- Research Article
- Open access
- Published:
Generalized Bihari Type Integral Inequalities and the Corresponding Integral Equations
Journal of Inequalities and Applications volume 2009, Article number: 409809 (2009)
Abstract
We study some special nonlinear integral inequalities and the corresponding integral equations in measure spaces. They are significant generalizations of Bihari type integral inequalities and Volterra and Fredholm type integral equations. The kernels of the integral operators are determined by concave functions. Explicit upper bounds are given for the solutions of the integral inequalities. The integral equations are investigated with regard to the existence of a minimal and a maximal solution, extension of the solutions, and the generation of the solutions by successive approximations.
1. Introduction and the Main Results
In this paper we study integral inequalities of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ1_HTML.gif)
and the corresponding integral equations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ2_HTML.gif)
where
is a measure space;
is a function from
into
such that the following properties hold:
for every
,
if , then
,
is
-measurable;
is a function from
into
with the following conditions:
is concave,
;
the functions and
belong to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ3_HTML.gif)
It may be noted that under the condition () the function
is increasing (see Lemma 2.1 for the justification).
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_IEq29_HTML.gif)
always represents a -algebra in
. The
-integrable functions over a measurable set
are considered to be
-almost measurable on
. The product of the measure space
with itself is understood as in [1], and it is denoted by
.
By we designate the set of nonnegative integers.
Special cases of (1.1) seem first to have been investigated by Lasalle [2] and Bihari [3]. Bihari's classical result gives an explicit upper bound for the solutions of the integral inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ4_HTML.gif)
where ,
and
are nonnegative continuous functions on
, and
is a positive continuous and increasing function on
. A group of inequalities is now associated with Bihari's name. Results for the various forms of such inequalities and references to different works in this topic can be found in [4–6]. Bihari type inequalities have been widely studied because they can be applied in the theory of difference, differential and integral equations. Riemann or classical Lebesgue integral is used in most of the theorems in this area. There are relatively few papers using other types of integral. For generalizations to abstract Lebesgue integral; see [7–9]. The linear version of (1.1) is given in [7]. The special case
,
,
of (1.1) is considered in [8], while the special case
,
,
of (1.1) is discussed in [9]. It turns out to be useful to study Bihari type inequalities with abstract Lebesgue integral. It is motivated proceeding in this direction as follows. We can get new facts about the nature of Bihari type inequalities even in the finite dimensional environment; the results can be applied in the study of certain new classes of differential and integral equations (see [7–11]).
The traditional treatment assumes not only that , but also that the sets
,
are intervals, while the present treatment (it should be emphasized that the methods employed to establish our results are not usual in this topic) makes it possible to consider more general sets (examples for functions satisfying (A
) and (A
) can be found in [11]). Such results are not quite so easy to find in literature, although they can be used as powerful tools in many fields of mathematics.
Besides , the following function spaces will play an important role.
Definition 1.1.
If is a nonempty subset of
such that
for every
, then let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ5_HTML.gif)
Next, the basic concepts of the solutions of the inequalities (1.1) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ6_HTML.gif)
and the equation (1.2) are defined.
Definition 1.2.
We say that a function is a solution of (1.1), (1.6), or (1.2) if
(i) is a nonempty subset of
such that
for every
,
(ii),
(iii) satisfies (1.1), (1.6), or (1.2) for each
.
It is easily verified (see Lemma 2.5) that if is a solution of (1.1), (1.6), or (1.2), then
is
-integrable over
for all
.
After these preparations we set ourselves the task of obtaining an upper bound for the solutions of (1.1). The following definition will be useful.
Definition 1.3.
-
(a)
For every
with
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ7_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ8_HTML.gif)
By (),
is a nonnegative real number for every
.
Now we are in a position to formulate the first main result.
Theorem 1.4.
Assume the conditions ()–(
).
(a)Every solution of (1.1) satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ9_HTML.gif)
(b)The function defined on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ10_HTML.gif)
belongs to .
In the second main result we test the scope of the previous theorem by applying it to prove the existence of a maximal and a minimal solution of the integral equation (1.2). At the same time, we show that every solution has maximal domain of existence , and we apply the method of successive approximations to (1.2). Moreover, the behavior of the solutions is studied in a special case. The considered integral equations are in a very general form, there are classical Volterra and Fredholm type integral equations among them.
Theorem 1.5.
Suppose the conditions ()–(
).
There exists a solution of (1.2), which is minimal in the sense that
,
whenever
is a solution of (1.6).
There exists a solution of (1.2), which is maximal in the sense that
,
whenever
is a solution of (1.1).
If is a solution of (1.2), then
has an extension
to
that is a solution of (1.2) on
.
(c)Let be a solution of (1.1). Then the successive approximations determined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ11_HTML.gif)
are well defined, ,
; the sequence
is increasing and converges pointwise on
to a solution of (1.2).
(d)Let be a solution of (1.6). Then the successive approximations (1.11) determined by
are well defined,
,
, and the sequence
is decreasing. Moreover, if either
is continuous (at
) or
, then they converge pointwise on
to a solution of (1.2).
(e)If in addition and
are bounded on
for all
, then every solution
of (1.2) is bounded on
for all
.
We conclude this section with some remarks.
Remark 1.6.
The next example shows that the concavity of alone does not imply neither the existence of an upper bound for the solutions of the integral inequality (1.1) nor the existence of a solution of the integral equation (1.2). Consider the integral inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ12_HTML.gif)
and the corresponding integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ13_HTML.gif)
where is the unit mass at
defined on the
-algebra of Borel subsets of
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ14_HTML.gif)
Then the conditions ()–(
) are satisfied without (
). It is obvious that the functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ15_HTML.gif)
are solutions of (1.12), showing that there are no either global or local upper bounds for the solutions of (1.12). It is easy to check that (1.13) has no solution.
Remark 1.7.
It is illustrated by an example that under the conditions ()–(
) the maximal domain of existence of a solution of (1.1) may be a proper subset of
. Let
, let
be the Lebesgue measurable subsets of
, and let
be the Lebesgue measure on
. The function
is defined on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ16_HTML.gif)
The functions and
are defined on
by
. Suppose
,
. Then (
)–(
) are satisfied. Let
be a non (Lebesgue) measurable function. Then
is a solution of (1.1) which has no extension to
.
Remark.
The following example makes it clear that some extra conditions for are necessary in Theorem 1.5(d). Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ17_HTML.gif)
let be the power set of
, and let the measure
be defined on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ18_HTML.gif)
where the measure is the unit mass at
defined on
. We consider the integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ19_HTML.gif)
where ,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ20_HTML.gif)
The conditions ()–(
) can be immediately verified. Define the function
by
. Noting that
leads to the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ21_HTML.gif)
A few easy calculations imply that, for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ22_HTML.gif)
and therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ23_HTML.gif)
Since (1.19) has the unique solution ,
, then the successive approximations
do not converge to the solution of (1.19).
2. Preliminaries
This section is devoted to some preparatory results. In the following three lemmas we establish some useful properties of concave functions.
Lemma 2.1.
If the function is concave, then
is increasing.
Proof.
Suppose that there exist for which
. By the concavity of
, the points of the graph of
are below or on the ray from
through
for all
, and therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ24_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ25_HTML.gif)
it follows from (2.1) that if
is large enough. This contradicts the range of
.
Lemma 2.2.
Suppose the function is concave,
, and
. Associate to
the nonnegative real number
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ26_HTML.gif)
Then
(a) is strictly decreasing on
;
(b) for all
;
(c)If , and there is a
such that
, then
for all
.
Proof.
The hypotheses on (since
is concave,
is continuous on
) guarantee that exactly one of the following three cases holds:
(i) and
for all
;
(ii)there exists a such that
for all
and
for all
;
(iii)there is a unique such that
,
for all
and
for all
.
It follows that if (i) is satisfied, and
otherwise. At the same time (b) and (c) are proved.
It remains to show (a). If , then (i)–(iii) show that
. Assume
. By the concavity of
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ27_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ28_HTML.gif)
The proof is complete.
Lemma 2.3.
Suppose the function is concave, and
. Associate to
and to each of the nonnegative real numbers
,
the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ29_HTML.gif)
Then
(a) is concave, and
;
(b)If ,
, then
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ30_HTML.gif)
(c)If and
, then
uniformly on
as
in
;
(d)the function defined on
is upper semicontinuous.
Proof.
-
(a)
It is obvious.
By (a), Lemma 2.2(b) and (a) give the result.
The triangle inequality insures that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ31_HTML.gif)
Then from Lemma 2.1
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ32_HTML.gif)
and this gives the result.
To prove this, choose , and
. The definition of
and Lemma 2.2(a) imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ33_HTML.gif)
By (b), uniformly on
as
in
, and hence there exists a neighborhood
of
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ34_HTML.gif)
It now follows from Lemma 2.2 (a) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ35_HTML.gif)
and the proof is complete.
The next result was proved in [8, Lemma (b)].
Lemma 2.4.
Suppose that () and (
) hold. Let
such that
for every
. Suppose
is
-integrable over
,
is
-almost measurable on
, and there exists a measurable subset
of
such that
is
-finite and
for all
. Then the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ36_HTML.gif)
is -almost measurable on
.
Lemma 2.5.
Assume the conditions ()–(
). If
is a nonempty subset of
such that
for every
and
, then
.
Proof.
Let be fixed. Since
is increasing it is Borel measurable. Consequently, since
is
-almost measurable on
,
is
-almost measurable on
. By (A
), we can find
such that
for all
. Hence, note that
is increasing:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ37_HTML.gif)
It now follows from the definition of and from (
) that
is
-integrable over
. The proof is complete.
A consequence of the previous results that will be important later on is follows.
Lemma 2.6.
Suppose that ()–(
) hold. If
is a nonempty subset of
such that
for every
and
, then the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ38_HTML.gif)
belongs to .
Proof.
Let .
By Lemma 2.4, the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ39_HTML.gif)
is -almost measurable on
. Hence it follows from the
-integrability of
and
over
(the latter can be seen from Lemma 2.5), combined with the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ40_HTML.gif)
that the function (2.16) is -integrable over
. We conclude that the function (2.15) is
-integrable over
.
The proof is complete.
We need the concept of AL-space, which is of fundamental significance in the proof of Theorem 1.5.
Definition 2.7.
Suppose (), and let
be a nonempty set from
.
(a)Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ41_HTML.gif)
For a given , the symbol
is defined by
.
(b)Let , and let
. For every
, let
be the equivalence class containing
, and we set
.
(c)We introduce the canonical ordering on : for
,
, and
means that
-almost everywhere on
.
Remark 2.8.
-
(a)
is a complete pseudometric space.
is an
-normed Banach lattice, briefly,
-space (see [12]).
If is a function and
is a subset of the domain of
, then the restriction of
to
is denoted by
.
Lemma 2.9.
Suppose (), and let
and
be nonempty sets from
such that
. If
and
are nonempty majorized subsets of
such that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ42_HTML.gif)
Proof.
Since is an
-space, it is order complete (see [12]), and hence
and
exist. Let
and
. Then
-almost everywhere on
, thus the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ43_HTML.gif)
is an upper bound of . It follows that
, that is
-almost everywhere on
. An argument entirely similar to the preceding part gives that
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ44_HTML.gif)
and therefore -almost everywhere on
.
The proof is now complete.
The next result can be found in [11, Lemma ].
Lemma 2.10.
Assume that the hypotheses (), (
), (
), and (
) are satisfied. Let
. Suppose we are given solutions
,
of (1.2) such that
for each
,
with
. Then there exists exactly one solution
of (1.2) for which
,
.
3. Proofs of the Main Results
Consider now the proof of Theorem 1.4.
Proof.
-
(a)
If
such that
, then (1.9) follows directly from (1.1).
Now, fix a point with
. To estimate the second term on the right of (1.1), we can apply Jensen's inequality (see [13]), by (
):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ45_HTML.gif)
and therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ46_HTML.gif)
This inequality, together with Definition 1.3(a), implies that the expression
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ47_HTML.gif)
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ48_HTML.gif)
By Lemma 2.3 (b) and Definition 1.3 (b), , and hence (1.9) can be deduced from
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ49_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_IEq396_HTML.gif)
The properties defining are trivial if
with
. So assume
such that
.
First, we show that the function is
-almost measurable on
. It is an easy consequence of (
) and Lemma 2.4 that the functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ50_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ51_HTML.gif)
are -almost measurable on
. This means that there exists a measurable subset
of
such that
and (3.6), (3.7) are measurable on
. Further, since
is
-almost measurable on
, it can be supposed that
is measurable on
. Thus we need to show that the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ52_HTML.gif)
is measurable on . To prove this let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ53_HTML.gif)
The measurability of (3.6) on implies that
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ54_HTML.gif)
it is enough to show that (3.8) is measurable on . It follows from the definitions of
and
that the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ55_HTML.gif)
is measurable. Hence, by Lemma 2.3 (d), (3.8) is measurable on .
It is now clear that is
-almost measurable on
.
Next, we prove that is
-integrable over
.
To prove this, it is enough to show that the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ56_HTML.gif)
is bounded on . Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ57_HTML.gif)
we need to verify that (3.12) is bounded on . By (
), we can find a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ58_HTML.gif)
It therefore follows from
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ59_HTML.gif)
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ60_HTML.gif)
for all , and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ61_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ62_HTML.gif)
and therefore another application of (3.14) gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ63_HTML.gif)
and from this the claim follows. Consequently, is
-integrable over
, as required.
The result is completely proved.
Now we are in a position to prove Theorem 1.5.
Proof.
We begin with the proof of (c) and (d).
-
(c)
To prove that
we use induction on
. Clearly
belongs to
. Let
such that the assertion holds. Then Lemma 2.6 yields that
. We show now that the sequence
is increasing. By our hypotheses on
, it follows that
, and we again complete the proof by induction. Suppose
such that
. Then, by Lemma 2.1 and the induction hypothesis
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ64_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ65_HTML.gif)
Theorem 1.4(a) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ66_HTML.gif)
Because of (3.22) and the fact that is increasing, there exists a function
such that
converges pointwise on
to
. Then
is a consequence of
together with (3.22), Lemma 2.1(a) and, Theorem 1.4(b). If
with
, then according to the continuity of
on
,
converges to
. Let
with
. As we have seen
is increasing, and therefore
for all
. Consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ67_HTML.gif)
Now (1.11) and the monotone convergence theorem give that is a solution of (1.2).
-
(d)
We can see exactly as in the proof of (c) that
,
and the sequence
is decreasing. It follows from an easy induction argument that
is nonnegative for all
. Linking up with the foregoing, there exists a function
such that
converges pointwise on
to
.
can be shown as in the proof of (c).
The continuity of on
implies that
converges to
for every
.
Assume now that . If
such that
for every large enough
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ68_HTML.gif)
If such that
for every
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ69_HTML.gif)
which leads to . In both cases
converges to
for every
.
According to (1.11) and the monotone convergence theorem is a solution of (1.2).
() For convergence of the successive approximations
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ70_HTML.gif)
to a solution of (1.2) it suffices, in view of (c), to show that
is a solution of (1.1), which is evident. It remains to prove that if
is a solution of (1.6), then
for every
. To this end, it is enough to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ71_HTML.gif)
This is true for , since the functions
and
are nonnegative. Let
for which the result holds. Then, because of the nonnegativity of
and the fact that
is increasing,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ72_HTML.gif)
and the proof of the induction step is complete.
() Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ73_HTML.gif)
Choose . The set of the upper bounds from
for the solutions of (1.1) on
is denoted by
. By Theorem 1.4,
is not empty. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ74_HTML.gif)
Then is a minorized subset of
(the elements of
are nonnegative). Since this space is order complete (see [12]),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ75_HTML.gif)
exists. Let , and let
be a solution of (1.1) such that
.
,
gives that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ76_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ77_HTML.gif)
because is increasing. Since
the proof of Lemma 2.5 shows that
, and hence by (
), the function
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ78_HTML.gif)
belongs to , and therefore
. It now comes from (3.33) and the definition of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ79_HTML.gif)
Since is increasing,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ80_HTML.gif)
and thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ81_HTML.gif)
We can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ82_HTML.gif)
If we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ83_HTML.gif)
then (3.38) shows that is a solution of (1.2). (3.33) gives that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ84_HTML.gif)
for every solutions of (1.1) for which
.
Repeat the preceding construction for every . We thus obtain a set of functions
, each a solution of (1.2) on its domain. Moreover, if
is a solution of (1.1), then for every
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ85_HTML.gif)
Introduce the next functions: for every with
let the function
be defined on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ86_HTML.gif)
Obviously, these functions are also solutions of (1.2) on their domains.
Now let ,
such that
and
. Using (3.41), it is easy to verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ87_HTML.gif)
and therefore Lemma 2.10 is applicable to the solutions . This gives a unique solution
of (1.2) for which
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ88_HTML.gif)
It remains to prove that is maximal. Let
be a solution of (1.1), and let
. If
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ89_HTML.gif)
while if , then by (3.41)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ90_HTML.gif)
so that Lemma 2.1 implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ91_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_IEq552_HTML.gif)
We first show that every solution of (1.2) with
has an extension
that is a solution of (1.2) such that
is a proper subset of
. This follows from (c) as soon as it is realized that every solution
of (1.2) with
has an extension
that is a solution of (1.1) such that
is a proper subset of
. Really, in this case the successive approximations determined by
converge to a solution
of (1.2), which is obviously an extension of
.
That realization can be reached in finitely many steps.
Let be a solution of (1.2) with
, and let
.
-
(i)
Suppose
.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ92_HTML.gif)
is an appropriate solution.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ93_HTML.gif)
is a solution of (1.1) that agrees with on
.
-
(ii)
Suppose
and
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ94_HTML.gif)
If , then the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ95_HTML.gif)
is a solution of (1.1) (,
, thus
) that agrees with
on
.
If , then we introduce the functions
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ96_HTML.gif)
which all lie in , and they are all solutions of (1.1). By Theorem 1.4 (b),
is a majorized subset of
. Since this space is order complete (see [12]),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ97_HTML.gif)
exists. Choose . Then
-almost everywhere on
for every
, and therefore Lemma 2.1 yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ98_HTML.gif)
Since the proof of Lemma 2.5 shows that
, and hence by (A
), the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ99_HTML.gif)
belongs to too. It now follows from
and (3.54) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ100_HTML.gif)
We set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ101_HTML.gif)
By (3.56)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ102_HTML.gif)
Fix . From Lemma 2.9 (with
and
there being
and
resp.) we get that
-almost everywhere on
, and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ103_HTML.gif)
Consequently, ,
. If
with
, then
. We can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ104_HTML.gif)
By what we have already proved that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ105_HTML.gif)
is a solution of (1.1) that agrees with on
.
-
(iii)
Suppose
and
. By an argument entirely similar to that for the case (ii), we can get a solution
of (1.1) that agrees with
on
.
After these preparations, the proof can be concluded quickly. Let be a solution of (1.2), and let
be the set of all solutions of (1.2) which agree with
on
. Since
,
is not empty. Partially order
by declaring
to mean that the restriction of
to the domain of
agrees with
. By Hausdorff's maximality theorem, there exists a maximal totally ordered subcollection
of
. Let
be the union of the domains of all members of
, and define
by
, where
occurs in
. It is easy to check that
is well defined, and it is a solution of (1.2). If
were a proper subset of
, then the first part of the proof would give a further extension of
, and this would contradict the maximality of
.
-
(e)
Let
be a solution of (1.2), and let
.
If , then
, so that
is bounded on
.
Suppose . Since
and
are bounded on
and
is finite, Theorem 1.4 (a) implies that it is enough to prove the boundedness of the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ106_HTML.gif)
on . Moreover, we have only to observe that the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ107_HTML.gif)
is bounded on . To prove this, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F409809/MediaObjects/13660_2009_Article_1950_Equ108_HTML.gif)
If and
,
, then
and
for every
, and therefore by Lemma 2.3(d), the function (3.63) is bounded on
. If
, then the definition of the function
gives that
. The claim about
is therewith confirmed.
The proof of the theorem is now complete.
References
Horváth L: On the associativity of the product of measure spaces. Acta Mathematica Hungarica 2003,98(4):301–310. 10.1023/A:1022886211723
LaSalle J: Uniqueness theorems and successive approximations. Annals of Mathematics 1949, 50: 722–730. 10.2307/1969559
Bihari I: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Mathematica Academiae Scientiarum Hungaricae 1956, 7: 81–94. 10.1007/BF02022967
Bainov D, Simeonov P: Integral Inequalities and Applications, Mathematics and Its Applications (East European Series). Volume 57. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1992:xii+245.
Mitrinovic DS, Pecaric JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications (East European Series). Volume 53. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587.
Pachpatte BG: Integral and Finite Difference Inequalities and Applications, North-Holland Mathematics Studies. Volume 205. Elsevier Science, Amsterdam, The Netherlands; 2006:x+309.
Horváth L: Integral inequalities in measure spaces. Journal of Mathematical Analysis and Applications 1999,231(1):278–300. 10.1006/jmaa.1998.6251
Horváth L: Generalizations of special Bihari type integral inequalities. Mathematical Inequalities & Applications 2005,8(3):441–449.
Horváth L: Generalization of a Bihari type integral inequality for abstract Lebesgue integral. Journal of Mathematical Inequalities 2008,2(1):115–128.
Horváth L: Integral equations in measure spaces. Integral Equations and Operator Theory 2003,45(2):155–176. 10.1007/s000200300001
Horváth L: Nonlinear integral equations with increasing operators in measure spaces. Journal of Integral Equations and Applications 2005,17(4):413–437. 10.1216/jiea/1181075352
Schaefer HH: Banach Lattices and Positive Operators, Die Grundlehren der Mathematischen Wissenschaften, Band 21. Springer, New York, NY, USA; 1974:xi+376.
Hewitt E, Stromberg KR: Real and Abstract Analysis, Graduate Texts in Mathematics. Volume 25. Springer, Berlin, Germany; 1965.
Acknowledgment
This work was supported by Hungarian Foundation for Scientific Research Grant no. K73274.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Horváth, L. Generalized Bihari Type Integral Inequalities and the Corresponding Integral Equations. J Inequal Appl 2009, 409809 (2009). https://doi.org/10.1155/2009/409809
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/409809