Fixed point index theory for perturbation of expansive mappings by $k$-set contractions

Smaïl Djebali, Karima Mebarki



In this work, we develop a fixed point index theory for the sum of $k$-set contractions and expansive mappings with constant $h> 1$ when $0\le k< h-1$ as well as in the limit case $k=h-1$. After computing this new index, several fixed point theorems and recent results are derived, including Krasnosel'skii type theorems. Two examples of application illustrate the theoretical results.


Index; fixed point; k-set contraction; expansive mapping; Krasnosel'skii's Theorem; compression and expansion

Full Text:



H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal. 11 (1972), no. 4, 346–384.

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709.

R. Avery and D.R. Anderson, Fixed point theorem of cone expansion and compression of functional type, Special issue in honour of Professor Allan Peterson on the occasion of his 60th birthday, Part I, J. Difference Equ. Appl. 8 (2002), no. 11, 1073–1083.

R.I. Avery, D.R. Anderson and R.J. Krueger, An extension of the fixed point theorem of cone expansion and compression of functional type, Comm. Appl. Nonlinear Anal. 13 (2006), no. 1, 15–26.

J. Banas and Z. Knap, Measures of weak noncompactness and nonlinear integral equations of convolution type, J. Math. Anal. Appl. 146 (1990), no. 2, 353–362.

J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcel Dekker, Inc., New York, 1980.

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Translated from the Romanian. Editura Academiei Republicii Socialiste Romania, Bucharest; Noordhoff International Publishing, Leiden, 1976.

W.M. Bogdanowicz, Representations of linear continuous functionals on the space C(X, Y ) of continuous functions from compact X into locally convex Y , Proc. Japan Acad. 42 (1966), 1122–1127.

F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308.

T.A. Burton, A fixed-point theorem of Krasnosel’skiı̆, Appl. Math. Lett. 11 (1998), no. 1, 85–88.

Y.Q. Cheen, K.S. Ha and Y.J. Cho, The fixed point index for accretive mappings with k-set contraction perturbations in cones, J. Korean Math. Soc. 34 (1997), no. 1, 237–245.

G. Darbo, Punti uniti in transformationi a condominio non-compactto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92.

K. Deimling, Nonlinear Functional Analysis, Springer–Verlag, Berlin, Heidelberg, 1985.

S. Djebali, Fixed point theory for 1-set contractions: a survey, Applied Mathematics in Tunisia, Proceedings of the International Conference on Advances in Applied Mathematics (ICAAM), Hammamet, Tunisia, December 2013; Springer Proceedings in Mathematics and Statistics, Vol. 131, Springer–Birkhäuser, 2015, pp. 53–100.

S. Djebali, F. Madjidi and K. Mebarki, Existence results for singular boundary value problems on unbounded Domains in Banach spaces, Mediterr. J. Math. 11 (2014), 45–74.

S. Djebali and K. Mebarki, Fixed point index on translates of cones and applications, Nonlinear Stud. 21 (2014), no. 4, 579–589.

G. Feltrin, A note on a fixed point theorem on topological cylinders, Ann. Mat. Pura Appl. 196 (2017), 1441–1458.

M. Feng, X. Zhang and W. Ge, Positive fixed point of strict set contraction operators on ordered Banach spaces and applications, Abstr. Appl. Anal. (2010), Art. ID 439137, 13 pp.

D. Guo, Y.I. Cho and J. Zhu, Partial Ordering Methods in Nonlinear Problems, Shangdon Science and Technology Publishing Press, Shangdon, 1985.

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, Boston, Mass, USA, 1988.

L. Guozhen, The fixed point index and the fixed point theorems of 1-set contraction mappings, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1163–1170.

M.A. Krasnosel’skiı̆, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk 10 (1955), 123–127.

M.A. Krasnosel’skiı̆, Fixed points of cone-compressing or cone-extending operators, Soviet Math. Dokl. 1 (1960), 1285–1288.

M.A. Krasnosel’skiı̆, Positive Solutions of Operator Equations, Noordhoff, Groningen, The Netherlands, 1964.

M.K. Kwong, On Krasnoselskiı̆’s cone fixed point theorem, Fixed Point Theory Appl. (2008), Art. ID 164537, 18 pp.

M.K. Kwong, The topological nature of Krasnoselskiı̆’s cone fixed point theorem, Nonlinear Anal. 69 (2008), 891–897.

textscR.D. Nussbaum, The fixed point index and asymptotic fixed point theorems for k-set-contractions, Bull. Amer. Math. Soc. 75 (1969), 490–495.

R.D. Nussbaum, The fixed point index and fixed point theorems. Topological methods for ordinary differential equations (Montecatini Terme, 1991), Lecture Notes in Math., vol. 1537, Springer, Berlin, 1993, 143–205

D. O’Regan, Fixed-point theory for the sum of two operators, Appl. Math. Lett. 9 (1996), no. 1, 1–8.

D. O’Regan and R. Precup, Compression-expansion fixed point theorem in two norms and applications, J. Math. Anal. Appl. 309 (2005), no. 2, 383–391.

W.V. Petryshyn, Fixed point theorems for various classes of 1-set-contractive and 1-ballcontractive mappings in Banach spaces, Trans. Amer. Math. Soc. 182 (1973), 323–3582.

B.N. Sadovskiı̆, On a fixed point principle, Funkc. Anal. Prilož. 1 (1967), no. 2, 74–76. (in Russian)

D.R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1980.

F. Wang, Fixed-point theorems for the sum of two operators under ω-condensing, Fixed Point Theory Appl. 102 (2013), 13 pp.

T. Xiang and R. Yuan, A class of expansive-type Krasnosel’skiı̆ fixed point theorems, Nonlinear Anal. 71 (2009), no. 7–8, 3229–3239.

T. Xiang, S.G. Georgiev, Noncompact-type Krasnosel’skiı̆ fixed-point theorems and their applications, Math. Methods Appl. Sci. 39 (2016), no. 4, 833–863.


  • There are currently no refbacks.

Partnerzy platformy czasopism