1 Introduction
Burton [1] investigated the convergence of solutions of nonlinear differential equations of the second order with constant delay, L:
\[\ddot{x} + f(t, x, \dot{x})\dot{x} + b(t)g(x(t-L)) = 0.\]
The author found sufficient conditions which guarantee that the solutions of the former equation satisfy \((x(t), \dot{x}(t)) \to 0\) as \(t \to \infty\).
Later, Pi [2,3] discussed the stability of the zero solution of the differential equation of the second order with variable delay, \(\tau(t)\):
\[\ddot{x} + f(t, x, \dot{x})\dot{x} + b(t)g(x(t - \tau(t))) = 0.\]
Using the fixed point theory and an exponential weighted metric, the author established sufficient conditions which guarantee that the zero solution of the equation considered is stable and asymptotic stable. Pi [4] is concerned with the integro-differential equation of the second order with variable delay, r(t):
\[\ddot{x} + f(t, x, \dot{x})\dot{x} + \int_{t-r(t)}^{t} a(t, s)g(x(s))ds = 0.\]
The author investigated the stability of the zero solution of this equation by means of the fixed point theory and an exponential weighted metric. In addition, for some related works on the qualitative behavior solutions of functional differential and integro-differential equations of the second order, we refer to the reader to the papers and books of Abdollahpour, et al. [5], Ardjouni and Djoudi [6], Burton [7,8], Graef and Tunç [9], Hale [10], Jin and Luo [11], Korkmaz and Tunç [12], Levin and Nohel [13,14], Pi [15], Tunç and Biçer [16], Tunç [17], Tunç and Tunç [18], Zhao, et al. [19], Zhao and Yuan [20] and the references therein.
Motivated by the works mentioned, we consider the integro-differential equation of the second order with variable delay, \(r(t) \ge 0\):
\[\ddot{x} + a_0(t)f(t, x, \dot{x})\dot{x} + b(t)h(x(t - r(t))) + \int_{t - r(t)}^{t} a(t, s)g(x(s))ds = 0, (1)\] where \(b: \Re^+ \to \Re\) is a bounded and continuous function, \(a: [0, \infty) \times [-r(0), \infty) \to \Re\), \(g: \Re \to \Re\), \(f: \Re^+ \times \Re \times \Re \to \Re^+\) and \(a_0, r: \Re^+ \to \Re\), \(\Re^+ = [0, \infty)\), are continuous functions.
We can state Eq. (1) as the following system:
\[\begin{cases} \dot{x} = y, \\ \dot{y} = -a_0(t)f(t, x, y)y - b(t)h(x(t - r(t)) - \int_{t - r(t)}^{t} a(t, s)g(x(s))ds. \end{cases}\] (2)
For each \(t_0 \ge 0\), we define \(m(t_0) = \inf\{s - r(s) : s \ge t_0\}\). Let \(C(t_0) = C([m(t_0), t_0], \Re)\) with the continuous function norm \(\|\cdot\|\), where \(\|\psi\| = \sup\{|\psi(s)| : m(t_0) \le s \le t_0\}\). We use \(\|\varphi\|\) as the supremum on \([m(t_0), \infty)\).
It is well known that for a given continuous function \(\phi\) and a number \(y_0\), there exists a solution of system Eq. (2) on an interval \([t_0, T]\), and if the solution remains bounded, then \(T = \infty\). By (x(t), y(t)) we denote the solution \((x(t,t_0,\varphi),y(t,t_0,\varphi))\).
Let \(A(t) = a_0(t) f(t, x(t), y(t))\). Then, from the system in Eq. (2), it follows that:
\[\begin{cases} \dot{x} = y, \\ \dot{y} = -A(t)y - b(t)h(x(t - r(t)) - \int_{t - r(t)}^{t} a(t, s)g(x(s))ds. \end{cases}\](3)
Throughout this paper it is assumed that t-r(t) is strictly increasing and \(\lim_{t\to\infty}(t-r(t))=\infty\). Let p(t) be the inverse of t-r(t). Suppose that \(0\le b(t)\le M_1\) for some constant \(M_1>0\) and let \(G(t,t)=\int\limits_t^{p(t)}a(u,t)du\) and \(G(t,s)=\int\limits_t^{p(s)}a(u,s)du\). There exists a constant \(M_2>0\) such that \(|G(t,t)|\le M_2\).
The following theorem is our main result.
2 Main result
Theorem 2.1 We assume that the following assumptions hold:
- (i) There exists a constant l > 0, \((l = \max\{l_1, l_2\})\) such that h(x) and g(x) satisfy the Lipschitz condition on \([-l_1, l_1]\) and \([-l_2, l_2]\) respectively. The function h(x) is odd and strictly increasing on \([-l_1, l_1]\) the function g(x) is odd and strictly increasing on \([-l_2, l_2]\), and x h(x) is non-decreasing on \([0, l_1]\), x g(x) is non-decreasing on \([0, l_2]\).
- (ii) There exist positive constants \(\alpha_1, \alpha_2 \in (0,1)\) and a continuous function \(\alpha_1(t):[0,\infty) \to [0,\infty)\) such that \(a_0(t)f(t,x,y) \ge a_1(t)\) for \(t \ge 0\). For all \(t \ge 0\), \(\int\limits_{t-r(t)}^t \left|G(t,v)\right| dv\) is increasing with respect to t, and \(\int\limits_{t-r(t)}^t \left|a(t,v)\right| dv\) is
\[2 \sup_{t \ge 0} \int_{t}^{p(t)} \int_{0}^{\infty} e^{-\int_{s}^{w+s} a_{1}(v)dv} b(s)dwds + 2 \sup_{t \ge 0} \int_{0}^{t} \int_{t-s}^{\infty} e^{-\int_{s}^{w+s} a_{1}(v)dv} b(s)dwds \le \alpha_{1},\] \[2 \int_{t}^{p(t)} \int_{0}^{\infty} e^{-\int_{s}^{w+s} a_{1}(v)dv} G(s,s)dwds + 2 \int_{0}^{t} \int_{t-s}^{\infty} e^{-\int_{s}^{w+s} a_{1}(v)dv} G(s,s)dwds\]
\[+2\int_{0}^{t}e^{-\int_{s}^{t}a_{1}(v)dv}\int_{t-s}^{\infty}\left|G(s,v)\right|dvds\leq\alpha_{2}.\]
(iii) There exist constants \(\alpha_2 > 0\) and Q > 0 such that, for each \(t \ge 0\), if \(J \ge Q\), then:
\[\int_{1}^{t+J} a_1(s)ds \ge \alpha_2 J.\]
Then there exists \(\delta \in (0,1)\) such that for each initial function \(\psi:[m(t_0),t_0] \to \Re\) and \(\dot{x}(t_0)\) satisfying \(|\dot{x}(t_0)| + ||\psi|| \le \delta\), there is a unique continuous function \(x:[m(t_0),\infty) \to \Re\) satisfying \(x(t)=\psi(t),\ t\in[m(t_0),t_0]\), which is a solution of Eq. (1) on \([t_0,\infty)\). Moreover, the zero solution of Eq. (1) is stable.
Before giving the proof of the former theorem, we need the following two lemmas.
Lemma 2.1 (Pi [14]) Let the function \(p:[-r(0),\infty) \to [0,\infty)\) denote the inverse of t-r(t). Then:
\[\dot{x}(t) = \int_{t-r(t)}^{t} a(t,s)g(x(s))ds\] is equivalent to
\[\dot{x}(t) = -G(t,t)g(x(t)) + \frac{d}{dt} \int_{t-r(t)}^{t} G(t,s)g(x(s))ds.\]
Lemma 2.2 Let \(\psi:[m(t_0),t_0] \to \Re\) be a given continuous function. If (x(t),y(t)) is the solution of system Eq. (2) satisfying \(x(t)=\psi(t),t\in[m(t_0),t_0]\) and \(y(t)=\dot{x}(t)\), then x(t) is the solution of the integral equation:
\[x(t) = \psi(t_0)e^{\int_{t_0}^{t} K(s)ds} + \int_{t_0}^{t} e^{\int_{u}^{t} K(s)ds} B(u)du + \int_{t_0}^{t} e^{\int_{u}^{t} K(s)ds} \hat{R}(u)[x(u) - h(x(u))]du\]\[+ \int_{t-r(t)}^{t} \hat{R}(s)h(x(s))ds - e^{\int_{t_0}^{t} K(s)ds} \int_{t_0-r(t_0)}^{t_0} \hat{R}(s)h(x(s))ds\]
\[-\int_{t_{0}}^{t} \left[ \int_{u-r(u)}^{u} \hat{R}(s)h(x(s))ds \right] K(u) e^{-\int_{u}^{t} K(s)ds} du + \int_{t_{0}}^{t} N(t,s)h(x(s-r(s)))ds\] \[-\int_{t_{0}}^{t} \left[ \int_{t_{0}}^{u} N(u,s)h(x(s-r(s)))ds \right] K(u) e^{-\int_{t_{0}}^{t} K(s)ds} du\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} H(u) [x(u) - g(x(u))] du + \int_{t-r(t)}^{t} H(s)g(x(s))ds\] \[-e^{-\int_{t_{0}}^{t} K(s)ds} \int_{u-r(u)}^{t_{0}} H(s)g(x(s))ds\] \[-\int_{t_{0}}^{t} \left[ \int_{u-r(u)}^{u} H(s)g(x(s))ds \right] K(u) e^{-\int_{u}^{t} K(s)ds} du\] \[+\int_{t_{0}}^{t} E(t,s)g(x(s-r(s)))ds - \int_{t_{0}}^{t} \left[ \int_{t_{0}}^{u} E(u,s)g(x(s-r(s)))ds \right] K(u) e^{-\int_{u}^{t} K(s)ds} du\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{u-r(u)}^{u} G(u,v)g(x(v))dv \right] du\] \[-\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{t_{0}-r(t_{0})}^{u} \int_{t_{0}-r(t_{0})}^{t} G(t_{0},v)g(x(v))dv \right] du\] \[-\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{t_{0}-r(t_{0})}^{u} \int_{s-r(s)}^{s} G(s,v)g(x(v))dv \right] A(s) e^{-\int_{s}^{u} A(v)dv} ds \right] du. \tag{4}\]
Throughout the proof of this lemma, we use the following notations:
\[C(t,u) = e^{\int_{u}^{t} A(s)ds} G(u,u), D(t) = \int_{t_0}^{\infty} C(u+t-t_0,t)du,\] \[\frac{D(t)}{1-\dot{r}(t)} = \tilde{D}(t), \ \tilde{D}(p(t)) = H(t),\] \[E(t,s) = \int_{t_0+t-s}^{\infty} C(u+s-t_0,s)du, M(t,u) = e^{\int_{u}^{t} A(s)ds} b(u),\]
\[N(t,s) = \int_{t-s+t_0}^{\infty} M(u+s-t_0,s)du,\] \[\int_{t_0}^{\infty} M(u+t-t_0,t)du = R(t), \frac{R(t)}{1-\dot{r}(t)} = \tilde{R}(t), \tilde{R}(p(t)) = \hat{R}(t).\]
Proof. Let:
\[B(t) = \dot{x}(t_0) \exp(-\int_{t_0}^{t} A(s)ds).\] (5)
It can be written from Eq. (3) and Eq. (5) that:
\[\dot{x}(t) = B(t) - \int_{t_0}^{t} e^{-\int_{t_0}^{t} A(s)ds} b(u)h(x(u - r(u)))du\] \[-\int_{t_0}^{t} e^{-\int_{s}^{t} A(v)dv} (\int_{s-r(s)}^{s} a(s,v)g(x(v))dv)ds.\] (6)
In view of Lemma 2.1 and Eq. (6), it follows that:
\[\dot{x}(t) = B(t) - \int_{t_0}^{t} e^{-\int_{t_0}^{t} A(s)ds} b(u)h(x(u - r(u)))du\] \[-\int_{t_0}^{t} e^{-\int_{s}^{t} A(v)dv} [G(s, s)g(x(s)) - \frac{d}{ds} \int_{s-r(s)}^{s} G(s, v)g(x(v))dv]ds. \tag{7}\]
Hence:
\[\dot{x}(t) = B(t) - \int_{t_0}^{t} e^{-\int_{t_0}^{t} A(s)ds} b(u)h(x(u - r(u)))du - \int_{t_0}^{t} C(t, s)g(x(s))ds\] \[+ \int_{t_0}^{t} e^{-\int_{s}^{t} A(v)dv} (\frac{d}{ds} \int_{s-r(s)}^{s} G(s, v)g(x(v))dv)ds.\] (8)
By \(|G(t,t)| \le M_2\), \(s \le t\), we get:
theorem, we have:
\[\left| \int_{t-s+t_{0}}^{\infty} C(u+s-t_{0},s) du \right| = \left| \int_{t-s+t_{0}}^{\infty} e^{-\int_{s}^{u+s-t_{0}} A(v) dv} G(s,s) du \right|\] \[\leq \int_{t-s}^{\infty} e^{-\int_{s}^{w+s} A(v) dv} |G(s,s)| dw\] \[= \int_{t-s}^{Q} e^{-\int_{s}^{w+s} A(v) dv} |G(s,s)| dw + \int_{Q}^{\infty} e^{-\int_{s}^{w+s} A(v) dv} |G(s,s)| dw,\] \[\int_{Q}^{\infty} e^{-\int_{s}^{w+s} A(v) dv} |G(s,s)| dw \leq \int_{Q}^{\infty} e^{-\int_{s}^{w+s} A(v) dv} M_{2} dw \leq M_{2} \frac{e^{-a_{2}Q}}{a_{2}}.\]
The former first relation implies that the integral \(\int_{t_0+t-s}^{\infty} C(u+s-t_0,s)du\) is convergent. Set \(u-t_0=w, s\leq t\). Then, in view of the assumptions of the
\[\int_{t-s+t_0}^{\infty} M(u+s-t_0,s) du = \int_{t-s}^{\infty} e^{-\int_{s}^{w+s} A(s) ds} dw \le \int_{t-s}^{\infty} e^{-\int_{s}^{w+s} a_1(s) ds} b(s) dw\] \[\le \int_{0}^{\infty} e^{-a_2 w} b(s) dw.\]
Since the function b is bounded, the integral \(\int_{t-s+t_0}^{\infty} M(u+s-t_0,s)du\) exists. Hence, Eq. (8) can be written as:
\[\dot{x}(t) = B(t) - h(x(t - r(t))) \int_{t_0}^{\infty} M(u + t - t_0, t) du\] \[+ \frac{d}{dt} \int_{t_0}^{t} N(t, s) h(x(s - r(s))) ds - g(x(t - r(t))) D(t)\] \[+ \frac{d}{dt} \int_{t_0}^{t} E(t, s) g(x(s - r(s))) ds\] \[+ \int_{t_0}^{t} e^{-\int_{s}^{t} A(v) dv} \left(\frac{d}{ds} \int_{s - r(s)}^{s} G(s, v) g(x(v)) dv \right) ds. \tag{9}\]
Then, we have:
\[\dot{x}(t) = B(t) - \tilde{R}(p(t))h(x(t)) + \frac{d}{dt} \int_{t-r(t)}^{t} \tilde{R}(p(s))h(x(s))ds + \frac{d}{dt} \int_{t_{0}}^{t} N(t,s)h(x(s-r(s)))ds - \tilde{D}(p(t))g(x(t)) + \frac{d}{dt} \int_{t-r(t)}^{t} \tilde{D}(p(s))g(x(s))ds + \frac{d}{dt} \int_{t_{0}}^{t} E(t,s)g(x(s-r(s)))ds + \int_{t}^{t} e^{-\int_{s}^{t} A(v)dv} (\frac{d}{ds} \int_{s-r(s)}^{s} G(s,v)g(x(v))dv)ds.\] (10)
Therefore, from Eq. (9) and Eq. (10), it follows that:
\[\dot{x}(t) = B(t) - \hat{R}(t)x(t) + \hat{R}(t)[x(t) - h(x(t))] + \frac{d}{dt} \int_{t-r(t)}^{t} \hat{R}(s)h(x(s))ds\] \[+ \frac{d}{dt} \int_{t_0}^{t} N(t,s)h(x(s-r(s)))ds - H(t)x(t) + H(t)[x(t) - g(x(t))]\] \[+ \frac{d}{dt} \int_{t-r(t)}^{t} H(s)g(x(s))ds + \frac{d}{dt} \int_{t_0}^{t} E(t,s)g(x(s-r(s)))ds.\]
For all \(t \in [t_0, T_1]\) by the variation of parameters formula, we obtain:
\[x(t) = x(t_0)e^{-\int_{t_0}^{t} K(s)ds} + \int_{t_0}^{t} e^{-\int_{u}^{t} K(s)ds} B(u)du\] \[+ \int_{t_0}^{t} e^{-\int_{u}^{t} K(s)ds} \hat{R}(u)[x(u) - h(x(u))]du + \int_{t_0}^{t} e^{-\int_{u}^{t} K(s)ds} H(u)[x(u) - g(x(u))]du\] \[+ \int_{t_0}^{t} e^{-\int_{u}^{t} K(s)ds} \left[\frac{d}{du} \int_{u-r(u)}^{u} \hat{R}(s)h(x(s))ds\right]du\] \[+ \int_{t_0}^{t} e^{-\int_{u}^{t} K(s)ds} \left[\frac{d}{du} \int_{t_0}^{u} N(u,s)h(x(s-r(s)))ds\right]du\] \[+ \int_{t_0}^{t} e^{-\int_{u}^{t} K(s)ds} \left[\frac{d}{du} \int_{u-r(u)}^{u} H(s)g(x(s))ds\right]du\]
\[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \frac{d}{du} \int_{t_{0}}^{u} E(u,s)g(x(s-r(s)))ds \right] du\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{t_{0}}^{u} e^{-\int_{s}^{u} A(v)dv} \left( \frac{d}{ds} \int_{s-r(s)}^{s} G(s,v)g(x(v))dv \right) ds \right] du. \tag{11}\]
If we use the integration by parts for the last five terms in Eq. (11), then we have:
\[x(t) = \psi(t_0)e^{-\int_{t_0}^{t}K(s)ds} + \int_{t_0}^{t}e^{-\int_{t}^{t}K(s)ds}B(u)du\] \[+ \int_{t_0}^{t}e^{-\int_{t_0}^{t}K(s)ds}\hat{R}(u)[x(u) - h(x(u))]du\] \[+ \int_{t_0}^{t}e^{-\int_{t_0}^{t}K(s)ds}\hat{R}(s)h(x(s))ds - e^{-\int_{t_0}^{t}K(s)ds}\int_{t_0-r(t_0)}^{t_0}\hat{R}(s)h(x(s))ds\] \[- \int_{t_0}^{t}[\int_{u-r(u)}^{u}\hat{R}(s)h(x(s))ds]K(u)e^{-\int_{u}^{t}K(s)ds}du + \int_{t_0}^{t}N(t,s)h(x(s-r(s)))ds\] \[- \int_{t_0}^{t}[\int_{t_0}^{u}N(u,s)h(x(s-r(s)))ds]K(u)e^{-\int_{u}^{t}K(s)ds}du\] \[+ \int_{t_0}^{t}e^{-\int_{t_0}^{t}K(s)ds}H(u)[x(u) - g(x(u))]du + \int_{t-r(t)}^{t}H(s)g(x(s))ds\] \[- e^{-\int_{t_0}^{t}K(s)ds}\int_{t_0-r(t_0)}^{t_0}H(s)g(x(s))ds - \int_{t_0}^{t}[\int_{u-r(u)}^{u}H(s)g(x(s))ds]K(u)e^{-\int_{u}^{t}K(s)ds}du\] \[+ \int_{t_0}^{t}E(t,s)g(x(s-r(s)))ds - \int_{t_0}^{t}[\int_{t_0}^{u}E(u,s)g(x(s-r(s)))ds]K(u)e^{-\int_{u}^{t}K(s)ds}du\] \[+ \int_{t_0}^{t}e^{-\int_{u}^{t}K(s)ds}[\int_{u-r(u)}^{u}G(u,v)g(x(v))dv]du\]
\[-\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ e^{-\int_{u}^{u} A(v)dv} \int_{t_{0}-r(t_{0})}^{t_{0}} G(t_{0},v)g(x(v))dv \right] du\] \[-\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{t_{0}}^{u} \int_{s-r(s)}^{s} G(s,v)g(x(v))dv \right] A(s)e^{-\int_{s}^{u} A(v)dv} ds \right] du.\]
Conversely, suppose that a continuous function \(x(t) = \psi(t), t \in [m(t_0), t_0]\) satisfies the integral equation given on \([t_0, T]\). Then this function is differentiable on \([t_0, T_2]\). We just need to differentiate the integral equation and then we obtain the relation Eq. (4). This completes the proof of Lemma 2.2.
Let \((C, \|.\|)\) be the Banach space of bounded continuous functions on \([m(t_0), \infty)\) with the supremum norm. For a given continuous initial function \(\psi: [m(t_0), t_0] \to \Re\), define the set \(C_{\psi} \subset C\) by:
\[C_{\psi} = \{ \varphi : [m(t_0), \infty) \to R | \varphi \in C, \varphi(t) = \psi(t), t \in [m(t_0), t_0] \},\]
\[\begin{split} &C_{\psi}^{l}=\{\varphi:[m(t_{0}),\infty)\to R\big|\varphi\in C, \varphi(t)=\psi(t), t\in[m(t_{0}),t_{0}], \big|\varphi(t)\big|\leq l, t\geq m(t_{0})\}\,,\\ &\text{where } \psi:[m(t_{0}),t_{0}]\to[-l,l] \text{ is given as initial function and } l \text{ is a positive constant. We also use } \|.\| \text{ to denote the supremum norm of the initial function.} \\ &\text{Let } P_{1} \text{ be a mapping defined on } C_{\psi}^{1} \text{ as follows: for } \varphi\in C_{\psi}^{l}, \text{ if } t\in[m(t_{0}),t_{0}],\\ &\text{then } (P_{1}\varphi)(t)=\psi(t). \text{ In addition, if } t>t_{0} \text{ , then:} \end{split}\]
\[(P_{1}\varphi)(t) = \psi(t_{0})e^{-\int_{t_{0}}^{t}K(s)ds} + \int_{t_{0}}^{t}e^{-\int_{u}^{t}K(s)ds}B(u)du + \int_{t_{0}}^{t}e^{-\int_{u}^{t}K(s)ds}\hat{R}(u)[\varphi(u) - h(\varphi(u))]du\] \[+ \int_{t-r(t)}^{t}\hat{R}(s)h(\varphi(s))ds - e^{-\int_{t_{0}}^{t}K(s)ds}\int_{t_{0}-r(t_{0})}^{t_{0}}\hat{R}(s)h(\psi(s))ds\] \[- \int_{t_{0}}^{t}[\int_{u-r(u)}^{u}\hat{R}(s)h(\varphi(s))ds]K(u)e^{-\int_{u}^{t}K(s)ds}du + \int_{t_{0}}^{t}N(t,s)h(\varphi(s-r(s)))ds\] \[- \int_{t_{0}}^{t}[\int_{t_{0}}^{u}N(u,s)h(\varphi(s-r(s)))ds]K(u)e^{-\int_{u}^{t}K(s)ds}du\]
\[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} H(u)[\varphi(u) - g(\varphi(u))]du + \int_{t-r(t)}^{t} H(s)g(\varphi(s))ds\] \[-e^{-\int_{t_{0}}^{t} K(s)ds} \int_{t_{0}-r(t_{0})}^{t_{0}} H(s)g(\psi(s))ds - \int_{t_{0}}^{t} \int_{u-r(u)}^{u} H(s)g(\varphi(s))ds]K(u)e^{-\int_{u}^{t} K(s)ds} du\] \[+\int_{t_{0}}^{t} E(t,s)g(\varphi(s-r(s)))ds - \int_{t_{0}}^{t} \int_{t_{0}}^{u} E(u,s)g(\varphi(s-r(s)))ds]K(u)e^{-\int_{u}^{t} K(s)ds} du\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{u-r(u)}^{u} G(u,v)g(\varphi(v))dv \right]du\] \[-\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ e^{-\int_{t_{0}}^{t} A(v)dv} \int_{t_{0}-r(t_{0})}^{t_{0}} G(t_{0},v)g(\psi(v))dv \right]du\] \[-\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{t_{0}-r(t_{0})}^{u} \int_{s-r(s)}^{s} G(s,v)g(\varphi(v))dv \right]A(s)e^{-\int_{s}^{u} A(v)dv} ds ds du.\]
Note that \(P_1\) may not be a contraction mapping. We solve this problem in Lemma 2.3 by introducing an exponential weighted metric.
Lemma 2.3 Suppose that there exist constants \(l_1 > 0\), \(l_2 > 0\), \(l = \max(l_1, l_2)\) such that h(x) and g(x) satisfy the Lipschitz condition on \([-l_1, l_1]\) and \([-l_2, l_2]\) respectively. Then there exists a metric d on \(C_{\psi}^l\) such that:
- (i) The metric space \((C_{uv}^l, d)\) is complete;
- (ii) \(P_2\) is a contraction mapping on \((C^l_{\psi},d)\) if \(P_2:C^l_{\psi}\to C^l_{\psi}\).
Proof.
(i) We change the supremum norm to an exponentially weighted norm \(|\phi|_f\), which is defined on \(C_{\psi}^l\). Let S be the space of all continuous functions \(\varphi:[m(t_0),\infty)\to R\) such that:
\[|\varphi|_f = \sup\{|\varphi(t)|e^{-f(t)}: t \in [m(t_0), \infty)\} < \infty,\] where \(f(t) = kL \int_{t_0}^{t} [\hat{R}(s) + R(s) + H(s) + D(s) + \int_{s-r(s)}^{s} |G(s,v)| dv] ds\), k is a constant and k > 12, and \(L_1\) is the common Lipschitz constant for x - h(x) and h(x), \(L_2\) is the common Lipschitz constant for x - g(x) and g(x), \(L = \max(L_1, L_2)\). Then \((S, | \cdot |_f)\) is a Banach space, Thus, (S, d) is a complete metric space, where d denotes the induced metric: \(d(\varphi, \eta) = |\varphi - \eta|_f\), where \(\varphi, \eta \in S\). Under this metric, the space \(C_{\psi}^l\) is a closed subset of S. Therefore, the metric space \((C_{\psi}^l, d)\) is complete.
(ii) Suppose that \(P_1: C_{\psi}^l \to C_{\psi}^l\). For \(\varphi, \eta \in C_{\psi}^l\), since \(\tilde{R}(t) \ge 0\), \(K(t) \ge 0\), \(N(t,s) \ge 0\) and \(E(t,s) \ge 0\), then:
\[\begin{aligned} & |(P_{1}\varphi)(t) - (P_{1}\eta)(t)| \leq \int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \hat{R}(u) |[\varphi(u) - h(\varphi(u))] - [\eta(u) - h(\eta(u))]| du \\ & + \int_{t_{0}}^{t} \hat{R}(s) |h(\varphi(s)) - h(\eta(s))| ds \\ & + \int_{t_{0}}^{t} K(u) e^{-\int_{u}^{t} K(s)ds} [\int_{u-r(u)}^{u} \hat{R}(s) |h(\varphi(s)) - h(\eta(s))| ds] du \\ & + \int_{t_{0}}^{t} N(t,s) |h(\varphi(s-r(s))) - h(\eta(s-r(s)))| ds \\ & + \int_{t_{0}}^{t} K(u) e^{-\int_{u}^{t} K(s)ds} [\int_{t_{0}}^{u} N(u,s) \times |h(\varphi(s-r(s))) - h(\eta(s-r(s)))| ds] du \\ & + \int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} H(u) |[(\varphi(u) - g(\varphi(u))] - [\eta(u) - g(\eta(u))]| du \\ & + \int_{t_{0}}^{t} H(s) |g(\varphi(s)) - g(\eta(s))| ds \\ & + \int_{t_{0}}^{t} K(u) e^{-\int_{u}^{t} K(s)ds} [\int_{u-r(u)}^{u} H(s) |g(\varphi(s)) - g(\eta(s))| ds] du \end{aligned}\]
\[+\int_{t_{0}}^{t} E(t,s) |g(\varphi(s-r(s))) - g(\eta(s-r(s)))| ds\] \[+\int_{t_{0}}^{t} K(u)e^{-\int_{u}^{t} K(s)ds} \left[\int_{t_{0}}^{u} E(u,s) \times |g(\varphi(s-r(s))) - g(\eta(s-r(s)))| ds\right] du\] \[+\int_{t_{0}}^{t} K(u)e^{-\int_{u}^{t} K(s)ds} \left[\int_{u-r(u)}^{u} H(s) |g(\varphi(s)) - g(\eta(s))| |ds\right] du\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[\int_{u-r(u)}^{u} |G(u,v)| |g(\varphi(v)) - g(\eta(v))| |dv\right] du\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[\int_{u-r(u)}^{u} |G(u,v)| |g(\varphi(v)) - g(\eta(v))| |dv\right] du\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[\int_{u-r(u)}^{u} |G(u,v)| |g(\varphi(v)) - g(\eta(v))| |dv\right] ds\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[\int_{u-r(u)}^{u} |A(s)e^{-\int_{s}^{u} A(v)dv} \times (\int_{s-r(s)}^{s} |G(s,v)| |g(\varphi(v)) - g(\eta(v))| |dv\right] ds\]
If we use the relation \(\hat{R}(t) \le K(t)\), \(H(t) \le K(t)(\hat{R}(t) + H(t) = K(t))\), then we have:
\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]
\[+\int_{t_{0}}^{t} K(u)e^{-\int_{u}^{t} K(s)ds} \left[ \int_{u-r(u)}^{u} H(s) |g(\varphi(s) - g(\eta(s))| ds \right] du\] \[+\int_{t_{0}}^{t} E(t,s) |g(\varphi(s-r(s))) - g(\eta(s-r(s)))| ds\] \[+\int_{t_{0}}^{t} K(u)e^{-\int_{u}^{t} K(s)ds} \left[ \int_{t_{0}}^{u} E(u,s) \times |g(\varphi(s-r(s))) - g(\eta(s-r(s)))| ds \right] du\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{u-r(u)}^{u} |G(u,v)| |g(\varphi(v)) - g(\eta(v))| |dv| du\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \int_{t_{0}}^{u} |G(u,v)| |g(\varphi(v)) - g(\eta(v))| |dv| du\] \[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \int_{t_{0}}^{u} |G(s,v)| |g(\varphi(v)) - g(\eta(v))| |dv| ds du\] \[\times \int_{s-r(s)}^{s} |G(s,v)| |g(\varphi(v)) - g(\eta(v))| |dv| ds du\] (12)
For \(u \le t, v \le t\) since \(D(t), H(t) \ge 0\), we have:
\[f(u) - f(t) = -kL \int_{u}^{t} [\hat{R}(s) + R(s) + H(s) + D(s) + \int_{s-r(s)}^{s} |G(s, v)| dv] ds,\] \[f(u) - f(t) \le -kL \int_{u}^{t} \hat{R}(s) ds, f(u) - f(t) \le -kL \int_{u}^{t} R(s) ds,\] \[f(u) - f(t) \le -kL \int_{u}^{t} H(s) ds, f(u) - f(t) \le -kL \int_{u}^{t} D(s) ds,\] \[f(u) - f(t) \le -kL \int_{u}^{t} (\hat{R}(s) + H(s)) ds = -kL \int_{u}^{t} K(s) ds,\] \[f(u) - f(t) \le -kL \int_{u}^{t} \int_{s-r(s)}^{s} |G(s, v)| dv ds.\]
For \(s \le t\), it is clear that:
\[f(s-\tau(s)) - f(t) \le -kL \int_{s}^{t} R(u)du, \ f(s-r(s)) - f(t) \le -kL \int_{s}^{t} D(u)du.\]
By \(N(t,s) \ge 0\), we have:
\[N(t,s) = \int_{t-s+t_0}^{\infty} M(u+s-t_0,s)du \le \int_{t_0}^{\infty} M(u+s-t_0,s)du = R(s) \Longrightarrow N(t,s) \le R(s).\]
Since \(E(t,s) \ge 0\), then:
\[E(t,s) = \int_{t-s+t_0}^{\infty} C(u+s-t_0,s)du \le \int_{t_0}^{\infty} C(u+s-t_0,s)du = D(s) \Rightarrow E(t,s) \le D(s).\]
For \(s \le u\), it is also clear that \(\int_{s-r(s)}^{s} |G(s,v)| dv \le \int_{u-r(u)}^{u} |G(s,v)| dv\). Hence, for the terms included in Eq. (12), we find:
\[\int_{t_{0}}^{t} e^{-\int_{t_{0}}^{t} \hat{R}(s)ds} \hat{R}(u) |[\varphi(u) - h(\varphi(u))] - [\eta(u) - h(\eta(u))]|e^{-f(t)}du\] \[\leq \int_{t_{0}}^{t} e^{-\int_{u}^{t} \hat{R}(s)ds} \hat{R}(u) L_{1} |\varphi(u) - \eta(u)|e^{-f(u)} e^{f(u) - f(t)}du\] \[\leq \int_{t_{0}}^{t} e^{-\int_{u}^{t} \hat{R}(s)ds} \hat{R}(u) e^{-kL\int_{u}^{t} \hat{R}(s)ds} du L_{1} |\varphi - \eta|_{f} \leq \frac{1}{kL} L_{1} |\varphi - \eta|_{f},\] \[\int_{t_{0}}^{t} \hat{R}(s) |h(\varphi(s) - h(\eta(s))|e^{-f(t)}ds\] \[\leq \int_{t_{0}-t}^{t} \hat{R}(s) L_{1} |\varphi(s) - \eta(s)|e^{-f(s)} e^{f(s) - f(t)} ds \leq \frac{1}{kL} L_{1} |\varphi - \eta|_{f},\] \[\int_{t_{0}}^{t} K(u) e^{-\int_{u}^{t} K(s)ds} [\int_{u-r(u)}^{u} \hat{R}(s) |h(\varphi(s)) - h(\eta(s))| ds] e^{-f(t)} du\] \[\leq \int_{t_{0}}^{t} K(u) e^{-\int_{u}^{t} K(s)ds} [\frac{1}{kL} e^{-kL\int_{u}^{t} \hat{R}(u)du} - \frac{1}{kL} e^{-kL\int_{u-r(u)}^{t} \hat{R}(u)du}] du L_{1} |\varphi - \eta|_{f}\] \[\leq \frac{1}{kL} \int_{t_{0}}^{t} K(u) e^{-\int_{u}^{t} K(s)ds} du L_{1} |\varphi - \eta|_{f} \leq \frac{1}{kL} L_{1} |\varphi - \eta|_{f},\] \[\int_{t}^{t} N(t,s) |h(\varphi(s - r(s)) - h(\eta(s - r(s)))|e^{-f(t)} ds\]
\[\leq \int_{t_0}^{t} R(s) e^{-it \int_{s}^{t} R(s) ds} ds L_1 |\varphi - \eta|_{f} \leq \frac{1}{kL} L_1 |\varphi - \eta|_{f},\] \[\int_{t_0}^{t} K(u) e^{-\int_{s}^{t} K(s) ds} \int_{t_0}^{u} N(t, s) |h(\varphi(s - r(s))) - h(\eta(s - r(s)))| ds |e^{-f(t)} du\] \[\leq \frac{1}{kL} \int_{t_0}^{t} K(u) e^{-\int_{s}^{t} K(s) ds} du L_1 |\varphi - \eta|_{f} \leq \frac{1}{kL} L_1 |\varphi - \eta|_{f},\] \[\int_{t_0}^{t} e^{-\int_{s}^{t} H(s) ds} H(u) ||\varphi(u) - g(\varphi(u))| - [(\eta(u) - g(\eta(u))]|e^{-f(t)} du\] \[\leq \int_{t_0}^{t} e^{-(kL+1)\int_{s}^{t} H(s) ds} H(u) du L_2 |\varphi - \eta|_{f} = [\frac{1}{kL+1} - \frac{1}{kL} e^{-(kL+1)\int_{s_0}^{t} H(s) ds}] L_2 |\varphi - \eta|_{f},\] \[\leq \frac{1}{kL} L_2 |\varphi - \eta|_{f},\] \[\int_{t_0}^{t} K(u) e^{-\int_{s}^{t} K(s) ds} ||f(s)| - f(t)|_{g} ds \leq \frac{1}{kL} L_2 |\varphi - \eta|_{f},\] \[\int_{t_0}^{t} K(u) e^{-\int_{s}^{t} K(s) ds} ||f(s)| - f(t)|_{g} ds \leq \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_2 ||f(s)||_{g} ds = \frac{1}{kL} L_\]
\[\leq \int_{t_{0}}^{t} K(u) e^{-\int_{u}^{t} K(s) ds} \left[ \int_{t_{0}}^{t} D(s) e^{-kL\int_{s}^{t} D(u) du} ds \right] du L_{2} \left| \varphi - \eta \right|_{f}\] \[\leq \frac{1}{kL} \int_{t_{0}}^{t} K(u) e^{-\int_{u}^{t} K(s) ds} du L_{2} \left| \varphi - \eta \right|_{f} \leq \frac{1}{kL} L_{2} \left| \varphi - \eta \right|_{f} ,\] \[\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s) ds} \left[ \int_{u-r(u)}^{u} \left| G(u,v) \right| \left| g(\varphi(v)) - g(\eta(v)) \right| \right] dv \right] e^{-f(t)} du\] \[\leq \frac{1}{kL} \int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s) ds} d\left( e^{-kL\int_{u-x-r(s)}^{t} \int_{s-r(s)}^{s} \left| G(s,w) \right| dw ds} \right) du L_{2} \left| \varphi - \eta \right|_{f} \leq \frac{1}{kL} L_{2} \left| \varphi - \eta \right|_{f} ,\] \[\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s) ds} \left[ \int_{t_{0}}^{u} A(s) e^{-\int_{s-r(s)}^{t} A(v) dv} \left( \int_{s-r(s)}^{s} \left| G(s,w) \right| \left| g(\varphi(v)) - g(\eta(v)) \right| \right) dv \right) ds \right] e^{-f(t)} du\] \[\leq \int_{t_{0}}^{t} e^{-\int_{s}^{t} K(s) ds} e^{f(u) - f(t)} \left( \int_{t_{0}}^{u} A(s) e^{-\int_{s-r(s)}^{u} A(v) dv} \left( \int_{u-r(u)}^{u} \left| G(u,v) \right| dv \right) du L_{2} \left| \varphi - \eta \right|_{f}\] \[\leq \int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s) ds} \int_{u-r(u)}^{u} \left| G(u,v) \right| dv \right] e^{-kL\int_{s}^{t} \int_{s-r(s)}^{s} \left| G(s,w) \right| dw ds} du L_{2} \left| \varphi - \eta \right|_{f}\] \[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]
In view of the discussion made, by an easy calculation, we get:
\[\begin{aligned} \big| (P_{1}\varphi)(t) - (P_{1}\eta)(t) \big| e^{-f(t)} &\leq \frac{5}{kL} L_{1} \big| \varphi - \eta \big|_{f} + \frac{7}{kL} L_{2} \big| \varphi - \eta \big|_{f} \\ &\leq \frac{12}{kL} \big| \varphi - \eta \big|_{f}, \ t > t_{0}. \end{aligned}\]
For \(t \in [m(t_0), t_0]\), we have \((P_1 \varphi)(t) = (P_1 \eta)(t) = \psi(t)\). Hence, \(d(P_1 \varphi - P_1 \eta) \le \frac{12}{k} d(\varphi - \eta), k > 12\). Thus, \(P_l\) is a contraction mapping on \((C_{\psi}^l, d)\).
We continue to prove the theorem. Choose \(\psi: [m(t_0), t_0] \to R\) and \(\dot{x}(t_0)\) such that:
\[(Q + \frac{e^{-a_2 Q}}{a_2}) |\dot{x}(t_0)| + \delta + h(\delta) \int_{t_0 - r(t_0)}^{t_0} \hat{R}(s) ds \le (1 - \alpha_1) h(l),\] \[(Q + \frac{e^{-a_2 Q}}{a_2}) \delta + g(\delta) \int_{t_0 - r(t_0)}^{t_0} H(s) ds \le (1 - \alpha_2) g(l) - l.\] (13)
Since (i) implies that \(g(\theta)=h(\theta)=0\) the \(g(l) \le l\), \(h(l) \le l\). In addition, since g(x) and h(x) satisfy the Lipschitz condition on [-l,l], then g(x) and h(x) are continuous on [-l,l] such that there exists a \(\delta\) with \(\delta < l\). By the expression for \((P_2\varphi)\) (t) we have:
\[(P_{2}\varphi)(t) = \psi(t_{0})e^{-\int_{t_{0}}^{t}K(s)ds} + \int_{t_{0}}^{t}e^{-\int_{u}^{t}K(s)ds}B(u)du + \int_{t_{0}}^{t}e^{-\int_{u}^{t}K(s)ds}\hat{R}(u)[\varphi(u) - h(\varphi(u))]du\] \[+ \int_{t-r(t)}^{t}\hat{R}(s)h(\varphi(s))ds - e^{-\int_{t_{0}}^{t}K(s)ds}\int_{t_{0}-r(t_{0})}^{t_{0}}\hat{R}(s)h(\psi(s))ds\] \[- \int_{t_{0}}^{t}[\int_{u-r(u)}^{u}\hat{R}(s)h(\varphi(s))ds]K(u)e^{-\int_{u}^{t}K(s)ds}du + \int_{t_{0}}^{t}N(t,s)h(\varphi(s-r(s)))ds\] \[- \int_{t_{0}}^{t}[\int_{t_{0}}^{u}N(u,s)h(\varphi(s-r(s)))ds]K(u)e^{-\int_{u}^{t}K(s)ds}du\] \[+ \int_{t_{0}}^{t}e^{-\int_{u}^{t}K(s)ds}H(u)[\varphi(u) - g(\varphi(u))]du + \int_{t-r(t)}^{t}H(s)g(\varphi(s))ds\] \[- e^{-\int_{t_{0}}^{t}K(s)ds}\int_{t_{0}-r(t_{0})}^{t_{0}}H(s)g(\psi(s))ds - \int_{t_{0}}^{t}[\int_{u-r(u)}^{u}H(s)g(\varphi(s))ds]K(u)e^{-\int_{u}^{t}K(s)ds}du\] \[+ \int_{t_{0}}^{t}E(t,s)g(\varphi(s-r(s)))ds - \int_{t_{0}}^{t}[\int_{t_{0}}^{u}E(u,s)g(\varphi(s-r(s)))ds]K(u)e^{-\int_{u}^{t}K(s)ds}du\]
\[+\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{u-r(u)}^{u} G(u,v)g(\varphi(v))dv \right] du\] \[-\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ e^{-\int_{u_{0}}^{t} A(v)dv} \int_{t_{0}-r(t_{0})}^{t_{0}} G(t_{0},v)g(\psi(v))dv \right] du\] \[-\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{t_{0}}^{u} \int_{s-r(s)}^{s} G(s,v)g(\varphi(v))dv \right] A(s)e^{-\int_{s}^{u} A(v)dv} ds \right] du.\]
Thus.
\[\begin{split} \big| (P_{l} \varphi)(t) \big| &\leq \delta + \int\limits_{t_{0}}^{t} e^{-\int\limits_{u}^{t} K(s) ds} \big| \dot{x}(t_{0}) \big| e^{-\int\limits_{t_{0}}^{u} A(s) ds} du + \int\limits_{t_{0}}^{t} e^{-\int\limits_{u}^{t} K(s) ds} \hat{R}(u) [l_{1} - h(l_{1})] du \\ &+ \int\limits_{t_{0}}^{t} \hat{R}(s) h(l_{1}) ds + \int\limits_{t_{0} - r(t_{0})}^{t_{0}} \hat{R}(s) h(\delta) ds \\ &+ \int\limits_{t_{0}}^{t} \big[ \int\limits_{u - r(u)}^{u} \hat{R}(s) h(l_{1}) ds \big] K(u) e^{-\int\limits_{u}^{t} K(s) ds} du \\ &+ \int\limits_{t_{0}}^{t} N(t, s) h(l_{1}) ds + \int\limits_{t_{0}}^{t} \big[ \int\limits_{u}^{u} N(t, s) h(l_{1}) ds \big] K(u) e^{-\int\limits_{u}^{t} K(s) ds} du \\ &+ \int\limits_{t_{0}}^{t} e^{-\int\limits_{u}^{t} K(s) ds} H(u) [l_{2} - g(l_{2})] du + \int\limits_{t - r(t)}^{t} H(s) g(l_{2}) ds \\ &+ \int\limits_{t_{0} - r(t_{0})}^{t_{0}} H(s) g(\delta) ds + \int\limits_{t_{0}}^{t} \big[ \int\limits_{u - r(u)}^{u} H(s) g(l_{2}) ds \big] K(u) e^{-\int\limits_{u}^{t} K(s) ds} du \\ &+ \int\limits_{t_{0}}^{t} E(t, s) g(l_{2}) ds + \int\limits_{t_{0}}^{t} \big[ \int\limits_{u - r(u)}^{u} E(u, s) g(l_{2}) ds \big] K(u) e^{-\int\limits_{u}^{t} K(s) ds} du \\ &+ \int\limits_{t_{0}}^{t} e^{-\int\limits_{u}^{t} K(s) ds} \big[ \int\limits_{u - r(u)}^{u} G(u, v) g(l_{2}) dv \big] du \\ &+ \int\limits_{t_{0}}^{t} e^{-\int\limits_{u}^{t} K(s) ds} \big[ e^{-\int\limits_{u}^{t} A(v) dv} \int\limits_{l_{0} - r(t_{0})}^{t_{0}} G(t_{0}, v) g(\delta) dv \big] du \end{split}\]
\[+\int_{l_0}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{l_0}^{u} \int_{s-r(s)}^{s} G(s,v)g(l_2)dv \right] A(s) e^{-\int_{s}^{u} A(v)dv} ds du.\] (14)
If we do the necessary calculations, for some terms included in Eq. (14), we get:
\[\int_{t_0}^{t} N(t,s)ds = \int_{t_0}^{t} \int_{t_0+t-s}^{\infty} M(u+s-t_0,s)duds = \int_{t_0}^{s} \int_{t_0+t-s}^{\infty} e^{-\int_{s}^{t} A(v)dv} b(s)duds\] \[\leq \sup_{t\geq 0} \int_{0}^{t} \int_{t-s}^{\infty} e^{-\int_{s}^{t} a_1(v)dv} b(s)duds,\] \[\int_{t-r(t)}^{t} \hat{R}(s)ds = \int_{t-r(t)}^{t} \tilde{R}(p(s))ds = \int_{t-r(t)}^{t} \frac{R(p(s))}{1-\dot{r}(s)}ds = \int_{t}^{p(t)} R(s)ds\] \[\leq \sup_{t\geq 0} \int_{t}^{t} \int_{0}^{\infty} e^{-\int_{s}^{t} a_1(v)dv} b(s)duds,\] \[\int_{t_0}^{t} e^{-\int_{s}^{t} K(s)ds} \hat{R}(u)(l_1-h(l_1))du \leq \int_{t_0}^{t} e^{-\int_{s}^{t} K(s)ds} K(u)(l_1-h(l_1))du, \quad (K(t)=\hat{R}(t)+H(t))\] \[= (l_1-h(l_1))(1-e^{-\int_{t_0}^{t} K(s)ds} )\] \[\leq (l_1-h(l_1)),\] \[\int_{t_0}^{t} e^{-\int_{t_0}^{t} K(s)ds} |\dot{x}(t_0)| e^{-\int_{t_0}^{t} A(s)ds} du \leq \int_{t_0}^{t} |\dot{x}(t_0)| e^{-\int_{t_0}^{t} A(s)ds} du,\] \[\int_{t_0}^{t} e^{-\int_{s}^{t} K(s)ds} du = \int_{t_0}^{t_0-t} e^{-\int_{t_0}^{t} A(s)ds} du + \int_{t_0+t_0}^{t} e^{-\int_{t_0}^{t} A(s)ds} du \leq Q + \frac{e^{-asQ}}{a_2},\] \[\int_{t_0}^{t} e^{-\int_{s}^{t} K(s)ds} H(u)[l_2-g(l_2)]du \leq \int_{t_0}^{t} e^{-\int_{s}^{t} K(s)ds} K(u)(l_2-g(l_2))du,\] \[= (l_2-g(l_2))\int_{t_0}^{t} \frac{d}{du} [e^{-\int_{s}^{t} K(s)ds}] du \leq (l_2-g(l_2)),\] \[\int_{t-r(t)}^{t} H(s)ds = \int_{t-r(t)}^{t} \tilde{D}(p_2(s))ds = \int_{t-r(t)}^{t} \frac{D(p_2(s))}{1-\dot{r}(s)} ds = \int_{t}^{p(t)} D(s)ds\]
\[\leq \int_{t_{0}}^{p(t)} \int_{0}^{\infty} e^{-\int_{s}^{w+s} a_{1}(v)dv} G(s,s)dwds,\] \[\int_{t_{0}}^{t} E(t,s)ds = \int_{t_{0}}^{t} \int_{t-s+t_{0}}^{\infty} C(u+s-t_{0},s)duds = \int_{t_{0}}^{t} \int_{t-s+t_{0}}^{\infty} e^{-\int_{s}^{u+s-t_{0}} A(s)ds} G(s,s)duds\] \[\leq \int_{0}^{t} \int_{t-s}^{\infty} e^{-\int_{s}^{u} a_{1}(v)dv} G(s,s)dwds,\] \[\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ e^{\int_{0}^{u} \int_{t_{0}-r(t_{0})}^{u} A(v)dv} \int_{t_{0}-r(t_{0})}^{t_{0}} G(t_{0},v)g(\psi(\delta))dv \right]du \leq \delta \int_{t_{0}}^{t} e^{-\int_{u}^{u} A(v)dv} du\] \[= \delta \left( \int_{t_{0}}^{t} e^{-\int_{u}^{u} A(s)ds} du + \int_{t_{0}+Q}^{t} e^{-\int_{u}^{u} A(s)ds} du \right) \delta \left( Q + \frac{e^{-a_{2}Q}}{a_{2}} \right),\] \[\int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \left[ \int_{u-r(u)}^{u} G(u,v)g(t_{2})dv \right]du\] \[+ \int_{t_{0}}^{t} e^{-\int_{u}^{t} K(s)ds} \int_{t_{0}}^{u} \left[ \int_{s-r(s)}^{u} G(s,v)g(t_{2})dv \right] A(s)e^{-\int_{s}^{u} A(v)dv} ds \right]du\] \[\leq 2g(t_{2}) \int_{0}^{t} e^{-\int_{u}^{t} a(v)dv} \int_{s-r(s)}^{s} \left| G(s,v) \right| dvdu.\]
So, we find:
\[\begin{aligned} \big| (P_{2}\varphi)(t) \big| &\leq \delta + (Q + \frac{e^{-a_{2}Q}}{a_{2}}) \big| \dot{x}(t_{0}) \big| + (l_{1} - h(l_{1})) + 2h(l_{1}) \\ &\times \sup_{t \geq 0} \int_{t}^{p(t)} \int_{0}^{\infty} e^{-\int_{s}^{u+s} a(v)dv} b(s) du ds + h(\delta) \int_{t_{0} - r(t_{0})}^{t_{0}} \hat{R}(s) ds \\ &+ 2h(l_{1}) \sup_{t \geq 0} \int_{0}^{t} \int_{t-s}^{\infty} e^{-\int_{s}^{u+s} a(v)dv} b(s) du ds + (l_{2} - g(l_{2})) \\ &+ 2g(l_{2}) \int_{t}^{p(t)} \int_{0}^{\infty} e^{-\int_{s}^{u+s} a(v)dv} G(s, s) dw ds + g(\delta) \int_{t_{0} - r(t_{0})}^{t_{0}} H(s) ds \end{aligned}\]
\[+2g(l_{2})\int_{0}^{t}\int_{t-s}^{\infty}e^{-\int_{s}^{u+s}a(v)dv}G(s,s)dwds +2g(l_{2})\int_{t_{0}}^{t}e^{-\int_{u}^{t}a(v)dv}\int_{s-r(s)}^{s}|G(s,v)|dvdu +\delta(Q+\frac{e^{-a_{2}Q}}{a_{2}}).\]
In view of assumption (ii), we can obtain:
\[|(P_{2}\varphi)(t)| \leq \delta + (Q + \frac{e^{-a_{2}Q}}{a_{2}})|\dot{x}(t_{0})| + (l_{1} - h(l_{1})) + \alpha_{1}h(l_{1}) + h(\delta) \int_{t_{0} - r(t_{0})}^{t_{0}} \hat{R}(s)ds + (l_{2} - g(l_{2})) + \alpha_{2}g(l_{2}) + g(\delta) \int_{t_{0} - r(t_{0})}^{t_{0}} H(s)ds + \delta(Q + \frac{e^{-a_{2}Q}}{a_{2}}).\]
Hence.
\[|(P_2\varphi)(t)| \le (1 - \alpha_1)h(l) + (l - h(l)) + \alpha_1h(l)\]\[+(l - g(l)) + \alpha_2g(l) + (1 - \alpha_2)g(l) - l = l.\]
Observe that if \(t \in [m(t_0), t_0]\), then \((P_2\varphi)(t) = \psi(t)\). We can obtain \(|(P_2\varphi)(t)| \le l\), \(t \in [m(t_0), \infty)\). Thus, \(P_2\varphi : C_{\psi}^l \to C_{\psi}^l\). We have proved \(P_2\) is a contraction mapping, hence \(P_2\) has a unique fixed point x(t) and \(|x(t)| \le l\).
From Eq. (6), we have:
\[|y(t)| \le |\dot{x}(t)| e^{-\int_{t_0}^{t} A(s)ds} + \int_{t_0}^{t} e^{-\int_{u}^{t} A(s)ds} b(u) |h(x(u-r(u)))| du\] \[+ \int_{t_0}^{t} e^{-\int_{s}^{t} A(v)dv} (\int_{s-r(s)}^{s} |a(s,v)| |g(x(v))| dv) ds.\]
Since for \(t \in [0, \infty)\), \(0 \le b(t) \le M_1\) and \(\int_{t-r(t)}^{t} |a(t, v)| dv\) is bounded, and there
exists a constant N > 0 such that \(\int_{t-r(t)}^{t} |a(t,v)| dv \le N\), then:
\[|y(t)| \le |\dot{x}(t)| + M_1 \int_{t_0}^t e^{-\int_u^t A(s)ds} |h(x(u-r(u)))| du + NLl \int_{t_0}^t e^{-\int_s^t A(v)dv} ds\]
\[\leq \left(1 + (M_1 + NL)(Q + \frac{\exp(-a_2Q)}{a_2})\right).\]
Thus, it follows that:
\[|x(t)| + |y(t)| \le l \left(2 + (M_1 + NL)(Q + \frac{e^{-a_2Q}}{a_2})\right).\]
To show the stability of zero solution, let \(\forall \varepsilon > 0\) be given; we only need to replace \(\varepsilon\) by l. This completes the proof of the theorem.
3 Conclusion
A functional integro-differential equation of the second order with variable delay was considered. The stability of the zero solution of this equation was discussed by the fixed point theory subject to an exponential weighted metric. Our result improves and includes some results found in the literature.
Acknowledgements
The authors thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper.