Solutions of some functional equations in a class of generalized Hölder functions
Maria Lupa
Journal of Applied Mathematics and Computational Mechanics |
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SOLUTIONS OF SOME FUNCTIONAL EQUATIONS IN A CLASS OF GENERALIZED HÖLDER FUNCTIONS
Maria Lupa
Institute of Mathematics, Częstochowa
University of Technology
Częstochowa, Poland
maria.lupa@im.pcz.pl
Received: 17 October
2016; accepted: 15 November 2016
Abstract. The
existence and uniqueness of solutions a nonlinear iterative equation
in the class of -times
differentiable functions with the
-derivative satisfying
a generalized Hölder condition is considered.
Keywords: iterative functional equation, generalized Hölder condition
1. Introduction
In [1, 2]
the space (
) of
times differentiable functions with the
-the derivative satisfying generalized
-Hölder condition was introduced
and some of its properties proved. In the present paper we examine the
existence and uniqueness of solutions of a nonlinear iterative functional
equation in this class of functions. We apply some ideas from Kuczma [3],
Matkowski [4, 5] (see also Kuczma, Choczewski, Ger [6]), where differentiable
solutions, Lipschitzian
solutions, bounded variation solutions of different type of itrerative
functional equations were investigated.
2. Preliminaries
Consider non-linear functional equation
![]() | (1) |
where are given and
is a unknown function.
We accept the following notation: ,
- is the Banach space
of the r-time differentiable functions defined on the interval
with values in
, such that, for some
; its r-th derivative satisfies the
following
-Hölder condition
![]() |
where a fixed function satisfies
the following condition (see [1, 2]):
(Γ)
![]() ![]() ![]() ![]() ![]() |
We assume that
(i)
(ii)
(iii) fulfils the Lipschitz
condition in
(iv) there exists such that
, where
is
the n-th iteration function
(v) is analityc function at , where
is the solution of
equation
We
define functions by the formula
|
|
Lemma 1. [4]
By assumptions (i)-(iii), defined by (2) are of
the form:
1. for
![]() | (3) |
2. for
![]() ![]() | (4) |
where
![]() ![]() | (5) |
and are of the class
in I, for all
numbers
such that
Remark 1.
If (i)-(iii) are fulfilled, then given by
![]() ![]() |
fulfill -Hölder condition for
and Lipschitz
condition with respect to
in
[
. It means, that there
are positive constants
and
![]() |
such that for ,
we have
![]() ![]() |
Define the functions by the following
formulas:
![]() | (6) |
Remark 2.
The functions defined by (6) fulfill
-Hölder condition with respect to variable x in
I and Lipschitz condition with respect to the variable
in each
set
Remark 3.
If satisfy the assumptions
(i)-(iii) and
is a solution of
equation (1) then the derivatives
satisfy the system of
equations
![]() |
If assumptions (i)-(iv) are fulfilled and is a solution of
equation (1) in
, then the
numbers
![]() | (7) |
satisfy the system of equations
![]() | (8) |
where are defined by (2).
Remark 4.
Let be a solution of the
equation (1). Present
in the following form
![]() | (9) |
where and
Define the functions
![]() ![]() |
and for ,
![]() |
It follows from above definitions and equation (9) that 𝜓 satisfies the following equation
![]() |
It is easy to prove, that if assumptions
(i)-(iv) are fulfilled and are the solution of
equations (8), then the function
satisfies the equation
(1) in
and the condition (7)
if and only if the function
given by (9) belongs
to
and satisfies
![]() |
Thus, we assume that and consider the equation (1) whose solution satisfies the
condition
![]() |
Then system of equations (8) takes the following form
![]() |
3. Main result
Theorem 1.
If assumptions (i)-(iii) are fulfilled, is a monotone function in the interval I, the conditions (iv) and
(v) are fulfilled for
and
![]() | (10) |
![]() | (11) |
then equation (1) has exactly one solution satisfying the
condition
![]() | (12) |
Moreover, there exists a neighbourhood of the point
and the number
such that for a
function
, satisfying the
condition (12) and the inequality
, a sequence of
functions
![]() |
converges to a solution of (1) according to the
norm in the space
Proof.
From (v) we have in some neighbourhood of the point
.
Denote by
the radius of
convergence of this series. From (11) and from
the continuity of functions
and
, from definition of the function
there
exists a neighbourhood
of the point
and
such that
![]() | (13) |
From Remark 1, definition of and from (13) there are positive constants
and
, that in
we have
![]() ![]() | (14) |
From Remark 2, definition of there are in
constants
,
such that
![]() | (15) |
We accept the following notation:
![]() | (16) |
![]() | (17) |
![]() ![]() ![]() ![]() ![]() | (18) |
![]() | (19) |
![]() | (20) |
By we denote the sum of
for all
such that
In view of Lemma 1, we have
![]() |
and, from (13), we get
![]() | (21) |
Let us take and
![]() |
Put
|
|
Then let’s take such that
and
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | (23) |
Choose . Of course
. We will select a
neighborhood of zero
such that
and
.
Consider the Banach space with the norm:
![]() |
Let us define the set
![]() |
Note that is a closed subset of
Banach space
and for
the norm is expressed
by the formula
|
|
Thus, the set with the metric
ϱ(
is a complete metric
space.
By the mean value theorem and by definition of
the number of c we have for
![]() | (25) |
and so
For define the
transformation
by the
formula
![]() |
We will show that
Based on Remarks 1 and 3 the function belongs to
from (iv) and (10),
(12) appears that
. Then using the
formulas (12), (13), (22), (25) and the assumption (i) we obtain
![]() ![]() ![]() |
Which means from (24) that . Thus
.
Now we prove that T is a contraction map. Let us
put ,
.
Basing on formulas (4)-(5) of Lemma 1 and from (24) we have
![]() ![]() |
Note, that if , then in view of the
mean value theorem, from
the definition of the number
and from (i) we have
the following inequalities
![]() | (26) |
![]() | (27) |
![]() | (28) |
![]() | (29) |
By induction on we
also obtain:
![]() | (30) |
From (v) and by selection of we have uniform and absolute
convergence of
the series
![]() ![]() |
Let's consider the expression:
![]() |
From (30) we obtain
![]() |
Note that a series
![]() ![]() |
converges, because the numbers have been selected in such a way that
![]() |
Therefore
![]() | (31) |
Similarly for we
get
![]() | (32) |
By induction and from (26)-(29) we have
![]() | (33) |
![]() | (34) |
Now from (33) and (34) we get
![]() ![]() | (35) |
From (6), by the mean value theorem and from (33) and (34) we get
![]() | (36) |
Now, from (15)-(22), (27)-(32) and (36) we get
![]() ![]() |
Putting and making use
of definition (24) of the norm in
we have
![]() |
which means that ,
where
in view on (23).
By the
Banach fixed point theorem, there is exactly one solution of (1) satisfying the condition (12). This solution is given
as the limit of series of successive approximations.
![]() |
where . This sequence converges in the sense of the norm of
.
By Lemma 4 in [7], there exists the unique extension
of
to
the whole interval
such that
for
and
satisfies the equation (1) in
. This completes the proof.
Conclusions
In this paper, applying the Banach
contraction principle, a theorem on the existence and uniqueness of -solutions of nonlinear iterative
functional equation (1) has been proved. The suitable unique solution is
determined as a limit of sequence of successive approximations.
References
[1] Lupa M., A special case of generalized Hölder functions, Journal of Applied Mathematics and Computational Mechanics 2014, 13(4), 81-89.
[2] Lupa M., On a certain property of generalized Hölder functions, Journal of Applied Mathematics and Computational Mechanics 2015, 14(4), 127-132.
[3] Kuczma M., Functional Equations in a Single Variable, PWN, Warszawa 1968.
[4] Matkowski J., On the uniqueness of differentiable solutions of a functional equation, Bulletin de l’Academie des Sciences, Serie des sciences math., astr. et phys. 1970, XVIII, 5, 253-255.
[5] Matkowski J., On the existence of differentiable solutions of a functional equation, Bulletin de l’Academie des Sciences, Serie des sciences math., astr. et phys. 1971, XIX, 1, 19-21.
[6] Kuczma M., Choczewski B., Ger R., Iterative Functional Equations, Cambridge University Press, Cambridge-New York-Port Chester-Melbourne-Sydney 1990.
[7] Lupa M., On solutions of a functional equation in a special class of functions, Demonstratio Mathematica 1993, XXVI, 1, 137-147.