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

when J = R + . 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 C([0, T]) and C(R + ). 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

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 ([0, ]) and (R + ).
Quadratic integral equations appear in theories of radiative transfer and neutron transport and in kinetic theory of gases (cf.[1][2][3][4]).Up to this time, a lot of papers have appeared on those equations [1][2][3][4][5][6][7][8][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.

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 Conv stand for the closure and convex closure of , respectively.The family of all nonempty and bounded subsets of  will be denoted by M  while its subfamily consisting of all relatively compact sets is denoted by N  .Following [5,8,10] we accept the following definition of a measure of noncompactness.
Definition 1.A mapping  : M  → R + is said to be a measure of noncompactness if it satisfies the following conditions.
Theorem 2. Let Ω be nonempty bounded closed convex subset of the space  and let  : Ω → Ω be continuous such that () ≤ () for any nonempty subset  of Ω, where  is a constant,  ∈ [0, 1).Then  has a fixed point in the set Ω.
In the sequel we will work in the Banach space ([0, ]) consisting of all real functions defined and continuous on [0, ].The space ([0, ]) 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 ([0, ]) and a positive number  > 0. For  ∈  and  > 0 let us denote by (, ) the modulus of continuity of the function  on the interval [0, ]; that is, Further, let us put It can be shown [9,10] that the function   0 is a measure of noncompactness in the space ([0, ]).
In what follows, we will also work in the space (R + ) consisting of all real functions defined and continuous on R + .The space (R + ) equipped with the family of seminorms becomes a Fréchet space furnished with the distance or equivalently A nonempty subset  ⊂ (R + ) is said to be bounded if for  ∈ N.
Further, let M  denote the family of all nonempty and bounded subsets of (R + ) and N  the family of all relatively compact subsets of (R + ).Obviously N  ⊂ M  .
We accept the following definition of the notion of a sequence of measures of noncompactness [12,13].Definition 3. A sequence of functions {  } ∈N , where   : M  → [0,∞), is said to be a sequence of measures of noncompactness in (R + ) if it satisfies the following conditions.
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  ([0,]) in the family N ([0,]) of nonempty and relatively compact subsets of the Banach space [0, ] and  (R + ) in the family N (R + ) of nonempty and relatively compact subsets of the Fréchet space (R + ) (with distance   ).Let (Λ, ) be arbitrary metric space, and let us consider a mapping  : Λ → N (R + ) .Since we will consider the continuity of such mappings (with respect to the distance  (R + ) in N (R + ) ), we need the following lemma.

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][6][7]13].The authors of papers [8,9,11,13] examined equations similar to (22); however, their considerations were conducted in the Banach space (R + ) 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][2][3][4].
Equation ( 22) will be considered under the following assumptions.
Let us define the operator  : (R + ) → (R + ) defined by the formula Notice that for  ∈ Ω we have in virtue of (  1 )-(  3 ) and ( 36)

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 (Λ, ),  = [0, ] or  = R + .
For fixed  ∈ Λ we will denote the operator   : () → () specified by formula Obviously Fix() is the set of all solutions of (40) and it is compact in view of condition (31), so Fix() ∈ N () .The aim of this paper is to provide the conditions concerning the functions involved in (40), which imply that the sets Fix() 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  = [0, ].Let us take the following assumptions.
The following example shows that conditions ( 1 )-( 4 ) are not enough for continuous dependence of the solutions on parameter .

Journal of Function Spaces
Example 9. Let us consider the equation where  ∈ Λ := [0, 1].The space Λ is equipped with the standard Euclidean metric.Obviously it is a particular case of (40), where  (, ) ≡ 0,  (, , ) ≡ 1,   ≡ 0, 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 Fix(0) = { 1 ,  2 }, where  1 () ≡ 0 and  2 () =  2 /4, but for  > 0 we have Fix() = {}, where () = (/4 + √ ).It is easy to check whether the sets Fix() do not change continuously in point  = 0 (in view of Hausdorff metric).It means the conditions ( 1 )-( 4 ), 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. ( Consider the following equation: where   1 : [0, ] → R + is unknown function.Standard calculations show that solution   1 of the above equation is given by formula Keeping in mind lim  1 →  ( 1 ) = 0 we infer that there exists sufficiently small number  > 0,  < 1 satisfying inequality Now, let us fix  1 ∈ (, ) and define the set We will prove for all  1 ,  2 ∈ Λ,  ∈ R + ,  ∈ R.