Functional differential inequalities of parabolic type
Abstract
We prove a theorem wich generalizes a result of J. Szarski (see [4];Theor. 2)concerning weak inequalities fora diagonal system of second order differential functional inequalities of hetype(Error rendering LaTeX formula)
, assuming that
is parabolic with respect to u for any
. After introducing the definition of left parabolic (or right parabolic) function with respect to another one, we obtain the over mentioned generalization (Theor. 2.2) as a consequence of a theorem about strong inequalities (Theor. 2.1) which is a generalization of Theor.1 of [4] in the case of left parabolic (or right parabolic) functions. These generalizations have been suggested by the following example in wich we have the assertion of Theor. 2 of [4] even if hypotheses of the theorem are not al1 verified. Consider the function f defined as (l)(Error rendering LaTeX formula) for(Error rendering LaTeX formula) belonging to the set of real and symmetried 2 x 2 matrices and z continuous function in Đ, with continuous in D partial second derivatives with respect to x as well as functions u and v defined assuming
and
for every
.
![i = 1,...,m](http://siba-ese.unile.it/plugins/generic/latexRender/cache/4a7a425bd2ba4b5f814a69572cb51191.png)
![f<sup>i</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/65b3eef07224d2ad290948513b006bc2.png)
![i=1,...,m](http://siba-ese.unile.it/plugins/generic/latexRender/cache/a406535f07f7a21d9353017ebc27b9cd.png)
![u(t,x) = t (x_{1}<sup>2</sup> + x_{2}<sup>2</sup> - 1)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/2fcdedcf6ed3a1d9a756eee5e5751c94.png)
![v(t,x) = 0](http://siba-ese.unile.it/plugins/generic/latexRender/cache/32e9318360c25cec84931f7decbd1c40.png)
![(t,x) ∈ D](http://siba-ese.unile.it/plugins/generic/latexRender/cache/cfe75582fab4a89d035974149520c481.png)
DOI Code:
10.1285/i15900932v9n2p173
Full Text: PDF