Convex Solutions of a Nonlinear Integral Equation of Urysohn Type
© Tiberiu Trif. 2009
Received: 4 August 2009
Accepted: 25 September 2009
Published: 5 November 2009
We study the solvability of a nonlinear integral equation of Urysohn type. Using the technique of measures of noncompactness we prove that under certain assumptions this equation possesses solutions that are convex of order for each , with being a given integer. A concrete application of the results obtained is presented.
where is a positive constant. In the special case when (or even ), the authors proved in  that if is positive and nondecreasing, is positive and nondecreasing in the first variable (when the other two variables are kept fixed), and they satisfy some additional assumptions, then there exists at least one positive nondecreasing solution to (1.1). A similar existence result, but involving a Volterra type integral equation, has been obtained by Banaś and Martinon .
It should be noted that both existence results were proved with the help of a measure of noncompactness related to monotonicity introduced by Banaś and Olszowy . The reader is referred also to the paper by Banaś et al. , in which another measure of noncompactness is used to prove the solvability of an integral equation of Urysohn type on an unbounded interval.
The main purpose of the present paper is twofold. First, we generalize the result from the paper  to the framework of higher-order convexity. Namely, we show that given an integer , if and are convex of order for each , then (1.1) possesses at least one solution which is also convex of order for each . Second, we simplify the proof given in  by showing that it is not necessary to make use of the measure of noncompactness related to monotonicity introduced by Banaś and Olszowy .
2. Measures of Noncompactness
Measures of noncompactness are frequently used in nonlinear analysis, in branches such as the theory of differential and integral equations, the operator theory, or the approximation theory. There are several axiomatic approaches to the concept of a measure of noncompactness (see, e.g., [9–11] or ). In the present paper the definition of a measure of noncompactness given in the book by Banaś and Goebel  is adopted.
Let be a real Banach space, let be the family consisting of all nonempty bounded subsets of , and let be the subfamily of consisting of all relatively compact sets. Given any subset of , we denote by and the closure and the convex hull of , respectively.
Definition 2.1 (see ).
A function is said to be a measure of noncompactness in if it satisfies the following conditions.
(1)The family ker (called the kernel of ) is nonempty and it satisfies .
(2) whenever satisfy .
(3) for all .
(4) for all and all .
(5)If is a sequence of closed sets from such that for each positive integer and if , then the set is nonempty.
Theorem 2.2 (see [12, Theorem 5.1]).
Let be a real Banach space, let be a measure of noncompactness in , and let be a nonempty bounded closed convex subset of . Further, let be a continuous operator such that for each subset of , where is a constant. Then has at least one fixed point in .
3. Convex Functions of Higher Orders
Note that a convex function of order is a nonnegative function, a convex function of order is a nondecreasing function, while a convex function of order is an ordinary convex function.
holds for every nonnegative integer and every for which .
for all and all such that . Due to (3.11), it is clear that every convex function of order is also Jensen convex of order . In general, the converse does not hold. However, under the additional assumption that is continuous, the two notions turn out to be equivalent.
Theorem 3.1 (see [13, page 387, Theorem 1]).
Let be a nondegenerate interval, let be an integer, and let be a continuous function. Then is convex of order if and only if it is Jensen convex of order .
Finally, we mention the following result concerning the difference of order of a product of two functions:
holds for every such that .
4. Main Results
Our first main result concerns the integral equation of Urysohn type (1.1) in which , , and are given functions, while is the unknown function. We assume that the functions , , and satisfy the following conditions:
is a given integer number;
is a continuous function which is convex of order for each ;
is convex of order for each whenever is convex of order for each ;
for all and all ;
is a continuous function such that the function is convex of order for each whenever and ;
If the conditions ( )–( ) are satisfied, then (1.1) possesses at least one solution which is convex of order for each .
Then whenever (see [5, the proof of Theorem 3.2]).
Therefore, the inequality holds for every in satisfying . This proves the continuity of at .
This inequality together with (4.16), ( ), and ( ) ensures that the function is Jensen convex of order for each . Since is continuous on , by Theorem 3.1 it follows that is convex of order for each . Taking into account (4.13), we conclude that maps into itself, as claimed.
by virtue of (2.5).
By ( ) and Theorem 2.2 we conclude the existence of at least one fixed point of in . This fixed point is obviously a solution of (1.1) which (in view of the definition of ) is convex of order for each .
Theorem 4.1 can be further generalized as follows. Given an integer number and a sequence , we denote by the set consisting of all functions with the property that for each the function is convex of order . For instance, if and , then consists of all functions in that are nonnegative, nonincreasing, and convex on .
for each and each integer . By [13, Theorem 6, page 392] it follows that if is an absolutely monotonic (resp., a completely monotonic) function, then belongs to every set with and (resp., ) for each .
Instead of the conditions ( ), ( ), ( ), and ( ) we consider the following conditions.
belongs to .
belongs to whenever
is a continuous function such that the function belongs to whenever and .
If the conditions ( )–( ), ( ), ( ), and ( )-( ) are satisfied, then (1.1) possesses at least one solution .
Consider the operator , defined on , as in the proof of Theorem 4.1. As we have already seen in the proof of Theorem 4.1 we have whenever and is continuous on .
The rest of the proof is similar to the corresponding part in the proof of Theorem 4.1 and we omit it.
5. An Application
in which is a given positive integer and is a positive real parameter. Note that (5.1) is similar to the Chandrasekhar equation, arising in the theory of radiative transfer (see, e.g., Chandrasekhar  or Banaś et al. , and the references therein).
If , then for all . If and , then , while if and , then . Note that .
Assume now that . Then we can select an sufficiently close to such that . Obviously, satisfies (5.3).
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