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ArticleOriginal scientific text

Title

Monotonic solutions for quadratic integral equations

Authors 1, 2

Affiliations

  1. Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
  2. Department of Mathematics, Faculty of Sciences, Alexandria University at Damanhour, 22511 Damanhour, Egypt

Abstract

Using the Darbo fixed point theorem associated with the measure of noncompactness, we establish the existence of monotonic integrable solution on a half-line ℝ₊ for a nonlinear quadratic functional integral equation.

Keywords

ENG
integral equation, monotonic solution, measure of noncompactness, Darbo fixed point theorem, superposition operator

Bibliography

  1. G. Anichini and G. Conti, Existence of solutions of some quadratic integral equations, Opuscula Mathematica 28 (2008), 433-440. doi: 10.7151/dmdico.1132
  2. J. Appell and P.P. Zabrejko, Continuity properties of the superposition operator, Preprint No. 131, Univ. Augsburg, 1986.
  3. I.K. Argyros, Quadratic equations and applications to Chandrasekhar's and related equations, Bull. Austral. Math. Soc. 32 (1985), 275-292. doi: 10.1017/S0004972700009953
  4. J. Banaś, On the superposition operator and integrable solutions of some functional equations, Nonlin. Anal. Th. Meth. Appl. 12 (1988), 777-784. doi: 10.1016/0362-546X(88)90038-7
  5. J. Banaś, Integrable solutions of Hammerstein and Urysohn integral equations, J. Autral. Math. Soc. 46 (1989), 61-68. doi: 10.1017/S1446788700030378
  6. J. Banaś and A. Chlebowicz, On integrable solutions of a nonlinear Volterra integral equation under Carathéodory conditions, Bull. London Math. Soc. 41 (2009), 1073-1084.
  7. J. Banaś and W.G. El-Sayed, Measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variation, J. Math. Anal. Appl. 167 (1992), 133-151. doi: 10.1016/0022-247X(92)90241-5
  8. J. Banaś and W.G. El-Sayed, Solvability of Functional and Integral Equations in some Classes of Integrable Functions, Politechnika Rzeszowska, Rzeszów, 1993.
  9. J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes in Math. 60, M. Dekker, New York-Basel, 1980.
  10. J. Banaś, J. Rocha and K.B. Sadarangani, Solvability of a nonlinear integral equation of Volterra type, J. Comp. Appl. Math. 157 (2003), 31-48. doi: 10.1016/S0377-0427(03)00373-X
  11. J. Banaś and B. Rzepka, On existence and asymptotic stability of a nonlinear integral equation, J. Math. Anal. Appl. 284 (2003), 165-173. empty
  12. J. Banaś and B. Rzepka, Monotonic solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl. 332 (2007), 1371-1379. doi: 10.1016/j.jmaa.2006.11.008
  13. J. Banaś and B. Rzepka, Nondecreasing solutions of a quadratic singular Volterra integral equation, Math. Comp. Model. 49 (2009), 488-496. doi: 10.1016/j.mcm.2007.10.021
  14. J. Banaś, M. Lecko and W.G. El-Sayed, Existence theorems of some quadratic integral equations, J. Math. Anal. Appl. 222 (1998), 276-285. doi: 10.1006/jmaa.1998.5941
  15. J. Banaś and T. Zając, Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity, Nonlin. Anal. Th. Meth. Appl. 71 (2009), 5491-5500. doi: 10.1016/j.na.2009.04.037
  16. J. Banaś and L. Olszowy, Measures of noncompactness related to monotonicity, Comment. Math. 41 (2001), 13-23.
  17. J. Caballero, A.B. Mingarelli and K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electr. J. Differ. Equat. 57 (2006), 1-11.
  18. S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960.
  19. G. Darbo, Punti uniti in trasformazioni a codominio noncompatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92.
  20. M.A. Darwish, On solvability of some quadratic functional-integral equation in Banach algebra, Commun. Appl. Anal. 11 (2007), 441-450.
  21. M.A. Darwish and J. Henderson, Existence and asymptotic stability of solutions of a perturbed quadratic fractional integral equations, Fract. Calculus Appl. Anal. 12 (2009), 71-86.
  22. M.A. Darwish and S.K. Ntouyas, Monotonic solutions of a perturbed quadratic fractional integral equation, Nonlin. Anal. Th. Meth. Appl. 71 (2009), 5513-5521. doi: 10.1016/j.na.2009.04.041
  23. N. Dunford and J. Schwartz, Linear Operators I, Int. Publ., Leyden, 1963.
  24. W.G. El-Sayed and B. Rzepka, Nondecreasing solutions of a quadratic integral equation of Urysohn type, Comp. Math. Appl. 51 (2006), 1065-1074. doi: 10.1016/j.camwa.2005.08.033
  25. S. Hu, M. Khavani and W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Analysis 34 (1989), 261-266. doi: 10.1080/00036818908839899
  26. M.A. Krasnosel'skii, P.P. Zabrejko, J.I. Pustyl'nik and P.E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden, 1976.
  27. M. Väth, A general theorem on continuity and compactness of the Urysohn operator, J. Integral Equations Appl. 8 (1996), 379-389. doi: 10.1216/jiea/1181075958
  28. M. Väth, Continuity of single- and multivalued superposition operators in generalized ideal spaces of measurable functions, Nonlin. Funct. Anal. Appl. 11 (2006), 607-646.
  29. M. Väth, Volterra and Integral Equations of Vector Functions, Marcel Dekker, New York-Basel, 2000.
  30. P.P. Zabrejko, A.I. Koshlev, M.A. Krasnosel'skii, S.G. Mikhlin, L.S. Rakovshchik and V.J. Stecenko, Integral Equations, Noordhoff, Leyden, 1975.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
2011/31/2
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Pages:
157-171
Main language of publication
English
Received
2010-12-04
Published
2011

DOI: 10.7151/dmdico.1132

Exact and natural sciences