On the problem of the uniqueness of fixed points and solutions for quadratic fractional-integral equations on Banach algebras
[ 1 ] Instytut Matematyki, Wydział Automatyki, Robotyki i Elektrotechniki, Politechnika Poznańska | [ P ] employee
2024
scientific article
english
- fixed point
- Hölder space
- Banach algebra
- seminorm
- uniqueness
- generalized fractional integral operator
- quadratic integral equation
EN In this paper, we study the problem of the uniqueness of fixed points for operators defined on subspaces of the space of continuous functions C[a, b] equipped with norms stronger than the supremum norm. We are particularly interested in Hölder spaces since they are natural ranges of integral operators of fractional order. Our goal is to preserve the expected regularity of the fixed points (solutions of the equations) when investigating their uniqueness, without assuming a contraction condition on the space under study. We claim some symmetry between the case of the obtained results and the case of the classical Banach fixed-point theorem in such spaces, even for operators which are not necessarily contractions in the sense of the norm of these subspaces. This result is of particular interest for the study of quadratic integral equations, and as an application example we prove the uniqueness theorem for such a kind equations with general fractional order integral operators, which are not necessarily contractions, in a suitably constructed generalized Hölder space.
1535-1 - 1535-32
Article number: 1535
CC BY (attribution alone)
open journal
final published version
at the time of publication
public
70
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