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

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: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(s)=\phi_{i}(s)+ \int_{a}^{b}K_{i}\bigl(s, r,u(r)\bigr) \,dr, $$\end{document}u(s)=ϕi(s)+∫abKi(s,r,u(r))dr, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s\in(a,b)\subseteq\mathbb{R}$\end{document}s∈(a,b)⊆R; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u, \phi_{i}\in C((a,b),\mathbb{R}^{n})$\end{document}u,ϕi∈C((a,b),Rn) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K_{i}:(a,b)\times(a,b)\times \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$\end{document}Ki:(a,b)×(a,b)×Rn→Rn, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i=1,2,\ldots,6 $\end{document}i=1,2,…,6 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(s)=p_{i}(s)+\lambda \int_{0}^{t}m(s, r)g_{i}\bigl(r,u(r) \bigr)\,dr+\mu \int_{0}^{\infty}n(s, r)h_{i}\bigl(r,u(r) \bigr)\,dr, $$\end{document}u(s)=pi(s)+λ∫0tm(s,r)gi(r,u(r))dr+μ∫0∞n(s,r)hi(r,u(r))dr, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s\in(0,\infty)$\end{document}s∈(0,∞), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda,\mu\in\mathbb{R}$\end{document}λ,μ∈R, u, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p_{i}$\end{document}pi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m(s, r)$\end{document}m(s,r), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n(s, r)$\end{document}n(s,r), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g_{i}(r,u(r))$\end{document}gi(r,u(r)) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h_{i}(r,u(r))$\end{document}hi(r,u(r)), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i=1,2,\ldots,6$\end{document}i=1,2,…,6, are real-valued measurable functions both in s and r on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(0,\infty)$\end{document}(0,∞).


Introduction and preliminaries
Mathematical 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 [-]. 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 [], G-metric spaces [], partial metric spaces [] and fuzzy metric spaces [] etc. After the proper introduction of cone metric space by Huang and Zhong [], 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. [] offered the conception of a complex-valued metric space for finding the fixed point results satisfying rational type contractive conditions. Definition . ([]) Let Y be non-empty set and C + = {c ∈ C : c }. Then the mapping d : Y × Y → C + is a complex-valued metric if it satisfies the following axioms: The set Y together with d is called a complex-valued metric space.
In this setting, Azam  Throughout this manuscript Y represents a complex-valued metric space, unless otherwise specified. For two self-maps f  and f  defined on a non-empty set Y , w ∈ Y is a common fixed point of f  and f  if f  w = f  w = w. To study common fixed points, Jungck [] initiated the concept of weak compatibility of maps thus: In the study of common fixed point results of weakly compatible mappings we often require the assumption of the continuity of mappings or the completeness of the underlying space. Regarding this Aamri and Moutawakil [] relaxed these conditions by introducing the notion of the (E.A)-property. In , the new notion of Common Limit in the Range property (for short (CLR)-property) was given by Sintunavarat Note that the (E.A)-property tolerates the condition of closeness of the range of subspaces of the involved mappings. However, the significance of the (CLR)-property reveals that closeness of the range of subspaces is not essential.
Sarwar and Bahadur Zada [] established the following common fixed point results. The aim of this manuscript is to study the existence of unique common solution for the systems of: • Urysohn integral equations in complex-valued metric spaces, • Volterra-Hammerstein integral equations in ordinary metric spaces.

Existence of unique common solution to the systems of Urysohn integral equations
Our plan is to apply Theorem . to the existence of a unique common solution to the following system: Assume that the following conditions hold: Now, we are in a position to formulate the existence results.
, and (f  , f  ) are weakly compatible. Then the system (.) of Urysohn integral equations has a unique common solution.
Proof Notice that the system (.) of Urysohn integral equations has a unique common solution if and only if the system (.) of operators has a unique common fixed point. Now, From condition () of Theorem ., we have Next, we need to show the weak compatibility of the pair (f  , f  ). For this, we have with the help of (C  ), we get . Thus (f  , f  ) is weakly compatible. In a similarly way one can easily show the weakly compatibility of the pairs (f  , f  ), (f  , f  ) and (f  , f  ). Also, from condition () of Theorem ., the pairs (f  , f  ) and (f  , f  ) satisfy the common (CLR f  )-property. Thus by Theorem . we can find a unique common fixed point of f  , f  , f  , f  , f  , and f  in Y , that is, the system (.) of Urysohn integral equations has a unique common solution in Y .
In the next result we use the common (E.A)-property and the proof is simple, so we omit it.

Existence of unique common solution to the systems of Volterra-Hammerstein integral equations
In this section, we present the real-valued metric version of Theorem . and Theorem . and the proof can easily be obtained, so we omit its proof here. Assume that Let Z = (L(, ∞), R) be an incomplete metric space with metric Define the six operators f  , f  , f  , f  , f  , and f  on Z by Now, we are in a position to formulate the existence results.

Theorem . Under the assumptions
.
Next, we need to show the weak compatibility of the pair (f  , f  ). For this purpose, with the help of (C *  ), we get f  f  z(s)f  f  z(s) = , which implies that f  f  z(s) = f  f  z(s), whenever f  z(s) = f  z(s). Thus the pair (f  , f  ) is weakly compatible. In a similar way one can easily show the weakly compatibility of the pairs (f  , f  ), (f  , f  ), and (f  , f  ). Also, from condition () of Theorem . the pairs (f  , f  ) and (f  , f  ) satisfy the common (CLR f  )-property. Thus by Corollary ., we can find a unique common fixed point of f  , f  , f  , f  , f  , and f  in Z, that is, the system (.) of Volterra-Hammerstein non-linear integral equations has a unique common solution in Z.
In the next theorem we use the common (E.A)-property.

Conclusions
In the current work, we studied the existence of unique common solution for the systems of Urysohn and Volterra-Hammerstein integral equations in incomplete spaces. Several problems that appear in applied mathematics, physical sciences, geology, mechanics, engineering, economics, and biology generate mathematical models described by integral equations.