Mixed integral inequalities and their applications
Lechosław Hącia 

Poznań University of Technology,  Institute of Mathematics  
e-mail: lhacia@math.put.poznan.pl 

Abstract 

The presented paper is connected with the special class of integral inequalities which can be applied to study behaviour properties of solutions for some boundary problems of hyperbolic differential equations. 

1. Main results 

Theorem.  If c,, , and  a, b  are continuous and  c is nondecreasing in , then inequality
  (1)
implies
,
 (a) for 
,
if a is nonincreasing and positive and b is nondecreasing and nonnegative;
(b) for 
,
if a is nondecreasing and nonnegative and  b nonincreasing and positive.
Here:                             
.
.
.
Sketch of the proof:  Denote a function , which ,by assumptions, is nonnegative, nondecreasing and continuous in D.
Then inequality (1) is of the form 
.
Treating it as a one-dimensional inequality with respect to the variable y for every we get
,
if a is nonincreasing and positive and b is nondecreasing and nonnegative.
Similarly, treating it as a one-dimensional inequality with respect to the variable x for every we obtain 
,
if a is nondecreasing and positive and  b nonincreasing and positive. 
Hence
,
or 
.

Using Gronwall inequality we get
.
and
.

2. Conclusions

    The presented theorem generalizes results of papers [1-10] and it can be applied to study 
behaviour properties (uniqueness, stability, boundedness) of solutions for partial differential equations of the hyperbolic type.

References

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[7]	Hącia L., On Volterrra integral inequalities and their applications, International Journal of Mathematics and Mathematical Sciences, Vol.3 (2004), 117-134.
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