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 nonlinear functional equation
(1) 
where are given and is a unknown function.
We accept the following notation: ,  is the Banach space of the rtime differentiable functions defined on the interval with values in , such that, for some ; its rth derivative satisfies the following Hölder condition
where a fixed function satisfies the following condition (see [1, 2]):
(Γ) is increasing and concave, γ(0) = 0,, 
We assume that
(i)
(ii)
(iii) fulfils the Lipschitz condition in
(iv) there exists such that, where is the nth 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) 
; is a Hölder constant of in  (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
for . 
Let's consider the expression:
From (30) we obtain
Note that a series
where 
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), 8189.
[2] Lupa M., On a certain property of generalized Hölder functions, Journal of Applied Mathematics and Computational Mechanics 2015, 14(4), 127132.
[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, 253255.
[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, 1921.
[6] Kuczma M., Choczewski B., Ger R., Iterative Functional Equations, Cambridge University Press, CambridgeNew YorkPort ChesterMelbourneSydney 1990.
[7] Lupa M., On solutions of a functional equation in a special class of functions, Demonstratio Mathematica 1993, XXVI, 1, 137147.