Existence of fixed points and measures of weak noncompactness. (English) Zbl 1194.47060

The author proves the existence of fixed points of an operator \(A\) which is defined on a closed convex subset \(M\) of a Banach space \(X\) into itself and satisfies the following properties: (a) the measure of weak noncompactness of \(A(C)\) where \(C\subset M\) is not relatively weakly compact is strictly less than the measure of weak compactness of \(C\); (b) the sequence \(\{Ax_n\}\) has a strongly convergent subsequence whenever \(\{x_n\}\) is a weakly convergent sequence in \(M\); and (c) there exist \(x_0\in M\) and \(R>0\) such that \(Ax-x_0\neq\lambda(x-x_0)\) for all \(\lambda>1\) and all \(x\in\{z\in M:\|z-x_0\|=R\}\). As a result, the existence of fixed points for the sum of two operators and the solvability of a variant of Hammerstein’s integral equation are obtained.


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


[1] Darbo, G., Punti uniti in transformazioni a codomio non compatto, Rend. Sem. Mat. Univ. Padova, 24, 84-92 (1955) · Zbl 0064.35704
[2] Krasnoselskii, Some problems of nonlinear analysis, M.A., Amer. Math. Soc. Transl., 10, 2, 345-409 (1958) · Zbl 0080.10403
[3] Sadovskii, B. N., On a fixed point principle, Funkt. Anal., 4, 2, 74-76 (1967) · Zbl 0165.49102
[4] Petryshyn, W. V., Structure of the fixed points sets of \(k\)-set-contractions, Arch. Ration. Mech. Anal., 40, 312-328 (1970/71) · Zbl 0218.47028
[5] Reich, S., Fixed points in locally convex spaces, Math. Z., 125, 17-31 (1972) · Zbl 0216.17302
[6] Reich, S., Fixed points of condensing functions, J. Math. Anal. Appl., 41, 460-4677 (1973) · Zbl 0252.47062
[7] De Blasi, F. S., On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Math. Roumanie, 21, 259-262 (1977) · Zbl 0365.46015
[8] Banas, J., Applications of measures of weak noncompactness and some classes of operators in the theory of functional equations in the Lebesgue space, Nonlinear Anal., 30, 6, 3283-3293 (1997) · Zbl 0894.47040
[9] O’Regan, D., Fixed-point theory for weakly sequentially continuous mappings, Math. Comput. Modelling, 27, 5, 1-14 (1998) · Zbl 1185.34026
[10] 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
[11] Latrach, K.; Aziz Taoudi, M.; Zeghal, A., Some fixed point theorems of Schauder and the Krasnosel’skii type and application to nonlinear transport equations, J. Differential Equations, 221, 256-271 (2006) · Zbl 1091.47046
[12] J. Garcia-Falset, Existence of fixed points for the sum of two operators, Math. Nachr. (in press); J. Garcia-Falset, Existence of fixed points for the sum of two operators, Math. Nachr. (in press) · Zbl 1221.47101
[13] Latrach, K.; Aziz Taoudi, M., Existence results for a generalized nonlinear Hammerstein equation on \(L_1\) spaces, Nonlinear Anal., 66, 2325-2333 (2007) · Zbl 1128.45006
[14] Barbu, V., Nonlinear semigroups differential equations in Banach spaces (1976), Noordhoff International Publishing · Zbl 0328.47035
[15] Ph. Bénilan, Equations d’évolution dans un espace de Banach quelconque et applications, Thesis, Orsay, 1972; Ph. Bénilan, Equations d’évolution dans un espace de Banach quelconque et applications, Thesis, Orsay, 1972
[16] Appell, J.; De Pascale, E., Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Unione Mat. Ital. B, 3, 6, 497-515 (1984) · Zbl 0507.46025
[17] Lê, C. H., Dérivabilité d’un semigroup engendré par un opérateur m-accrétif dans \(L^1\) et accrétif dans \(L^\infty \), C.R. Acad. Sci. Paris, 283, 469-472 (1976) · Zbl 0336.47038
[18] Barroso, C. S.; O’Regan, D., Measure of weak compactness and fixed point theory, Fixed point Theory, 6, 2, 247-255 (2005) · Zbl 1118.47042
[19] Arino, O.; Gautier, S.; Penot, J. P., A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkcial. Ekvac., 27, 3, 273-279 (1984) · Zbl 0599.34008
[20] Garcia-Falset, J.; Reich, S., Zeroes of accretive operators and the asymptotic behavior of nonlinear semigroups, Houston J. Math., 32, 1197-1225 (2006) · Zbl 1116.47049
[21] Andreu, F.; Mazón, J. M.; Rossi, J. D.; Toledo, J., A nonlocal \(p\)-Laplacian evolution equation with Neumann boundary conditions, J. Math. Pure Appl., 90, 201-227 (2008) · Zbl 1165.35322
[22] Appell, J.; Zabrejko, P. P., Nonlinear Superposition Operators (1990), Cambridge Univ. Press · Zbl 0701.47041
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