Solvability of functional quadratic integral equations with perturbation

• Autorzy: Metwali M. A. M.
• Języki publikacji: EN

Abstrakty

EN

We study the existence of solutions of the functional quadratic integral equation with a perturbation term in the space of Lebesgue integrable functions on an unbounded interval by using the Krasnoselskii fixed point theory and the measure of weak noncompactness.

Słowa kluczowe

EN

quadratic integral equation; measure of noncompactness; Krasnoselskii fixed point theorem; superposition operators

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2013
• Tom: Vol. 33, no. 4
• Strony: 725--739
• Opis fizyczny: Bibliogr. 35 poz.

Twórcy

autor

Metwali M. A. M.

• Faculty of Mathematics and Computer Science A. Mickiewicz University Umultowska 87, 61-614 Poznan, Poland

Bibliografia

• [1] J. Appell, P.P. Zabrejko, Nonlinear Superposition Operators, Cambridge University Press, Cambridge, 1990.
• [2] J. Appell, E. De Pascale, H.T. Ngyêñ, P.P. Zabrejko, Nonlinear integral inclusions of Hammerstein type, Topol. Methods Nonlinear Anal. 5 (1995), 111–124.
• [3] I.K. Argyros, On a class of quadratic integral equations with perturbations, Functiones et Approximatio 20 (1992), 51–63.
• [4] J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory conditions, Nonlinear Anal. 70 (2009), 3172–3179.
• [5] J. Banas, Z. Knap, Measures of weak noncompactness and nonlinear integral equations of convolution type, J. Math. Anal. Appl. 146 (1990), 353–362.
• [6] J. Banas, Z. Knap, Integrable solutions of a functional-integral equation, Rev. Mat. Univ. Complut. Madrid 2 (1989), 31–38.
• [7] J. Banas, M. Lecko, W.G. El-Sayed, Existence theorems for some quadratic integral equations, J. Math. Anal. Appl. 222 (1998), 276–285.
• [8] J. Banas, J.Rivero, On measures of weak noncompactness, Ann. Mat. Pura Appl. 151 (1988), 213–224.
• [9] J. Banas, T. Zajac, Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity, Nonlinear Anal. 71 (2009), 5491–5500.
• [10] J. Caballero, A.B. Mingarelli, K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electron. J. Differential Equations 57 (2006), 1–11.
• [11] S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960.
• [12] M. Cichon, M. Metwali, Monotonic solutions for quadratic integral equations, Discuss. Math. Diff. Incl. 31 (2011), 157–171.
• [13] M. Cichon, M. Metwali, On quadratic integral equations in Orlicz spaces, J. Math. Anal. Appl. 387 (2012), 419–432.
• [14] F.S. De Blasi, On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Sci. Math. R. S. Roumanie 21 (1977), 259–262.
• [15] K. Deimling, Ordinary Differential Equations in Banach Spaces, Lect. Notes in Mathematics 596, Springer-Verlag, Berlin, 1977.
• [16] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
• [17] J. Dieudonné, Sur les espaces de Köthe, J. Anal. Math. (1951), 81–115.
• [18] D. Dinculeanu, Vector Measures, Pergamon Press, 1967.
• [19] N. Dunford, J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers,New York, 1958.
• [20] S. Djebali, Z. Sahnoun, Nonlinear alternatives of Schauder and Krasnoselskij types with applications to Hammerstein integral equations in L1 spaces, J. Differential Equations 249 (2010), 2061–2075.
• [21] W.G. El-Sayed, B. Rzepka, Nondecreasing solutions of a quadratic integral equation of Urysohn type, Comp. Math. Appl. 51 (2006), 1065–1074.
• [22] G. Emmanuele, About the existence of integrable solutions of functional-integral equation, Rev. Mat. Univ. Complut. Madrid 4 (1991), 65–69.
• [23] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, Berlin, 1977.
• [24] A.N. Kolmogorov, S.V. Fomin, Introductory Real Analysis, Prentice-Hall Inc., 1970.
• [25] M.A. Krasnoselskii, On the continuity of the operator Fu(x) = f(x, u(x)), Dokl. Akad. Nauk SSSR 77 (1951), 185–188.
• [26] M.A. Krasnoselskii, Two remarks on the method of successive approximations, Uspehi.Mat. Nauk. 10 (1955), 123–127.
• [27] M.A. Krasnoselskii, P.P. Zabrejko, E.I. Pustylnik, P.E. Sobolevskii, Integral Operators In Spaces of Summable Functions, Nauka, Moscow, 1966. English Translation: Noordhoff, Leyden, 1976.
• [28] M. Kunze, On a special class of nonlinear integral equations, J. Integral Equations Appl.7 (1995), 329–350.
• [29] K. Latrach, M.A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L1 spaces, Nonlinear Anal. 66 (2007), 2325–2333.
• [30] W. Pogorzelski, Integral Equations and Their Applications, Pergamon Press and PWN, Oxford, Warszawa, 1966.
• [31] G. Scorza Dragoni, Un teorema sulle funzioni continue rispetto ad une e misarubili rispetto ad un altra variable, Rend. Sem. Mat. Univ. Padova 17 (1948), 102–106.
• [32] M.A. Taoudi, Integrable solutions of a nonlinear functional integral equation on an unbounded interval, Nonlinear Anal. 71 (2009), 4131–4136.
• [33] M. Väth, A general theorem on continuity and compactness of the Urysohn operator, J. Integral Equations Appl. 8 (1996), 379–389.
• [34] P.P. Zabrejko, A.I. Koshlev, M.A. Krasnoselskii, S.G. Mikhlin, L.S. Rakovshchik,V.J. Stecenko, Integral Equations, Noordhoff, Leyden, 1975.
• [35] H. Zhu, On a nonlinear integral equation with contractive perturbation, Advances in Difference Equations, Vol. (2011), Article ID 154 742, 10 pp.