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.
1. Introduction and the Main Results
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.
and the equation (1.2) are defined.
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.
Now we are in a position to formulate the first main result.
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.
(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).
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.
This section is devoted to some preparatory results. In the following three lemmas we establish some useful properties of concave functions.
The proof is complete.
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)].
A consequence of the previous results that will be important later on is follows.
By Lemma 2.4, the function
The proof is complete.
We need the concept of AL-space, which is of fundamental significance in the proof of Theorem 1.5.
is an -normed Banach lattice, briefly, -space (see ).
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 , .
3. Proofs of the Main Results
Consider now the proof of Theorem 1.4.
Now, fix a point with . To estimate the second term on the right of (1.1), we can apply Jensen's inequality (see ), by ( ):
To prove this, it is enough to show that the function
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
and the proof of the induction step is complete.
Obviously, these functions are also solutions of (1.2) on their domains.
That realization can be reached in finitely many steps.
is an appropriate solution.
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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