Affiliation:
1. Cadi Ayyad University , National School of Applied Sciences , Morocco .
Abstract
Abstract
In this paper, we prove the following generalization of the classical Darbo fixed point principle : Let X be a Banach space and µ be a montone measure of noncompactness on X which satisfies the generalized Cantor intersection property. Let C be a nonempty bounded closed convex subset of X and T : C → C be a continuous mapping such that for any countable set Ω ⊂ C, we have µ(T(Ω)) ≤ kµ(Ω), where k is a constant, 0 ≤ k < 1. Then T has at least one fixed point in C. The proof is based on a combined use of topological methods and partial ordering techniques and relies on the Schauder and the Knaster-Tarski fixed point principles.
Reference20 articles.
1. H. Amann, Order structures and fixed points. In: SAFA 2 - Atti del 2 Seminario di Anal. Funzionale e Appl., Cosenza, 28. Settembre - 6. Ottobre 1977, 1-51. Universit di Calabria, (1979).
2. J. Appell, M. Vath, A. Vignoli, Compactness and existence results for ordinary differential equations in Banach spaces. Z. Anal. Anwendungen 18 (1999), no. 3, 569–584.
3. D. Ariza-Ruiz, J. Garcia-Falset, Abstract measures of noncompactness and fixed points for nonlinear mappings. Fixed Point Theory 21 (2020), no. 1, 47–65.
4. J. M. Ayerbe Toledano, T. Dominguez Benavides, G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory. Operator Theory: Advances and Applications, 99, Birkhuser Verlag, Basel, 1997.
5. S. Banach, Sur les opérations dans les ensembles abstraits et leur application auxéquations intégrales. Fund. Math. 3 (1922), 133–181.