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# Existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces

*Advances in Difference Equations*
**volumeÂ 2018**, ArticleÂ number:Â 247 (2018)

## Abstract

In this paper, the existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces are established under more general conditions by constructing a special Banach space and using cone theory and the Banach contraction mapping principle. The results obtained herein improve and generalize some well-known results.

## 1 Introduction

Nonlinear integro-differential equations in abstract spaces arise in different fields of physical sciences, engineering, biology, and applied mathematics. The theory of nonlinear integro-differential equations in abstract spaces is a fast growing field with important applications to a number of areas of analysis as well as other branches of science [1]. In recent years, there has been a significant development in nonlinear integro-differential equations (see [1â€“20] and the references therein).

Using the upper and lower solutions method and monotone iterative technique, Guo, Liu and Zheng et al. [2â€“4] studied the existence and uniqueness of solutions for the first order integro-differential equations

in Banach space *E*, where \(u_{0}\in E, f:I\times E\times E\times E\to E\), for any \(u\in C[I,E], g(t)=f (t,u(t),(Tu)(t),(Su)(t) ): I\to E \text{ is continuous}\), and *T* is a Volterra integral operator defined by

*S* is a Hammerstein integral operator defined by

where \(k\in C[D_{1},R], h\in C[D_{2},R], D_{1}=\{(t,s)\in I\times I: t\geq s\}, D_{2}=\{(t,s)\in I\times I\}\).

In [5], the authors studied the second order integro-differential equations

in Banach space *E*, where \(u_{0}, u_{1}\in E, f:I\times E\times E\to E\), for any \(u\in C[I,E], g(t)=f (t,u(t),(Tu)(t) ): I\to E \text{ is continuous}\), and *T* is a Volterra integral operator defined by

where \(k\in C[D,R], D=\{(t,s)\in I\times I: t\geq s\}\). By using an iteration method, the existence and uniqueness results for second order integro-differential equations (1.1) are obtained without demanding the existence of upper and lower solutions and monotonicity conditions.

In [1], Guo, Lakshmikantham and Liu studied the IVP for first order integro-differential equations of Volterra type on an infinite interval in Banach space *E*:

where \(u_{0}\in E, f:J\times E\times E\to E\), for any \(u\in C[J,E], g(t)=f (t,u(t),(Tu)(t) ): J\to E \text{ is continuous}\), and *T* is a Volterra integral operator defined by

where \(k\in C[D,R], D=\{(t,s)\in J\times J: t\geq s\}\). By using the Banach contraction mapping principle, the authors obtained the following results.

### Theorem 1.1

*If the following conditions are satisfied*:

- (H
_{1}): -
\(\|f(t,u,v)-f(t,\overline{u},\overline{v})\| \leq\beta(t)(a\|u-\overline{u}\|+b\|v-\overline{v}\|), \forall t\in J, u,v,\overline{u},\overline{v}\in E\),

*where constants*\(a\geq0, b\geq0, \beta\in C[J,R^{+}]\); - (H
_{2}): -
$$\begin{aligned} & k^{*}=\sup_{t\in J} \int_{0}^{t} \bigl\vert k(t,s) \bigr\vert \,ds< \infty, \\ & \beta^{*}= \int_{0}^{\infty}\beta(t)\,dt< \infty, \\ & a^{*}= \int_{0}^{\infty} \bigl\Vert f(t,\theta,\theta) \bigr\Vert \, dt< \infty, \end{aligned}$$
*here**Î¸**denotes the zero element of**E*; - (H
_{3}): -
\(c_{0}=(a+bk^{*})\beta^{*}<1\),

*then IVP*(1.2)*has a unique solution*\(x^{*}(t)\)*in*\(C^{1}[J,E]\cap BC[J,E]\).

### Remark 1.1

Clearly, the assumptions imposed in TheoremÂ 1.1 are too strict. As is well known, there is no research focused on the improvement of the assumptions in TheoremÂ 1.1. By constructing a special Banach space and using cone theory and the Banach contraction mapping principle, this paper improves TheoremÂ 1.1 without assumptions (H_{2}) and (H_{3}).

## 2 Preliminaries and lemmas

Let \((E, \| \cdot\|)\) be a real Banach space and *P* be a cone in *E* which defines a partial ordering in *E* by \(x \leq y\) if and only if \(y-x\in P\). *Î¸* denotes the zero element in *E*. A cone *P* is said to be normal if there exists a constant \(N> 0\) such that, for any \(x, y\in E\), \(\theta\leq x\leq y \) implies \(\|x\|\leq N\|y\|\). A cone *P* is said to be generating if \(E=P-P\), i.e., every element \(x\in E\) can be represented in the form \(x=y-z\), where \(y, z\in P\). A cone is called solid if it contains interior points, i.e., \(\mathring {P}\neq{\O}\). An operator \(T: E\to E\) is said to be a positive operator if \(x\geq\theta\) implies \(Tx\geq\theta\). The detailed contents of the cone and partial ordering may be found in [6â€“8].

Let \(\beta(t)\) be a nonnegative continuous function on *J*, \(k(t,s)\) be continuous on \(D=\{(t,s) | 0\leq s\leq t<\infty\}\). Set

Then \(C_{\beta1}\) is a Banach space with norm \(\|\cdot\|_{\beta1}\), and \(C_{\beta2}\) is a Banach space with norm \(\|\cdot\|_{\beta2}\).

Let *P* be a cone of *E*, \(P_{\beta2}= \{u\in C_{\beta2} | u\geq\theta, t\in J \}\). Clearly, \(P_{\beta2}\) is a cone of \(C_{\beta2}\).

### Lemma 2.1

*Let*

*then*

### Proof

Direct calculations shows that, for \(u\in C_{\beta1}\) and \(t\in J\),

â€ƒâ–¡

### Lemma 2.2

*Let*

*then*

### Proof

Direct calculations show that, for \(u\in C_{\beta2}\) and \(t\in J\),

â€ƒâ–¡

### Lemma 2.3

([6])

*Let*
*P*
*be a generating normal cone and*
*B*
*a positive linear operator whose spectral radius satisfies*
\(r(B)<1\). *If the operator A satisfies the estimate*

*then*
*A*
*has a unique fixed point in*
*E*, *and for each initial approximation*
\(x_{0}\in E\), *let*
\(x_{n}=Ax_{n-1} (n=1,2,\ldots)\), *then we have*
\(x_{n}\to x^{*}\ (n\to\infty)\).

## 3 Main results

### 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

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

It follows from (H_{1}) that

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

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 LemmaÂ 2.1,

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 \(C_{\beta1}\).â€ƒâ–¡

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

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*

*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:

Define operators *A* and *B* by

By LemmaÂ 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 \(u,v\in C_{\beta2}, u\geq v\),

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 LemmaÂ 2.1.2 in [1] we see that \(P_{\beta2}\) is also a solid cone in \(C_{\beta2}\), and so, from LemmaÂ 1.4.1 in [7], we know that \(P_{\beta2}\) is a generating cone in \(C_{\beta2}\). Hence all the conditions of LemmaÂ 2.3 are satisfied, and the conclusion of TheoremÂ 3.2 holds.â€ƒâ–¡

### 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 [1â€“20].

## References

Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic Publishers, Dordrecht (1996)

Guo, D.: Solutions of nonlinear integro-differential equations of mixed type in Banach spaces. J. Appl. Math. Simul.

**2**(1), 1â€“11 (1989)Liu, L., Wu, C., Guo, F.: Existence theorems of global solutions of initial value problems for nonlinear integro-differential equations of mixed type in Banach spaces and applications. Comput. Math. Appl.

**7**(1), 13â€“22 (2004)Zheng, X., Wang, J.: Initial value problems for a nonlinear integro-differential equation of mixed type in Banach spaces. J. Comput. Anal. Appl.

**21**(5), 835â€“847 (2016)Zhang, P., Liu, L.: Existence and uniqueness of fixed points for mixed non-monotone binary operators in ordered Banach spaces and its applications. Acta Math. Sin.

**53**(1), 55â€“60 (2010) (in Chinese)Krasnoselâ€™skii, M.A., Zabreiko, P.P.: Geometrical Methods of Nonlinear Analysis. Springer, Berlin (1984)

Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Boston (1988)

Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

Liu, L., Wu, C., Guo, F., Wu, Y.: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. J. Math. Anal. Appl.

**309**(2), 638â€“649 (2005)Hao, X., Liu, L.: Mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces. Math. Methods Appl. Sci.

**40**(13), 4832â€“4841 (2017)Du, S.W., Lakshmikantham, V.: Monotone iterative technique for differential equations in a Banach space. J. Math. Anal. Appl.

**87**(2), 454â€“459 (1982)Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities, Vol. I. Academic Press, New York (1969)

Liu, L., Guo, F., Kim, J.K.: Global solutions of initial value problems for second order nonlinear integro-differential equations of mixed type in Banach spaces. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal.

**9**(4), 511â€“522 (2002)Guo, D.: Second order integro-differential equations of Volterra type on unbounded domains in Banach spaces. Nonlinear Anal.

**41**(3â€“4), 465â€“476 (2000)Amann, H.: On the number of solutions of nonlinear equations in ordered Banach spaces. J. Funct. Anal.

**11**(3), 346â€“384 (1972)Liu, L.: Iterative method for solutions and coupled quasi-solutions of nonlinear Fredholm integral equations in ordered Banach spaces. Indian J. Pure Appl. Math.

**27**(10), 959â€“972 (1996)Sun, Y.: Positive solutions for nonlinear integro-differential equations of mixed type in Banach spaces. Abstr. Appl. Anal.

**2013**, 787033 (2013)Wang, Z., Liu, L., Wu, Y.: The unique solution of boundary value problems for nonlinear second-order integral-differential equations of mixed type in Banach spaces. Comput. Math. Appl.

**54**(9â€“10), 1293â€“1301 (2007)Su, H., Liu, L., Zhang, X., Wu, Y.: Global solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. J. Math. Anal. Appl.

**330**(2), 1139â€“1151 (2007)Su, H., Liu, L., Zhang, X.: The solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. Nonlinear Anal.

**66**(5), 1025â€“1036 (2007)

### Acknowledgements

The authors would like to thank the referees for their pertinent comments and valuable suggestions. This paper was completed during the first author visited School of Mathematical Sciences, Peking University. We express our sincere gratitude to Professor Baoxiang Wang for valuable suggestions on the paper and the fund of the Key Laboratory of Mathematics and Applied Mathematics, Peking University.

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## Funding

This work is supported financially by the National Natural Science Foundation of China (11501318, 11371221, 11571296) and the Natural Science Foundation of Shandong Province of China (ZR2014AM032).

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Zhang, P., Hao, X. Existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces.
*Adv Differ Equ* **2018**, 247 (2018). https://doi.org/10.1186/s13662-018-1681-0

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DOI: https://doi.org/10.1186/s13662-018-1681-0

### MSC

- 47H07
- 47H10
- 47G20
- 34G20

### Keywords

- Fixed points of operator
- Banach contraction mapping principle
- Integro-differential equation