Rocky Mountain Journal of Mathematics

Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type

Mohamed Abdalla Darwish and Johnny Henderson

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Source
Rocky Mountain J. Math. Volume 42, Number 2 (2012), 545-566.

Dates
First available: 23 April 2012

Permanent link to this document
http://projecteuclid.org/euclid.rmjm/1335187165

Digital Object Identifier
doi:10.1216/RMJ-2012-42-2-545

Zentralblatt MATH identifier
06035450

Mathematical Reviews number (MathSciNet)
MR2915506

Subjects
Primary: 45G10: Other nonlinear integral equations 45M99: None of the above, but in this section 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.

Keywords
Quadratic integral equation monotonic solutions measure of noncompactness Urysohn-Stieltjes Darbo's fixed point theorem

Citation

Darwish, Mohamed Abdalla; Henderson, Johnny. Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type. Rocky Mountain Journal of Mathematics 42 (2012), no. 2, 545--566. doi:10.1216/RMJ-2012-42-2-545. http://projecteuclid.org/euclid.rmjm/1335187165.


Export citation

References

  • G. Anichini and G. Conti, Existence of solutions of some quadratic integral equations, Opuscula Math. 28 (2008), 433-440.
  • J. Appell and P.P. Zabrejko, Nonlinear superposition operators, Cambridge Tracts Math. 95, Cambridge University Press, Cambridge, 1990.
  • I.K. Argyros, On a class of quadratic integral equations with perturbation, Funct. Approx. Comment. Math. 20 (1992), 51-63.
  • –––, Quadratic equations and applications to Chandrasekhar's and related equations, Bull. Austral. Math. Soc. 32 (1985), 275-292.
  • F.V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, 1964.
  • J. Banaś, Existence results for Volterra-Stieltjes quadratic integral equations on an unbounded interval, Math. Scand. 98 (2006), 143-160.
  • –––, Some properties of Urysohn-Stieltjes integral operators, Inter. J. Math. Math. Sci. 21 (1998), 78-88.
  • J. Banaś, J. Caballero, J. Rocha and K. Sadarangani, Monotonic solutions of a class of quadratic integral equations of Volterra type, Comput. Math. Appl. 49 (2005), 943-952.
  • J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces,
  • J. Banaś, M. Lecko and W.G. El-Sayed, Existence theorems of some quadratic integral equations, J. Math. Anal. Appl. 222 (1998), 276-285.
  • J. Banaś and A. Martinon, Monotonic solutions of a quadratic integral equation of Volterra type, Comput. Math. Appl. 47 (2004), 271-279.
  • J. Banaś and L. Olszowy, Measures of noncompactness related to monotonicity, Comment. Math. 41 (2001), 13-23.
  • J. Banaś and D. O'Regan, On existence and local attractivity of solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl. 345 (2008), 573-582.
  • –––, Volterra-Stieltjes integral operators, Math. Comput. Model. 41 (2005), 335-344.
  • J. Banaś, J. Rocha and K. Sadarangani, Solvability of a nonlinear integral equation of Volterra type, J. Comput. Appl. Math. 157 (2003), 31-48.
  • J. Banaś and B. Rzepka, On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation, Appl. Math. Comput. 213 (2009), 102-111.
  • –––, Nondecreasing solutions of a quadratic singular Volterra integral equation, Math. Comput. Model. 49 (2009), 488-496.
  • M. Benchohra and M.A. Darwish, On monotonic solutions of a quadratic integral equation of Hammerstein type, Inter. J. Appl. Math. Sci., to appear.
  • V.C. Boffi and G. Spiga, Nonlinear removal effects in time-dependent particle transport theory, Z. Angew. Math. Phys. 34(3) (1983), 347-357.
  • –––, An equation of Hammerstein type arising in particle transport theory, J. Math. Phys. 24 (1983), 1625-1629.
  • L.W. Busbridge, The mathematics of radiative transfer, Cambridge University Press, Cambridge, 1960.
  • J. Caballero, B. López and K. Sadarangani, Existence of nondecreasing and continuous solutions for a nonlinear integral equation with supremum in the kernel, Z. Anal. Anwend. 26 (2007), 195-205.
  • –––, On monotonic solutions of an integral equation of Volterra type with supremum, J. Math. Anal. Appl. 305 (2005), 304-315.
  • J. Caballero, D. O'Regan and K. Sadarangani, On monotonic solutions of some integral equations, Arch. Math. (Brno) 41 (2005), 325-338.
  • –––, On solutions of an integral equation related to traffic flow on unbounded domains, Arch. Math. (Basel) 82 (2004), 551-563.
  • J. Caballero, J. Rocha and K. Sadarangani, On monotonic solutions of an integral equation of Volterra type, J. Comput. Appl. Math. 174 (2005), 119-133.
  • J. Caballero, J. Rocha and K.B. Sadarangani, On monotonic solutions of an integral equation related with the Chandrasekhar equation,
  • K.M. Case and P.F. Zweifel, Linear transport theory, Addison-Wesley, Reading, MA 1967.
  • S. Chandrasekher, Radiative transfer, Dover Publications, New York, 1960.
  • M. Crum, On an integral equation of Chandrasekhar, Quart. J. Math., Oxford, 18 (1947), 244-252.
  • M.A. Darwish, Existence and asymptotic behaviour of solutions of a fractional integral equation, Appl. Anal. 88 (2009), 169-181.
  • –––, On a quadratic fractional integral equation with linear modification of the argument, Canad. Appl. Math. Quart. 16 (2008), 45-58.
  • –––, On monotonic solutions of an integral equation of Abel type, Math. Bohem. 133 (2008), 407-420.
  • –––, On monotonic solutions of a singular quadratic integral equation with supremum, Dynam. Syst. Appl. 17 (2008), 539-550.
  • –––, On a singular quadratic integral equation of Volterra type with supremum, IC/2007/071, Trieste, Italy, 2007.
  • –––, On solvability of some quadratic functional-integral equation in Banach algebra, Commun. Appl. Anal. 11 (2007), 441-450.
  • –––, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311 (2005), 112-119.
  • M.A. Darwish and J. Henderson, Existence and asymptotic stability of solutions of a perturbed quadratic fractional integral equation, Fract. Calc. Appl. Anal. 12 (2009), 71-86.
  • M.A. Darwish and S.K. Ntouyas, Existence of monotone solutions of a perturbed quadratic integral equation of Urysohn type, Nonlinear Stud. 18 (2011), 155-165.
  • –––, Monotonic solutions of a perturbed quadratic fractional integral equation, Nonlinear Anal. 71 (2009), 5513-5521.
  • M.A. Darwish and K. Sadarangani, On existence and asymptotic stability of solutions of a functional-integral equation of fractional order, J. Conv. Anal. 17 (2010), 413-426.
  • K. Deimling, Nonlinear fuctional analysis, Springer-Verlag, Berlin, 1985.
  • N. Dunford and J. Schwartz, Linear operators I, Inter. Publ., Leyden, 1963.
  • A. Granas and J. Dugundji, Fixed point theory, Springer-Verlag, New York, 2003.
  • S. Hu, M. Khavani and W. Zhuang, Integral equations arrising in the kinetic theory of gases, Appl. Anal. 34 (1989), 261-266.
  • X. Hu and J. Yan, The global attractivity and asymptotic stability of solution of a nonlinear integral equation, J. Math. Anal. Appl. 321 (2006), 147-156.
  • C.T. Kelly, Approximation of solutions of some quadratic integral equations in transport theory, J. Integral Equations Appl. 4 (1982), 221-237.
  • R.W. Leggett, A new approach to the H-equation of Chandrasekher, SIAM J. Math. Anal. 7 (1976), 542-550.
  • Z. Liu and S. Kang, Existence of monotone solutions for a nonlinear quadratic integral equation of Volterra type, Rocky Mountain J. Math. 37 (2007), 1971-1980.
  • I. Natanson, Theory of functions of real variables, Ungar, New York, 1960.
  • L.K. Nowosad and B.R. Saltzberg, On the solutions of a quadratic integral equation arising in signal design, J. Franklin Inst. 281 (1966), 437-454.
  • G. Spiga, R.L. Bowden and V.C. Boffi, On the solutions of a class of nonlinear integral equations arising in transport theory, J. Math. Phys. 25 (1984), 3444-3450.
  • C.A. Stuart, Existence theorems for a class of nonlinear integral equations, Math. Z. 137 (1974), 49-66.