Banach Journal of Mathematical Analysis

On the existence of at least a solution for functional integral equations via measure of noncompactness

Calogero Vetro and Francesca Vetro

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In this article, we use fixed-point methods and measure of noncompactness theory to focus on the problem of establishing the existence of at least a solution for the following functional integral equation

u(t)=g(t,u(t))+0tG(t,s,u(s))ds,t[0,+[, in the space of all bounded and continuous real functions on R+, under suitable assumptions on g and G. Also, we establish an extension of Darbo’s fixed-point theorem and discuss some consequences.

Article information

Banach J. Math. Anal., Volume 11, Number 3 (2017), 497-512.

Received: 18 June 2016
Accepted: 28 July 2016
First available in Project Euclid: 19 April 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47H08: Measures of noncompactness and condensing mappings, K-set contractions, etc.
Secondary: 45N05: Abstract integral equations, integral equations in abstract spaces 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]

Banach space functional integral equation measure of noncompactness


Vetro, Calogero; Vetro, Francesca. On the existence of at least a solution for functional integral equations via measure of noncompactness. Banach J. Math. Anal. 11 (2017), no. 3, 497--512. doi:10.1215/17358787-2017-0003.

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  • [1] R. P. Agarwal, M. Meehan, D. O’Regan, Fixed Point Theory and Applications, Cambridge Tracts in Math. 141, Cambridge Univ. Press, Cambridge, 2001.
  • [2] A. Aghajani, J. Banaś, and N. Sabzali, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 2, 345–358.
  • [3] A. Aghajani, M. Mursaleen, and A. Shole Haghighi, Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness, Acta Math. Sci. Ser. B Engl. Ed. 35 (2015), no. 3, 552–566.
  • [4] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, Oper. Theory Adv. Appl., Birkhäuser, Basel, 1992.
  • [5] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math. 60, Dekker, New York, 1980.
  • [6] J. Banaś and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New Delhi, 2014.
  • [7] J. Banaś and B. Rzepka, An application of a measure of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (2003), no. 1, 1–6.
  • [8] J. Banaś and K. Sadarangani, Compactness conditions in the study of functional, differential and integral equations, Abstr. Appl. Anal. 2013 (2013), no. 819315.
  • [9] M. Berzig, Generalization of the Banach contraction principle, preprint, arXiv:1310.0995v1 [math. CA].
  • [10] L. S. Cai and J. Liang, New generalizations of Darbo’s fixed point theorem, Fixed Point Theory Appl. 2015 (2015), no. 156.
  • [11] J. Chen and X. Tang, Generalizations of Darbo’s fixed point theorem via simulation functions with application to functional integral equations, J. Comput. Appl. Math. 296 (2016), 564–575.
  • [12] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Semin. Mat. Univ. Padova 24 (1955), 84–92.
  • [13] K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math. 596, Springer, Berlin, 1977.
  • [14] B. C. Dhage, S. B. Dhage, and H. K. Pathak, A generalization of Darbo’s fixed point theorem and local attractivity of generalized nonlinear functional integral equations, Differ. Equ. Appl. 7 (2015), no. 1, 57–77.
  • [15] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, Amer. Math. Soc., Providence, 1998.
  • [16] K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), no. 1, 301–309. JFM56.1124.04.
  • [17] A. Samadi and M. B. Ghaemi, An extension of Darbo’s theorem and its application, Abstr. Appl. Anal. 2014, no. 852324.
  • [18] M. Väth, Volterra and Integral Equations of Vector Functions, Pure Appl. Math. 224, Dekker, New York, 2000.