Open Access Research

Analysis of Abel-type nonlinear integral equations with weakly singular kernels

JinRong Wang12, Chun Zhu2 and Michal Fečkan34*

Author Affiliations

1 School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou, 550018, P.R. China

2 Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, P.R. China

3 Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynská dolina, Bratislava, 842 48, Slovakia

4 Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, Bratislava, 814 73, Slovakia

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Boundary Value Problems 2014, 2014:20  doi:10.1186/1687-2770-2014-20

Dedicated to Professor Ivan Kiguradze

Published: 17 January 2014


In this paper, we investigate Abel-type nonlinear integral equations with weakly singular kernels. Existence and uniqueness of nontrivial solution are presented in an order interval of a cone by using fixed point methods. As a byproduct of our method, we improve a gap in the proof of Theorem 5 in Buckwar (Nonlinear Anal. TMA 63:88-96, 2005). As an extension, solutions in closed form of some Erdélyi-Kober-type fractional integral equations are given. Finally theoretical results with three illustrative examples are presented.

MSC: 26A33, 45E10, 45G05.

Abel-type nonlinear integral equations; weakly singular kernels; existence; numerical solutions