Abstract
In the present paper, we investigate the construction of compact sets of bounded continuous functions on unbounded set \({\Omega}\) of \({\mathbb{R}^n}\), and then introduce a measure of noncompactness on this space. As an application, we study the existence of solutions for a class of nonlinear functional integral equations of Volterra type by using some fixed point theorems associated with this new measure of noncompactness. Obtained results generalize numerous other ones. We will also include some examples which show that our results are applicable where the previous ones are not.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aghajani A., Allahyari R., Mursaleen M.: A generalization of Darbo’s theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 260, 68–77 (2014)
Aghajani, A., Banaś, J., Jalilian, Y.: Existence of solutions for a class of nonlinear Volterra singular integral equations. Comput. Math. Appl. 62(3), 1215–1227 (2011)
Aghajani A., Banaś J., Sabzali N.: Some generalizations of Darbo fixed point theorem and applications. Bull. Belg. Math. Soc. Simon Stevin 20(2), 345–358 (2013)
Aghajani A., Jalilian Y.: Existence and global attractivity of solutions of a nonlinear functional integral equation, Commun. Nonlin. Sci. Numer. Simul. 15, 3306–3312 (2010)
Aghajani, A., Mursaleen, M., Shole Haghighi, A.: Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness. Acta Math. Sci. (Accepted)
Aghajani, A., Sabzali, N.: Existence of coupled fixed points via measure of noncompactness and applications. J. Nonlinear Convex Anal. 15(5), 941–952 (2014)
Aghajani A., Shole Haghighi A.: Existence of solutions for a class of functional integral equations of Volterra type in two variables via measure of noncompactness. IJST 38A1, 1–8 (2014)
Arab, R., Allahyari, R., Shole haghighi, A.: Existence of solutions of infinite systems of integral equations in two variables via measure of noncompactness. Appl. Math. Comput. (2014). doi:10.1016/j.amc.2014.08.023
Banaś J.: Measures of noncompactness in the space of continuous tempered functions. Demonstratio Math. 14, 127–133 (1981)
Banaś J., Dhage B.C.: Global asymptotic stability of solutions of a functional integral equation. Nonlin. Anal. 69, 1945–1952 (2008)
Banaś, J., Goebel, K.: Measures of noncompactness in Banach spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Dekker, New York (1980)
Banaś J., Lecko M.: An existence theorem for a class of infinite systems of integral equations. Math. Comput. Model. 34, 533–539 (2001)
Banaś J., O’Regan D.: On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order. J. Math. Anal. Appl. 345, 573–582 (2008)
Banaś J., Rzepka R.: An application of a measure of noncompactness in the study of asymptotic stability. Appl. Math. Lett. 16, 1–6 (2003)
Banaś J., Sadarangani K.: Solvability of Volterra-Stieltjes operator-integral equations and their applications. Comput. Math. Appl. 41, 1535–1544 (2001)
Darbo G.: Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Univ. Padova 24, 84–92 (1955)
Ha-Olsen H., Holden H.: The Kolmogorov–Riesz compactness theorem. Expos. Math. 28, 385–394 (2010)
Liu L., Guo F., Wu C., Wu Y.: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. J. Math. Anal. Appl. 309, 638–649 (2005)
Mursaleen, M.: Application of measure of noncompactness to infinite systems of differential equations. Can. Math. Bull. (In press)
Mursaleen M., Mohiuddine S.A.: Applications of measures of noncompactness to the infinite system of differential equations in l p spaces. Nonlinear Anal. 75(4), 2111–2115 (2012)
Olszowy, L.: On some measures of noncompactness in the Fréchet spaces of continuous functions. Nonlinear Anal. 71(11), 5157–5163 (2009)
Olszowy L.: Solvability of infinite systems of singular integral equations in Fréchet space of continuous functions. Comput. Math. Appl. 59, 2794–2801 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arab, R., Allahyari, R. & Shole Haghighi, A. Construction of a Measure of Noncompactness on \({BC(\Omega)}\) and its Application to Volterra Integral Equations. Mediterr. J. Math. 13, 1197–1210 (2016). https://doi.org/10.1007/s00009-015-0547-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-015-0547-x