Fixed point results for multivalued contractions on gauge spaces. (English) Zbl 1013.47013

Agarwal, Ravi P. (ed.) et al., Set valued mappings with applications in nonlinear analysis. London: Taylor & Francis. Ser. Math. Anal. Appl. 4, 175-181 (2002).
This article deals with some fixed point theorems for multivalued operators in (sequentially) complete gauge spaces (spaces with generalized metric taking values from \([0,\infty)^\Lambda\)). The author formulates an analogue of the Banach-Caccioppoli fixed point principle for admissible multivalued contractions in such spaces (this analogue is a generalization of theorems by Covitz-Nadler and Cain-Nashed) and an analogue of the Krasnosel’skij-Zabreiko theorem about homotopies of contractions for admissible multivalued contractions in such spaces. As an application, the Cauchy problem \[ x'(t)\in f(t,x(t))\text{ a.e. }t\in [0,\infty),\quad x(0) = 0 \] in a Hilbert space \(H\) is considered.
For the entire collection see [Zbl 0996.00018].


47H04 Set-valued operators
47H10 Fixed-point theorems
34G20 Nonlinear differential equations in abstract spaces