Properties of entire solutions of some linear PDE's
Andriy Bandura
,Oleh Skaskiv
,Petro Filevych
Journal of Applied Mathematics and Computational Mechanics |
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PROPERTIES OF ENTIRE SOLUTIONS OF SOME LINEAR PDE'S
Andriy Bandura1, Oleh Skaskiv2, Petro Filevych3
1Department
of Advanced Mathematics, Ivano-Frankivsk National
Technical University of Oil
and Gas, Ivano-Frankivsk, Ukraine
2Department of Function Theory and Theory of Probability, Ivan
Franko National University of Lviv Lviv, Ukraine
3Department of Information Technologies, Vasyl Stephanyk
Precarpathional National University Ivano-Frankivsk, Ukraine
andriykopanytsia@gmail.com, olskask@gmail.com, filevych@mail.ru
Received: 22 April 2017; accepted: 15 May 2017
Abstract. In
this paper, there are improved sufficient conditions of boundedness of the
-index in a direction for entire
solutions of some linear partial differential equations. They are new even for
the one-dimensional case and
Also, we found a
positive continuous function
such that entire
solutions of the homogeneous linear differential equation with arbitrary fast
growth have a bounded
-index and estimated its
growth.
MSC 2010: 34M05, 34M10, 35B08, 35B40, 32A15, 32A17
Keywords: linear partial
differential equation, entire function, bounded -index
in direction, bounded
-index, homogeneous linear
differential equation, growth of solutions
1. Introduction
Let be a continuous function. An entire function
,
, is
called [1-4] a function of
bounded
-index in a direction
, if there exists
such that
(1)
for every and
every
, where
The
least such integer
is called the
-index in the direction
of the entire function
and is denoted by
In the case
we
obtain the definition of an entire function of one
variable of bounded
-index (see [5,
6]). And the value of the
-index
is denoted by
This paper is devoted to three problems in
theory of partial differential equations in and
differential equations in a complex plane.
At first, we consider the partial differential equation
![]() | (2) |
where
are entire functions in
There
are known sufficient conditions [1, 2, 4] of boundedness of the
-index in the
direction for entire solutions of (2). In particular, some inequalities must be
satisfied outside discs of any radius. Replacing the
universal quantifier by the existential, we relax the
conditions.
Also the ordinary differential equation
![]() | (3) |
is considered. Shah, Fricke,
Sheremeta, Kuzyk [6-8] did not
investigate an index boundedness of the entire solution
of (3) because the right hand side of (3)
is a function of two variables. But now in view of entire function theory
of bounded -index in direction,
it is natural to pose and to consider the following
question.
Problem 1 [3,
Problem 4]. Let
be
a function of bounded
-index in directions
and
What
is a function
such that an entire solution
of equation (3) has a bounded
-index?
Finally, we consider the linear homogeneous differential equation of the form
![]() | (4) |
which is obtained from (2), if
There is
a known result of Kuzyk and Sheremeta [5] about the growth of the entire
function of the bounded
-index. Later Kuzyk, Sheremeta [6] and Bordulyak [9] investigated the
boundedness of the
-index of entire solutions of equation
(4) and its growth.
Meanwhile, many mathematicians such as
Kinnunen, Heittokangas, Korhonen, Rättya, Cao, Chen, Yang, Hamani,
Belaїdi [10-14] used the
iterated orders to study the growth of solutions (4). Lin, Tu and Shi [15] proposed a more flexible scale to study the growth of solutions. They
used -order. But, the
iterated orders and
-orders
do not cover arbitrary growth (see example in [16]). There is considered a more general
approach to describe the relations between the growth of entire coefficients
and entire solutions of (4). In view of results from [16], the authors raise the question: what is a positive continuous function
such
that entire solutions of (4) with arbitrary fast growth have bounded
-index? We provide an answer to the question.
2. Auxiliary propositions and notations
For
and
positive continuous function
we define
By
we
denote a class of functions
which
satisfiy the condtion
For simplicity, we also
use a notation .
Theorem A [1, 4]. Let An entire in
function
is
of bounded
-index in direction
if and only if there exist numbers
and
,
and
such
that for all
![]() |
Let us to write
,
- zeros of the function
for a given
If
for all
then
we put
Theorem B [1,
4]. Let be an entire function of the bounded
-index in the
direction
Then
for every
and for every
there
exists
such that for all
.
Theorem C [1,
4]. Let be an entire function in
Then
the function
is of bounded
-index
in the direction
if and only if the following
conditions hold: 1) for every
there exists
such
that for each
2) for every
there exists
such that for every
Theorem D [1,
4]. Let . An entire function
has a
bounded
-index in direction
if and only if there exist
and
such
that
for each
.
Theorem E [17]. Let be a bounded closed domain in
be
a continuous function,
be an entire
function. Then there exists
such that for all
and for all
![]() |
Theorem F [5]. Let be a positive continuously
differentiable function
of real
Suppose that
as
where
If
an entire function
has a
bounded
-index then
3. Boundedness of L-index in direction of entire solutions of some linear partial differential equations
Denote
where
is a zero set of the function
The following theorem is valid.
Theorem 1. Let
, and
be entire
functions of the bounded
-index
in the direction
Suppose that there exist
and
such
that for each
and
![]() | (5) |
Then an entire
function satisfying (2) has bounded
-index
in the direction
Proof: Theorem C provides that and
Denote
Suppose
that
Theorem B and
inequality (5) imply that there exist
and
such that for all
![]() |
![]() |
By equation (2), we evaluate the derivative in the direction
![]() |
The obtained equality implies that for all :
![]() |
Thus, there exists
such that for all
![]() | (6) |
If then there exists a sequence of points
satisfying
(6) and such that
with as
Substituting
in
(6) and taking the limit as
we obtain that this
inequality is valid for all
If
(i.e. all zeros of
belong to
)
then by Theorem D the entire
function satisfying (2) has a bounded
-index in the direction
Otherwise,
. Since
and
then there exists
such that
Let
be an arbitrary point from
and
Since
the entire functions
have
a bounded
-index in the
direction
by Theorem C the set
contains at most
zeros of the functions or
Let
be
zeros of the slice function
(i.e.
) such that
where
Since
we have
Obviously,
![]() |
Thus, if then
(6) holds. Hence, for these points
the inequality
and (6) imply
![]() |
![]() | (8) |
where and
Let be
the sum of the diameters of
Then
Therefore, there exist numbers
and
such
that if
then
We choose arbitrary points
and
and
connect them by
a smooth curve
such that
and
This
curve can be selected such that
Then on
inequality (7) holds. It is easy to prove that the function
is continuous on
and continuously differentiable except
a finite number of points. Moreover, for a complex-valued function of real
variable the inequality
holds except
points, where
Then, in view of (7), we have
![]() ![]() |
where Integrating
over the variable
we deduce
i.e.
We
can choose
such that
Hence,
![]() | (9) |
Since and
for all
by
Cauchy's inequality in variable
we obtain
that is
(10)
Inequalities (9) and (10) imply that
![]() |
where Hence,
![]() ![]() |
Therefore, by Theorem A, the function
has a bounded
-index
in the direction
And by Theorem 3 from [1] the function
is of the bounded
-index in the direction
too.
Remark 1. We
require validity of (5) for
some but nor for all positive
Thus, Theorem 1 improves the
corresponding theorem from [1, 4]. The proposition is new even in the
one-dimensional case (see results for the bounded
-index
in [6] and bounded index in [8]).
4. Boundedness of l-index of entire
solutions of the equation
We denote
Theorem 2. Let
be
an entire function of bounded
-index in the
directions
for every
If
there exist
and
such
that
for all
, then any entire solution w(z) of (3) has a bounded l-index.
Proof: Differentiating (3) in variable and using Theorem B we obtain that for all
![]() |
Hence, This inequality is
similar to (6). Repeating arguments from Theorem 1, we
deduce that
has a bounded l-index.
Theorem 2 is proved.
As application of the theorem we consider the differential equation:
![]() ![]() | (11) |
Corollary 1.
Let be entire function of bounded
-index,
Then every entire function satisfying (11)
has
a bounded
-index.
5. The linear homogeneous differential equation with fast growing coefficients
As in [10], let be a
strictly increasing positive unbounded function on
be an inverse function to
We
define the order of the growth of an entire function
and the function
where
,
is chosen such that
And also we need the greater function
where
is chosen such that
Let
be the class of positive
continuously differentiable on
functions
such that
as
We need the following proposition of
Bordulyak:
Theorem G [9]. Let
and entire functions
satisfy the condition
for
all
If
an entire function
is a solution of (4) then
is of the bounded
-index and
![]() |
Theorem 3. Let
be a strictly increasing positive
unbounded function on
If
every entire function
has a bounded
-index (
)
then every entire function satisfying (4) has a bounded
-index. If, in addition,
is a
continuously differentiable function of real variable
then
![]() | (12) |
for every entire transcendental
function satisfying (4).
Proof: Since , the following inequalities hold
for
arbitrary
and
It means that
Denote
Hence, for
one
has
![]() |
i.e. (5) is valid for By Theorems D and 1 entire solutions of (4) have
a bounded
-index. It is easy to prove
that for all
Thus,
by Theorem 3 from [1], an entire function
satisfying (4) is of the
bounded
-index, too. The
function
is a strictly increasing and continuously differentiable function of a real
variable. Then
Furthermore,
as
Using
Theorem F we obtain (12).
Theorem 4. Let
be a strictly increasing positive
unbounded and continuously differentiable function on
If
as
then
every entire function satisfying (4) has a bounded
-index and
![]() |
Proof: At first, we prove that Indeed,
![]() |
where as
As
above, one has
Hence,
Thus,
and
satisfy
conditions of Theorem G with
Therefore, every entire function
satisfying (4) has a bounded
-index.
These theorems are a refinement of results of
M. Bordulyak, A. Kuzyk and
M. Sheremeta [6, 9]. Unlike these authors, we define the specific function such that entire solutions have a bounded
-index.
But the function
depends of the function
Below, we will
construct functions
and
for the entire transcendental function
of infinite order.
Theorem 5. For
an arbitrary continuous right differentiable on function
such that
there
exists a convex on
function
with the properties (i)
,
; (ii)
for
an unbounded from above set of values
.
Proof: For a given we
put
and
Clearly, the function
is continuous on
and
is fully
contained in a range of this function. For every
there exists
such
that
and
for
all
. Given the above, it is easy to justify
the existence of increasing to
sequence
, for which: 1)
; 2) a sequence
is increasing to
, where
for
every
; 3)
for all
and
every
.
Let and
for
.
Clearly, that
is a nondecreasing on
function. Hence, a function
is convex on
. For this function
we have
and for every
i.e. (ii) holds. If for
some
, then we obtain
(i):
![]() |
This follows from Theorem 5 that ,
.
Theorem 6. For
an arbitrary entire transcendental function of
infinite order there exists a convex on
function
such that 1)
2)
for
an unbounded from the above set of values
3)
Proof: We put ,
.
Since
is of infinite order, it follows
.
Let
be a function constructed for the
function
in Theorem 5. Denote
. Then
. It means that
the function
is a convex increasing on half-bounded interval
, where
. We
put
for
and
for
. By Theorem 5
assumptions 1) and 2) hold. We also obtain
. Therefore, 3) is true.
Let where
is chosen such that
Theorem 7. For
an arbitrary entire transcendental function of
infinite order there exists a strictly increasing positive unbounded and
continuously differentiable function
on
with
And
if
where
then
Proof: In view of Theorem 6
we choose where
is
an inverse function to
Then
It is obvious that the function
is a strictly increasing positive unbounded and
Besides,
is a
continuously differentiable function except for the
points of discontinuity of
We
estimate a logarithmic derivative of
:
![]() |
where as
It implies that
.
6. Conclusions
Note that a concept of the bounded -index in a direction has a few
advantages in the comparison with traditional approaches to study the
properties of entire solutions of differential equations. In particular, if an
entire solution has a bounded index, then it immediately yields its growth
estimates, a uniform in a some sense distribution of its zeros, a certain
regular behavior of the solution, etc.
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