Journal of Integral Equations and Applications

Solvability of a general nonlinear integral equation in $L^1$ spaces by means of a measure of weak noncompactness

Fuli Wang

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This paper is concerned with existence results for a quite general nonlinear functional integral equation in $L^1$ spaces. For this purpose, making use of the De Blasi measure of weak noncompactness, we first establish a new fixed point theorem of the nonautonomous superposition operators. After that, our theorem is applied to prove the solvability of the mentioned nonlinear functional integral equation.

Article information

J. Integral Equations Applications, Volume 27, Number 2 (2015), 273-287.

First available in Project Euclid: 9 September 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47H08: Measures of noncompactness and condensing mappings, K-set contractions, etc. 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05]

Superposition operators measure of weak noncompactness fixed points nonlinear integral equations Banach algebras


Wang, Fuli. Solvability of a general nonlinear integral equation in $L^1$ spaces by means of a measure of weak noncompactness. J. Integral Equations Applications 27 (2015), no. 2, 273--287. doi:10.1216/JIE-2015-27-2-273.

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  • J. Appell and E. De Pascale, Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Un. Mat. Ital. 3 (1984), 497–515.
  • J. Appell and P.P. Zabrejko, Nonlinear superposition operators, Cambr. Tracts Math. 95, Cambridge University Press, Cambridge, 1990.
  • J. Banaś and J. Rivero, On measures of weak noncompactness, Ann. Mat. Pura. Appl. 151 (1988), 213–224.
  • J. Banaś and M.A. Taoudi, Fixed points and solutions of operator equations for the weak topology in Banach algebras, Taiwan J. Math. 18 (2014), 871–893.
  • A. Ben Amar, S. Chouayekh and A. Jeribi, New fixed point theorems in Banach algebras under weak topology features and applications to nonlinear integral equations, J. Funct. Anal. 259 (2010), 2215–2237.
  • D. Boyd and J.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464.
  • F.S. De Blasi, On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Sci. Math. Roum. 21 (1977), 259–262.
  • B.C. Dhage, A fixed point theorem in Banach algebras involving three operators with applications, Kyungpook Math. J. 44 (2004), 145–155.
  • S. Djebali and Z. Sahnoun, Nonlinear alternatives of Schauder and Krasnosel'skii types with applications to Hammerstein integral equations in $L^1$ spaces, J. Differ. Equat. 249 (2010), 2061–2075.
  • N. Dunford and J.T. Schwartz, Linear operators, Part I: General theory, Interscience, New York, 1958.
  • K. Latrach and M.A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on $L^1$ spaces, Nonlin. Anal. 66 (2007), 2325–2333.
  • R. Lucchetti and F. Patrone, On Nemytskij's operator and its application to the lower semicontinuity of integral functionals, Indiana Univ. Math. J. 29 (1980), 703–735.
  • M.A. Krasnosel'skii, On the continuity of the operator $Fu(x)= f(x,u(x))$, Dokl. Akad. Nauk 77 (1951), 185–188.
  • M.A. Krasnosel'skii, Topological methods in the theory of nonlinear integral equations, Pergamon Press, New York, 1964.
  • Fuli Wang, Fixed points theorems for the sum of two operators under $\omega$-condensing, Fixed Point Theor. Appl. (2013), doi:10.1186/1687-1812-2013-102.