#### Abstract

We establish sufficient conditions for the existence and uniqueness of random solutions of nonlinear Volterra-Fredholm stochastic integral equations of mixed type by using admissibility theory and fixed point theorems. The results obtained in this paper generalize the results of several papers.

#### 1. Introduction

Random or stochastic integral equations are important in the study of many physical phenomena in life sciences, engineering, and technology [1–13]. Currently there are two basic versions of stochastic integral equations being studied by mathematical statisticians and probabilists namely, those integral equations involving Ito-Doob type of stochastic integrals and those which can be formed as probabilistic analogues of classical deterministic integral equations whose formulation involves the usual Lebesgue integral. Equations of the later category have been studied extensively by several authors [4, 10, 14–40]. Many papers have been appeared on the problem of existence of solutions of nonlinear random integral equations and the results are established by applying various fixed point techniques. These methods are broadly classified into three categories:

(i)admissibility theory, ([2, 7, 24, 27, 41–47]), (ii)random contractor method, ([17, 21, 35, 47–52]), (iii)measure of noncompactness method, ([11, 53–61]).All these methods are effectively used to study the existence of solutions for stochastic integral equations. Further asymptotic behaviour and stability of solutions of stochastic integral equations are discussed in the papers [33, 42, 50, 54, 55, 59, 61–63]. In this paper we will study the existence of random solutions of nonlinear stochastic integral equations of mixed type.

Consider a nonlinear stochastic integral equation of the form

where is a stochastic process and

(a-i), the supporting set of the complete probability measure space , with the -algebra and probability measure , (a-ii) is the unknown random function for , the nonnegative real numbers, (a-iii) is a scalar function defined for and , the real line, (a-iv) and are stochastic kernels defined for and satisfying , (a-v) is the stochastic kernel defined for and in , (a-vi) are scalar functions defined for and , the real line.The first and the second part of the stochastic integral (1.1) are to be understood as an ordinary Lebesque integral with probabilistic characterization, while the third part is an Ito-Doob stochastic integral. Our aim is to investigate the existence as well as uniqueness of random solutions of the stochastic integral equation (1.1) by making use of “admissibility theory” that was first introduced by Tsokos [40] and fixed point theorems due to Krasnoselskii and Banach. The results generalize the previous results of [2, 7, 24, 27, 41–46].

#### 2. Preliminaries

Let be the random process. We will assume that for each , a minimal -algebra , , is such that is measurable with respect to . In addition, we will assume that the minimal -algebra is an increasing family such that

(H1) the random process is a real martingale (H2) there is a real continuous nondecreasing function, , such that for we have - a.e. where denotes the expected value of the random process.In the definitions that follow, we will assume that is measurable and that , for each . Also we denote

*Definition 2.1. *Denote by the linear space of all mean square continuous maps on and define a topology on by means of the following family of seminorms.
It is known that such a topology is metrizable and that the metric space is complete.

*Definition 2.2. *Define to be the space of all maps on such that
where , a constant and , a continuous function on . The norm in the space is defined by

*Definition 2.3. *Let be the space of maps on with , for some . The norm in space is defined by

*Definition 2.4. *The pair of Banach spaces with is called admissible with respect to the operator if .

*Definition 2.5. *We will call a random solution of the stochastic integral equation (1.1) if for each and satisfies equation (1.1) -a.e. for all .

*Definition 2.6. *The Banach space is said to be stronger than , if every sequence which converges in the topology of converges also in the topology of .

Finally, let be Banach spaces and a linear operator from into . The following lemma is well known [13].

Lemma 2.7. *Let be a continuous operator from into . If and are Banach spaces in stronger than and if the pair is admissible with respect to , then is a continuous operator from into . **Let us define the operators **
for . **We state the following assumptions for our use. **The functions , and are continuous functions of with values in .**For each and in has values in the space and the functions and for each and such that has values in the space .**The stochastic kernels and are essentially a bounded function with respect to for every and such that and continuous as maps from into .**The stochastic kernel is essentially a bounded function with respect to for every and in and continuous as maps from into . **
Define for ,
**
The assumptions imply that if , then for each ,
**
Because of the continuity assumptions on and it follows from the above inequality that
**
which together with (H1) and (H2) implies that the integral in (2.8) is well defined.*

Lemma 2.8. *Under the assumptions , (H1) and (H2), , and are continuous linear operators from into provided
*

*Proof. *It is easy to show that and are linear maps from into . The continuity of and are also easy to prove [8, 13]. We will prove that is continuous.

Let . Then

Hence, on compact intervals
where is a constant depends upon . This proves the continuity of . The linearity of is obvious.

To show that maps into . Let . Then

The right-hand side of the above inequality goes to zero as , since . Thus, this proves that maps into . The proof of the continuity of is similar to that of .

Let the operators , and be as defined in (2.6), (2.7), and (2.8) and let the assumptions of Lemma 2.8 hold. Then it follows from Lemma 2.7 that, if and are Banach spaces stronger than and the pair is admissible with respect to the operators and , then , and are continuous from into . Thus, there exist positive constants , and such that

The constants are the bounds of the operator .

Theorem 2.9 (Krasnoselskii Theorem). *Let be a closed, bounded and convex subset of a Banach space and let and be operators on satisfying the following conditions: *(i)* whenever , *(ii)* is a contraction operator on , *(iii)* is completely continuous. **Then there is at least one point such that .*

#### 3. Main Results

In this section we will prove the main result of this paper.

Theorem 3.1. *For the stochastic integral equation (1.1) assume the following conditions *(i)* and are Banach spaces in , stronger than , such that is admissible with respect to the operators , and defined by (2.6), (2.7), and (2.8);*(ii)* for some ; *(iii)* is a continuous map from
with values in satisfying
for and a constant;*(iv)* is a completely continuous map from into ; *(v)* is a continuous map from with values in satisfying
for and a constant;*(vi)* is a continuous map from into such that
for and a constant. **Then there exists a unique random solution of (1.1) in provided **
where , and are defined by (2.16).*

*Proof. *The set closed, bounded, and convex in . Let . Then define the operator by
We will show that is a contraction mapping and that . Let . Then
From our assumption it is clear that and . Furthermore
Since , is a contraction operator. Next we show that . From (3.6), we have
Since , by hypothesis, we have which implies that .

Let us define the operator as

It is clear that is composition of continuous map and completely continuous map . Hence is completely continuous. Furthermore, if , we have
This shows that if , then . Hence, applying Krasnoselskii's fixed point theorem, we can conclude that there exists a random solution of (1.1) in the set .

We will now consider the case under which the stochastic integral equation (1.1) possesses a unique solution. This will be achieved by using the Banach contraction mapping principle.

Theorem 3.2. *For the stochastic integral equation (1.1) assume the following conditions *(i)* and are Banach spaces in , stronger than , such that is admissible with respect to the operators and defined by (2.6), (2.7), and (2.8); *(ii)* for some ; *(iii)* is a continuous map from
with values in satisfying
for and a constant;*(iv)* is a continuous map from with values in satisfying
for and a constant;*(v)* is a continuous map from with values in satisfying
for and a constant;*(vi)* is a continuous map from into such that
for and a constant.**Then there exists a unique random solution of (1.1) in provided **
where , and are defined by (2.16).*

*Proof. *Define the operator as follows

We will show that is a contraction operator on and that . Let . Then as and is a Banach space. Also

Thus, in view of (2.16), we have
Since , is a contraction operator on .

We will now show that . For any , we have

Since , it follows that
Using the condition that
we have from (3.18)
Hence for all or . Thus the condition of Banach's fixed point theorem is satisfied and hence there exists a fixed point such that . That is,

#### 4. Applications

In this section we will give some application of Theorem 3.2.

Theorem 4.1. *Suppose the stochastic integral equation (1.1) satisfies the following conditions: *(i)*there exists a constant and a continuous function , such that
*(ii)* are continuous functions on , such that and for and *(iii)* is a continuous functions on , such that for and **Then there exists a unique random solution of (1.1) such that **
provided are small enough.*

*Proof. *It is easy to show that the hypothesis of Theorem 3.2 are satisfied by simply showing the pair of spaces is admissible with respect to the operators , and . This follows from Lemma 2.8.

Corollary 4.2. *Suppose the stochastic integral equation (1.1) satisfies the following conditions: *(i)(ii)* are continuous functions on , such that and for and *(iii)* is a continuous functions on , such that for and **Then there exists a unique random solution of (1.1) such that **
provided are small enough.*

*Proof. *Take in Theorem 4.1.

Corollary 4.3. *Suppose the stochastic integral equation (1.1) satisfies the following conditions: *(i)*, and *(ii)*same as conditions in Theorem 3.2. **Then there exists a unique random solution of (1.1) provided and for small enough.*

*Proof. *We will show that the pair is admissible with respect to the operator . Let . Then
which implies that the pair is admissible. Similarly we can show that the pair is admissible with respect to the operators . It is easy to check the other conditions of Theorem 3.2 and hence there exists a unique random solution of equation of the stochastic integral equation (1.1).

*Remark 4.4. *Using the same argument one can establish the existence of a unique random solution of the following general stochastic integral equation
where , and satisfy appropriate conditions. This general case is treated in a separate paper.

#### 5. Example

Consider the following nonlinear stochastic integral equation:

where is a stochastic process. This equation is a particular case of general stochastic integral equation occurring in mathematical biology and chemotherapy [10–13]. The above equation takes the form of (1.1) with

Take and . It is easy to see that , , , , , and . Further and by taking , the other condition of Theorem 3.2 is satisfied. It is clear that (5.1) satisfies assumptions (i) to (vi) of Theorem 3.2. Hence there exists a unique random solution for (5.1).