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A new contraction condition and its application to weakly singular Volterra integral equations of the second kind

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Abstract

In this paper, we prove some results concerning the existence and uniqueness of solutions for a large class of nonlinear Volterra integral equations of the second kind, especially singular Volterra integral equations, in the Banach space \(X:=C([0,1])\) consisting of real functions defined and continuous on the interval [0, 1]. The main idea used in the proof is that by using a new contraction condition we can construct a Cauchy sequence in the complete metric space X such that it is convergent to a unique element of this space. Finally, we present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results.

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Acknowledgements

The authors would like to thank the anonymous referee(s) for careful corrections, which helped them to improve the manuscript. They would like to express hearty thanks to the associate editor, Prof. Simeon Reich, for his support during the review process.

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Authors and Affiliations

  1. Department of Applied Mathematics Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran

    Omid Baghani

  2. Department of Mathematics Faculty of Mathematics, University of Sistan and Baluchestan, P.O. Box 98135-674, Zahedan, Iran

    Hamid Baghani

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  1. Omid Baghani
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  2. Hamid Baghani
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Correspondence to Omid Baghani.

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Baghani, O., Baghani, H. A new contraction condition and its application to weakly singular Volterra integral equations of the second kind. J. Fixed Point Theory Appl. 19, 2601–2615 (2017). https://doi.org/10.1007/s11784-017-0445-x

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