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# Some new results for BVPs of first-order nonlinear integro-differential equations of volterra type

- Yepeng Xing
^{1}Email author and - Yi Fu
^{1}

**2011**:14

https://doi.org/10.1186/1687-1847-2011-14

© Xing and Fu; licensee Springer. 2011

**Received:**7 December 2010**Accepted:**22 June 2011**Published:**22 June 2011

## Abstract

In this work we present some new results concerning the existence of solutions for first-order nonlinear integro-differential equations with boundary value conditions. Our methods to prove the existence of solutions involve new differential inequalities and classical fixed-point theorems.

**MR(2000)Subject Classification**. 34D09,34D99.

## Keywords

- Boundary value problems
- integro-differential equations
- fixed-point motheds

## 1. Introduction and preliminaries

As is known, integro-differential equations find many applications in various mathematical problems, see Cordunean's book [1], Guo et al.'s book [2] and references therein for details. For the recent developments involving existence of solutions to BVPs for integro-differential equations and impulsive integro-differential equations we can refer to [3–17]. So far the main method appeared in the references to guarantee the existence of solutions is the method of upper and low solutions. Motivated by the ideas in the recent works [18, 19], we come up with a new approach to ensure the existence of at least one solution for certain family of first-order nonlinear integro-differential equations with periodic boundary value conditions or antiperiodic boundary value conditions. Our methods involve new differential inequalities and the classical fixed-point theory.

with *k*
_{
i
} (*t*, *s*) : [0, 1] × [0, 1] → [0, +∞) continuous for *i* = 1, 2, ⋯, *n*; *A* and *B* are *n* × *n* matrices with real valued elements, *θ* is the zero vector in ℝ ^{
n
} . For *A* = (*a*
_{
ij
} )_{
n × n
}, we denote ||*A*|| by
. In what follows, we assume function *f* : [0, 1] × ℝ ^{
n
} × ℝ ^{
n
} → ℝ ^{
n
} is continuous, and det (*A* + *B*) ≠ 0.

Noticing that det (*A*+*B*) ≠ 0, conditions *Ax*(0)+*Bx*(1) = *θ* do not include the periodic conditions *x*(0) = *x*(1). Furthermore, if *A* = *B* = *I*, where *I* denotes *n* × *n* identity matrix, then *Ax*(0)+*Bx*(1) = *θ* reduces to the so-called "anti-periodic" conditions *x*(0) = *-x*(1). The authors of [20–24] consider this kind of "anti-periodic" conditions for differential equations or impulsive differential equations. To the best of our knowledge it is the first article to deal with integro-differential equations with "anti-periodic" conditions so far.

*f*: [0, 1] × ℝ

^{ n }× ℝ

^{ n }× ℝ

^{ n }→ ℝ

^{ n }is continuous, (

*Lx*) (

*t*) denotes

with *l*
_{
i
} (*t*, *s*) : [0, 1] × [0, 1] → ℝ, *i* = 1, 2, ⋯, *n* being continuous.

This article is organized as follows. In Sect. 1 we give some preliminaries. Section 2 presents some existence theorems for PVPs (1.1), (1.3) and a couple of examples to illustrate how our newly developed results work. In Sect. 3 we focus on the existence of solutions for (1.2) and also an example is given.

*x*,

*y*∈ ℝ

^{ n }, then 〈

*x*,

*y*〉 denotes the usual inner product and ||

*x*|| denotes the Euclidean norm of

*x*on ℝ

^{ n }. Let

The following well-known fixed-point theorem will be used in the proof of Theorem 3.3.

is bounded, then H has at least one fixed-point.

## 2. Existence results for periodic conditions

where *g* : [0, 1] × ℝ ^{
n
} × ℝ ^{
n
} → ℝ ^{
n
} and *m* : [0, 1] → ℝ are both continuous functions, with *m* having no zeros in [0, 1].

**Proof**. The result can be obtained by direct computation.

**Theorem 2.1**. Let

*g*and

*m*be as in Lemma 2.1. Assume that there exist constants

*R >*0,

*α*≥ 0 such that

where *M*(*R*) is a positive constant depending on *R*, *B*
_{
R
} = {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}. Then PBVP (2.1) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

**Proof**. Let

*C*=

*C*([0, 1],

*R*

^{ n }) and Ω = {

*x*(

*t*) ∈

*C*, ||

*x*(

*t*)||

_{ C }<

*R*}. Define an operator by

for all *t* ∈ [0, 1].

*g*is continuous, see that

*T*is also a continuous map. It is easy to verify the operator

*T*is compact by the Arzela-Ascoli theorem. Indeed, for the ball Ω,

*H*

_{ λ }=

*I - λT*,

*λ*∈ [0, 1], where

*I*is the identity. So if (2.5) is true, then from the homotopy principle of Schauder degree [[25], Chap.4.], we have

Therefore, it follows from the non-zero property of Leray-Schauder degree that *H*
_{1}(*x*) = *x - Tx* = 0 has at least one *x* ∈ Ω.

*r*(

*t*) = ||

*x*(

*t*)||

^{2},

*t*∈ [0, 1], where

*x*(

*t*) is a solution of (2.7). Then

*r*(

*t*) is differentiable and we have by the product rule

*x*be a solution of (2.6) with We now show that

*x*∉ ∂Ω. From (2.5) and (2.3) we have, for each

*t*∈ [0, 1] and each

*λ*∈ [0, 1],

Then it follows from (2.2) that *x* ∉ *∂* Ω. Thus, (2.5) is true and the proof is completed.

**Corollary 2.1**. Let

*g*and

*m*be as in Lemma 2.1 with

*m*(

*t*)

*<*0,

*t*∈ [0, 1]. If there exist constants

*R >*0,

*α*≥ 0 such that

where *M*(*R*) is a positive constant depending on *R*, *B*
_{
R
} = {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}, then PBVP (2.1) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

It completes the proof.

where *M*(*R*) is a positive constant dependent on *R*, *B*
_{
R
} = {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}, then PBVP (1.1) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

**Proof**. Consider the PVPB (2.9), which is of the form (2.1) with

*m*(

*t*) ≡

*-*1 and

*g*(

*t*,

*x*, (

*Kx*)(

*t*)) =

*f*(

*t*,

*x*, (

*Kx*)(

*t*))

*- x*. Clearly,

So, (2.2) reduces to (2.10). Besides, (2.3) reduces to (2.11). Hence the result follows from Theorem 2.1.

where *M* (*R*) is a positive constant dependent on *R*, *B*
_{
R
} = {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}. Then PBVP (1.1) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

we have (2.11) is true if (2.13) is true. Then the proof is completed.

Now an example is provided to show how our theorems work.

Let us show (2.14) has at least one solution (*x*(*t*), *y*(*t*))^{⊤} with
, ∀*t* ∈ [0, 1].

It is clear that (2.14) has no constant solution. Let *u* = (*x*, *y*)^{⊤},
and
. First note that for
.

It is easy to see that any number in satisfies (2.15). Then our conclusion follows from Corollary 2.2.

In what follows we focus on the first-order integro-differential equations of mixed type in the form of (1.2). The results presented in the following three statements are similar to Theorem 2.1, Theorem 2.2 and Corollary 2.2, respectively. So we omit all the proofs here.

where *g* : [0, 1] × ℝ ^{
n
} × ℝ ^{
n
} × ℝ ^{
n
} → ℝ ^{
n
} and *m* : [0, 1] → ℝ are both continuous functions, with *m* having no zeros in [0, 1].

where *M*(*R*) is a positive constant depending on *R*, *B*
_{
R
} = {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}. Then PBVP (2.16) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

where *M*(*R*) is a positive constant depending on *R*, *B*
_{
R
}= {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}. Then PBVP (1.2) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

where *M*(*R*) is a positive constant dependent on *R*, *B*
_{
R
}= {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}, then PBVP (1.3) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

Now we give an example to illustrate how to apply our theorems.

We prove that (2.19) has at least one solution (*x*(*t*), *y*(*t*))^{⊤} with
, ∀*t* ∈ [0, 1].

First note that (2.19) has no constant solution. Let *u* = (*x*, *y*)^{⊤},
and
.

We compute directly . Then our conclusion follows from Corollary 2.3.

Now we modify Theorem 2.1 and Corollary 2.2 to obtain some new results.

where *M*(*R*) is a positive constant dependent on *R*, *B*
_{
R
} = {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}. Then PBVP (2.1) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

**Proof**. The proof is similar to that of Theorem 2.1 except choosing *r*(*t*) = - ||*x*(*t*) ||^{2} instead.

where *M*(*R*) is a positive constant depending on *R*, *B*
_{
R
} = {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}. Then PBVP (1.1) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

**Proof**. Consider PVPB (2.21), which is in the form from (2.1) with

*m*(

*t*) ≡ 1 and

*g*(

*t*,

*x*, (

*Kx*)(

*t*)) =

*f*(

*t*,

*x*, (

*Kx*)(

*t*)) +

*x*. Clearly,

Then the conclusion follows from Theorem 2.5.

**Remark 2.1**. Corollary 2.4 and Corollary 2.2 differ in sense that Corollary 2.4 may apply to certain problems, whereas Corollary 2.2 may not apply, and vice-versa.

**Example 2.3**3. Let us prove that the PBVP

has at least one solution *x*(*t*) with |*x*(*t*)| *<* 1, ∀*t* ∈ [0, 1].

It is not difficult to check that
if *R* ∈ [0.5, 2]. So the conclusion follows from Corollary 2.4.

**Remark 2.2** Since the coefficient of *x*
^{3} is negative, it appears impossible to find two constants *R >* 0 and *α* ≥ 0 satisfying (2.12) and (2.13) at the same time.

## 3. Existence results for "non-periodic" conditions

In this section we study the problem of existence of solutions for BVP (1.2).

**Proof**. The result can be obtained by direct computation.

**Theorem 3.1**. Assume det

*B*≠ 0 and ||

*B*

^{-1}

*A*|| ≤ 1. Suppose there exist constants

*R >*0,

*α*≥ 0 such that

where *M*(*R*) is a positive constant depending on *R*, *BR =* {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}. Then BVP (1.2) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

**Proof**. Let

*C*=

*C*([0, 1],

*R*

^{ n }) and Ω = {

*x*(

*t*) ∈

*C*, ||

*x*(

*t*)||

_{ C }

*< Rg*. Define an operator by

*f*is continuous, we see that

*T*is also a continuous map. It is easy to verify that the operator

*T*is compact by the Arzela-Ascoli theorem. It is sufficient to prove

*r*(

*t*) = ||

*x*(

*t*)||

^{2},

*t*∈ [0, 1], where

*x*(

*t*) is a solution of (3.6). By the product rule we have

*x*be a solution of (3.5) with . We now show that

*x*∉

*∂*Ω. From (3.2) and (3.3) we obtain, for each

*t*∈ [0, 1] and each

*λ*∈ [0, 1],

Then it follows from (3.1) that *x ∉ ∂* Ω. Thus, (3.4) is true and the proof is completed.

**Corollary 3.1**Let

*f*be a scalar-valued function in (1.1). and assume there exist constants

*R >*0,

*α*≥ 0 such that

*M*(

*R*) is a positive constant depending on

*R*,

*B*

_{ R }= {

*x*∈ ℝ

^{ n }, |

*x*| ≤

*R*}. Then anti-periodic boundary value problem

has at least one solution *x* ∈ *C*[0, 1] with |*x*(*t*)| *< R*, *t* ∈ [0, 1].

**Proof**. Since *A* = *B* = 1, we have
, *B*
^{-1}
*A* = 1,
. Then the conclusion follows from Lemma 3.1.

has at least one solution *x*(*t*) with |*x*(*t*)| *<* 1, ∀*t* ∈ [0, 1].

It is not difficult to check that . So, the conclusion follows from Corollary 3.1.

Now we modify Theorem 3.1 to include another class of *f*.

**Theorem 3.2**. Assume det

*B*≠ 0 and ||

*A*

^{-1}

*B*|| ≤ 1. Suppose there exist constants

*R >*0,

*α*≥ 0 such that

where *M*(*R*) is a positive constant depending on *R*, *B*
_{
R
} = {*x* ∈ ℝ ^{
n
} , ||*x*|| ≤ *R*}. Then BVP (1.2) has at least one solution *x* ∈ *C* with ||*x*|| _{
C
} *< R*.

Introducing the function *r*(*t*) = -||*x*(*t*) ||^{2}, *t* ∈ [0, 1], where *x*(*t*) is a solution of (3.6), for the rest part of the proof we proceed as in the proof of Theorem 3.1.

**Corollary 3.2**Let

*f*be a scalar-valued function in (1.1). If there exist constants

*R >*0,

*α*≥ 0 such that

*M*(

*R*) is a positive constant dependent on

*R*,

*B*

_{ R }= {

*x*∈ ℝ

^{ n }, |

*x*| ≤

*R*}. Then anti-periodic boundary value problem

has at least one solution *x* ∈ *C*[0, 1] with |*x*(*t*)| *< R*, *t* ∈ [0, 1].

**Proof**. Since *A* = *B* = 1, we have
, *A*
^{-1}
*B* = 1,
. Then the conclusion follows from Lemma 3.2.

where nonnegative functions *p*, *q*, *r* ∈ *L*
^{1}[0, 1]. We denote
for any function *x* ∈ *L*
_{1} [0, 1].

where . Then (1.2) has at least one solution.

*T*is compact. Taking into account that the family of BVP (1.2) is equivalent to the family of problem

*x*=

*Tx*, our problem is reduced to show that

*T*has a least one fixed point. For this purpose, we apply Schaefer's Theorem by showing that all potential solutions of

*a priori*, with the bound being independent of

*λ*. With this in mind, let

*x*be a solution of (3.14). Note that

*x*is also a solution of (3.6). We have, for ∀

*t*∈ [0, 1] and ∀

*λ*[0, 1],

The proof is completed.

We omit it here because it is trivial.

## Declarations

### Acknowledgements

Research is supported by National Natural Science Foundation of China (10971139), Shanghai municipal education commission(No. 10YZ72)and Shanghai municipal education commission(No. 09YZ149).

## Authors’ Affiliations

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