Open Access

Global existence and boundedness of solutions of a certain nonlinear integro-differential equation of second order with multiple deviating arguments

Journal of Inequalities and Applications20162016:46

DOI: 10.1186/s13660-016-0987-2

Received: 7 September 2015

Accepted: 21 January 2016

Published: 5 February 2016

Abstract

In this paper, we consider the global existence and boundedness of solutions for a certain nonlinear integro-differential equation of second order with multiple constant delays. We obtain some new sufficient conditions which guarantee the global existence and boundedness of solutions to the considered equation. The obtained result complements some recent ones in the literature. An example is given of the applicability of the obtained result. The main tool employed is an appropriate Lyapunov-Krasovskii type functional.

Keywords

integro-differential equation second order global existence boundedness Lyapunov-Krasovskii type functional

1 Introduction

In recent years, there have been several papers written on the global existence and boundedness of solutions for certain nonlinear differential and integro-diffrential equations of second order with and without delays; see, for example, Ahmad and Rama Mohana Rao [1], Baxley [2], Burton [3], Constantin [4], Driver [5], Fujimoto and Yamaoka [6], Grace and Lalli [7], Graef and Tunç [8], Kalmanovskii and Myshkis [9], Krasovskii [10], Miller [11, 12], Mustafa and Rogovchenko [13, 14], Napoles Valdes [15], Ogundare et al. [16], Reissig et al. [17], Tidke [18], Tiryaki and Zafer [19], Tunç [2025], Tunç and Tunç [26], Yoshizawa [27], Wu et al. [28], Yin [29] and the references therein.

It should be noted that Napoles Valdes [15] dealt with the ordinary integro-differential equation of second order:
$$x^{{\prime\prime}}+a ( t ) f \bigl( t,x,x^{{\prime}} \bigr) x^{{\prime}}+g \bigl( t,x^{{\prime}} \bigr) +h ( x ) = \int_{0}^{t}C ( t, \tau ) x^{{\prime}} ( \tau )\,d\tau. $$
The author investigated extendibility, boundedness, stability, and square integrability of solutions to the considered equation. The method of proof consists of the use of a suitable Lyapunov function.
In a recent paper, Graef and Tunç [8] discussed the continuability, boundedness, and square integrability of solutions to the second order functional integro-differential equation with multiple delays:
$$\begin{aligned} &x^{{\prime\prime}}+a ( t ) f \bigl( t,x,x^{{\prime}} \bigr) x^{{\prime}}+g \bigl( t,x,x^{{\prime}} \bigr) +\sum_{i=1}^{n}h_{i} \bigl( x ( t-\tau_{i} ) \bigr) \\ &\quad= \int_{0}^{t}C ( t,s ) x^{{\prime}} ( s )\,ds. \end{aligned}$$
The proof of the results in [8] involves the definition of a Lyapunov-Krasovskii type functional.
In this paper, instead of the mentioned integro-differential equations discussed in [8, 15], we consider the following nonlinear and non-autonomous integro-differential equation of second order with multiple constant delays:
$$\begin{aligned} & \bigl( p(x)x^{{\prime}} \bigr) ^{{\prime}}+a ( t ) f \bigl( t,x,x^{{\prime}} \bigr) x^{{\prime}}+b ( t ) g \bigl( t,x^{{\prime}} \bigr) +\sum_{i=1}^{n}c_{i} ( t ) h_{i} \bigl( x ( t-\tau_{i} ) \bigr) \\ &\quad= \int_{0}^{t}C ( t,s ) x^{{\prime}} ( s )\,ds, \end{aligned}$$
(1)
which can be written in the system form as
$$\begin{aligned} &x^{\prime}=\frac{y}{p(x)}, \\ &y^{\prime}= \int_{0}^{t}C ( t,s ) \frac{y ( s ) }{p(x(s))}\,ds-a ( t ) f \biggl( t,x,\frac{y}{p(x)} \biggr) \frac {y}{p(x)}-b ( t ) g \biggl( t,\frac{y}{p(x)} \biggr) \\ &\hphantom{y^{\prime}=}{}-\sum_{i=1}^{n}c_{i} ( t ) h_{i} \bigl( x ( t ) \bigr) +\sum_{i=1}^{n}c_{i} ( t ) \int_{t-\tau_{i}}^{t}h_{i}^{{\prime }} \bigl( x ( s ) \bigr) \frac{y ( s ) }{p(x(s))}\,ds, \end{aligned}$$
(2)
where \(\tau_{i} \) (\(i=1,2,\ldots,n\)) are positive constants, \(a,b,c_{i}:\Re ^{+}\rightarrow\Re^{+}\), \(\Re^{+}=(0,\infty)\), \(f:\Re^{+}\times \Re^{2}\rightarrow\Re^{+}\), and \(g:\Re^{+}\times\Re\rightarrow \Re^{+} \) are continuous functions, \(h_{i}\in C^{1}(\Re,\Re)\), \(p\in C^{1} ( \Re , ( 0,\infty ) ) \), and \(C ( t,s ) \) is a continuous function for \(0\leq t\leq s\leq\infty\).

It is worth mentioning that the global existence and boundedness of solutions of equation (1) have not yet been discussed in the literature. The aim of this paper is to give some sufficient conditions to guarantee the global existence and boundedness of solutions of equation (1). This case shows the novelty and originality of the present paper. The result to be obtained complements and improves the results in the literature (Graef and Tunç [8], Napoles Valdes [15]). This paper may also be useful for researchers working on the qualitative behavior of solutions of functional integro-differential equations.

We assume that there are positive constants \(\delta_{i}\), \(\beta _{i}\), \(\gamma_{i}\), λ, \(p_{1}\), m, M, \(g_{0}\), \(g_{1}\), \(c_{i}\), \(C_{i}\), R, and \(\tau^{\ast}\) such that the following conditions hold:
  1. (A1)

    \(1\leq p(x)\leq p_{1}\), \(\int_{-\infty}^{+\infty} \vert p^{\prime}(u)\vert \,du<\infty\),

     
  2. (A2)

    \(0< m\leq b(t)\leq a(t)\leq M\), \(0< c_{i}\leq c_{i}(t)\leq C_{i}\), \(c_{i}^{\prime}(t)\leq0\),

     
  3. (A3)

    \(g(t,0)=0\) and \(0< g_{0}\leq\frac{g(t,y)}{y}\leq g_{1}\) (\(y\neq 0\)),

     
  4. (A4)

    \(h_{i}(0)=0\), \(0<\delta_{i}\leq\frac{h_{i}(x)}{x}\leq \beta _{i}\) (\(x\neq0\)), \(\vert h_{i}^{\prime}(x)\vert \leq\gamma_{i}\),

     
  5. (A5)

    \(\max ( \int_{0}^{t}\vert C ( t,s ) \vert \,ds+\int_{t}^{\infty} \vert C ( u,t ) \vert \,du ) \leq R\),

     
  6. (A6)

    \(R+2\lambda\tau^{\ast}\leq\frac{m}{p_{1}}(f(t,x,y)+g_{0})\) for all t, x and y.

     

2 Main result

What follows is our main theorem.

Theorem

Suppose that conditions (A1)-(A6) hold. Then all solutions of system (2) are continuable and bounded.

Proof

We define a Lyapunov-Krasovskii functional by
$$ W(t)=W \bigl(t,x(t),y(t) \bigr)=e^{-\frac{\gamma(t)}{\mu}}V_{0} \bigl(t,x(t),y(t) \bigr), $$
(3)
where μ a positive constant,
$$\begin{aligned} \gamma(t) =& \int_{0}^{t} \bigl\vert \theta(s) \bigr\vert \,ds= \int_{0}^{t} \bigl\vert x^{\prime}(s)p^{\prime} \bigl(x(s) \bigr) \bigr\vert \,ds \\ =& \int_{\alpha_{1}(t)}^{\alpha_{2}(t)} \bigl\vert p^{\prime } \bigl(x(u) \bigr) \bigr\vert \,du\leq \int_{-\infty}^{+\infty} \bigl\vert p^{\prime } \bigl(x(u) \bigr) \bigr\vert \,du< \infty, \end{aligned}$$
for \(\theta(t)=x^{\prime}(t)p^{\prime}(x(t))\), \(\alpha_{1}(t)=\min \{ x(0),x(t) \} \), \(\alpha_{2}(t)=\max \{ x(0),x(t) \}\), and
$$\begin{aligned} V_{0}(t) =&V_{0} \bigl(t,x(t),y(t) \bigr)= \frac{1}{2}y^{2}+p(x)\sum_{i=1}^{n}c_{i}(t)\int_{0}^{x}h_{i}(s)\,ds \\ &{}+\sum_{i=1}^{n}\lambda_{i} \int_{-\tau _{i}}^{0} \int_{t+s}^{t}y^{2}(u)\,du\,ds+ \int_{0}^{t} \int_{t}^{\infty} \bigl\vert C ( u,s ) \bigr\vert y^{2}(s)\,du\,ds. \end{aligned}$$
(4)
From the assumptions (A1), (A2), and (A4) it follows that
$$\begin{aligned} V_{0}(t) \geq&\frac{1}{2}y^{2}+p(x)\sum _{i=1}^{n}c_{i}(t) \int _{0}^{x}h_{i}(s)\,ds \\ \geq&\frac{1}{2}y^{2}+\frac{1}{2}\sum _{i=1}^{n}c_{i}\delta_{i}x^{2} \\ \geq&k \bigl( x^{2}+y^{2} \bigr), \end{aligned}$$
(5)
where \(k=\frac{1}{2}\min \{1,\sum_{i=1}^{n}c_{i}\delta_{i} \} \).
Let \(( x(t),y(t) ) \) be a solution of (2). Calculating the time derivative of the functional \(V_{0}(t)\), we obtain
$$\begin{aligned} V_{0}^{\prime}(t) =&y \int_{0}^{t}C ( t,s ) \frac{y ( s ) }{p(x(s))}\,ds-a ( t ) f \biggl( t,x,\frac{y}{p(x)} \biggr) \frac{y^{2}}{p(x)}-b ( t ) g \biggl( t, \frac{y}{p(x)} \biggr) y \\ &{}+y\sum_{i=1}^{n}c_{i} ( t ) \int_{t-\tau _{i}}^{t}h_{i}^{{\prime }} \bigl( x ( s ) \bigr) \frac{y ( s ) }{p(x(s))}\,ds+\theta (t)\sum_{i=1}^{n}c_{i}(t) \int_{0}^{x}h_{i}(s)\,ds \\ &{}+p(x)\sum_{i=1}^{n}c_{i}^{{\prime }}(t) \int_{0}^{x}h_{i}(s)\,ds+\sum _{i=1}^{n} ( \lambda_{i}\tau _{i} ) y^{2}-\sum_{i=1}^{n} \lambda_{i} \int_{t-\tau _{i}}^{t}y^{2} ( s )\,ds \\ &{}+y^{2} \int_{t}^{\infty} \bigl\vert C ( u,t ) \bigr\vert \,du- \int_{0}^{t} \bigl\vert C ( t,s ) \bigr\vert y^{2}(s)\,ds. \end{aligned}$$
By the assumptions (A1)-(A6) and the inequality \(2\vert ab\vert \leq ( a^{2}+b^{2} ) \), the following estimates can be verified:
$$\begin{aligned}& \begin{aligned}[b] y \int_{0}^{t}C ( t,s ) \frac{y ( s ) }{p(x(s))}\,ds &\leq \int_{0}^{t} \bigl\vert C ( t,s ) \bigr\vert \bigl\vert y(t) \bigr\vert \bigl\vert y(s) \bigr\vert \,ds \\ &\leq y^{2} \int_{0}^{t} \bigl\vert C ( t,s ) \bigr\vert \,ds+ \int_{0}^{t} \bigl\vert C ( t,s ) \bigr\vert y^{2} ( s )\,ds, \end{aligned} \\& -a ( t ) f \biggl( t,x,\frac{y}{p(x)} \biggr) \frac{y^{2}}{p(x)}-b ( t ) g \biggl( t,\frac{y}{p(x)} \biggr) y\leq-\frac{m}{p_{1}} \biggl(f \biggl(t,x,\frac{y}{p(x)} \biggr)+g_{0} \biggr)y^{2}, \\& \begin{aligned}[b] y\sum_{i=1}^{n}c_{i} ( t ) \int_{t-\tau _{i}}^{t}h_{i}^{{\prime }} \bigl( x ( s ) \bigr) \frac{y ( s ) }{p(x(s))}\,ds &\leq \sum_{i=1}^{n}C_{i} \gamma_{i} \int_{t-\tau_{i}}^{t} \bigl\vert y ( t ) y ( s ) \bigr\vert \,ds \\ &\leq\sum_{i=1}^{n} ( C_{i} \gamma_{i}\tau_{i} ) y^{2}+\sum _{i=1}^{n}C_{i}\gamma_{i} \int_{t-\tau_{i}}^{t}y^{2} ( s )\,ds, \end{aligned} \\& \theta(t)\sum_{i=1}^{n}c_{i}(t) \int_{0}^{x}h_{i}(s)\,ds\leq \frac{\vert \theta(t)\vert }{2}\sum_{i=1}^{n} ( C_{i}\beta_{i} ) x^{2}, \\& p(x)\sum_{i=1}^{n}c_{i}^{{\prime}}(t) \int_{0}^{x}h_{i}(s)\,ds\leq0. \end{aligned}$$
From these estimates we obtain, quite readily,
$$\begin{aligned} V_{0}^{\prime}(t) \leq& \biggl( R-\frac{m}{p_{1}} \biggl(f \biggl(t,x,\frac{y}{p(x)} \biggr)+g_{0} \biggr) \biggr) y^{2} \\ &{}+\sum_{i=1}^{n} ( C_{i} \gamma_{i}\tau_{i} ) y^{2}+\sum _{i=1}^{n}C_{i}\gamma_{i} \int_{t-\tau_{i}}^{t}y^{2} ( s )\,ds \\ &{}+\sum_{i=1}^{n} ( \lambda_{i} \tau_{i} ) y^{2}-\sum_{i=1}^{n} \lambda_{i} \int_{t-\tau_{i}}^{t}y^{2} ( s )\,ds \\ &{}+\frac{\vert \theta(t)\vert }{2}\sum_{i=1}^{n} ( C_{i}\beta_{i} ) x^{2}. \end{aligned}$$
Let
$$\tau^{\ast}=\max \{ \tau_{1},\tau_{2},\ldots, \tau_{n} \} $$
and
$$\lambda=\sum_{i=1}^{n}\lambda_{i}= \sum_{i=1}^{n}C_{i} \gamma_{i}. $$
Hence, in view of the discussion and (A6), we can conclude that
$$\begin{aligned} V_{0}^{\prime}(t) \leq& \biggl( R+2\lambda \tau^{\ast}- \frac {m}{p_{1}} \biggl(f \biggl(t,x, \frac{y}{p(x)} \biggr)+g_{0} \biggr) \biggr) y^{2}+ \frac{\vert \theta (t)\vert }{2}\sum_{i=1}^{n} ( C_{i} \beta_{i} ) x^{2} \\ \leq&\frac{\vert \theta(t)\vert }{2}\sum_{i=1}^{n} ( C_{i}\beta_{i} ) x^{2}. \end{aligned}$$
(6)
It is now clear that the time derivative of the functional \(W ( t ) \) defined by (3) along any solution of system (2) leads that
$$W^{\prime}(t)=e^{-\frac{\gamma(t)}{\mu}} \biggl( -\frac{\vert \theta (t)\vert }{\mu}V_{0} \bigl(t,x ( t ),y ( t ) \bigr)+\frac {d}{dt}V_{0} \bigl(t,x ( t ) ,y ( t ) \bigr) \biggr) . $$
Therefore, using (5), (6), and taking \(\mu=\frac{2k}{\sum_{i=1}^{n}C_{i}\beta_{i}}\), we obtain
$$W^{\prime}(t)\leq e^{-\frac{\gamma(t)}{\mu}} \Biggl( -\frac{\vert \theta(t)\vert }{2}\sum _{i=1}^{n} ( C_{i}\beta_{i} ) x^{2}+\frac{\vert \theta(t)\vert }{2}\sum_{i=1}^{n} ( C_{i}\beta _{i} ) x^{2} \Biggr) =0. $$
This implies that \(W^{\prime}(t)\leq0\). Since all the functions appearing in equation (1) are continuous, it is obvious that there exists at least a solution of equation (1) defined on \([ t_{0},t_{0}+\delta ) \) for some \(\delta>0\). We need to show that the solution can be extended to the entire interval \([ t_{0},\infty ) \). We assume on the contrary that there is a first time \(T<\infty\) such that the solution exists on \([ t_{0},T ) \) and
$$\lim_{t\rightarrow T^{-}} \bigl( \bigl\vert x(t) \bigr\vert + \bigl\vert y(t) \bigr\vert \bigr) =\infty. $$
Let \(( x(t),y(t) ) \) be such a solution of system (2) with initial condition \(( x_{0},y_{0} ) \). Since the Lyapunov-Krasovskii type functional \(W(t)\) is positive definite and decreasing, \(W^{\prime }(t)\leq0\), along the trajectories of system (2), we can say that \(W(t)\) is bounded \([ t_{0},T ] \). We have
$$W\bigl(T,x(T),y(T)\bigr)\leq W(t_{0},x_{0},y_{0})=W_{0}. $$
Hence, it follows from (3) and (5) that
$$x^{2}(T)+y^{2}(T)\leq\frac{W_{0}}{D}, $$
where \(D=k\exp ( -\gamma ( T ) \mu^{-1} ) \). This inequality implies that \(\vert x(t)\vert \) and \(\vert y(t)\vert \) are bounded as \(t\rightarrow T^{-}\). Thus, we can conclude that \(T<\infty\) is not possible, we must have \(T=\infty\). This completes the proof of the theorem. □

Example

We consider the following nonlinear integro-differential equation of second order with two constants delays, \(\tau_{1}>0\) and \(\tau_{2}>0\):
$$\begin{aligned} & \biggl( \biggl( 2+\frac{\sin x}{1+x^{2}} \biggr) x^{\prime} \biggr) ^{{\prime }}+ \biggl( 1+\frac{2}{1+t^{2}} \biggr) \bigl( e^{-t}+\sin x+\cos x^{{\prime }}+6 \bigr) x^{\prime} \\ &\quad{}+ \biggl( 1+\frac{1}{1+t^{2}} \biggr) \biggl( 3x^{\prime}+ \frac{x^{\prime }}{1+x^{\prime^{2}}} \biggr) +2 \bigl( e+e^{-t} \bigr) x ( t-\tau _{1} ) \\ &\quad{}+2 \bigl( e^{2}+e^{-t} \bigr) x ( t- \tau_{2} ) = \int_{0}^{t}\frac{s}{ ( 1+2t ) ^{2}}x^{\prime}(s)\,ds. \end{aligned}$$
(7)
When we compare equation (7) with equation (1), the existence can be seen of the following estimates:
$$\begin{aligned}& p ( x ) =2+\frac{\sin x}{1+x^{2}},\quad 1\leq p ( x ) \leq3, \\& \int_{-\infty}^{\infty} \bigl\vert p^{\prime}(u) \bigr\vert \,du\leq \int_{-\infty}^{\infty} \biggl( \biggl\vert \frac{\cos u}{1+u^{2}} \biggr\vert + \biggl\vert \frac{2u\sin u}{ ( 1+u^{2} ) ^{2}} \biggr\vert \biggr)\,du\leq\pi, \\& a ( t ) =1+\frac{2}{1+t^{2}},\qquad b ( t ) =1+\frac{1}{1+t^{2}}, \\& m=1\leq b(t)\leq a(t)\leq3=M, \\& c_{1} ( t ) =e+e^{-t},\qquad c_{2} ( t ) =e^{2}+e^{-t}, \\& c_{1}=e\leq c_{1}(t)\leq e+1=C_{1}, \\& c_{2}=e^{2}\leq c_{2}(t)\leq e^{2}+1=C_{2}, \\& c_{1}^{{\prime}}(t)\leq0,\qquad c_{2}^{{\prime}} (t) \leq0,\quad t \geq0, \\& f \bigl( t,x,x^{{\prime}} \bigr) =e^{-t}+\sin x+\cos x^{{\prime }}+6,\qquad 4\leq f \bigl( t,x,x^{{\prime}} \bigr) \leq9, \\& g \bigl( t,x^{{\prime}} \bigr) =3x^{\prime}+\frac{x^{\prime }}{1+x^{\prime ^{2}}}, \\& g ( t,0 ) =0,\qquad g_{0}=3\leq\frac{g ( t,x^{{\prime }} ) }{x^{{\prime}}} \leq4=g_{1} \quad( y \neq0 ), \\& h_{1} ( x ) = h_{2} ( x ) =2x, \\& h_{1} ( 0 ) =h_{2} ( 0 ) =0,\qquad\frac{h_{1} ( x ) }{x}= \frac{h_{2} ( x ) }{x}=2 \quad( x\neq0 ) , \\& \bigl\vert h_{1}^{{\prime}} ( x ) \bigr\vert = \bigl\vert h_{2}^{{\prime}} ( x ) \bigr\vert =2, \\& \int_{0}^{t} \bigl\vert C ( t,s ) \bigr\vert \,ds+ \int_{t}^{\infty} \bigl\vert C ( u,t ) \bigr\vert \,du \\& \quad= \int_{0}^{t} \biggl\vert \frac {s}{ ( 1+2t ) ^{2}} \biggr\vert \,ds+ \int_{t}^{\infty} \biggl\vert \frac{t}{ ( 1+2u ) ^{2}} \biggr\vert \,du\leq\frac{3}{8}=R. \end{aligned}$$

Thus, all the assumptions of the theorem hold. So we can conclude that all solutions of (7) are continuable and bounded.

Declarations

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences, Yüzüncü Yıl University
(2)
Department of Primary School-Mathematics, Faculty of Education, Siirt University

References

  1. Ahmad, S, Rama Mohana Rao, M: Theory of Ordinary Differential Equations. With Applications in Biology and Engineering. Affiliated East-West Press Pvt. Ltd., New Delhi (1999) Google Scholar
  2. Baxley, JV: Global existence and uniqueness for second-order ordinary differential equations. J. Differ. Equ. 23(3), 315-334 (1977); Stanford University Press, Stanford, Calif. (1963) View ArticleMathSciNetMATHGoogle Scholar
  3. Burton, TA: Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Corrected version of the 1985 original. Dover, Mineola (2005) MATHGoogle Scholar
  4. Constantin, A: Global existence of solutions for perturbed differential equations. Ann. Mat. Pura Appl. (4) 168, 237-299 (1995) View ArticleMathSciNetGoogle Scholar
  5. Driver, RD: Existence and stability of solutions of a delay-differential system. Arch. Ration. Mech. Anal. 10, 401-426 (1962) View ArticleMathSciNetMATHGoogle Scholar
  6. Fujimoto, K, Yamaoka, N: Global existence and nonexistence of solutions for second-order nonlinear differential equations. J. Math. Anal. Appl. 411(2), 707-718 (2014) View ArticleMathSciNetGoogle Scholar
  7. Grace, SR, Lalli, BS: Asymptotic behavior of certain second order integro-differential equations. J. Math. Anal. Appl. 76(1), 84-90 (1980) View ArticleMathSciNetMATHGoogle Scholar
  8. Graef, J, Tunç, C: Continuability and boundedness of multi-delay functional integro-differential equations of the second order. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 109(1), 169-173 (2015) View ArticleMATHGoogle Scholar
  9. Kalmanovskii, V, Myshkis, A: Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and Its Applications, vol. 463. Kluwer Academic, Dordrecht (1999) View ArticleGoogle Scholar
  10. Krasovskii, NN: Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford (1963) MATHGoogle Scholar
  11. Miller, RK: Asymptotic stability properties of linear Volterra integro-differential equations. J. Differ. Equ. 10, 485-506 (1971) View ArticleMATHGoogle Scholar
  12. Miller, RK, Michel, AN: Ordinary Differential Equations. Academic Press, New York (1982) MATHGoogle Scholar
  13. Mustafa, GO, Rogovchenko, YV: Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations. Nonlinear Anal., Theory Methods Appl. 51(2), 339-368 (2002) View ArticleMathSciNetMATHGoogle Scholar
  14. Mustafa, GO, Rogovchenko, YV: Global existence of solutions for a class of nonlinear differential equations. Appl. Math. Lett. 16(5), 753-758 (2003) View ArticleMathSciNetMATHGoogle Scholar
  15. Napoles Valdes, JE: A note on the boundedness of an integro-differential equation. Quaest. Math. 24(2), 213-216 (2001) View ArticleMathSciNetMATHGoogle Scholar
  16. Ogundare, BS, Ngcibi, S, Murali, V: Boundedness and stability properties of solutions to certain second-order differential equation. Adv. Differ. Equ. Control Process. 5(2), 79-92 (2010) MathSciNetMATHGoogle Scholar
  17. Reissig, R, Sansone, G, Conti, R: Non-linear Differential Equations of Higher Order. Translated from the German. Noordhoff International Publishing, Leyden (1974) MATHGoogle Scholar
  18. Tidke, HL: Global existence of solutions for nonlinear integral equations of second order. J. Appl. Funct. Anal. 5(1), 113-120 (2010) MathSciNetMATHGoogle Scholar
  19. Tiryaki, A, Zafer, A: Global existence and boundedness for a class of second-order nonlinear differential equations. Appl. Math. Lett. 26(9), 939-944 (2013) View ArticleMathSciNetMATHGoogle Scholar
  20. Tunç, C: Boundedness results for solutions of certain nonlinear differential equations of second order. J. Indones. Math. Soc. 16(2), 115-126 (2010) MathSciNetMATHGoogle Scholar
  21. Tunç, C: Some new stability and boundedness results of solutions of Liénard type equations with a deviating argument. Nonlinear Anal. Hybrid Syst. 4(1), 85-91 (2010) View ArticleMathSciNetMATHGoogle Scholar
  22. Tunç, C: Stability and boundedness of solutions of non-autonomous differential equations of second order. J. Comput. Anal. Appl. 13(6), 1067-1074 (2011) MathSciNetMATHGoogle Scholar
  23. Tunç, C: Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations. Appl. Comput. Math. 10(3), 449-462 (2011) MathSciNetMATHGoogle Scholar
  24. Tunç, C: On the stability and boundedness of solutions of a class of nonautonomous differential equations of second order with multiple deviating arguments. Afr. Math. 23(2), 249-259 (2012) View ArticleMATHGoogle Scholar
  25. Tunç, C: A note on the stability and boundedness of non-autonomous differential equations of second order with a variable deviating argument. Afr. Math. 25(2), 417-425 (2014) View ArticleMATHGoogle Scholar
  26. Tunç, C, Tunç, O: A note on certain qualitative properties of a second order linear differential system. Appl. Math. Inf. Sci. 9(2), 953-956 (2015) MathSciNetGoogle Scholar
  27. Yoshizawa, T: Stability Theory by Liapunov’s Second Method. Publications of the Mathematical Society of Japan, vol. 9. The Mathematical Society of Japan, Tokyo (1966) Google Scholar
  28. Wu, C, Hao, S, Xu, C: Global existence and boundedness of solutions to a second-order nonlinear differential system. J. Appl. Math. 2012, Article ID 603783 (2012) MathSciNetGoogle Scholar
  29. Yin, Z: Global existence and boundedness of solutions to a second order nonlinear differential system. Studia Sci. Math. Hung. 41(4), 365-378 (2004) MATHGoogle Scholar

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© Tunç and Ayhan 2016