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Fixed points for some non-obviously contractive operators. (English) Zbl 1002.47031
Given a Banach space $E$ and a closed set $F\subseteq C[(0, T], E)$, let $A: F\to F$ be a nonlinear operator which satisfies for some $\alpha,\beta\in [0,1)$ and $\gamma\ge 0$ the contraction-type condition $$\|Au(t)- Av(t)\|\le \beta\|u(t)- v(t)\|+ {\gamma\over t^\alpha} \int^t_0\|u(s)- v(s)\|ds$$ ($u,v\in F$, $0< t\le T$). It was shown by {\it B. Lou} [Proc. Am. Math. Soc. 127, No. 8, 2259-2264 (1999; Zbl 0918.47046)] that $A$ has then a unique fixed point. In the present paper, the authors prove that this is equivalent to, and even weaker than, the Banach contraction principle in the setting of $K$-normed spaces. Moreover, they show by means of a counterexample that the term $\beta\|u(t)- v(t)\|$ above cannot be replaced by $\beta\|u-v\|_{C([0, T],E)}$.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
45D05Volterra integral equations
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