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In this paper, based on the so-called degree of nondensifiability (DND), we introduce the concept of DND-convex-power condensing mapping which generalizes that of the DND-condensing one. A fixed point result for this new class of mappings is proved, and with some examples we evidence the differences between it and other fixed point theorems of the same type. As application of our results, we prove under suitable conditions the existence of fixed point for the superposition operator, defined in the Banach spaces of the continuous functions from [ 0 , 1 ] and with values in a Banach space. [ABSTRACT FROM AUTHOR]

Let H(D) be the set of all analytic functions on the open unit disk D of C, u є H(D) and Φ an entire function on C. In this paper, we characterize the boundedness and compactness of the weighted differentiation superposition operator Dum SΦ from H∞ to the nth weighted-type space. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if a globally Lipschitz non-autonomous superposition operator maps the space of functions of bounded second $ \kappa $-variation into itself then its generator function satisfies a Matkowski condition. We also provide conditions for the existence and uniqueness of solutions of the Hammerstein and Volterra equations in this space.

Let φ be an entire function, the superposition operator is defined by S φ (f) = φ ∘ f . In this paper, we characterize the entire functions φ that transform weighted Bloch spaces of analytic functions B μ into another space of the same kind B ν by superposition. Both μ and ν are normal functions or both belong to a certain class of functions. We also obtain several results about the boundedness of superposition operators acting between Logarithmic-type Bloch spaces, weight Banach spaces among others. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if a globally Lipschitz non-autonomous superposition operator maps the space of functions of bounded second 1-variation into itself then its generator function satisfies a Matkowski condition. We also provide conditions for the existence and uniqueness of solutions of the Hammerstein and Volterra equations in this space. [ABSTRACT FROM AUTHOR]

In this paper we characterize the nonlinear superposition operators mapping the space \mathcal {Q}_s into the Dirichlet-type space \mathcal {D}^p_\alpha and answer basically the remaining open cases provided by Galanopoulos, Girela, and Auxiliadora Márquez [J. Math. Anal. Appl. 463 (2018), pp. 659–680]. Meanwhile, we give some sufficient-necessary conditions for the superposition operators between \mathcal {Q}_s spaces in all cases. [ABSTRACT FROM AUTHOR]

In this article, the author studies the superposition operator and Strauss lemma in Besov spaces with logarithmic smoothness. As consequences, the author generalizes the corresponding classical results, especially in critical case. [ABSTRACT FROM AUTHOR]

Let Ω be an open subset of ℝn of finite measure. Let f be a Borel measurable function from ℝ to ℝ. We prove necessary and sufficient conditions on f in order that the composite function Tf[g] = f o g belongs to the Grand Lebesgue space Lp),θ(Ω) whenever g belongs to Lp),θ(Ω). We also study continuity, uniform continuity, Hölder and Lipschitz continuity of the composition operator Tf[·] in Lp),θ(Ω). [ABSTRACT FROM AUTHOR]

Let Ω be an open subset of Rn. Let f be a Borel measurable function from R to R. We prove necessary and sufficient conditions on f in order that the composite function Tf OEg? D f ı g belongs to a generalized Morrey space Mw p Ω whenever g belongs to Mw p .Ω. Then we prove necessary conditions and sufficient conditions on f in order that the composition operator Tf OE be continuous, uniformly continuous, Hölder continuous and Lipschitz continuous in Mw p .Ω. We also consider its 'vanishing' generalized Morrey subspace M w;0 p .Ω and prove the related results for the composition operator as operator acting from M w;0 p .Ω to Mw p .Ω and also between the spaces M w;0 p.Ω. For the uniform, Hölder and Lipschitz continuity, we also have conditions that are both necessary and sufficient. We also have both necessary and sufficient conditions for the continuity under certain additional natural assumptions. We also consider the most commonly used Morrey classes that are related to power-type weights in the context of a discussion of some of the conditions that we impose on the weights. [ABSTRACT FROM AUTHOR]

Our goal of this article is to introduce a new product operator that will be called $ {D^n_u} S_{\phi} $ the product of weighted differentiation and superposition operators from $ {H}^{\infty} $ to Zygmund spaces. Moreover, we characterize a necessary and sufficient conditions for $ {D^n_u} S_{\phi} $ operators from $ {H}^{\infty} $ to Zygmund spaces to be bounded and compact.

A certain class Φ of functions of two variables is introduced in connection with the general superposition operator and with the extended notion of sup-measurability. Some cases are considered where there exist sup-measurable functions of two variables which do not belong to the class Φ. [ABSTRACT FROM AUTHOR]

We prove that a function in several variables is in the local Zygmund class Zm,1 if and only if its composite with every smooth curve is of class Zm,1. This complements the well-known analogous result for local Hölder-Lipschitz classes Cm;?, which we reprove along the way. We demonstrate that these results generalize to mappings between Banach spaces and use them to study the regularity of the superposition operator f*: g ↬ f o g acting on the global Zygmund space Δm+1Rd. We prove that, for all integers m; k ≥ 1, the map f*:Δm+1Rd is of Lipschitz class Ck-1;1 if and only if f 2 Zm+k,1.R. [ABSTRACT FROM AUTHOR]

We study the Hardy space of translated Dirichlet series H + . It consists on those Dirichlet series ∑ a n n - s such that for some (equivalently, every) 1 ≤ p < ∞ , the translation ∑ a n n - (s + 1 σ) belongs to the Hardy space H p for every σ > 0 . We prove that this set, endowed with the topology induced by the seminorms ‖ · ‖ 2 , k k ∈ N (where ‖ ∑ a n n - s ‖ 2 , k is defined as ‖ ∑ a n n - (s + 1 k) ‖ H 2 ), is a Fréchet space which is Schwartz and non nuclear. Moreover, the Dirichlet monomials { n - s } n ∈ N are an unconditional Schauder basis of H + . We also explore the connection of this new space with spaces of holomorphic functions on infinite-dimensional spaces. In the spirit of Gordon and Hedenmalm's work, we completely characterize the composition operator on the Hardy space of translated Dirichlet series. Moreover, we study the superposition operators on H + and show that every polynomial defines an operator of this kind. We present certain sufficient conditions on the coefficients of an entire function to define a superposition operator. Relying on number theory techniques we exhibit some examples which do not provide superposition operators. We finally look at the action of the differentiation and integration operators on these spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of a.e. monotonic solutions of some general delay integral problems for both fractional and integer orders in the space of Lebesgue integrable functions on the interval R+ = [0;1) and in the space of locally integrable functions L1loc (R+). In particular, the uniqueness of solutions for considered problems is obtained.

In this paper, we study the existence of a.e. monotonic solutions of some general delay integral problems for both fractional and integer orders in the space of Lebesgue integrable functions on the interval R+ = [0, ∞) and in the space of locally integrable functions Lloc1 (R+). In particular, the uniqueness of solutions for considered problems is obtained. [ABSTRACT FROM AUTHOR]

We will consider the so-called superposition operator in the space CC(ℝ+) of real functions defined, continuous on the real half-axis ℝ+ and converging to finite limits at infinity.

Some inequalities connected to measures of noncompactness in the space of regulated function R(J, E) were proved in the paper. The inequalities are analogous of well known estimations for Hausdorff measure and the space of continuous functions. Moreover two sufficient and necessary conditions that superposition operator (Nemytskii operator) can act from R(J, E) into R(J, E) are presented. Additionally, sufficient and necessary conditions that superposition operator Ff : R(J, E) → R(J, E) was compact are given.

In this paper we study the existence of monotonic solutions of fractional nonlinear quadratic integral equations in the space of Lebesgue integrable functions on [0,τ]. The uniqueness of the solution is also discussed. In addition an example is given to illustrate our abstract results. [ABSTRACT FROM AUTHOR]

We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series H ∞ (C + 2) . We also show how the composition operators of this space of Dirichlet series are related to the composition operators of the corresponding spaces of holomorphic functions. Finally, we give a characterization of the superposition operators in H ∞ (C +) and in the spaces H p . [ABSTRACT FROM AUTHOR]

Using concentration-compactness arguments, we prove a variant of the Brézis–Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest.

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a proper étale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current groupoid as a current algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators Cℓ(K, f): Cℓ (K,M) → Cℓ (K,N), γ → f o γ between manifolds of Cℓ-functions. Under natural hypotheses, Cℓ(K, f) turns out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map f : M → N. These results are new in their generality and of independent interest. [ABSTRACT FROM AUTHOR]

It is shown that integral operators of the fully nonlinear type K (x) (t) = ∫ Ω k (t , s , x (t) , x (s)) d s exhibit similar degeneracy phenomena in a large class of spaces as superposition operators F (x) (t) = f (t , x (t)). In particular, K is Fréchet differentiable in L p only if it is affine with respect to the " x (t) " argument. Similar degeneracy results hold if K satisfies a local Lipschitz or compactness condition. Also vector functions, infinite measure spaces, and a much richer class of function spaces than only L p are considered. As a side result, degeneracy assertions for superposition operators are obtained in this more general setting, complementing the known results for scalar functions. As a particular example, it is shown that the operators arising in continuous limits of coupled Kuramoto oscillators fail everywhere to be Fréchet differentiable or locally compact. [ABSTRACT FROM AUTHOR]

Using the notion of the integral weight, we characterize all entire functions that transform a Bloch-type space B μ 1 into another space of the same kind B μ 2 by superposition for very general weights μ 1 and μ 2 , satisfying a growth condition. [ABSTRACT FROM AUTHOR]

In this paper we investigate the problem of uniform continuity of nonautonomous superposition operators acting between spaces of functions of bounded Λ-variation. In particular, we give the sufficient conditions for nonautonomous superposition operators to continuously map a space of functions of bounded Λ-variation into itself. The conditions cover the generators being functions of C1-class (in view of two variables), but also allow for less regular functions, including discontinuous generators. [ABSTRACT FROM AUTHOR]

We characterize all entire functions ϕ that maps a logarithmic Bloch-type space Blogβα into another of the same kind by superposition. As consequences of our study, we obtain several results about the boundedness of superposition operators acting between α-Bloch spaces, Bloch-Orlicz spaces among others. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study properties of the numerical range W(T) of some nonlinear operators in Banach spaces of sequences ℓp. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of “function of second bounded variation” in the sense of Shiba, and we show that if a superposition operator applies the space of all such functions on itself and it is uniformly bounded, then its generating function satisfies a Matkowski condition. // En este trabajo introducimos la noción de función de segunda variación acotada en el sentido de Shiba y mostramos que si un operador de superposición aplica el espacio de todas estas funciones en sí mismo y es uniformemente acotado, entonces su función generadora satisface una condición de Matkowski.

In this paper we introduce the notion of “function of second bounded variation” in the sense of Shiba, and we show that if a superposition operator applies the space of all such functions on itself and it is uniformly bounded, then its generating function satisfies a Matkowski condition.

We prove that if a superposition operator maps a subset of the space of all metric-vector-space-valued-functions of bounded n-dimensional Φ-variation into another such space, and is uniformly continuous, then the generating function of the operator is an affine function in the functional variable. [ABSTRACT FROM AUTHOR]

We construct a distinct category of nonclassical quantum states by applying a superposition of products (SUP) of field annihilation ( $\hat {a}$ ) and creation ( $\hat {a}^{\dagger }$ ) operators of the type ( $s\hat {a}\hat {a}^{\dagger }+t\hat {a}^{\dagger }\hat {a}$ ), with $s^{2}+t^{2}=1$ , upon thermal and even coherent states. We allow these SUP operated states to undergo a decoherence process and then describe the nonclassical features of the resulted field by using the entanglement potential (EP) and the negativity of the Wigner distribution function. Our analysis reveals that both the measures are reduced in the linear loss process. The partial negativity of the Wigner function disappears when losses exceed 50% but EP exists always. [ABSTRACT FROM AUTHOR]

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We prove that if a superposition operator maps a subset of the space of all metric-vector-space-valued-functions of bounded n-dimensional Φ-variation into another such space, and is uniformly continuous, then the generating function of the operator is an affine function in the functional variable

In this paper we investigate autonomous as well as nonautonomous superposition operators acting between spaces of functions of bounded $$\Lambda $$ -variation. A particular emphasis is put on acting conditions as well as on continuity problems for such operators. In particular, we give necessary and sufficient conditions for nonautonomous superposition operators to map a space of functions of bounded $$\Lambda $$ -variation into itself. Moreover, we prove the continuity of certain autonomous superposition operators acting between various spaces of functions of bounded $$\Lambda $$ -variation. We also examine the existence of $$\Lambda BV$$ -solutions as well as the topological structure of such solution sets to classical nonlinear integral equations. [ABSTRACT FROM AUTHOR]

Given a set-function $$\psi $$ defined on bounded subsets of a Banach space with certain properties, necessary and sufficient criteria for $$\psi (A(U))=0$$ are given, when A is positively homogeneous of some order and U is bounded. The results are applied to give necessary and sufficient criteria for the compactness and weak compactness of a Fréchet derivative (in some point or at $$\infty $$ ) and when an operator is improving. [ABSTRACT FROM AUTHOR]