Locally Lipschitz Composition Operators in Space of the Functions of Bounded κΦ-Variation

This operator is also called superposition operator or substitution operator or Nemytskij operator associated with h. In what follows, wewill refer to (1) as the autonomous case and to (2) as the nonautonomous case. For an extensive treatment of composition operator and function spaces we refer to the monographs Appell et al. [1], Appell and Zabrejko [2], and Runst and Sickel [3]. In 1984, Sobolevskij [4] proved the following statement: “the autonomous composition operator associated with h : R → R is locally Lipschitz in the space Lip[a, b] if and only if the derivative h󸀠 exists and is locally Lipschitz.” In recent articles Appell et al. [5] and Merentes et al. [6] obtained several results of the Sobolevskij type. As the authors explain in the introduction, the significance of these results lies in the fact that inmost applications tomany nonlinear problems it is sufficient to impose a local Lipschitz condition, instead of a global Lipschitz condition. In fact they proved that Sobolevskij’s result is valid in the spacesBV φ [a, b],HBV[a, b],


Introduction
( More generally, given ℎ : [, ]×R → R, we consider the operator , defined by  () := ℎ (,  ()) , ( ∈ [, ]) . ( This operator is also called superposition operator or substitution operator or Nemytskij operator associated with ℎ.In what follows, we will refer to (1) as the autonomous case and to (2) as the nonautonomous case.For an extensive treatment of composition operator and function spaces we refer to the monographs Appell et al. [1], Appell and Zabrejko [2], and Runst and Sickel [3].
In 1984, Sobolevskij [4] proved the following statement: "the autonomous composition operator associated with ℎ : R → R is locally Lipschitz in the space Lip[, ] if and only if the derivative ℎ  exists and is locally Lipschitz." In recent articles Appell et al. [5] and Merentes et al. [6] obtained several results of the Sobolevskij type.As the authors explain in the introduction, the significance of these results lies in the fact that in most applications to many nonlinear problems it is sufficient to impose a local Lipschitz condition, instead of a global Lipschitz condition.In fact they proved that Sobolevskij's result is valid in the spaces   [, ], [, ],   [, ], and Φ[, ].
Motivated by the work done in the papers [5,6], we establish a similar result to the one given by Sobolevskij, in the space of functions Φ[, ].
Although the composition operator (or Nemytskij operator) is very simple, it turns out to be one of the most interesting and important operators studied in nonlinear functional analysis; the behavior of this operator exhibits many surprising and even pathological features in various function spaces.For example, about 35 years ago Dahlberg [7] proved the following: for 1 ≤  ≤ ∞ and 1+(1/) <  < / integer, if  maps the Sobolev space    (R  ) into itself, then ℎ is a linear function.Among these pathologies there is one called degeneracy phenomenon, which states that the global Lipschitz condition necessarily leads to affine functions in various functions spaces.This property was first proved in [8] for the space Lip[, ].Additional information about the degeneracy phenomena can be found in [9,10].
This paper is organized as follows: Section 2 contains definitions, notations, and necessary background about the class of functions of bounded Φ-variation in the sense of Schramm-Korenblum; Section 3 contains the main theorem.Also in this section we state and prove a Helly-type theorem, which plays a crucial role in the demonstration of our Sobolevskij-type result.

Some Function Spaces
The concept of functions of bounded variation has been well known since C. Jordan gave the complete characterization of functions of bounded variation as a difference of two increasing functions in 1881.This class of functions exhibits so many interesting properties that it makes a suitable class of functions in a variety of contexts with wide applications in pure and applied mathematics [1,11].This notion of a function of bounded variation has been generalized by several authors.One of these generalized versions was given by Korenblum in 1975 [12].He considered a new kind of variation, called -variation, and introduced a function  for distorting the expression |  −  −1 | in the partition itself, rather than the expression |(  ) − ( −1 )| in the range.One advantage of this alternative approach is that a function of bounded -variation may be decomposed into the difference of two simpler functions called -decreasing functions.
where the supremum is taken over all partitions  of the interval [, ].In the case (; [, ]) < ∞ one says that  has bounded -variation on [, ] and one will denote by [, ] the space of functions of bounded -variation on [, ].
We will say that Φ is a Φ * -sequence if  +1 () ≤   () for all  and  and a Φ-sequence if in addition ∑    () diverges for  > 0.
From now on, all sequences considered in this work will be Φ-sequences.We will consider a nonoverlapping family of subintervals We may define, for  of bounded Φ-variation, the total Φ-variation of  by where the supremum is taken over all {  },   ⊆ [, ].
Hernández and Rivas (see [14]) showed that if Φ = {  } ≥1 is a Φ-sequence and Φ satisfies condition   2 , then  Φ [, ] is a linear space.We denote by Φ[, ] the collection of all functions  such that  is of bounded Φ-variation for some  > 0. S. K. Kim and J. Kim in 1986 [15] considered a bounded Φ-variation as follows.Let us consider  Φ () as a function of variable .
With this in mind, we define a norm in the space Φ 0 = { ∈ Φ | () = 0} as follows: We will consider the following norm in the space Φ[, ]: where By the above definition, we have the following.
that is, the Luxemburg norm is lower semicontinuous on Φ 0 .
The following lemma is basic for our main result.

Main Results
In the proof of the main result of this paper, we will employ a compactness result, for instance, Helly's selection principle or second Helly's theorem.Helly's theorem for functions of generalized variation has been of some importance for a long time.Helly's selection principle has been the subject of intensive research, and many applications, generalizations, and improvements of them can be found in the literature (see, e.g., [19][20][21] and the references therein).
In this part we will state and prove our main results.In the proof of our main result we make use of a Helly-type selection theorem for a Φ-decreasing function.
In the paper [22] Cyphert and Kelingos proved the same result for an arbitrary infinite family of functions defined on [0, 1] which is both uniformly bounded and uniformly decreasing.
Theorem 16 (Helly-type selection theorem).An arbitrary infinite family of functions defined on [0, 1] which is both uniformly bounded and uniformly Φ-decreasing contains a subsequence which converges at every point of [0, 1] to a Φdecreasing function.
Using (17) we can, by means of the standard Cantor diagonalization technique, find a sequence of functions   in F which converges pointwise at each rational point of [0, 1], to a function .Since each   satisfies (18), so does , for all rational numbers ,  ∈ [0, 1].Define  at irrational points  by The existence of this limit can be seen as follows: Let {  } and {   } be two sequences of rational points converging to , arranged so that  1 <   1 <  2 <   2 < ⋅ ⋅ ⋅ <  and such that (  ) →  and (   ) →  as  → ∞.Then Then   ( − ) = 0, and hence  = .
From (19) we obtain, by taking limits of rational points in inequality (18), that  satisfies (18) for all pairs of positive real numbers; that is,  is Φ-decreasing with constant  on [0, 1].By Theorem 13  is of bounded Φ-variation and  is continuous.Hence, by another Cantor diagonalization process, a convergent subsequence of the functions   can be found.Now, let us consider 0 <  < 1 and  > 0.Then, we fix two rational numbers  1 and  2 with  1 <  <  2 such that Since the sequence {  },  = 1, 2, . .., converges to  in the rational numbers, there exists  > 0 such that Now, from ( 22) and ( 23) we obtain We are now in a position to formulate and prove our main result.

Theorem 17. Let us suppose that the composition operator 𝐻 associated with ℎ maps the space 𝜅Φ𝐵𝑉[𝑎, 𝑏] into itself. Then 𝐻 is locally Lipschitz if and only if ℎ 󸀠 exists and is locally Lipschitz in R.
Proof.First let us assume that ℎ  is locally Lipschitz in R. Given  ∈ Φ[, ], for  > 0, we denote by  1 () the minimal Lipschitz constant of ℎ  and by  2 () the supremum of |ℎ  | on the bounded set The finiteness of  2 () implies that  satisfies a local Lipschitz condition with respect to the norm ‖ ⋅ ‖ ∞ , so we only have to prove a local Lipschitz condition for  with respect to the Φ-variation norm.We will prove this by applying twice the mean value theorem.