Nonlinear alternatives of Schauder and Krasnosel’skij types with applications to Hammerstein integral equations in \(L^1\) spaces. (English) Zbl 1208.47044

In this interesting paper, by using the De Blasi measure of weak noncompactness, the authors establish some new variants of nonlinear alternatives of Leray-Schauder and Krasnosel’skij type involving the weak topology of Banach spaces. In addition, the authors apply their abstract results to a nonlinear Hammerstein integral equation in \(L^{1}\) spaces.
The main results of this paper complement some recent ones due to K.Latrach, M.A.Taoudi and A.Zeghal [J. Differential Equations 221, 256–2710 (2006; Zbl 1091.47046)] and K.Latrach and M.A.Taoudi [Nonlinear Anal.66, 2325–2333 (2007; Zbl 1128.45006)].
Reviewer: Long Wei (Jiangxi)


47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
47H10 Fixed-point theorems
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI


[1] Agarwal, R. A.; Meehan, M.; O’Regan, D., Fixed Point Theory and Applications, Cambridge Tracts in Math., vol. 141 (2001), University Press: University Press Cambridge · Zbl 0960.54027
[2] Agarwal, R. P.; O’Regan, D.; Liu, X., A Leray-Schauder alternative for weakly-strongly sequentially continuous weakly compact maps, Fixed Point Theory Appl., 1, 1-10 (2005) · Zbl 1098.47046
[3] Akhmerov, R. R.; Kamenskij, M. I.; Potapov, A. S.; Rodkina, A. E.; Sadovskij, B. N., Measures of Noncompactness and Condensing Operators (1992), Birkhäuser: Birkhäuser Basel · Zbl 0748.47045
[4] Appell, J., The superposition operator in function spaces - a survey, Expo. Math., 6, 209-270 (1988) · Zbl 0648.47041
[5] Appell, J.; De Pascale, E., Su alcuni parametri connessi con la misura di non compatteza di Hausdorff in spazi di funzioni misurabili, Boll. Unione Mat. Ital. Sez. B (6), 3, 497-515 (1984) · Zbl 0507.46025
[6] Arino, O.; Gautier, S.; Penot, J. P., A fixed point theorem for sequentially continuous mappings with applications to ordinary differential equations, Funkcial. Ekvac., 27, 273-279 (1984) · Zbl 0599.34008
[7] Banas, J.; Knap, Z., Measure of noncompactness and nonlinear integral equations of convolution type, J. Math. Anal. Appl., 146, 353-362 (1990) · Zbl 0699.45002
[8] Barroso, C. S., Krasnosel’skij fixed point theorem for weakly continuous maps, Nonlinear Anal., 55, 25-31 (2003) · Zbl 1042.47035
[9] Barroso, C. S.; Teixeira, E. V., A topological and geometric approach to fixed points results for sum of operators and applications, Nonlinear Anal., 60, 625-650 (2005) · Zbl 1078.47014
[10] Ben Amar, A.; Jeribi, A.; Mnif, M., On a generalization of the Schauder and Krasnosel’skij fixed point theorems on Dunford-Pettis spaces and applications, Math. Methods Appl. Sci., 28, 1737-1756 (2005) · Zbl 1186.47043
[11] Brézis, H., Analyse Fonctionnelle. Théorie et Applications (1993), Masson: Masson Paris · Zbl 0511.46001
[12] Burton, T. A., A fixed point theorem of Krasnosel’skij, Appl. Math. Lett., 11, 85-88 (1998) · Zbl 1127.47318
[13] De Blasi, F. S., On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 21, 259-262 (1997) · Zbl 0365.46015
[14] Dhage, B. C., On some nonlinear alternatives of Leray-Schauder type and functional integral equations, Arch. Math. (Brno), 42, 1-23 (2006) · Zbl 1164.47357
[15] Dhage, B. C.; Ntouyas, S. K., Existence results for nonlinear functional integral equations via a fixed point theorem of Krasnosel’skij-Schaefer type, Nonlinear Stud., 9, 3, 307-317 (2002) · Zbl 1009.47054
[16] Dhage, B. C.; O’Regan, D., A fixed point theorem in Banach algebras with applications to functional integral equations, Funct. Differ. Equ., 7, 3-4, 259-267 (2000) · Zbl 1040.45003
[17] Dunford, N.; Schwartz, J. T., Linear Operators, Part I: General Theory (1988), Interscience: Interscience New York
[18] Edwards, R., Functional Analysis. Theory and Applications (1965), Holt-Rinehart-Winston: Holt-Rinehart-Winston New York · Zbl 0182.16101
[19] Emmanuele, G., An existence theorem for Hammerstein integral equations, Port. Math., 51, 4, 607-611 (1994) · Zbl 0823.45004
[20] Floret, K., Weakly Compact Sets, Lecture Notes in Math., vol. 801 (1980), Springer: Springer Berlin · Zbl 0442.46001
[21] Granas, A.; Dugundji, J., Fixed Point Theory, Springer Monogr. Math. (2003), Springer: Springer New York · Zbl 1025.47002
[22] Kamenskii, M.; Obukhovskii, V.; Zecca, P., Condensing Multi-valued Maps and Semilinear Differential Inclusions in Banach Spaces (2001), Walter de Gruyter & Co.: Walter de Gruyter & Co. Berlin · Zbl 0988.34001
[23] Krasnosel’skij, M. A., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen, The Netherlands · Zbl 0121.10604
[24] Krasnosel’skij, M. A., Integral Operators in Space of Summable Functions (1976), Noordhoff: Noordhoff Leyden, The Netherlands · Zbl 0355.70020
[25] Latrach, K.; Jeribi, A., A nonlinear boundary value problem arising in growing cell populations, Nonlinear Anal., 36, 843-862 (1999) · Zbl 0935.35170
[26] Latrach, K.; Taoudi, M. A., Existence results for a generalized nonlinear Hammerstein equation on \(L_1\) spaces, Nonlinear Anal., 66, 2325-2333 (2007) · Zbl 1128.45006
[27] Latrach, K.; Taoudi, M. A.; Zeghal, A., Some fixed point theorems of the Schauder and the Krasnosel’skij type and application to nonlinear transport equations, J. Differential Equations, 221, 256-271 (2006) · Zbl 1091.47046
[28] Lucchetti, R.; Patrone, F., On Nemytskij’s operator and its application to the lower semicontinuity of integral functionals, Indiana Univ. Math. J., 29, 5, 703-735 (1980) · Zbl 0476.47049
[29] Moreira, D. R.; Teixeira, E. V.O., Weak convergence under nonlinearities, Ann. Braz. Acad. Sci., 75, 1, 9-19 (2003) · Zbl 1042.47040
[30] Ngoc, L. T.P.; Long, N. T., On a fixed point theorem of Krasnosel’skij type and application to integral equations, Fixed Point Theory Appl., 1-24 (2006), Article ID 30847 · Zbl 1353.47121
[31] Ntouyas, S. K.; Tsamatos, P. G., A fixed point theorem of Krasnosel’skij nonlinear alternative type with applications to functional integral equations, Differential Equations Dynam. Systems, 7, 2, 139-146 (1999) · Zbl 0978.45003
[32] O’Regan, D., Fixed-point theory for the sum of two operators, Appl. Math. Lett., 9, 1-8 (1996) · Zbl 0858.34049
[33] O’Regan, D., Fixed-point theory for weakly sequentially continuous mappings, Math. Comput. Modelling, 27, 1-14 (1998) · Zbl 1185.34026
[34] O’Regan, D.; Meehan, M., Existence Theory for Nonlinear Integral and Integrodifferential Equations (1998), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0932.45010
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