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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@article{Bandura_2017, doi = {10.17512/jamcm.2017.2.02}, url = {https://doi.org/10.17512/jamcm.2017.2.02}, year = 2017, publisher = {The Publishing Office of Czestochowa University of Technology}, volume = {16}, number = {2}, pages = {17--28}, author = {Andriy Bandura and Oleh Skaskiv and Petro Filevych}, title = {Properties of entire solutions of some linear PDE's}, journal = {Journal of Applied Mathematics and Computational Mechanics} }
TY - JOUR DO - 10.17512/jamcm.2017.2.02 UR - https://doi.org/10.17512/jamcm.2017.2.02 TI - Properties of entire solutions of some linear PDE's T2 - Journal of Applied Mathematics and Computational Mechanics JA - J Appl Math Comput Mech AU - Bandura, Andriy AU - Skaskiv, Oleh AU - Filevych, Petro PY - 2017 PB - The Publishing Office of Czestochowa University of Technology SP - 17 EP - 28 IS - 2 VL - 16 SN - 2299-9965 SN - 2353-0588 ER -
Bandura, A., Skaskiv, O., & Filevych, P. (2017). Properties of entire solutions of some linear PDE's. Journal of Applied Mathematics and Computational Mechanics, 16(2), 17-28. doi:10.17512/jamcm.2017.2.02
Bandura, A., Skaskiv, O. & Filevych, P., 2017. Properties of entire solutions of some linear PDE's. Journal of Applied Mathematics and Computational Mechanics, 16(2), pp.17-28. Available at: https://doi.org/10.17512/jamcm.2017.2.02
[1]A. Bandura, O. Skaskiv and P. Filevych, "Properties of entire solutions of some linear PDE's," Journal of Applied Mathematics and Computational Mechanics, vol. 16, no. 2, pp. 17-28, 2017.
Bandura, Andriy, Oleh Skaskiv, and Petro Filevych. "Properties of entire solutions of some linear PDE's." Journal of Applied Mathematics and Computational Mechanics 16.2 (2017): 17-28. CrossRef. Web.
1. Bandura A, Skaskiv O, Filevych P. Properties of entire solutions of some linear PDE's. Journal of Applied Mathematics and Computational Mechanics. The Publishing Office of Czestochowa University of Technology; 2017;16(2):17-28. Available from: https://doi.org/10.17512/jamcm.2017.2.02
Bandura, Andriy, Oleh Skaskiv, and Petro Filevych. "Properties of entire solutions of some linear PDE's." Journal of Applied Mathematics and Computational Mechanics 16, no. 2 (2017): 17-28. doi:10.17512/jamcm.2017.2.02
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,
Denoting we obtain |
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.
References
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