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SOME FIXED POINT THEOREMS IN GENERALIZED DARBO FIXED POINT THEOREM AND THE EXISTENCE OF SOLUTIONS FOR SYSTEM OF INTEGRAL EQUATIONS

  • Arab, Reza (Department of Mathematics, Sari Branch Islamic Azad University)
  • Received : 2014.04.05
  • Published : 2015.01.01

Abstract

In this paper we introduce the notion of the generalized Darbo fixed point theorem and prove some fixed and coupled fixed point theorems in Banach space via the measure of non-compactness, which generalize the result of Aghajani et al. [6]. Our results generalize, extend, and unify several well-known comparable results in the literature. One of the applications of our main result is to prove the existence of solutions for the system of integral equations.

Keywords

measure of noncompactness;fixed point;coupled fixed point;system of integral equations

References

  1. R. P. Agarwal, M. Benchohra, and D. Seba, On the application of measure of noncom-pactness to the existence of solutions for fractional differential equations, Results Math. 55 (2009), no. 3-4, 221-230. https://doi.org/10.1007/s00025-009-0434-5
  2. A. Aghajani, J. Banas, and Y. Jalilian, Existence of solution for a class of nonlinear Voltrra sigular integral equations, Comput. Math. Appl. 62 (2011), no. 3, 1215-1227. https://doi.org/10.1016/j.camwa.2011.03.049
  3. R. P. Agarwal and D. O'Regan, Singular Volterra integral equations, Appl. Math. Lett. 13 (2000), no. 1, 115-120.
  4. R. P. Agarwal and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, 2004.
  5. A. Aghajani, R. Allahyari, and M. Mursaleen, A generalization of Darbo's theorem with application to the solvability of systems of integral equations, J. Comput. Appl. Math. 260 (2014), 68-77. https://doi.org/10.1016/j.cam.2013.09.039
  6. J. Banas, Measures of noncompactness in the space of continuous tempered functions, Demonstratio Math. 14 (1981), no. 1, 127-133.
  7. A. Aghajani, J. Banas, and N. Sabzali, Some generalizations of Darbo fixed point theo-rem and applications, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 2, 345-358.
  8. A. Aghajani and N. Sabzali, A coupled fixed point theorem for condensing operators with application to system of integral equations, Journal of Nonlinear and Convex Analysis (To appear).
  9. I. Altun and D. Turkoglu, A fixed point theorem for mappings satisfying a general contractive condition of operator type, J. Comput. Anal. Appl. 9 (2007), no. 1, 9-14.
  10. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, Dekker, New York, 1980.
  11. S. S. Chang and Y. J. Huang, Coupled Fixed Point Theorems with Applications, J. Korean Math. Soc. 33 (1996), no. 3, 575-585.
  12. M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604-608. https://doi.org/10.1090/S0002-9939-1973-0334176-5
  13. X. Hu and Y. Yan, The global attractivity and asymptotic stability of solution of a nonlinear integral equation, J. Math. Anal. Appl. 321 (2006), no. 1, 147-156. https://doi.org/10.1016/j.jmaa.2005.08.010
  14. Z. Liu and S. M. Kang, Existence and asymptotic stability of solutions to functional-integral equation, Taiwanese J. Math. 11 (2007), no. 1, 187-196. https://doi.org/10.11650/twjm/1500404645
  15. L. Liu, F. Guo, C. Wu, and Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309 (2005), no. 2, 638-649. https://doi.org/10.1016/j.jmaa.2004.10.069
  16. M. Mursaleen and A. Alotaibi, Infinite system of differential equations in some spaces, Abstr. Appl. Anal. (2012); doi:10.1155/2012/863483. https://doi.org/10.1155/2012/863483
  17. M. Mursaleen and S. A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in $l_p$ spaces, Nonlinear Anal. 75 (2012), no. 4, 2111-2115. https://doi.org/10.1016/j.na.2011.10.011

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  2. Existence of solutions for a class of system of functional integral equation via measure of noncompactness vol.313, 2017, https://doi.org/10.1016/j.cam.2016.09.011
  3. The behaviour of measures of noncompactness in $$L^\infty ({\mathbb {R}}^n)$$ L ∞ ( R n ) with application to the solvability of functional integral equations 2017, https://doi.org/10.1007/s13398-017-0397-4
  4. Darbo type fixed and coupled fixed point results and its application to integral equation vol.77, pp.1, 2018, https://doi.org/10.1007/s10998-017-0223-y
  5. Existence of solutions to nonlinear functional-integral equations via the measure of noncompactness vol.20, pp.2, 2018, https://doi.org/10.1007/s11784-018-0546-1