# Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations

Volume 10, Issue 7, pp 3364--3380 Publication Date: July 20, 2017
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### Authors

Xinqiu Zhang - School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China.
Lishan Liu - School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China.
Yonghong Wu - Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia.

### Abstract

In this article, we study the existence and the uniqueness of iterative positive solutions for a class of nonlinear singular integral equations in which the nonlinear terms may be singular in both time and space variables. By using the fixed point theorem of mixed monotone operators in cones, we establish the conditions for the existence and uniqueness of positive solutions to the problem. Moreover, we derive various properties of the positive solutions to the equation and establish their dependence on the model parameter. The theorem obtained in this paper is more general and complements many previous known results including singular and nonlinear cases. Application of the results to the study of differential equations are also given in the article.

### Keywords

• Mixed monotone operator
• fixed point theorem
• iterative positive solution
• singular integral equations
• boundary value problem
• cone.

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