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Convex Solutions of a Nonlinear Integral Equation of Urysohn Type
Fixed Point Theory and Applications volume 2009, Article number: 917614 (2009)
Abstract
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.
1. Introduction
Existence of solutions of differential and integral equations is subject of numerous investigations (see, e.g., the monographs [1–3] or [4]). Moreover, a lot of work in this domain is devoted to the existence of solutions in certain special classes of functions (e.g., positive functions or monotone functions). We merely mention here the result obtained by Caballero et al. [5] concerning the existence of nondecreasing solutions to the integral equation of Urysohn type
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ1_HTML.gif)
where is a positive constant. In the special case when
(or even
), the authors proved in [5] 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 [6].
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 [7]. The reader is referred also to the paper by Banaś et al. [8], 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 [5] 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 [5] by showing that it is not necessary to make use of the measure of noncompactness related to monotonicity introduced by Banaś and Olszowy [7].
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 [12]). In the present paper the definition of a measure of noncompactness given in the book by Banaś and Goebel [12] 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 [12]).
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.
An important and very convenient measure of noncompactness is the so-called Hausdorff measure of noncompactness, defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ2_HTML.gif)
The importance of this measure of noncompactness is given by the fact that in certain Banach spaces it can be expressed by means of handy formulas. For instance, consider the Banach space consisting of all continuous functions
, endowed with the standard maximum norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ3_HTML.gif)
Given ,
, and
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ4_HTML.gif)
be the usual modulus of continuity of . Further, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ5_HTML.gif)
and . Then it can be proved (see Banaś and Goebel [12, Theorem 7.1.2]) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ6_HTML.gif)
For further facts concerning measures of noncompactness and their properties the reader is referred to the monographs [9, 11] or [12]. We merely recall here the following fixed point theorem.
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
Let be a nondegenerate interval. Given an integer
, a function
is said to be convex of order
or
-convex if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ7_HTML.gif)
for any system of
points in
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ8_HTML.gif)
is called the divided difference of at the points
. With the help of the polynomial function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ9_HTML.gif)
the previous divided difference can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ10_HTML.gif)
An alternative way to define the divided difference is to set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ11_HTML.gif)
whenever . Finally, we mention a representation of the divided difference by means of two determinants. It can be proved that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ12_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ13_HTML.gif)
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.
Let be a nondegenerate interval, let
be an arbitrary function, and let
. The difference operator
with the span
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ14_HTML.gif)
for all for which
. The iterates
of
are defined recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ15_HTML.gif)
It can be proved (see, e.g., [13, page 368, Corollary 3]) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ16_HTML.gif)
for every for which
. On the other hand, the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ17_HTML.gif)
holds for every nonnegative integer and every
for which
.
Let be a nondegenerate interval. Given an integer
, a function
is called Jensen convex of order
or Jensen
-convex if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ18_HTML.gif)
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:
Lemma 3.2.
Let be a nondegenerate interval, and let
be a nonnegative integer. Given two functions
, the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ19_HTML.gif)
holds for every such that
.
4. Main Results
Throughout this section is a positive real number. In the space
, consisting of all continuous functions
, we consider the usual maximum norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ20_HTML.gif)
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 a continuous function such that
for all
and the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ21_HTML.gif)
is convex of order for each
whenever
is convex of order
for each
;
there exists a continuous function which is nondecreasing in each variable and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ22_HTML.gif)
for all and all
;
is a continuous function such that the function
is convex of order
for each
whenever
and
;
there exists a continuous nondecreasing function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ23_HTML.gif)
there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ24_HTML.gif)
Theorem 4.1.
If the conditions ()–(
) are satisfied, then (1.1) possesses at least one solution
which is convex of order
for each
.
Proof.
Consider the operator , defined on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ25_HTML.gif)
Then whenever
(see [5, the proof of Theorem 3.2]).
We claim that is continuous on
. To this end we fix any
in
and prove that
is continuous at
. Let
, and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ26_HTML.gif)
Further, let . The uniform continuity of
on
as well as that of
on
ensures the existence of a real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ27_HTML.gif)
for all and all
satisfying
. Then for every
such that
and every
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ28_HTML.gif)
Therefore, the inequality holds for every
in
satisfying
. This proves the continuity of
at
.
Next, let be the positive real number whose existence is assured by (
), and let
be the subset of
, consisting of all functions
such that
and
is convex of order
for each
. Obviously,
is a nonempty bounded closed convex subset of
. We claim that
maps
into itself. To prove this, let
be arbitrarily chosen. For every
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ29_HTML.gif)
Since is convex of order
(i.e., nonnegative), according to (
) and (
) we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ30_HTML.gif)
This inequality and () yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ31_HTML.gif)
Taking into account that , by (
), (
), and (
) we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ32_HTML.gif)
On the other hand, for every we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ33_HTML.gif)
where are the functions defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ34_HTML.gif)
respectively. According to Lemma 3.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ35_HTML.gif)
for every and every
such that
. But
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ36_HTML.gif)
where . By virtue of (
) we have
, whence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ37_HTML.gif)
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.
Finally, we prove that the operator satisfies the Darbo condition with respect to the Hausdorff measure of noncompactness
. To this end let
be an arbitrary nonempty subset of
and let
. Further, let
and let
be such that
. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ38_HTML.gif)
Letting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ39_HTML.gif)
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ40_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ41_HTML.gif)
whence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ42_HTML.gif)
Taking into account that is uniformly continuous on
,
is uniformly continuous on
and
is uniformly continuous on
, we have that
,
and
as
. So letting
we obtain
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ43_HTML.gif)
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
.
Recall (see, e.g., Roberts and Varberg [14, pages 233-234]) that a function is called absolutely monotonic (resp., completely monotonic) if it possesses derivatives of all orders on
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ44_HTML.gif)
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.
is a given integer number and
is a sequence such that either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ45_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ46_HTML.gif)
belongs to
.
is a continuous function such that
for all
and the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ47_HTML.gif)
belongs to whenever
is a continuous function such that the function
belongs to
whenever
and
.
Theorem 4.2.
If the conditions ()–(
), (
), (
), and (
)-(
) are satisfied, then (1.1) possesses at least one solution
.
Proof.
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
.
Instead of the set , considered in the proof of Theorem 4.1, we take now
to be the subset of
consisting of all functions
such that
. Then
is a nonempty bounded closed convex subset of
. We claim that
maps
into itself. Indeed, according to (4.13) we have
whenever
satisfies
. On the other hand,
admits the representation (4.14), where
are defined by (4.15). Given any
, note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ48_HTML.gif)
whence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ49_HTML.gif)
for every such that
. By proceeding as in the proof of Theorem 4.1 one can show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ50_HTML.gif)
Therefore .
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
As an application of the results established in the previous section, in what follows we study the solvability of the integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ51_HTML.gif)
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 [15] or Banaś et al. [16], and the references therein).
We are going to prove that if , then (5.1) possesses at least one continuous nonnegative solution, which is nonincreasing and convex. To this end, we apply Theorem 4.2 for
and
. Take
,
,
and
. It is immediately seen that all the conditions (
)–(
), (
), (
), and (
) are satisfied if the functions
and
are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ52_HTML.gif)
respectively. It remains to show that () is satisfied, too. Taking into account the expressions of
and
, condition (
) is equivalent to the following statement. If
, then there exists an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ53_HTML.gif)
Clearly, such an must satisfy
. Let
be the functions defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ54_HTML.gif)
respectively. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ55_HTML.gif)
one can see that attains a maximum at
, the maximum value being
. On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F917614/MediaObjects/13663_2009_Article_1192_Equ56_HTML.gif)
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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Trif, T. Convex Solutions of a Nonlinear Integral Equation of Urysohn Type. Fixed Point Theory Appl 2009, 917614 (2009). https://doi.org/10.1155/2009/917614
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DOI: https://doi.org/10.1155/2009/917614