### Theorem 3.1

*Suppose that* (H_{1}) *holds*. *Then IVP* (1.2) *has a unique solution*
\(u\in C_{\beta1}\).

### Proof

It is well known that *u* is a solution of IVP (1.2) if and only if

$$\begin{aligned} u(t)=u_{0}+ \int_{0}^{t}f \bigl(s,u(s),(Tu) (s) \bigr)\,ds,\quad t\in J. \end{aligned}$$

Define the operator \(A:C_{\beta1}\to C_{\beta1}\) by

$$\begin{aligned} (Au) (t)=u_{0}+ \int_{0}^{t}f \bigl(s,u(s),(Tu) (s) \bigr)\,ds,\quad t\in J. \end{aligned}$$

(3.1)

It follows from (H_{1}) that

$$\begin{aligned} \bigl\Vert f(t,u,v) \bigr\Vert \leq \bigl\Vert f(t,\theta, \theta) \bigr\Vert +\beta(t) \bigl(a \Vert u \Vert +b \Vert v \Vert \bigr),\quad \forall t\in J, u,v\in E. \end{aligned}$$

(3.2)

For any \(u\in C_{\beta1}\), by (H_{1}), (3.1) and (3.2),

$$\begin{aligned} \bigl\Vert (Au) (t) \bigr\Vert & \leq \Vert u_{0} \Vert + \int_{0}^{t} \bigl[ \bigl\Vert f(s,\theta,\theta) \bigr\Vert +\beta(s) \bigl(a \bigl\Vert u(s) \bigr\Vert +b \bigl\Vert (Tu) (s) \bigr\Vert \bigr) \bigr] \,ds \\ & \leq \Vert u_{0} \Vert +\Phi_{1}(t) + \frac{1}{2} \Vert u \Vert _{\beta1}e^{4\Phi_{1}(t)}, \quad\forall t\in J, \end{aligned}$$

then \(Au\in C_{\beta1}\), so \(A:C_{\beta1}\to C_{\beta1}\).

On the other hand, for any \(u,v\in C_{\beta1}\), by (H_{1}) and Lemma 2.1,

$$\begin{aligned} \bigl\Vert Au(t)-Av(t) \bigr\Vert & \leq \int_{0}^{t} \beta(s) \bigl(a \bigl\Vert u(s)-v(s) \bigr\Vert +b \bigl\Vert (Tu) (s)-(Tv) (s) \bigr\Vert \bigr) \,ds \\ & \leq\frac{1}{2} \Vert u-v \Vert _{\beta1}e^{4\Phi_{1}(t)}, \end{aligned}$$

then \(\|Au-Av\|_{\beta1}\leq\frac{1}{2}\|u-v\|_{\beta1}\). Thus the Banach contraction mapping principle implies that *A* has a unique fixed point in \(C_{\beta1}\). □

In the following, we consider the second order nonlinear integro-differential equations of Volterra type on an infinite interval,

$$ \textstyle\begin{cases} u''=f (t,u(t),(Tu)(t) ),\quad t\in J=[0,\infty),\\ u(0)=u_{0},\qquad u'(0)=u_{1}. \end{cases} $$

(3.3)

Suppose \(f:J\times E\times E\to E\), for any \(u\in C[J,E], g(t)=f (t,u(t),(Tu)(t) ): J\to E\) is continuous, \(u_{0},u_{1}\in E\).

### Theorem 3.2

*Let*
*P*
*be a normal solid cone of*
*E*. *Assume that there exists*
\(\beta \in C[J,R^{+}]\)
*such that*, *for any*
\(y_{1},y_{2},\overline{y}_{1}, \overline{y}_{2},\in E, y_{1}\geq\overline{y}_{1},y_{2}\geq\overline{y}_{2}\), *we have*

$$\begin{aligned} {-}\beta(t)\bigl[a(y_{1}-\overline{y}_{1})+b(y_{2}- \overline{y}_{2})\bigr] & \leq f (t,y_{1},y_{2} ) -f (t,\overline{y}_{1},\overline{y}_{2} ) \\ & \leq\beta(t)\bigl[a(y_{1}-\overline{y}_{1})+b(y_{2}- \overline{y}_{2})\bigr]. \end{aligned}$$

(3.4)

*Then IVP* (3.3) *has a unique solution in*
\(C_{\beta2}\).

### Proof

It is clear that *u* is a solution of IVP (3.3) if and only if *u* is a solution of the following integral equation:

$$\begin{aligned} u(t)=u_{0}+tu_{1}+ \int_{0}^{t}(t-s)f \bigl(s,u(s),(Tu) (s) \bigr)\,ds,\quad t \in J. \end{aligned}$$

Define operators *A* and *B* by

$$\begin{aligned} &(Au) (t)=u_{0}+tu_{1}+ \int_{0}^{t}(t-s)f \bigl(s,u(s),(Tu) (s) \bigr)\,ds,\quad t \in J. \\ &(Bu) (t)= \int_{0}^{t}(t-s)\beta(s) \bigl(au(s)+b(T_{1}u) (s)\bigr)\,ds,\quad t\in J. \end{aligned}$$

By Lemma 2.2, it is easy to see that *B* is a positive linear bounded operator on \(C_{\beta2}\), and \(\|B\|<\frac{1}{2}\), then \(r(B)<\frac{1}{2}\). By (3.4), for any \(u,v\in C_{\beta2}, u\geq v\),

$$\begin{aligned} -B\bigl(u(t)-v(t)\bigr)\leq(Au) (t)-(Av) (t)\leq B\bigl(u(t)-v(t)\bigr),\quad \forall t\in J. \end{aligned}$$

Since *P* is a normal cone of *E*, it is easy to show that \(P_{\beta2}\) is normal in \(C_{\beta2}\). Since *P* is a solid cone of *E*, by Lemma 2.1.2 in [1] we see that \(P_{\beta2}\) is also a solid cone in \(C_{\beta2}\), and so, from Lemma 1.4.1 in [7], we know that \(P_{\beta2}\) is a generating cone in \(C_{\beta2}\). Hence all the conditions of Lemma 2.3 are satisfied, and the conclusion of Theorem 3.2 holds. □

### Remark 3.1

In most of the early work, for example [1], the conditions (H_{2}) and (H_{3}) play an important role in the proof of the main results. Undoubtedly, it is interesting and important to remove these conditions, which is very helpful for the applications of IVPs (1.2) and (3.3). In this paper, the existence and uniqueness of solutions for a class of nonlinear integro-differential equations of mixed type on unbounded domains in Banach spaces are established under more general conditions. The restrictive conditions (H_{2}) and (H_{3}) are removed; this implies that our results in essence improve and generalize the corresponding conclusions of [1–20].