Continuous Dependence of the Solutions of Nonlinear Integral Quadratic Volterra Equation on the Parameter

Abstract

We prove results on the existence and continuous dependence of solutions of a nonlinear quadratic integral Volterra equation on a parameter. This dependence is investigated in terms of Hausdorff distance. The considerations are placed in the Banach space and the Fréchet space.

1. Introduction

In this paper we investigate the following nonlinear quadratic integral Volterra equation: both on a bounded interval (i.e., ) and on an unbounded one (), where , , and are given functions. We will study (1) in the Banach space when and in the Fréchet space when .

The main aim of the paper is to formulate assumptions that guarantee continuous dependence of solutions of (1) on parameter. In our considerations we do not assume the uniqueness of solutions, while dependence of the set of solutions on a parameter will be expressed in terms of Hausdorff distance of the spaces and .

Quadratic integral equations appear in theories of radiative transfer and neutron transport and in kinetic theory of gases (cf. [1–4]). Up to this time, a lot of papers have appeared on those equations [1–9]; however, to the best of our knowledge, there are no papers on continuous dependence of solutions of this kind of equations on parameter.

Existence results for (1) have been obtained with the help of fixed point theorems expressed in terms of measures of nonconpactness.

2. Notation and Auxiliary Facts

In this section we collect some definitions and results which will be needed later. Assume that is a real Banach space with the norm and the zero element . Denote by the closed ball centered at and with radius . The ball will be denoted by . If is a subset of , then the symbols and stand for the closure and convex closure of , respectively. The family of all nonempty and bounded subsets of will be denoted by while its subfamily consisting of all relatively compact sets is denoted by . Following [5, 8, 10] we accept the following definition of a measure of noncompactness.

Definition 1. A mapping is said to be a measure of noncompactness if it satisfies the following conditions. )The family is nonempty and .(). ().() for .()If is a sequence of closed sets from such that    and if  , then the intersection set is nonempty.

For our purposes we will only need the following fixed point theorem [6, 8–11].

Theorem 2. Let be nonempty bounded closed convex subset of the space and let be continuous such that for any nonempty subset of , where is a constant, . Then has a fixed point in the set .

In the sequel we will work in the Banach space consisting of all real functions defined and continuous on . The space is furnished with the standard norm

Now we recollect the definition of the measure of noncompactness which will be used further on. This measure was introduced in [8, 10]. Fix a nonempty bounded subset of and a positive number . For and let us denote by the modulus of continuity of the function on the interval ; that is, Further, let us put It can be shown [9, 10] that the function is a measure of noncompactness in the space .

In what follows, we will also work in the space consisting of all real functions defined and continuous on . The space equipped with the family of seminorms becomes a Fréchet space furnished with the distance or equivalently

A nonempty subset is said to be bounded if for .

Further, let denote the family of all nonempty and bounded subsets of and the family of all relatively compact subsets of . Obviously .

We accept the following definition of the notion of a sequence of measures of noncompactness [12, 13].

Definition 3. A sequence of functions , where , is said to be a sequence of measures of noncompactness in if it satisfies the following conditions. ()The family is nonempty and .() for .() for .()If is a sequence of closed sets from such that    and if   for each , then the intersection set is nonempty.

We have the following two facts (see [12, 13]).

Theorem 4. The family of mappings , where , satisfies the conditions from Definition 3 and, moreover, .

Theorem 5. Let be a nonempty, bounded, closed, and convex subset of the space and let be continuous mapping. Suppose that there exists a sequence of numbers such that for nonempty and . Then has at least one fixed point in the set .

Let be an arbitrary metric space. For any two nonempty and bounded subsets of we define their Hausdorff distance by formula where In the next chapters, we will consider Hausdorff distance in the family of nonempty and relatively compact subsets of the Banach space and in the family of nonempty and relatively compact subsets of the Fréchet space (with distance ).

Let be arbitrary metric space, and let us consider a mapping . Since we will consider the continuity of such mappings (with respect to the distance in ), we need the following lemma.

Lemma 6. A mapping is continuous at a point if and only if

Proof. From (6) we derive that for we have where The continuity of the mapping at a point means We will show that conditions (12) and (13) are equivalent to conditions (16) and (17).
First we suppose that (12) and (13) are fulfilled. Let us fix . The condition (12) guarantees that there exist numbers , , such that for .
Let us put . Applying (14) we get In a similar way we show that (17) is also satisfied.
Now we assume that (16) and (17) are fulfilled. We will prove conditions (12) and (13). Let us fix and . There exists a number such that and . Condition (16) implies that there is a number such that Hence, using (14) we obtain that and therefore (16) is confirmed. In a similar way we show that condition (17) is also satisfied.

3. Existence Result

In this section we give an existence result for the following nonlinear integral Volterra equation: In the last years there have been published a few dozen papers on nonlinear quadratic integral equations. From among papers cited here, majority of them was concerned with different kinds of equations from (22); see [5–7, 13]. The authors of papers [8, 9, 11, 13] examined equations similar to (22); however, their considerations were conducted in the Banach space and therefore their assumptions were too restrictive. In the paper [12] we investigated similar equations to (22) with different kinds of assumptions than these given below.

This theorem will be a starting point of our further investigations on the continuous dependence of solutions on parameter.

Observe that the above equation includes several classes of functional, integral, and functional integral equations considered in the literature [1–4].

Equation (22) will be considered under the following assumptions.(H₁)The function is continuous on .(H₂)The function is continuous and there exists a constant such that for any and for all .(H₃)The function is continuous. Moreover, there exists a function being continuous on such that for all .(H₄)Consider

Then we can formulate our existence result.

Theorem 7. Under assumptions , (22) has at least one solution in the space .

Proof. Consider the operator defined by the formula Using assumptions and reasoning similarly as in [5–11], we can show that the operator is well defined and is continuous on . Next let us put Notice that for we have in virtue of for and therefore the operator transforms the ball into itself.
Now, let us take a nonempty subset of the ball and fix . Choose such that . Without loss of generality we can assume that . Then we obtain where we denoted From the above estimate we derive the following one: Observe that , , and as , which are simple consequences of the uniform continuity of the functions , , and on the sets , , and , respectively. Hence we get Since in view of assumption (), from the above estimate and Theorem 2 we deduce that the operator has a fixed point in the set . Obviously is a solution of the functional integral equation (22).

Our next result is concerned with (22) on the real half-axis . Let us consider equation

Equation (32) will be considered under the following assumptions.The function is continuous on .The function is continuous and there exists a constant such that for any and for all .The function is continuous. Moreover, there exists a function being continuous on such that for all , .Consider

Then we can formulate the next existence result.

Theorem 8. Under assumptions , (32) has at least one solution in the space .

Proof. In contrast to papers [5, 6, 8, 9, 11], we place our considerations in the Fréchet space , instead of the Banach space .
Let us define the operator defined by the formula Using assumptions and standard arguments, we can show that the operator is well defined and is continuous on . Next let us put Notice that for we have in virtue of ()–( and (36) and therefore the operator transforms into itself.
Applying (31) with ,  , we get for Since in view of assumption (), from the above estimate and Theorem 5, we deduce that the operator has a fixed point in the set . Obviously is a solution of the functional integral equation (22).

4. Continuous Dependence of Solutions on Parameter and the Examples

In this section we will investigate (1) depending on parameter ; that is, we will consider equation of the type where is an element of metric space , or .

For fixed we will denote the operator specified by formula and put Obviously is the set of all solutions of (40) and it is compact in view of condition (31), so .

The aim of this paper is to provide the conditions concerning the functions involved in (40), which imply that the sets are nonempty and change continuously in the space with respect to parameter (in view of Hausdorff metric ).

First we consider case of the bounded set . Let us take the following assumptions. is a function such that the function is continuous on for arbitrarily fixed and moreover there exists a nondecreasing function , continuous at 0 and , such that for , . is a function such that the function is continuous and bounded on the set for arbitrarily fixed and there are a constant and a nondecreasing function , continuous at 0 and , such that for any , , , and for all , , . is a function such that the function is continuous on the set for arbitrarily fixed and there are a continuous function and a nondecreasing function , continuous at 0 and , such that for all , , , and for all , , .Consider 

The following example shows that conditions are not enough for continuous dependence of the solutions on parameter .

Example 9. Let us consider the equation where . The space is equipped with the standard Euclidean metric. Obviously it is a particular case of (40), where Notice that if is a solution of (49), then which shows that (49) may be represented by simpler form Using standard calculations we obtain that , where and , but for we have , where . It is easy to check whether the sets do not change continuously in point (in view of Hausdorff metric). It means the conditions , which guarantee the existence of the solutions together with Theorem 7, are not sufficient for continuous dependence of the solutions on the parameter . Thus, we decide to take an additional assumption. There is an integrable function such that for , , .

Theorem 10. Under assumptions , (40) has at least one solution in the space for all parameters and moreover the set of all solutions depends continuously (with respect to Hausdorff metric ) on the parameter .

Proof. Theorem 7 implies that (40) has at least one solution for all .
Let us fix arbitrarily and  . Since we want to prove that is continuous in point with respect to Hausdorff metric , we should show that there exists such that Let us fix and choose arbitrarily . Then, for any we obtain the following estimation: where
Summarizing, for fixed   and   and for arbitrary , , we derive where we denoted
Consider the following equation: where is unknown function. Standard calculations show that solution of the above equation is given by formula Keeping in mind we infer that there exists sufficiently small number , satisfying inequality Now, let us fix and define the set We will prove Indeed, for and , in view of (58) and (60) we obtain and it confirms (64).
If is a fixed point in then, in view of (62), we obtain which confirms (54). The proof of estimate (55) is analogous.

Finally we will consider (40) on the real half-axis . First we express the necessary assumptions. is a function such that the function is continuous on for arbitrarily fixed and moreover there exists a nondecreasing function , continuous at 0 and , such that for . is a function such that the function is continuous and bounded on the set for arbitrarily fixed and there are a constant and a nondecreasing function , continuous at 0 and , such that for any , and for all , , . is a function such that the function is continuous on the set for arbitrarily fixed and there are a continuous function and a nondecreasing function , continuous at 0 and , such that for all and for all , , .Consider There is an integrable function such that for , , .

Under the above assumptions we can formulate a theorem analogous to the previous one.

Theorem 11. Under assumptions , (40) has at least one solution in the space for all parameters and, moreover, the set of all solutions depends continuously (with respect to Hausdorff metric ) on the parameter .

Proof. The proof of this theorem uses Lemma 6 and it can be done in the same way as the proof of Theorem 10. We omit details.

Remark 12. Notice that, in hypotheses in Theorem 10, we have assumed uniform continuity of the functions , , and with respect to variable . Therefore, there appears a natural question.
Question. Is Theorem 10 true in the case when we assume only continuity of the functions , , and instead of uniform continuity?

Finally, we provide an example of an integral equation of the form (40) for which the assumptions of Theorem 10 are satisfied.

Example 13. Consider the following functional integral equation: where . The space is equipped with the standard Euclidean metric. Observe that (74) is a special case of (40), where , , and . Moreover that is, the function for is integrable.
Joining all above facts we infer that assumptions of Theorem 10 are satisfied. Finally, applying Theorem 10 we conclude that the set of all solutions of (74) depends continuously on the parameter .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.