Abstract
In this paper, we study the existence of fixed points for the product of nonlinear operators. This kind of fixed point theorems is necessary in consideration of quadratic differential and integral problems. We emphasize a possible extension of the applicability of obtained theorems and consequently we prove the existence of fixed points for operators acting on some function spaces that are not necessarily Banach algebras. This result can be successfully applied to many quadratic problems.
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Agarwal R., O’Regan D., Wong P.: Constant-sign solutions of a system of Volterra integral equations in Orlicz spaces. J. Integral Equations Appl. 20, 337–378 (2008)
R. Agarwal, D. O’Regan and P. Wong, Constant-Sign Solutions of Systems of Integral Equations. Springer, New York, 2013.
Appell J.: Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator. J. Math. Anal. Appl. 83, 251–263 (1981)
Appell J.: The importance of being Orlicz. Banach Center Publ. 64, 21–28 (2004)
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. B 3, 497–515 (1984)
J. Appell and P. P. Zabreiko, Nonlinear Superposition Operators. Cambridge University Press, Cambridge, 1990.
Argyros I.K.: On a class of quadratic integral equations with perturbation. Funct. Approx. Comment. Math. 20, 51–63 (1992)
Banaś J.: Integrable solutions of Hammerstein and Urysohn integral equations. J. Aust. Math. Soc. Ser. A 46, 61–68 (1989)
Banaś 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, 3283–3293 (1997)
Banaś J., El-Sayed W.G.: Measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variation. J. Math. Anal. Appl. 167, 133–151 (1992)
J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces. Lect. Notes Math. 60, Marcel Dekker, New York, 1980.
J. Bananś and M. Lecko, Fixed points of the product of operators in Banach algebra. Panamer. Math. J. 12 (2002), 101–109.
Banaś J., Sadarangani K.: Solutions of some functional-integral equations in Banach algebras. Math. Comput. Modelling 38, 245–250 (2003)
Caballero J., Mingarelli A.B., Sadarangani K.: Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer. Electron. J. Differential Equations 57, 1–11 (2006)
S. Chandrasekhar, Radiative Transfer. Dover Publications, New York, 1960.
Chistyakov V.V.: A Banach algebra of functions of several variables of finite total variation and Lipschitzian superposition operators. I. Nonlinear Anal 62, 559–578 (2005)
Cichoń M., Metwali M.: On quadratic integral equations in Orlicz spaces. J. Math. Anal. Appl. 387, 419–432 (2012)
Cichoń M., Metwali M.: On monotonic integrable solutions for quadratic functional integral equations. Mediterr. J. Math. 10, 909–926 (2013)
M. Cichoń and M. Metwali, Existence of monotonic \({L_{\varphi}}\) -solutions for quadratic Volterra functional-integral equations. Electron. J. Qual. Theory Differ. Equ. (2015), No. 13, 1–16.
Dhage B.C.: Remarks on two fixed-point theorems involving the sum and the product of two operators. Comput. Math. Appl. 46, 1779–1785 (2003)
Dhage B.C., Kumpulainen M.: Nonlinear functional boundary value problems involving the product of two nonlinearities. Appl. Math. Lett. 21, 537–544 (2008)
Erzakova N.: On measures of non-compactness in regular spaces. Z. Anal. Anwend. 15, 299–307 (1996)
Erzakova N.: Compactness in measure and measure of noncompactness. Sib. Math. J. 38, 926–928 (1997)
Erzakova N.: On measure-compact operators. Russian Math. 55, 37–42 (2011)
Florescu L.C.: The centennial of convergence in measure. Sci. Stud. Res. Ser. Math. Inform. 19, 221–239 (2009)
L. C. Florescu and Ch. Godet-Thobie, Young Measures and Compactness in Measure Spaces. Walter de Gruyter, Berlin, 2012.
M. A. Krasnosel’skii, P. P. Zabreiko, J. I. Pustyl’nik and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions. Noordhoff, Leyden, 1976.
M. A. Krasnosel’skii and Yu. Rutitskii, Convex Functions and Orlicz Spaces. Noordhoff, Groningen, 1961.
Krzyż J.: On monotonicity-preserving transformations. Ann. Univ. Mariae Curie-Skłodowska Sect. A 6, 91–111 (1952)
L. Maligranda, Orlicz Spaces and Interpolation. Departamento de Matemática, Universidade Estadual de Campinas, Campinas, 1989.
R. Płuciennik, The superposition operator in Musielak-Orlicz spaces of vectorvalued functions. In: Proceedings of the 14th Winter School on Abstract Analysis, Circolo Matematico di Palermo, 1987, 411–417.
M. Väth, Ideal Spaces. Lect. Notes Math. 1664, Springer, Berlin, 1997.
M. Väth, A Compactness criterion of mixed Krasnoselskii-Riesz type in regular ideal spaces of vector functions. Z. Anal. Anwend. 18 (1999), 713–732.
M. Väth, Volterra and Integral Equations of Vector Functions. Marcel Dekker, New York, 2000.
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Cichoń, M., Metwali, M.M.A. On a fixed point theorem for the product of operators. J. Fixed Point Theory Appl. 18, 753–770 (2016). https://doi.org/10.1007/s11784-016-0319-7
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DOI: https://doi.org/10.1007/s11784-016-0319-7