| \bigm\| =
1
4
\mathrm{e}\mathrm{x}\mathrm{p}( - 2t+ s), 0 \leq s \leq t,
and \bigm\| \bigm\| K(u, t)
\bigm\| \bigm\| =
1
4
\mathrm{e}\mathrm{x}\mathrm{p}( - 2u+ t), 0 \leq t \leq u.
Let
P =
\Biggl(
2 0
0 1
\Biggr)
.
Then, it is clear that the matrix P is positive definite and symmetric, and \| P\| =
\surd
5 = K2.
Moreover, we can see that
1
2
\bigl\{
AT (t)P + PA(t)
\bigr\}
=
\Biggl(
- 2 + 3 \mathrm{e}\mathrm{x}\mathrm{p}(t) 0
0 2 \mathrm{e}\mathrm{x}\mathrm{p}(t)
\Biggr)
,
so that \lambda 1(t, P,A) = - 2 + 3 \mathrm{e}\mathrm{x}\mathrm{p}(t) and \lambda 2(t, P,A) = 2 \mathrm{e}\mathrm{x}\mathrm{p}(t).
In addition, we can obtain
\lambda M (t, P, - A) = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
2 - 3 \mathrm{e}\mathrm{x}\mathrm{p}(t), - 2 \mathrm{e}\mathrm{x}\mathrm{p}(t)
\bigr\}
=
\left\{ 2 - 3 \mathrm{e}\mathrm{x}\mathrm{p}(t), 0 \leq t \leq \mathrm{l}\mathrm{n} 2,
- 2 \mathrm{e}\mathrm{x}\mathrm{p}(t), t > \mathrm{l}\mathrm{n} 2,
\lambda m(t, P, - A) = \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
2 - 3 \mathrm{e}\mathrm{x}\mathrm{p}(t), - 2 \mathrm{e}\mathrm{x}\mathrm{p}(t)
\bigr\}
=
\left\{ - 2 \mathrm{e}\mathrm{x}\mathrm{p}(t), 0 \leq t \leq \mathrm{l}\mathrm{n} 2,
2 - 3 \mathrm{e}\mathrm{x}\mathrm{p}(t), t > \mathrm{l}\mathrm{n} 2,
\lambda m(t, P, - A) \leq \lambda M (t, P, - A),
t\int
0
\bigm\| \bigm\| K(t, s)
\bigm\| \bigm\| ds = 1
4
t\int
0
\mathrm{e}\mathrm{x}\mathrm{p}( - 2t+ s)ds =
1
4
\bigl(
\mathrm{e}\mathrm{x}\mathrm{p}( - t) - \mathrm{e}\mathrm{x}\mathrm{p}( - 2t)
\bigr)
< \infty ,
\infty \int
t
\bigm\| \bigm\| K(u, t)
\bigm\| \bigm\| du =
1
4
\infty \int
t
\mathrm{e}\mathrm{x}\mathrm{p}( - 2u+ t)du \leq 1
8
< \infty , 0 \leq t \leq u,
and
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1716 C. TUNÇ, S. A. MOHAMMED
t\int
0
\infty \int
t
\| K(u, s)\| duds = 1
4
t\int
0
\infty \int
t
\mathrm{e}\mathrm{x}\mathrm{p}( - 2u+ s)duds =
=
1
8
t\int
0
\mathrm{e}\mathrm{x}\mathrm{p}( - 2t+ s)ds =
1
8
\bigl[
\mathrm{e}\mathrm{x}\mathrm{p}( - t) - \mathrm{e}\mathrm{x}\mathrm{p}( - 2t)
\bigr]
.
Let us define a function F by
F (t) =
\bigl[
\mathrm{e}\mathrm{x}\mathrm{p}( - t) - \mathrm{e}\mathrm{x}\mathrm{p}( - 2t)
\bigr]
, t > 0.
It can be easily seen that the function F takes its maximum value at t = \mathrm{l}\mathrm{n} 2, which is
1
4
. In this
case, we can conclude that
F (t) =
\bigl[
\mathrm{e}\mathrm{x}\mathrm{p}( - t) - \mathrm{e}\mathrm{x}\mathrm{p}( - 2t)
\bigr]
\leq 1
4
, t > 0,
so that
t\int
0
\infty \int
t
\| K(u, s)\| duds = 1
4
t\int
0
\infty \int
t
\mathrm{e}\mathrm{x}\mathrm{p}( - 2u+ s)duds \leq 1
32
<
1
16
, LK - 1
2 =
1
16
.
From the above discussion, we can arrive at
2\lambda m(t, P,A) - K2
t\int
0
\| K(t, s)\| ds - K2
\infty \int
t
\bigm\| \bigm\| K(u, t)
\bigm\| \bigm\| du =
= 2 \mathrm{e}\mathrm{x}\mathrm{p}(t) -
\surd
5
4
\bigl(
\mathrm{e}\mathrm{x}\mathrm{p}( - t) - \mathrm{e}\mathrm{x}\mathrm{p}( - 2t)
\bigr)
-
\surd
5
8
\mathrm{e}\mathrm{x}\mathrm{p}( - t) \geq 16 - 3
\surd
5
8
= K6, t \geq 0.
Thus, all assumptions of (A1), (C1) – (C3) hold. Hence, we can conclude that the zero solution of the
given Volterra integro-differential equation is asymptotic stable, and it is also stable.
The asymptotic stability of the zero solution for the considered Volterra integro-differential equa-
tion is shown by Figs. 1 – 3.
Example 4.2. Consider the nonlinear Volterra integro-differential equation in Example 4.1 for
the case h(t, x) \not = 0 as given below:
h(t, x) =
\Biggl(
(1 + t2 + x21)
- 1 \mathrm{s}\mathrm{i}\mathrm{n} t
(1 + t2 + x22)
- 1 \mathrm{c}\mathrm{o}\mathrm{s} t
\Biggr)
, where t \geq 0 and x =
\Biggl(
x1
x2
\Biggr)
\in \Re 2.
Then
\| h(t, x)\| =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\Biggl(
(1 + t2 + x21)
- 1 \mathrm{s}\mathrm{i}\mathrm{n} t
(1 + t2 + x22)
- 1 \mathrm{c}\mathrm{o}\mathrm{s} t
\Biggr) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq 1
1 + t2
= \Omega (t),
\infty \int
0
1
1 + t2
dt =
\pi
2
.
Thus, it is clear that \Omega \in L1(0,\infty ).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
ON ASYMPTOTICALLY STABILITY, UNIFORMLY STABILITY AND BOUNDEDNESS . . . 1717
Fig. 1. Trajectory of x1(t) for Example 4.1.
Fig. 2. Trajectory of x2(t) for Example 4.1.
Fig. 3. Numerical plots of solutions x1(t) and x2(t) for Example 4.1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1718 C. TUNÇ, S. A. MOHAMMED
Fig. 4. Trajectory of x1(t) for Example 4.2.
Fig. 5. Trajectory of x2(t) for Example 4.2.
Fig. 6. Numerical plots of solutions x1(t) and x2(t) for Example 4.2.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
ON ASYMPTOTICALLY STABILITY, UNIFORMLY STABILITY AND BOUNDEDNESS . . . 1719
In view of the proof of the boundedness theorem, that is, the proof of Theorem 3.1, we can
reconsider the inequality
xTPx \leq V0(t, x(t)) \leq K5 \mathrm{e}\mathrm{x}\mathrm{p}
\left( \infty \int
0
\Omega (s)ds
\right) , K5 > 0.
For the choice of P =
\biggl(
2 0
0 1
\biggr)
, it follows that
xTPx = [x1, x2]
\Biggl(
2 0
0 1
\Biggr) \Biggl[
x1
x2
\Biggr]
= 2x21 + x22.
By noting the above relations, we have
2x21 + x22 \leq V0(t, x) \leq K5 \mathrm{e}\mathrm{x}\mathrm{p}
\left( \infty \int
0
1
1 + s2
ds
\right) = K5
\Bigl[
\mathrm{e}\mathrm{x}\mathrm{p}
\Bigl( \pi
2
\Bigr)
- 1
\Bigr]
.
Hence, in the particular case, we can conclude that the boundedness of the solutions of the considered
Volterra integro-differential equation.
Further, the boundedness of the solutions of the considered Volterra integro-differential equation
is shown by Figs. 4 – 6.
Hence, it is true to say that the solutions of the Volterra integro-differential equation considered
are bounded.
5. Conclusion. We pay our attention to a class of first order nonlinear Volterra integro-
differential equation. We establish new sufficient conditions for the asymptotic stability, uniform
stability and boundedness of solutions to the considered Volterra integro-differential equations by
defining two new Lyapunov functionals. That is, by Theorems 2.1 and 3.1, we improve and extend
the stability and uniformly stability results from linear Volterra integro-differential equations to non-
linear Volterra integro-differential equations (see [19], Theorems 1 and 2). In addition, boundedness
of solutions is not discussed in Wang [19]. However, by Theorem 4.1 we communicate a new result
on the boundedness of solutions of Volterra integro-differential equation (1.2). By means of the ob-
tained results, we improved and extended the previous results that can be found in the literature from
linear cases to their nonlinear cases and give an additional result on the boundedness of solutions
with specific applications.
References
1. L. C. Becker, Uniformly continuous L1 solutions of Volterra equations and global asymptotic stability, Cubo, 11,
№ 3, 1 – 24 (2009).
2. T. A. Burton, Volterra integral and differential equations, Second ed., Math. Sci. and Eng., Vol. 202, Elsevier B. V.,
Amsterdam (2005).
3. T. Furumochi, S. Matsuoka, Stability and boundedness in Volterra integro-differential equations, Mem. Fac. Sci. Eng.
Shimane Univ. Ser. B, Math. Sci., 32, 25 – 40 (1999).
4. J. R. Graef, C. Tunç, S. Şevgin, Behavior of solutions of non-linear functional Volterra integro-differential equations
with multiple delays, Dynam. Systems and Appl., 25, № 1-2, 39 – 46 (2016).
5. W. E. Mahfoud, Boundedness properties in Volterra integro-differential systems, Proc. Amer. Math. Soc., 100, № 1,
37 – 45 (1987).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1720 C. TUNÇ, S. A. MOHAMMED
6. Y. Raffoul, Boundedness in nonlinear functional differential equations with applications to Volterra integro-differential
equations, J. Integral Equat. and Appl., 16, № 4, 375 – 388 (2004).
7. M. Rahman, Integral equations and their applications, WIT Press, Southampton (2007).
8. M. Rama Mohana Rao, P. Srinivas, Asymptotic behavior of solutions of Volterra integro-differential equations, Proc.
Amer. Math. Soc., 94, № 1, 55 – 60 (1985).
9. C. Tunç, A note on the qualitative behaviors of nonlinear Volterra integro-differential equation, J. Egyptian Math.
Soc., 24, № 2, 187 – 192 (2016).
10. C. Tunç, New stability and boundedness results to Volterra integro-differential equations with delay, J. Egyptian
Math. Soc., 24, № 2, 210 – 213 (2016).
11. C. Tunç, Properties of solutions to Volterra integro-differential equations with delay, Appl. Math. Inf. Sci., 10, № 5,
1775 – 1780 (2016).
12. C. Tunç, Qualitative properties in nonlinear Volterra integro-differential equations with delay, J. Taibah Univ. Sci.,
11, № 2, 309 – 314 (2017).
13. C. Tunç, On the qualitative behaviors of a functional differential equation of second order, Appl. Appl. Math., 12,
№ 2, 813 – 842 (2017).
14. C. Tunç, S. A. Mohammed, A remark on the stability and boundedness criteria in retarded Volterra integro-differential
equations, J. Egyptian Math. Soc., 25, № 4, 363 – 368 (2017).
15. C. Tunç, O. Tunç, On the exponential study of solutions of Volterra integro-differential equations with time lag,
Electron. J. Math. Anal. and Appl., 6, № 1, 253 – 265 (2018).
16. C. Tunç, O. Tunç, New results on the stability, integrability and boundedness in Volterra integro-differential equations,
Bull. Comput. Appl. Math., 6, № 1, 41 – 58 (2018).
17. C. Tunç, O. Tunç, On behaviors of functional Volterra integro-differential equations with multiple time-lags, J. Taibah
Univ. Sci., 12, № 2, 173 – 179 (2018).
18. Ke Wang, Uniform asymptotic stability in functional-differential equations with infinite delay, Ann. Different. Equat.,
9, № 3, 325 – 335 (1993).
19. Q. Wang, The stability of a class of functional differential equations with infinite delays, Ann. Different. Equat., 16,
№ 1, 89 – 97 (2000).
20. A. M. Wazwaz, Linear and nonlinear integral equations, Methods and applications. Higher Education Press, Beijing;
Springer, Heidelberg (2011).
21. C. Tunç, A remark on the qualitative conditions of nonlinear IDEs, Int. J. Math. Comput. Sci., 15, № 3, 905 – 922
(2020).
22. C. Tunç, Asymptotic stability and boundedness criteria for nonlinear retarded Volterra integro-differential equations,
J. King Saud Univ.-Sci., 30, № 4, 3531 – 3536 (2018).
23. O. Tunç, On the qualitative analysis of integro-differential equations with constant time lag, Appl. Math. Inf. Sci.,
14, № 1, 57 – 63 (2020).
24. C. Tunç, O. Tunç, New qualitative criteria for solutions of Volterra integro-differential equations, Arab J. Basic and
Appl. Sci., 25, № 3, 158 – 165 (2018).
25. C. Tunç, O. Tunç, A note on the qualitative analysis of Volterra integro-differential equations, J. Taibah Univ. Sci.,
13, № 1, 490 – 496 (2019).
Received 25.01.18,
after revision — 07.06.18
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
|
| id | umjimathkievua-article-6037 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:37Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f7/d0e200cec1d3037ab6f9d996b4f04ef7.pdf |
| spelling | umjimathkievua-article-60372025-03-31T08:49:28Z On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations Tunç, C. Mohammed , S. A. Tunç, C. Mohammed , S. A. Non-linear first order Non-linear first order UDC 517.9 In this paper, two new Lyapunov functionals are defined. We apply these functionals to get sufficient conditions guaranteeing the asymptotic stability, uniform stability, and boundedness of solutions of certain nonlinear Volterra integro-differential equations of the first order. The results obtained are improvements and extensions of known results that can be found in literature. We also suggest examples to show the applicability of our results and for the sake of illustrations. Using MATLAB-Simulink, in particular cases we clearly show the behavior of orbits of Volterra integro-differential equations under consideration. UDC 517.9 Про асимптотичну стiйкiсть, рiвномiрну стiйкiсть та обмеженiсть розв’язкiв нелiнiйних iнтегро-диференцийних рiвнянь ВольтерраНаведено означення двох нових функціоналів Ляпунова. Ці функціонали використано для отримання достатніх умов, що гарантують асимптотичну стійкість, рівномірну стійкість та обмеженість розв'язків нелінійних інтегро-диференціальних рівнянь Вольтерра першого порядку. Отримані результати удосконалюють та розширюють відомі результати, що вже були опубліковані. Наведено приклади застосування отриманих результатів. За допомогою MATLAB-Simulink в окремих випадках показано поведінку орбіт розглянутих інтегро-диференціальних рівнянь Вольтерра. Institute of Mathematics, NAS of Ukraine 2020-12-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6037 10.37863/umzh.v72i12.6037 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 12 (2020); 1708-1720 Український математичний журнал; Том 72 № 12 (2020); 1708-1720 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6037/8878 |
| spellingShingle | Tunç, C. Mohammed , S. A. Tunç, C. Mohammed , S. A. On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations |
| title | On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations |
| title_alt | On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations |
| title_full | On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations |
| title_fullStr | On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations |
| title_full_unstemmed | On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations |
| title_short | On asymptotically stability, uniformly stability and boundedness of solutions of nonlinear Volterra integro-differential equations |
| title_sort | on asymptotically stability, uniformly stability and boundedness of solutions of nonlinear volterra integro-differential equations |
| topic_facet | Non-linear first order Non-linear first order |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6037 |
| work_keys_str_mv | AT tuncc onasymptoticallystabilityuniformlystabilityandboundednessofsolutionsofnonlinearvolterraintegrodifferentialequations AT mohammedsa onasymptoticallystabilityuniformlystabilityandboundednessofsolutionsofnonlinearvolterraintegrodifferentialequations AT tuncc onasymptoticallystabilityuniformlystabilityandboundednessofsolutionsofnonlinearvolterraintegrodifferentialequations AT mohammedsa onasymptoticallystabilityuniformlystabilityandboundednessofsolutionsofnonlinearvolterraintegrodifferentialequations |