Positive solutions of fractional integral equations by the technique of measure of noncompactness

Hemant Kumar Nashine; Reza Arab; Ravi P Agarwal; Manuel De la Sen

doi:10.1186/s13660-017-1497-6

Positive solutions of fractional integral equations by the technique of measure of noncompactness

J Inequal Appl. 2017;2017(1):225. doi: 10.1186/s13660-017-1497-6. Epub 2017 Sep 15.

Authors

Hemant Kumar Nashine¹, Reza Arab², Ravi P Agarwal¹, Manuel De la Sen³

Affiliations

¹ Department of Mathematics, Texas A&M University, Kingsville, Texas 78363-8202 USA.
² Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran.
³ Institute of Research and Development of Processes IIDP Faculty of Science and Technology, University of the Basque Country, Leioa, Spain.

Abstract

In the present study, we work on the problem of the existence of positive solutions of fractional integral equations by means of measures of noncompactness in association with Darbo's fixed point theorem. To achieve the goal, we first establish new fixed point theorems using a new contractive condition of the measure of noncompactness in Banach spaces. By doing this we generalize Darbo's fixed point theorem along with some recent results of (Aghajani et al. (J. Comput. Appl. Math. 260:67-77, 2014)), (Aghajani et al. (Bull. Belg. Math. Soc. Simon Stevin 20(2):345-358, 2013)), (Arab (Mediterr. J. Math. 13(2):759-773, 2016)), (Banaś et al. (Dyn. Syst. Appl. 18:251-264, 2009)), and (Samadi et al. (Abstr. Appl. Anal. 2014:852324, 2014)). We also derive corresponding coupled fixed point results. Finally, we give an illustrative example to verify the effectiveness and applicability of our results.

Keywords: Darbo’s fixed point theorem; coupled fixed point; integral equations; measures of noncompactness.