Some inequalities and superposition operator in the space of regulated functions. (English) Zbl 1435.47055
The superposition operator \(F\) generated by some function \(f\) has been studied by several authors in the space of regular functions, e.g., in the autonomous case \((Fx)(t)=f(x(t))\) by W. Aziz et al. [Z. Anal. Anwend. 33, No. 1, 119–123 (2014; Zbl 1312.47073)], in the nonautonomous case \((Fx)(t)=f(t,x(t))\) by A. Michalak [Z. Anal. Anwend. 35, No. 3, 285–308 (2016; Zbl 1362.47051)].
In the present paper, the authors study this operator in the more general case when all functions take values in a Banach space \(E\). Apart from considering acting and continuity conditions, they give three conditions under which a certain operator \(F\) is continuous and compact. Remarkably, these conditions are not only sufficient, but in the case of a separable space \(E\) also necessary. Their compactness arguments build on the construction of a new measure of noncompactness (too technical to be recalled here) in the space of regulated Banach space valued functions on an interval.
Reviewer’s remark: The authors seem to be unaware of the papers [N. Viloria, Divulg. Mat. 12, No. 2, 149–153 (2004; Zbl 1110.47311); W. Aziz, Rev. Mat. Teor. Apl. 21, No. 1, 11–20 (2014; Zbl 1316.47027)], which have some overlap with their results.
In the present paper, the authors study this operator in the more general case when all functions take values in a Banach space \(E\). Apart from considering acting and continuity conditions, they give three conditions under which a certain operator \(F\) is continuous and compact. Remarkably, these conditions are not only sufficient, but in the case of a separable space \(E\) also necessary. Their compactness arguments build on the construction of a new measure of noncompactness (too technical to be recalled here) in the space of regulated Banach space valued functions on an interval.
Reviewer’s remark: The authors seem to be unaware of the papers [N. Viloria, Divulg. Mat. 12, No. 2, 149–153 (2004; Zbl 1110.47311); W. Aziz, Rev. Mat. Teor. Apl. 21, No. 1, 11–20 (2014; Zbl 1316.47027)], which have some overlap with their results.
Reviewer: Jürgen Appell (Würzburg)
MSC:
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
| 46E40 | Spaces of vector- and operator-valued functions |
| 47H08 | Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. |
Keywords:
space of regulated functions; measure of noncompactness; integral inequalities; superposition operatorReferences:
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