An Existence Result For A Class Of Nonlinear Volterra Functional Integral Equations


Neda Khodabakhshi - Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran
Adrian Petrusel - bFaculty of Mathematics and Computer Science, Babe-Bolyai University Cluj-Napoca, Koglniceanu Street, No. 1, Cluj-Napoca, Romania.


Recently [N. Khodabakhshi, S. M. Vaezpour, Fixed Point Theory, to appear] provides sufficient conditions for the existence of common fixed point for two commuting operators using the technique associated to an abstract measure of non-compactness in Banach spaces. In this paper, we develope their work with further applicative investigation. More precisely, we give suitable assumptions in order to obtain the existence of solutions for a nonlinear integral equation.


Coincidence point, common fixed point, measure of noncompactness


[1] R. P. Agarwal, M. Meehan, D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, (2004). 2
[2] A. Aghajani, J. Bana´s, N. Sabzali, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin., 20 (2013), 345–358. 1
[3] A. Aghajani, Y. Jalilian, Existence and global attractivity of solutions of a nonlinear functional integral equation, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 3306–3312. 1
[4] J. Bana´s, On measures of noncompactness in Banach spaces, Comment. Math. Univ. Carolinae., 21 (1980), 131–143. 1
[5] J. Bana´s, Measures of noncompactness in the space of continuous tempered functions, Demonstratio Math., 14 (1981), 127–33. 3
[6] J. Bana´s, B. C. Dhage, Global asymptotic stability of solutions of a functional integral equation, Nonlinear Anal. 69 (2008), 1945–1952. 1
[7] J. Bana´s, K. Goebel, Measures of Noncompactness in Banach spaces, Lect. Notes Pure Appl. Math., vol. 60, Dekker, New York, (1980). 2, 2
[8] J. Bana´s, B. Rzepka, An application of a measure of noncompactness in the study of asymptotic stability, Appl. Math. Lett., 16 (2003), 1–6. 1
[9] J. Dugundji, A. Granas, Fixed Point Theory, vol. 1, PWN, Warszawa, (1982). 2.4
[10] W. G. El-Sayed, Nonlinear functional integral equations of convolution type. Port. Math., 54 (1997), 449–456
[11] A. Hajji, E. Hanebaly, Commuting mappings and α-compact type fixed point theorems in locally convex spaces, Int. J. Math. Anal., 1 (2007), 661–680. 1
[12] A. Hajji, A generalization of Darbo’s fixed point and common solutions of equations in Banach spaces, Fixed Point Theory Appl., 2013 (2013), 9 pages .
[13] X. Hu, J. Yan, The global attractivity and asymptotic stability of solution of a nonlinear integral equation, J. Math. Anal. Appl., 321 (2006), 147–156
[14] N. Khodabakhshi, S. M. Vaezpour, Common fixed point theorems via measure of noncompactness, Fixed Point Theory, to appear.
[15] Z. Liu, S. Kang, J. Ume, Solvability and asymptotic stability of an nonlinear functional-integral equation, Appl. Math. Lett., 24 (2011), 911–917
[16] Y. C. Wong, Introductory Theory of Topological Vector Spaces, Marcel Dekker, Inc., New York, (1992