A generalized coincidence point index


  • Nasreddine Mohamed Benkafadar University of Constantine
  • M. C. Benkara-Mostefa University of Constantine




Point, Concidence point, Index, Degree, Multi-valued mapping


The paper is devoted to build for some pairs of continuous single-valued maps a coincidence point index. The class of pairs (f, g) satisfies the condition that f induces an epimorphism of the Cech homology groups with compact supports and coefficients in the field of rational numbers Q. Using this concept one defines for a class of multi-valued mappings a fixed point degree. The main theorem states that if the general coincidence point index is different from {0}, then the pair (f, g) admits at least a coincidence point. The results may be considered as a generalization of the above Eilenberg-Montgomery theorems [12], they include also, known fixed-point and coincidence-point theorems for single-valued maps and multi-valued transformations.


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Author Biographies

Nasreddine Mohamed Benkafadar, University of Constantine

Department of Mathematics, Faculty of Sciences

M. C. Benkara-Mostefa, University of Constantine

Department of Mathematics, Faculty of Sciences


Y. G. Borisovitch, Topological characteristics and the investigation of solvability for nonlinear problems, Izvestiya VUZ'ov, Mathematics 2 (1997), 3-23.

Y. G. Borisovitch, Topological characteristics of infinite-dimensional mappings and the solvability of nonlinear boundary value problems, Proceedings of the Steklov Institute of Mathematics 3 (1993), 43-50.

K. Borsuk, Theory of retracts, Monografie Matematyczne 44 (Polska Academia NAUK, Warszawa, 1967).

K. Borsuk, A. Kosinski, On connections between the homology properties of a set and its frontiers, Bull. Acad. Pol. Sc., 4 (1956), 331-333.

E. G. Begle, The Vietoris mapping theorem for bicompact spaces, Ann. of Math. 2 (1950), 534-543. https://doi.org/10.2307/1969366

J. Bryszewski, On a class of multi-valued vector fields in Banach spaces, Fund. Math. 2 (1977), 79-94.

N. M. Benkafadar, B. D. Gel'man, On some generalized local degrees, Topology Proceedings 25 summer 2000 ( 2002 ), 417-433.

N. M. Benkafadar, B. D. Gel'man, On a local degree of one class of multivalued vector fields in infinite-dimensional Banach spaces, Abstract And Applied Analysis 4 (1996), 381-396. https://doi.org/10.1155/S1085337596000206

A. Dold, Fixed point index and fixed point theorems for euclidean neighborhood retracts, Topology 4 (1965), 1-8. https://doi.org/10.1016/0040-9383(65)90044-3

A. Dold, Lectures on Algebraic Topology, (Springer-Verlag, Berlin, 1972). https://doi.org/10.1007/978-3-662-00756-3

Z. Dzedzej, Fixed point index theory for a class of nonacyclic multivalued maps, Rospr. Math. 25, 3 (Warszawa, 1985).

S. Eilenberg, D. Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 58 (1946), 214-222. https://doi.org/10.2307/2371832

S. Eilenberg, N. Steenrod, Foundations of Algebraic Topology, (Princeton, 1952). https://doi.org/10.1515/9781400877492

A. Granas, The Leray-Shauder index and fixed point theory for arbitrary ANR-s, Bull. Soc. Math. Fr. 100 (1972), 209-228. https://doi.org/10.24033/bsmf.1737

L. Gorniewiecz, A. Granas, Some general theorems in coincidence Theory I., J. Math. pures et appl. 61 ( 1981 ), 361-373.

A. Granas, Sur la notion du degré topologique pour une certaine classe de transformations multivalentes dans des espaces de Banach, Bull. Acad. polon. Sci. 7 (1959), 181-194.

A. Granas, J. W. Jaworowski, Some theorems on multi-valued maps of subsets of the Euclidean space, Bull. Acad. Polon. Sci. 6 ( 1965 ), 277-283.

L. Gorniewicz, Homological methods in fixed point theory of multi-valued maps, Dissert. Math. 129 ( Warszawa, 1976 ).

B. D. Gel'man, Topological characteristic for multi-valued mappings and fixed points, Dokl. Acad. Naouk 3 (1975), 524-527.

B. D. Gel 'man, Generalized degree for multi-valued mappings, Lectures notes in Math. 1520, ( 1992 ), 174-192. https://doi.org/10.1007/BFb0084721

S. Kakutani, A Generalization of Brouwer's fixed point theorem, Duke Mathematical Journal, 8 (1941), 457-459. https://doi.org/10.1215/S0012-7094-41-00838-4

Z. Kucharski, A coincidence index, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Et Phys. 4 ( 1976 ), 245-252.

Z. Kucharski, Two consequences of the coincidence index, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. et Phys. 6 ( 1976 ), 437-444.

K. Kuratowski, Topology, Vol. I, II, ( Academic Press New York And London 1966 )

W. Kryszewski, Topological and approximation methods of degree theory of set-valued maps, Dissert. Math. 336, ( Warszawa, 1994 ).

A. Lasota, Z.Opial, An approximation theorem for multi-valued mappings, Podst. Sterow. 1 (1971), 71-75.

M. Powers, Lefschetz fixed point Theorems for a new class of multi-valued maps, Pacific J. Math. 68 (1970), 619-630. https://doi.org/10.1017/S030500410007660X

Z. Siegberg, G. Skordev, Fixed point index and chain approximation, Pacific J. Math. 2 (1982), 455-486. https://doi.org/10.2140/pjm.1982.102.455

E. H. Spanier, Algebraic Topology, (McGraw-Hill, 1966). https://doi.org/10.1007/978-1-4684-9322-1_5

A. D. Wallace, A fixed point theorem for trees, Bulletin of American Mathematical Society, 47 (1941), 757-760. https://doi.org/10.1090/S0002-9904-1941-07556-7

J. Warga, Optimal control of differential and functional equations, (Acad. Press, New York and London, 1975).


How to Cite

N. M. Benkafadar and M. C. Benkara-Mostefa, “A generalized coincidence point index”, Appl. Gen. Topol., vol. 6, no. 1, pp. 87–100, Apr. 2005.



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