ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Дифференциальные Уравнения
Процессы Управления

On Erdelyi-Kober Cubic Fractional Integral Equation of Urysohn-Volterra Type


Hamed Kamal Awad

Department of Mathematics,
Faculty of Sciences, Damanhour University, Egypt

Mohamed Abdalla Darwish

Department of Mathematics,
Faculty of Sciences, Damanhour University, Egypt


Cubic integral equations is the general form of the quadratic integral equations which have several applications in the theory of radiative transfer, in the traffic theory, in the theory of particle transport and in the kinetic theory of gases. In this paper, we present a result on the existence of solutions of the perturbed Erdelyi-Kober fractional cubic integral equation of Uryshon-Volterra type in the Banach space of real functions which are defined, continuous and bounded on an unbounded interval. In order to prove our main result we use the Darbo fixed point theorem and a measure of noncompactness.

Ключевые слова


  1. Alamo, J. A., Rodrí guez, J.; Operational calculus for modified Erdé lyi-Kober operators. Serdica 20, 351-363 (1994)
  2. Appell, J., Zabrejko, P. P.; Nonlinear Superposition Operators. Cambridge Tracts in Mathematics 95, Cambridge University Press. (1990)
  3. Awad, H. K., Darwish, M. A.; On monotonic solutions of a cubic Urysohn Integral equation with linear modification of the argument. Adv. Dyn. Syst. Appl. 13, 91-99 (2018)
  4. Awad, H. K., Darwish, M. A., Metwali, M. M. A.; On a cubic integral equation of Urysohn type with linear perturbation of second kind. Journal of Mathematics and Applications 41, 29-38 (2018)
  5. Banaś J., Goebel, K. Measures of Noncompactness in Banach Spaces. Lecture Notes inPure and Applied Mathematics, 60, Marcel Dekker, New York, (1980)
  6. Banaś, J.; Measures of noncompactness in the space of continuous tempered functions. Demonstratio Math. 14, 127-133 (1981)
  7. Banaś, J., Olszowy, L.; Measures of noncompactness related to monotonicity. Comment. Math. 41, 13-23 (2001)
  8. Caballero, J., O'Regan, D., Sadarangani, K.; On nondecreasing solutions of cubic integral equations of Urysohn type. Comment. Math. (Prace Mat. ) 44, 39-53 (2004)
  9. Darbo, G.; Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Univ. Padova 24, 84-92 (1955)
  10. Darwish, M. A.; On quadratic integral equation of fractional orders. J. Math. Anal. Appl. 311, 112-119 (2005)
  11. Darwish, M. A., Rzepka, B.; Asymptotically stable solutions of a generalized fractional quadratic functional-integral equation of Erdé lyi-Kober type. J. Funct. Spaces, Art. ID 192542, 9 pp (2014)
  12. Darwish, M. A., Sadarangani, K.; On Erdé lyi-Kober type quadratic integral equation with linear modification of the argument. Appl. Math. Comput. 238, 30-42 (2014)
  13. Darwish, M. A., Sadarangani, K.; On a quadratic integral equation with supremum involving Erdé lyi-Kober fractional order. Math. Nachr. 228, 566-576 (2015)
  14. Darwish, M. A.; On Erdé lyi -Kober fractional Urysohn-Volterra quadratic integral equations. Appl. Math. Comput. 273, 562-569 (2016)
  15. Darwish, M. A., Samet, B.; On Erdé lyi-Kober quadratic functional-integral equation in Banach algebra. Numer. Funct. Anal. Optim. 39, 276-294 (2018)
  16. Garra, R., Giusti, A., Mainardi, F., Pagnini, G.; Fractional relaxation with time-varying coefficient. Fract. Calc. Appl. Anal. 17, 424-439 (2014)
  17. Herrmann, R.; Reflection symmetric Erdé lyi-Kober type operators-a quasi-particle interpretation. Fract. Calc. Appl. Anal. 17, 1215-1228 (2014)
  18. Kiryakova, V.; Convolutions of Erdé lyi-Kober fractional integration operators. Complex analysis and applications '87 (Varna, 1987), 273-283, Publ. House Bulgar. Acad. Sci., Sofia. (1989)
  19. Kiryakova, V.; Generalized Fractional Calculus and Applications. Longman Sci. & Techn., Harlow and J. Wiley \& Sons, N. York. (1994)
  20. Kiryakova, V., Al-Saqabi, B. N.; Transmutation method for solving Erdé lyi-Kober fractional differintegral equations. J. Math. Anal. Appl. 211, 347-364 (1997)
  21. Pagnini, G.; Erdé lyi-Kober fractional diffusion. Fract. Calc. Appl. Anal. 15, 117-127 (2012)
  22. Sneddon, I. N.; Mixed Boundary Value Problems in Potential Theory, North-Holland Publ., Amsterdam. (1966)
  23. Stuart, C. A.; Existence theorems for a class of nonlinear integral equations. Math. Z. 137, 49-66 (1974)

Полный текст (pdf)