On different type of fixed point theorem for multivalued mappings via measure of noncompactness. (English) Zbl 1435.47053

Summary: In this paper, by using the measure of noncompactness concept, we present new fixed point theorems for multivalued maps. Further, we introduce a new class of mappings which are more general than Meir-Keeler mappings. Finally, we use these results to investigate the existence of weak solutions to an evolution differential inclusion with lack of compactness.


47H10 Fixed-point theorems
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
47H04 Set-valued operators
34G25 Evolution inclusions
Full Text: DOI arXiv Euclid


[1] N. U. Ahmed, Semigroup theory with applications to systems and control, Harlow John Wiley & Sons Inc. New York, 1991. · Zbl 0727.47026
[2] P. R. Akmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiĭ, Measures of noncompactness and condensing operators, Translated from the 1986 Russian original by A. Iacob. Operator Theory: Advances and Applications, 55. Birkhäuser Verlag, Basel, 1992.
[3] J. Banas and K. Goebel, Measure of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, 60. Marcel Dekker, Inc., New York, 1980. · Zbl 0441.47056
[4] C. Castaing and M. Valadier, Convex analysis and measurable multifunction, Lecture Notes in Math. 580, Springer-Verlag New York, 1977. · Zbl 0346.46038
[5] K. Deimling, Multivalued differential equations, De Gruyter Series in Nonlinear Analysis and Applications, 1, Walter de Gruyter & Co., Berlin, 1992.
[6] B. C. Dhage, Some generalizations of multivalued version of Schauder’s fixed point theorem with applications, Cubo 12 (2010), no. 3, 139-151. · Zbl 1226.47057
[7] K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. · Zbl 0952.47036
[8] K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sei. U.S.A. 38 (1952), 121-126. · Zbl 0047.35103
[9] L. Górniewicz, Topological fixed point theory of multivalued mappings, Mathematics and its Applications 495, Kluwer Academic Publishers Dordrecht, 1999. · Zbl 0937.55001
[10] M. Kamenskii, V. Obukhovskii, and P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces, Walter de Gruyter, Berlin-New York, 2001. · Zbl 0988.34001
[11] M. Kisielewicz, Differential inclusions and optimal control, Kluwer Dordrecht The Netherlands, 1991.
[12] A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329. · Zbl 0194.44904
[13] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag New York, 1983. · Zbl 0516.47023
[14] A. Samadia and M. B. Ghaemia, An extension of Darbo fixed point theorem and its applications to coupled fixed point and integral equations, Filomat 28 (2014), no. 4, 879-886.
[15] H. Thiems, Integrated semigroup and integral solutions to abstract Cauchy problem, J. Math. Anal. Appl. 152 (1990), 416-447. · Zbl 0738.47037
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