- Research Article
- Open Access
Generalized Bihari Type Integral Inequalities and the Corresponding Integral Equations
© László Horváth. 2009
- Received: 2 February 2009
- Accepted: 23 June 2009
- Published: 14 July 2009
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.
- Integral Equation
- Successive Approximation
- Banach Lattice
- Concave Function
- Measurable Subset
is a measure space;
is a function from into such that the following properties hold:
for every ,
if , then ,
is a function from into with the following conditions:
the functions and belong to
It may be noted that under the condition ( ) the function is increasing (see Lemma 2.1 for the justification).
By we designate the set of nonnegative integers.
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 . The special case , , of (1.1) is considered in , while the special case , , of (1.1) is discussed in . 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 ). 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.
If is a nonempty subset of such that for every , then let
and the equation (1.2) are defined.
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 ,
(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.
For every with , let
By ( ), is a nonnegative real number for every .
Now we are in a position to formulate the first main result.
Assume the conditions ( )–( ).
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.
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 .
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.
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.
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 .
Since (1.19) has the unique solution , , then the successive approximations do not converge to the solution of (1.19).
This section is devoted to some preparatory results. In the following three lemmas we establish some useful properties of concave functions.
If the function is concave, then is increasing.
it follows from (2.1) that if is large enough. This contradicts the range of .
(a) is strictly decreasing on ;
(b) for all ;
(c)If , and there is a such that , then for all .
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 ,
The proof is complete.
(a) is concave, and ;
(c)If and , then uniformly on as in ;
(d)the function defined on is upper semicontinuous.
It is obvious.
By (a), Lemma 2.2(b) and (a) give the result.
and this gives the result.
and the proof is complete.
The next result was proved in [8, Lemma (b)].
is -almost measurable on .
Assume the conditions ( )–( ). If is a nonempty subset of such that for every and , then .
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.
belongs to .
By Lemma 2.4, the function
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.
Suppose ( ), and let be a nonempty set from .
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 .
is an -normed Banach lattice, briefly, -space (see ).
If is a function and is a subset of the domain of , then the restriction of to is denoted by .
and therefore -almost everywhere on .
The proof is now complete.
The next result can be found in [11, Lemma ].
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 , .
Consider now the proof of Theorem 1.4.
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 ), by ( ):
First, we show that the function is -almost measurable on . It is an easy consequence of ( ) and Lemma 2.4 that the functions
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
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.
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
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
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
and the proof of the induction step is complete.
( ) Let
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
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
It remains to prove that is maximal. Let be a solution of (1.1), and let . If , then
That realization can be reached in finitely many steps.
If , then
is an appropriate solution.
If , then
Suppose and . Let
If , then the function
is a solution of (1.1) ( , , thus ) that agrees with on .
If , then we introduce the functions , ,
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 .
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
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.
This work was supported by Hungarian Foundation for Scientific Research Grant no. K73274.
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