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

Volume 10, Issue 7, pp 3364--3380
Publication Date: July 20, 2017 Submission Date: October 11, 2016
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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. - Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia. 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.

### Share and Cite

##### ISRP Style

Xinqiu Zhang, Lishan Liu, Yonghong Wu, Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 7, 3364--3380

##### AMA Style

Zhang Xinqiu, Liu Lishan, Wu Yonghong, Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations. J. Nonlinear Sci. Appl. (2017); 10(7):3364--3380

##### Chicago/Turabian Style

Zhang, Xinqiu, Liu, Lishan, Wu, Yonghong. "Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations." Journal of Nonlinear Sciences and Applications, 10, no. 7 (2017): 3364--3380

### Keywords

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

•  34B16
•  34B18

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