Structure of the Fixed Point of Condensing Set-Valued Maps

Document Type: Original Scientific Paper


Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, 87317-53153, Iran


In this paper, we present structure of the fixed point set results for condensing set-valued map. Also, we prove a generalization of the Krasnosel'skii-Perov connectedness principle to the case of condensing set-valued maps.


Main Subjects

[1] F. E. Browder, A further generalization of the Schauder fixed point theorem, Duke Math. J. 32 (1965) 575–578.

[2] D. Bugajewski, Weak solutions of integral equations with weakly singular kernel in Banach spaces, Comment. Math. (Prace Mat.) 34 (1994) 49–58.
[3] G. Darbo, Punti uniti in transformazioni a condiminio non compatto, Rend. Sem. Math. Univ. Padova. 24 (1955) 84–92.
[4] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin – Heidelberg, 1985.
[5] J. Dugundji and A. Granas, Fixed Point Theory, Volume 1, Serie, Monografie Matematyczne, PWN, Warsaw 1982.
[6] M. Fakhar, Z. Soltani and J. Zafarani, The Lefschetz fixed point theorem and its application to asymptotic fixed point theorem for set-valued mappings, J. Fixed Point Theory Appl. 17 (2015) 287–300.
[7] B. D. Gel’man, Topological properties of the set of fixed points of a multivalued map, Sb. Math. 188(12) (1997) 1761-1782.

[8] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Springer, New York, USA, 2006.
[9] M. A. Krasnosel’ski and A. I. Perov, On the existence of solutions of certain non-linear operator equations, Dokl. Akad. Nauk. SSSR 126 (1959) 15–18.
[10] P. K. F. Kuhfittig, The mean-value iteration for set-valued mappings, Proc. Amer. Math. Soc. 80(3) (1980) 401–405.

[11] T. W. Ma, Topological degrees of set-valued compact vector fields in locally convex spaces, Dissertiones Math. Rozprawy Mat. 92 (1972) 1–43.
[12] J. Mallet-Paret and R. D. Nussbaum, Asymptotic fixed point theory and the beer barrel theorem, J. Fixed Point Theory Appl. 4 (2008) 203–245.
[13] R. D. Nussbaum, Some asymptotic fixed point theorems, Trans. Amer. Math. Soc. 171 (1972) 349–375.
[14] W. V. Petryshyn and P. M. Fitzpatrick, A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings, Trans. Amer. Math. Soc. 194 (1974) 1–25.
[15] B. N. Sadovski˘ı, On a fixed point principle, (Russian) Funkcional. Anal. i Priložen. 1 (1967) 74–76.
[16] J. Schauder, Der fixpunktsatz in funktionalräumen, Studia Math. 2 (1930) 171–180.
[17] V.  Seda, A remark to the schauder fixed point theorem, Dedicated to Juliusz Schauder, 1899?1943, Topol. Methods Nonlinear Anal. 15 (2000) 61–73.