## 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

#### Abstract

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

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

Dates
First available in Project Euclid: 9 September 2015

https://projecteuclid.org/euclid.jiea/1441790289

Digital Object Identifier
doi:10.1216/JIE-2015-27-2-273

Mathematical Reviews number (MathSciNet)
MR3395971

Zentralblatt MATH identifier
1323.47085

#### Citation

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. https://projecteuclid.org/euclid.jiea/1441790289

#### References

• 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.