Abstract
The interaction between measures of weak noncompactness and fixed point theory is really strong and fruitful. In particular, measures of weak noncompactness play a significant role in topological fixed point problems. The purpose of this chapter is to exhibit the importance of the use of measures of weak noncompactness in topological fixed point theory and to demonstrate how the theory of measures of weak noncompactness will be applied in integral and partial differential equations. The theory of measures of weak noncompactness was initiated by De Blasi in the paper [28], where he introduced the first measure of weak noncompactness. De Blasi’s measure can be regarded as a counterpart of the classical Hausdorff measure of noncompactness. Unfortunately, it is not easy to construct formulas which allow to express the measure of weak noncompactness in a convenient form. For this reason, measures of weak noncompactness have been axiomatized [12] allowing thus several authors to construct measures of weak noncompactness in several Banach spaces [7, 11, 47, 48]. Measures of weak noncompactness have been successfully applied in operator theory, differential equations and integral equations. In particular, they enabled several authors to dispense with the lack of weak compactness in many practical situations. The material is far from exhausting the subject and basically we do not go into profound applications.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Agarwal, R.P., O’Regan, D., Taoudi, M.-A.: Fixed point theorems for ws-compact mappings in Banach spaces. Fixed Point Theory Appl. 2010(183596), 13 (2010)
Agarwal, R.P., O’Regan, D., Taoudi, M.-A.: Browder-Krasnoselskii-type fixed point theorems in Banach spaces. Fixed Point Theory Appl. 2010(243716), 20 (2010)
Agarwal, R.P., O’Regan, D., Taoudi, M.-A.: Fixed point theorems for condensing multivalued mappings under weak topology features. Fixed Point Theory 12(2), 247–254 (2011)
Agarwal, R.P., O’Regan, D., Taoudi, M.-A.: Fixed point theorems for convexpower condensing operators relative to the weak topology and applications to Volterra integral equations. J. Int. Eq. Appl. 24(2), 167–181 (2012)
Alekhno, E.A., Zabrejko, P.P.: On the weak continuity of the superposition operator in the space \(L_\infty \). Vestsi Nats. Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk 2, 17-23 (2005)
Ambrosetti, A.: Un teorema di esistenza per le equazioni differenziali negli spazi di Banach. Rendiconti del Seminario Matematico della Universita di Padova 39, 349361 (1967)
Angosto, C., Cascales, B.: Measures of weak noncompactness in Banach spaces. Topology Appl. 156(7), 1412–1421 (2009)
Appell, J., De Pascale, E.: Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Un. Mat. Ital. B(6) 3, 497-515 (1984)
Arino, O., Gautier, S., Penot, J.P.: A fixed point theorem for sequentially continuous mappings with applications to ordinary differential equations. Funkc. Ekvac. 27, 273–279 (1984)
Banaś, J., Hajnosz, A., Wȩdrychowicz, S.: On the equation \(x^{\prime } =f(t,\, x)\) in Banach spaces. Comment Math. Univ. Carolin. 23(2), 233–247 (1982)
Banaś, J., Knap, Z.: Measure of weak noncompactness and nonlinear integral equations of convolution type. J. Math. Anal. Appl. 146, 353–362 (1990)
Banaś, J., Rivero, J.: On measures of weak noncompactness. Ann. Mat. Pura Appl. 151, 213–224 (1988)
Banaś, J., Taoudi, M.-A.: Fixed points and solutions of operator equations for the weak topology in Banach algebras. Taiwanese J. Math. 18(3), 871-893 (2014)
Barroso, C.S.: Krasnosel’skii’s fixed point theorem for weakly continuous maps. Nonlinear Analysis 55(1), 25–31 (2003)
Barroso, C.S., Teixeira, E.V.: A topological and geometric approach to fixed points results for sum of operators and applications. Nonlin. Anal. 60(4), 625–650 (2005)
Boudourides, M.A.: An existence theorem for ordinary differential equations in Banach spaces. Bull. Austral. Math. Soc. 22(03), 457–463 (1980)
Burton, T.A.: Integral equations, implicit functions and fixed points. Proc. Amer. Math. Soc. 124, 2383–2390 (1996)
Burton, T.A.: A fixed point theorem of Krasnosel’skii. Appl. Math. Lett. 11, 85–88 (1998)
Burton, T.A., Furumochi, T.: Krasnosel’skii’s fixed point theorem and stability. Nonlin. Anal. 49, 445–454 (2002)
Bellour, A., O’Regan, D., Taoudi, M.-A.: On the existence of integrable solutions for a nonlinear quadratic integral equation. J. Appl. Math. Comput. 46(1–2), 67–77 (2014)
Bellour, A., Bousselsal, M., Taoudi, M.-A.: Integrable solutions of a nonlinear integral equation related to some epidemic models. Glasnik Matematicki 49(69), 395406 (2014)
Ben Amar, A., Chouayekh, S., Jeribi, A.: New fixed point theorems in Banach algebras under weak topology features and applications to nonlinear integral equation. J. Funct. Anal. 259(9), 2215–2237 (2010)
Cardinali, T., Rubbioni, P.: Multivalued fixed point theorems in terms of weak topology and measure of weak noncompactness. J. Math. Anal. Appl. 405(2), 409–415 (2013)
Cichoń, M.: Weak solutions of differential equations in Banach spaces. Discuss. Math. Differential Incl. 15, 5–14 (1995)
Cichoń, M., Kubiaczyk, I.: Existence theorems for the Hammerstein integral equation. Discuss. Math. Differential Incl. 16, 171–177 (1996)
Cichoń, M.: On solutions of differential equations in Banach spaces. Nonlin. Anal. 60(4), 651–667 (2005)
Daher, S.J.: On a fixed point principle of Sadovskii. Nonlinear Anal. 2(5), 643–645 (1978)
De Blasi, F.S.: On a property of the unit sphere in Banach spaces. Bull. Math. Soc. Sci. Math. Roumanie 21, 259–262 (1977)
Diestel, J.: A survey of results related to the Dunford-Pettis property. Contemp. Math. 2, 15–60 (1980)
Djebali, S., Sahnoun, Z.: Nonlinear alternatives of Schauder and Krasnosel’skij types with applications to Hammerstein integral equations in \(L^1\) spaces. J. Differential Equations 249(9), 2061–2075 (2010)
Djebali, S.: Fixed point theory for 1-set contractions: a survey. In: Applied Mathematics in Tunisia, 53–100, Springer Proc. Math. Stat., 131, Springer, Cham, 2015
Dhage, B.C.: Remarks on two fixed point theorems involving the sum and the product of two operators. Comp. Math. Appl. 46, 1779–1785 (2003)
Dhage, B.C.: On a fixed point theorem in Banach algebras with applications. Appl. Math. Lett. 18, 273–280 (2005)
Dobrakov, I.: On representation of linear operators on \(C_0(T, X)\). Czech. Math. J. 21, 13–30 (1971)
Dunford, N., J.T. Schwartz, J.T.: Linear Operators, Part I: General Theory. Interscience Publishers, New York (1958)
Garcia-Falset, J.: Existence of fixed points and measure of weak noncompactness. Nonlin. Anal. 71, 2625–2633 (2009)
Garcia-Falset, J.: Existence of fixed points for the sum of two operators. Math. Nachr. 283(12), 1736–1757 (2010)
Garcia-Falset, J., Latrach, K.: Krasnoselskii-type fixed-point theorems for weakly sequentially continuous map. Bull. Lond. Math. Soc. 44, 2538 (2012)
Garcia-Falset, J., Latrach, K., Moreno-Galvez, E., Taoudi, M.-A.: Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness. J. Differential Equations 252(5), 3436–3452 (2012)
Garcia-Falset, J., Muniz-Perez, O.: Fixed point theory for 1-set contractive and pseudocontractive mappings. Appl. Math. Comp. 219, 6843–6855 (2013)
Geitz, R.F.: Pettis integration. Proc. Amer. Math. Soc. 82, 81–86 (1981)
Grothendieck, A.: Espaces Vectroiels Topologiques. Sociedade de Matematica de Sao Paulo, Sao Paulo (1958)
Hencl, S., Kolar, J., Pangrac, O.: Integral functionals that are continuous with respect to the weak topology on \(W_0^{1,p}(0,1)\). Nonlin. Anal. 63, 81–87 (2005)
Hussain, N., Taoudi, M.-A.: Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations. Fixed Point Theory Appl. 2013, 196.27 (2013)
Krasnosel’skii, M.A., Zabrejko, P.P., Pustyl’nik, J.I., Sobolevskii, P.J.: Integral Operators in Spaces of Summable Functions. Noordhoff, Leyden (1976)
Krasnosel’skii, M.A.: On the continuity of the operator \(Fu(x)=f(x, u(x))\). Dokl. Akad. Nauk. SSSR 77, 185–188 (1951)
Kryczka, A., Prus, S., Szczepanik, M.: Measure of weak noncompactness and real interpolation of operators. Bull. Austr. Math Soc. 62, 389–401 (2000)
Kryczka, A., Prus, S.: Measure of weak noncompactness under complex interpolation. Studia Math. 147, 89–102 (2001)
Kubiaczyk, I., Szufla, S.: Kneser’s theorem for weak solutions of ordinary differential equations in Banach spaces. Publ. Inst. Math. (Beograd) 46, 99–103 (1982)
Latrach, K., Mokhtar-Kharroubi, M.; On an unbounded linear operator arising in the theory of growing cell population. J. Math. Anal. Appl. 211(1), 273–294 (1997)
Latrach, K., Taoudi, M.-A., Zeghal, A.: Some fixed point theorems of the Schauder and Krasnosel’skii type and application to nonlinear transport equations. J. Differential Equations 221(1), 256–271 (2006)
Latrach, K., Taoudi, M.-A.: Existence results for a generalized nonlinear Hammerstein equation on \(L^1\)-spaces. Nonlin. Anal. 66, 2325–2333 (2007)
Le Dret, H.: Equations aux Dérivées Partielles Elliptiques. Springer (2013)
Liu, L., Guo, F., Wu, C., Wu, Y.: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. J. Math. Anal. Appl. 309(2), 638–649 (2005)
Mitchell, A.R., Smith, C.K.L: An existence theorem for weak solutions of differential equations in Banach spaces. In: Lakshmikantham, V. (ed.) Nonlinear Equations in Abstract Spaces, pp. 387–404. Academic Press (1978)
O’Regan, D.: Integral equations in reflexive Banach spaces and weak topologies. Proc. Amer. Math. Soc. 124, 607–614 (1996)
O’Regan, D.: Fixed point theory for weakly sequentially continuous mappings. Math. Comput. Modelling 27(5), 1–14 (1998)
O’Regan, D.: Operator equations in Banach spaces relative to the weak topology. Arch. Math. 71, 123–136 (1998)
O’Regan, D.: Weak solutions of ordinary differential equations in Banach spaces. Appl. Math. Lett. 12, 101–105 (1999)
O’Regan, D., Taoudi, M.-A.: Fixed point theorems for the sum of two weakly sequentially continuous mappings. Nonlin. Anal. 73, 283–289 (2010)
Papageorgiou, N.S.: Weak solutions of differential equations in Banach spaces. Bull. Austral. Math. Soc. 33, 407–418 (1986)
Radstrom, H.: An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3, 165–169 (1952)
Roubicek, T.: Nonlinear Partial Differential Equations with Applications. Birkhuser Verlag Basel, Boston, Berlin (2005)
Ryan, R.: Dunford-Pettis properties. Bull. Acad. Polon. Sci. Math. 27, 373–379 (1979)
Salhi, N., Taoudi, M.-A.: Existence of integrable solutions of an integral equation of Hammerstein type on an unbounded interval. Mediterr. J. Math. doi:10.1007/s00009-011-0147-3 (2011)
Scorza Dragoni, G.: Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad unaltru variabili. Red. Sem. Mat. Univ. Padova 17, 102–106 (1948)
Shragin, V.I.: On the weak continuity of the Nemytskii operator. Uchen. Zap. Mosk. Obl. Ped. Inst. 75, 73–79 (1957)
Taoudi, M.-A.: Integrable solutions of a nonlinear functional integral equation on an unbounded interval. Nonlin. Anal. 71, 4131–4136 (2009)
Taoudi, M.-A.: Krasnosel’skii type fixed point theorems under weak topology features. Nonlin. Anal. doi:10.1016/j.na.2009.06.086 (2009)
Taoudi, M.-A., Salhi, N., Ghribi, B.: Integrable solutions of a mixed type operator equation. Appl. Math. Comput. 216(4), 1150–1157 (2010)
Taoudi, M.-A., Xiang, T.: Weakly noncompact fixed point results of the Schauder and the Krasnoselskii type. Mediterr. J. Math. 11(2), 667–685 (2014)
Zhang, G., Zhang, T., Zhang, T.: Fixed point theorems of Rothe and Altman types about convex-power condensing operator and application. Appl. Math. Comput. 214, 618–623 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Chlebowicz, A., Taoudi, MA. (2017). Measures of Weak Noncompactness and Fixed Points. In: Banaś, J., Jleli, M., Mursaleen, M., Samet, B., Vetro, C. (eds) Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness. Springer, Singapore. https://doi.org/10.1007/978-981-10-3722-1_6
Download citation
DOI: https://doi.org/10.1007/978-981-10-3722-1_6
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-3721-4
Online ISBN: 978-981-10-3722-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)