Uniformly bounded Nemytskij operators acting between the Banach spaces of generalized Hölder functions
Maria Lupa
,Małgorzata Wróbel
Journal of Applied Mathematics and Computational Mechanics 
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@article{Lupa_2017, doi = {10.17512/jamcm.2017.4.04}, url = {https://doi.org/10.17512/jamcm.2017.4.04}, year = 2017, publisher = {The Publishing Office of Czestochowa University of Technology}, volume = {16}, number = {4}, pages = {3745}, author = {Maria Lupa and Małgorzata Wróbel}, title = {Uniformly bounded Nemytskij operators acting between the Banach spaces of generalized Hölder functions}, journal = {Journal of Applied Mathematics and Computational Mechanics} }
TY  JOUR DO  10.17512/jamcm.2017.4.04 UR  https://doi.org/10.17512/jamcm.2017.4.04 TI  Uniformly bounded Nemytskij operators acting between the Banach spaces of generalized Hölder functions T2  Journal of Applied Mathematics and Computational Mechanics JA  J Appl Math Comput Mech AU  Lupa, Maria AU  Wróbel, Małgorzata PY  2017 PB  The Publishing Office of Czestochowa University of Technology SP  37 EP  45 IS  4 VL  16 SN  22999965 SN  23530588 ER 
Lupa, M., & Wróbel, M. (2017). Uniformly bounded Nemytskij operators acting between the Banach spaces of generalized Hölder functions. Journal of Applied Mathematics and Computational Mechanics, 16(4), 3745. doi:10.17512/jamcm.2017.4.04
Lupa, M. & Wróbel, M., 2017. Uniformly bounded Nemytskij operators acting between the Banach spaces of generalized Hölder functions. Journal of Applied Mathematics and Computational Mechanics, 16(4), pp.3745. Available at: https://doi.org/10.17512/jamcm.2017.4.04
[1]M. Lupa and M. Wróbel, "Uniformly bounded Nemytskij operators acting between the Banach spaces of generalized Hölder functions," Journal of Applied Mathematics and Computational Mechanics, vol. 16, no. 4, pp. 3745, 2017.
Lupa, Maria, and Małgorzata Wróbel. "Uniformly bounded Nemytskij operators acting between the Banach spaces of generalized Hölder functions." Journal of Applied Mathematics and Computational Mechanics 16.4 (2017): 3745. CrossRef. Web.
1. Lupa M, Wróbel M. Uniformly bounded Nemytskij operators acting between the Banach spaces of generalized Hölder functions. Journal of Applied Mathematics and Computational Mechanics. The Publishing Office of Czestochowa University of Technology; 2017;16(4):3745. Available from: https://doi.org/10.17512/jamcm.2017.4.04
Lupa, Maria, and Małgorzata Wróbel. "Uniformly bounded Nemytskij operators acting between the Banach spaces of generalized Hölder functions." Journal of Applied Mathematics and Computational Mechanics 16, no. 4 (2017): 3745. doi:10.17512/jamcm.2017.4.04
UNIFORMLY BOUNDED NEMYTSKIJ OPERATORS ACTING BETWEEN THE BANACH SPACES OF GENERALIZED HÖLDER FUNCTIONS
Maria Lupa, Małgorzata Wróbel
Institute of Mathematics, Czestochowa
University of Technology
Czestochowa, Poland
maria.lupa@im.pcz.pl, malgorzata.wrobel@im.pcz.pl
Received: 23 October 2017;
Accepted: 13 November 2017
Abstract. We investigate the Nemytskij (composition, superposition) operators acting between Banach spaces of times differentiable functions defined on the closed intervals of the real line with the derivatives satisfying a generalized Hölder condition. The main result says that if such a Nemytskij operator is uniformly bounded (in a special case uniformly continuous) then its generator is an affine function with respect to the second variable, i.e., the Matkowski representation holds. This extends an earlier result where an operator is assumed to be Lipschitzian.
MSC 2010: 47H30, 26A11
Keywords: Nemytskij (composition, superposition) operator, generalized Hölder condition, uniformly bounded mapping, uniformly continuous mapping
1. Introduction
The Nemytskij operators occur frequently in the theory of integral equations, differential equations, variational calculus or in optimization theory. There is vast literature devoted to applications of the results connected with the theory of the composition operators (see, for instance [1], where such operators occur in the problems of option pricing within the BlackScholes model for timedependent volatility).
Let be an interval of the real line () and let denote the set of all functions For a given function , the mapping defined by
, 
is called a composition (Nemytskij or superposition) operator of a generator.
In 1982 J. Matkowski [2] proved that if H maps a Banach space of Lipschitzian functions with the classical norm into itself and is globally Lipschitzian, i.e., if for some constant ,
then
(1) 
for some functions i.e., h is an affine function with respect to the second variable. Analogous results for representations of the generators of globally Lipschitzian operators have been proved for some other function spaces [37]. Later, it was observed that these results remain true if the Lipschitz normcontinuity of H is replaced by its uniform continuity [8, 9].
In 2011, J. Matkowski [10] proved, under very general assumptions, that for the function spaces including the Hölder spaces as a special case, the uniform continuity of the operator H can be replaced by a much weaker condition of the uniform boundedness (which is weaker than normboundedness), see also [11, 12].
The purpose of this paper is to show that if maps the space of times differentiable functions with the derivative satisfying a generalized Hölder condition into , where and is uniformly bounded, then the Matkowski representation holds, i.e., the generator of must be of the form (1) for some . This extends the main result of [5, 6] where is assumed to be Lipschitzian.
2. Preliminaries
Given a closed interval ( put . Let following condition:
(Γ) is increasing and concave, γ(0) = 0, ,
Given , denote by the set of all rtimes differentiable functions such that their rth derivatives satisfy the following condition (so called the generalized Hölder condition or the Hölder condition): there exists a constant such that
(2) 
where satisfy condition (Γ).
It is known that the space endowed with the norm


is a Banach space [3, 4].
Let , stand for the class of all rtimes differentiable functions , whose rth derivatives satisfy the Lipschitz condition on .
Remark 2.1. Observe, that if with the constant , then fulfils (2) with the constant .
Indeed, by , we get the monotonicity of the function

and, consequently, we have
for all
Remark 2.2 ([6], Remark 1). If and is finite, denotes the right derivative of γ at then
Remark 2.3. Let us notice that the function defined by , where , , fulfils the assumption and .
Definition 2.1. Let and a function be fixed. The mapping given by
, 
is said to be a composition (Nemytskij or superposition) operator. The function h is referred to as the generator of the operator H.
Assume now, that the function h fulfils the following condition:
(i) is times differentiable in , , and all its rth order partial derivatives fulfill the generalized Hölder conditions with respect to the first variable and Lipschitz condition with respect to the second variable.
Define, by recurrence, functions as follows:


Before formulating the main results of our paper, we will need the following two lemmas.
Lemma 2.1 ([6], Lemma 1). If the assumption (i) is fulfilled, then the functions , defined by (4), are times continuously differentiable in and, for , we have

where


and is a polynomial of the variables whose coefficients are functions of the variables and are of the class with respect to and .
Lemma 2.2 ([6], Lemma 2). If the assumption (i) is fulfilled, and
=, , 
then the derivatives of fulfil the following system of equations:
, 
and
3. Main results
Theorem 3.1. Let be fixed and let a function be such that for any the function is rtimes differentiable and its rth derivatives satisfy the Lipschitz condition on If the composition operator H of the generator h maps the space into and satisfies the inequality
(6) 
for some function then there exist and such that
Proof. For every fixed the function , , as a constant, is times differentiable and its nth derivatives satisfy the generalized Hölder condition, so, by assumption, Thus, taking into account the assumption, it follows that the function h fulfils (i).
Let us fix and define the function
It is obvious that with the constant , so, by Remark 1, satisfy the generalized Hölder condition with the constant
Let us take arbitrary , . Putting


we get


and
(9) 
for all
Hence, , , and, by the definition of the function , for all , we obtain
, ,  (10) 
and
, ,  (11) 
Since, by (8),
and, by (9),
therefore


Setting , where are defined in Lemma 1, inequality (6) has the following form





for all
Omitting the sign supremum on the left hand side of the above inequality and putting , by (10), (11), we get




and, consequently,



Hence, letting in so obtained inequality, by the righthand side continuity in zero of and (12), we have
whence,
Thus is linear with respect to the second variable and, consequently, there exist functions such that
(13) 
Since for all (13),
therefore , which completes the proof.
In the sequel the following definition plays a crucial role:
Definition 3.1 ([10], Definition 1). Let and be two metric spaces. We say that a mapping is uniformly bounded if, for any, there exists a real number such that for any nonempty set , we have
. 
The main result of this paper reads as follows:
Theorem 3.2. Let be fixed and let a function be such that for any the function is rtimes differentiable and its rth derivatives satisfy the Lipschitz condition on If the composition operator of the generator maps the space into and is uniformly bounded, then there exist and such that
and
, ( 
Proof. Take any and arbitrary such that
. 
Since by the uniform boundedness of , we have,
that is
and the result follows from Theorem 1.
Remark 3.1. If the function in Definition 3.1 is right continuous at and (or if only ), then, clearly, the uniform boundedness of the involved operator reduces to its uniform continuity.
Remark 3.2. Taking in Theorem 3.2 such that is finite, , for some constant , we get the main result of Knop [5].
Remark 3.3. Theorem 3.2 also extends the result of Lupa [6], where is such that (0) , ( and .
4. Conclusions
We show that if a Nemytskij composition operator mapping the space of times differentiable functions with the th derivative satisfying a generalized Hölder condition into is uniformly bounded or uniformly continuous, then its generator is an affine function with respect to the second variable. This significantly extends the result of [5, 6] where much stronger Lipschitz norm continuity of is assumed.
References
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