Existence of unique common solution to the system of non-linear integral equations via fixed point results in incomplete metric spaces

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• 作者：Mian Bahadur Zada ; Muhammad Sarwar…
• 关键词：common fixed point ; weakly compatible maps ; common \((CLR)\) ; property ; common \((E.A)\) ; property ; Urysohn integral equations ; Volterra ; Hammerstein integral equations
• 刊名：Journal of Inequalities and Applications
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1535KB
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X
• 卷排序：2017

文摘

In this article, we apply common fixed point results in incomplete metric spaces to examine the existence of a unique common solution for the following systems of Urysohn integral equations and Volterra-Hammerstein integral equations, respectively: $$u(s)=\phi_{i}(s)+ \int_{a}^{b}K_{i}\bigl(s, r,u(r)\bigr) \,dr, $$ where \(s\in(a,b)\subseteq\mathbb{R}\); \(u, \phi_{i}\in C((a,b),\mathbb{R}^{n})\) and \(K_{i}:(a,b)\times(a,b)\times \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\), \(i=1,2,\ldots,6 \) and $$u(s)=p_{i}(s)+\lambda \int_{0}^{t}m(s, r)g_{i}\bigl(r,u(r) \bigr)\,dr+\mu \int_{0}^{\infty}n(s, r)h_{i}\bigl(r,u(r) \bigr)\,dr, $$ where \(s\in(0,\infty)\), \(\lambda,\mu\in\mathbb{R}\), u, \(p_{i}\), \(m(s, r)\), \(n(s, r)\), \(g_{i}(r,u(r))\) and \(h_{i}(r,u(r))\), \(i=1,2,\ldots,6\), are real-valued measurable functions both in s and r on \((0,\infty)\).Keywordscommon fixed pointweakly compatible mapscommon \((CLR)\)-propertycommon \((E.A)\)-propertyUrysohn integral equationsVolterra-Hammerstein integral equationsMSC47H1054H251 Introduction and preliminariesMathematical models are very powerful and important parts of the mathematical analysis with numerous applications to real world problems. Several problems that appear in applied mathematics, physical sciences, geology, mechanics, engineering, economics, and biology generate mathematical models interpreted by functional equations, integral equations, matrix equations, and differential equations etc. There are multifarious and advanced methods, focusing on the existence of unique solutions to these models. To handle the existence of unique solution to such equations, one of these methods is the fixed point method; for example, refer to [1–4]. In metric fixed point theory the first remarkable result was given by Banach, usually known as the Banach contraction principle. This principle is a prominent tool for solving problems in non-linear analysis. Several mathematicians improved and extended this principle by modifying the interpretation and pattern of the metric function for instance: cone metric spaces [5], G-metric spaces [6], partial metric spaces [7] and fuzzy metric spaces [8] etc. After the proper introduction of cone metric space by Huang and Zhong [5], there was a drawback that fixed point results under rational type contractions are unsubstantial in a cone metric space as it is a vector-valued metric. Azam et al. [9] offered the conception of a complex-valued metric space for finding the fixed point results satisfying rational type contractive conditions.