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

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

About History Editorial Page Addresses Scope Editorial Staff Submission Review For Authors Publication Ethics Issues Русская версия

On a Coupled System of Urysohn-Stieltjes Integral Equations in Reflexive Banach Space

Author(s):

Ahmed M. A. El-Sayed

Prof. Dr, Professor of Mathematics, Department of Mathematics,
Faculty of Science, Alexandria University, Egypt.

amasayed@alexu.edu.eg

Masouda M. A. Al-Fadel

M.SC.(Mathematics)
Faculty of Science, Omar Al-Mukhtar University, Libya.

najemeoe1234@gmail.com

Abstract:

Urysohn-Stieltjes integral operators and Urysohn-Stieltjes integral equations have been studied by some authors. In this paper we prove the existence of at least one weak solution of a coupled system of Urysohn-Stieltjes integral equations in the reflexive Banach space. We used the O'Regan fixed point theorem and some propositions. As an application, the coupled system of Hammerstien-Stieltjes integral equations is also studied.

Keywords

References:

  1. J. Banas, Some properties of Urysohn-Stieltjes integral operators, Intern. J. Math. and Math. Sci. 21(1998) 78-88
  2. J. Banas, J. R. Rodriguez and K. Sadarangani, On a class of Urysohn-Stieltjes quadratic integral equations and their applications, J. Comput. Appl. Math. I13(2000) 35-50
  3. J. Banas and J. Dronka, Integral operators of Volterra-Stieltjes type, their properties and applications, Math. Comput. Modelling. 32(2000) (11-13)1321-1331
  4. J. Banas and K. Sadarangani, Solvability of Volterra-Stieltjes operator-integral equations and their applications, Comput Math. Appl. 41(12)(2001) 1535-1544
  5. J. Banas, J. C. Mena, Some Properties of Nonlinear Volterra-Stieltjes Integral Operators, Comput Math. Appl. 49(2005) 1565-1573
  6. J. Banas, D. O'Regan, Volterra-Stieltjes integral operators, Math. Comput. Modelling. 41(2005) 335-344
  7. C. W. Bitzer, Stieltjes-Volterra integral equations, Illinois J. Math. 14(1970) 434-451
  8. J. M. Ball, Weak continuity properties of mappings and semigroups, Proc. Royal. Soc. Edinbourgh Sect. A 72(1973-1974), 275-280. MR 53]1354
  9. S. Chen, Q. Huang and L. H. Erbe, Bounded and zero-convergent solutions of a class of Stieltjes integro-differential equations, Proc. Amer. Math. Soc. 113(1991) 999-1008
  10. J. Diestel, J. J. Uhl Jr., Vector Measures, in: Math. Surveys, vol. 15, Amer. Math. Soc, Providence, RI, (1977)
  11. N. Dunford, J. T. Schwartz, Linear operators, Interscience, Wiley, New York. (1958)
  12. A. M. A. EL-Sayed, H. H. G. Hashem, Weak maximal and minimal solutions for Hammerstein and Urysohn integral equations in reflexive Banach spaces, Differential Equation and Control Processes. 4(2008) 50-62
  13. A. M. A. El-Sayed, H. H. G. Hashem, Existence Results for Coupled Systems of Quadratic Integral Equations of Fractional Orders, Optimization Letters, 7(2013) 1251-1260
  14. A. M. A. El-Sayed, H. H. G. Hashem, Solvability of coupled systems of fractional order integro-differential equations, J. Indones. Math. Soc. 19(2)(2013) 111121
  15. R. F. Geitz, Pettis integration, Proc. Amer. Math. Soc. 82(1981) 81-86
  16. H. H. G. Hashem, On successive approximation method for coupled systems of Chandrasekhar quadratic integral equations, Journal of the Egyptian Mathematical Society. 23(2015) 108112
  17. E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. Providence, R. I. (1957)
  18. J. S. Macnerney, Integral equations and semigroups, Illinois J. Math. 7(1963) 148-173
  19. A. B. Mingarelli, Volterra-Stieltjes integral equations and generalized ordinary differential expressions, Lecture Notes in Math., 989, Springer (1983)
  20. I. P. Natanson, Theory of Functions of a Real Variable, Ungar, New York. (1960)
  21. D. O'Regan, Fixed point theory for weakly sequentially continuous mapping, Math. Comput. Modeling. 27(1998) 1-14
  22. H. A. H. Salem, Quadratic integral equations in reflexive Banach spaces, Discuss. Math. Di_er. Incl. Control Optim. 30(2010) 61-69
  23. A. Szep, Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar. 6(1971) 197-203
  24. S. Szua, Kneser's theorem for weak solutions of ordinary differential equation in reflexive Banach spaces, Bull. Polish Acad. Sci. Math. 26(1978) 407-413

Full text (pdf)