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It is known that there are several approaches to define a Sobolev class for Banach valued functions. We compare the usual definition via weak derivatives with the Reshetnyak–Sobolev space and with the Newtonian space; in particular, we provide sufficient conditions when all three agree. Also, we revise the difference quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it is acting as a superposition operator.

We characterize all entire functions that transform a Bloch–Orlicz space into the α-Bloch space by superposition.

In this paper we prove the existence of integrable solutions of a generalized functional-integral equation, which includes many key integral and functional equations that arise in nonlinear analysis and its applications. This is achieved by means of an improvement of a Krasnosel’skii type fixed point theorem recently proved by K. Latrach and the author. The result presented in this paper extends the corresponding result of [J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Caratheodory condition, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.020 ]. An example which shows the importance and the applicability of our result is also included.

Let I , J ⊂ R be intervals. One of the main results says that if a superposition operator H generated by a two place h : I × J → R , H ( φ ) ( x ) : = h ( x , φ ( x ) ) , maps the set Lip α ( I , J ) of all Holder functions φ : I → J into the Banach space Lip α ( I , R ) and is uniformly continuous with respect to the Lip α -norm, then h ( x , y ) = a ( x ) y + b ( x ) , x ∈ I , y ∈ J , for some a , b ∈ Lip α ( I , R ) .

In this paper we introduce a brief historical note about the study of the existence of solutions of functional integral equations. Hence we prove the existence of monotonic solutions of the functional integral equation of Urysohn type. The technique rely on the concept of measure of noncompactness and its associated Darbo fixed point theorem.

For a given set A ⊂ R n a representation theorem for locally defined operators mapping the space C m ( A ) of Whitney differentiable functions into C 0 ( A ) is presented.

This paper is devoted to the study of the superposition operator T f ( g ) ≔ f ∘ g in the framework of Lizorkin–Triebel spaces F p , q s ( R ) and Besov spaces B p , q s ( R ) . For the case s > 1 + ( 1 / p ) , 1 p ∞ , 1 ≤ q ≤ ∞ , it is natural to conjecture the following: the operator T f takes F p , q s ( R ) to itself if and only if f ( 0 ) = 0 and f belongs locally to F p , q s ( R ) . We establish this conjecture for the following two cases: (1) s − [ s ] > 1 / p , (2) s − [ s ] ≤ 1 / p 3 / 4 . For the case p ≤ 4 / 3 and s − [ s ] ≤ 1 / p , the conjecture is also proved, but with a restriction on s , namely | s − [ s ] + 1 2 − 1 p | > 1 p − 3 4 . A similar result holds for Besov spaces B p , q s ( R ) , but with some extra restrictions involving q .

We establish some properties of the superposition operator which are associated with monotonicity. Those properties are expressed in terms of the notion of degree of decrease or degree of increase. An application of the obtained results to the study of solvability of a quadratic Volterra integral equation is also derived.

In this paper we consider Besov algebras on R , that is Besov spaces B p , q s ( R ) for s > 1 / p . For s > 1 + ( 1 / p ) , p > 4 / 3 , and q ⩾ p we prove that the above algebras have a maximal symbolic calculus in the following sense: for any function f belonging locally to B p , q s ( R ) and such that f ( 0 ) = 0 , the associated superposition operator T f ( g ) : = f ○ g takes B p , q s ( R ) to itself.

On etudie les equations x(t)=f(t,x(t)) et x(t)=f(t,x(φ(t))) dans l'espace L 1 (a,b) equipe de la topologie de la convergence faible. On montre que ces equations ont des solutions integrables qui sont additionnellement des fonctions monotones p.p.