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On continuous composition operators

100%

Smajdor W.

Annales Polonici Mathematici

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2010

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tom 98

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nr 3

273-282

EN

Let I ⊂ ℝ be an interval, Y be a normed linear space and Z be a Banach space. We investigate the Banach space Lip₂(I,Z) of all functions ψ: I → Z such that $M_{ψ}:= sup{||[r,s,t;ψ]||: r < s < t, r,s,t ∈ I} < ∞$, where [r,s,t;ψ]:= ((s-r)ψ(t)+(t-s)ψ(r)-(t-r)ψ(s))/((t-r)(t-s)(s-r)). We show that ψ ∈ Lip₂(I,Z) if and only if ψ is differentiable and its derivative ψ' is Lipschitzian. Suppose the composition operator N generated by h: I × Y → Z, (Nφ)(t):= h(t,φ(t)), maps the set 𝓐(I,Y) of all affine functions φ: I → Y into Lip₂(I,Z). We prove that if N is continuous and $M_{ψ} ≤ M$ for some constant M > 0, where ψ(t) = N(t,φ(t)), then h(t,y) = a(y) + b(t), t ∈ I, y ∈ Y, for some continuous linear a: Y → Z and b ∈ Lip₂(I,Z). Lipschitzian and Hölder composition operators are also investigated.

2

Commutativity of set-valued cosine families

64%

Smajdor A., Smajdor W.

Open Mathematics

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2014

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tom 12

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nr 12

1871-1881

EN

Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. If {F t: t ≥ 0} is a regular cosine family of continuous additive set-valued functions F t: K → cc(K) such that x ∈ F t(x) for t ≥ 0 and x ∈ K, then $F_t \circ F_s (x) = F_s \circ F_t (x)fors,t \geqslant 0andx \in K$.

3

Concave iteration semigroups of linear continuous set-valued functions

64%

Smajdor A., Smajdor W.

Open Mathematics

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2012

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tom 10

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nr 6

2272-2282

EN

Let {F t: t ≥ 0} be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F 0(x) − F t (x) exist for x ∈ K and t > 0, then D t F t (x) = (−1)F t ((−1)G(x)) for x ∈ K and t ≥ 0, where D t F t (x) denotes the derivative of F t (x) with respect to t and $$G(x) = \mathop {\lim }\limits_{s \to 0} {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} \mathord{\left/ {\vphantom {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} {\left( { - s} \right)}}} \right. \kern-\nulldelimiterspace} {\left( { - s} \right)}}$$ for x ∈ K.

4

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Local analytic solutions of the functional equation $Φ(z) = H(z; Φ[f_1(z)],...,Φ[f_m(z)])$

63%

Smajdor W.

Annales Polonici Mathematici

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1970-1971

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tom 24

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nr 1

39-43

5

A theorem of the Hahn-Banach type and its applications

51%

Gajda Z., Smajdor A., Smajdor W.

Annales Polonici Mathematici

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1992

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tom 57

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nr 3

243-252

EN

Let Y be a subgroup of an abelian group X and let T be a given collection of subsets of a linear space E over the rationals. Moreover, suppose that F is a subadditive set-valued function defined on X with values in T. We establish some conditions under which every additive selection of the restriction of F to Y can be extended to an additive selection of F. We also present some applications of results of this type to the stability of functional equations.

6

Multivalued solutions of a linear functional equation

50%

Smajdor W.

Annales Polonici Mathematici

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1985

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tom 45

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nr 3

253-259

7

On continuous solutions of the Schröder equation

44%

Smajdor W.

Annales Polonici Mathematici

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1976

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tom 32

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nr 2

111-118

Ograniczanie wyników

5 Annales Polonici Mathematici

2 Open Mathematics

7 Smajdor W.

3 Smajdor A.

1 Gajda Z.

1 2014

1 2012

1 2010

1 1992

1 1985

1 1976

1 1970

On continuous composition operators

Commutativity of set-valued cosine families

Concave iteration semigroups of linear continuous set-valued functions

Local analytic solutions of the functional equation $Φ(z) = H(z; Φ[f_1(z)],...,Φ[f_m(z)])$

A theorem of the Hahn-Banach type and its applications

Multivalued solutions of a linear functional equation

On continuous solutions of the Schröder equation