Rational Fuzzy Cone Contractions on Fuzzy Cone Metric Spaces with an Application to Fredholm Integral Equations
Saif Ur Rehman, Hassen Aydi
Abstract
This paper is aimed at proving some common fixed point theorems for mappings involving generalized rational-type fuzzy cone-contraction conditions in fuzzy cone metric spaces. Some illustrative examples are presented to support our work. Moreover, as an application, we ensure the existence of a common solution of the Fredholm integral equations: <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>μ</mi> <mfenced open="(" close=")"> <mrow> <mi>τ</mi> </mrow> </mfenced> <mo>=</mo> <msubsup> <mrow> <mo>∫</mo> </mrow> <mrow> <mn>0</mn> </mrow> <mrow> <mi>τ</mi> </mrow> </msubsup> <mi>Γ</mi> <mfenced open="(" close=")"> <mrow> <mi>τ</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>μ</mi> <mfenced open="(" close=")"> <mrow> <mi>v</mi> </mrow> </mfenced> </mrow> </mfenced> <mi>d</mi> <mi>v</mi> </math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>ν</mi> <mfenced open="(" close=")"> <mrow> <mi>τ</mi> </mrow> </mfenced> <mo>=</mo> <msubsup> <mrow> <mo>∫</mo> </mrow> <mrow> <mn>0</mn> </mrow> <mrow> <mi>τ</mi> </mrow> </msubsup> <mi>Γ</mi> <mfenced open="(" close=")"> <mrow> <mi>τ</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>ν</mi> <mfenced open="(" close=")"> <mrow> <mi>v</mi> </mrow> </mfenced> </mrow> </mfenced> <mi>d</mi> <mi>v</mi> </math> , for all <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>μ</mi> <mo>∈</mo> <mi>U</mi> </math> , <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>v</mi> <mo>∈</mo> <mfenced open="[" close="]"> <mrow> <mn>0</mn> <mo>,</mo> <mi>η</mi> </mrow> </mfenced> </math> , and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mn>0</mn> <mo><</mo> <mi>η</mi> <mo>∈</mo> <mi>ℝ</mi> </math> , where <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mi>U</mi> <mo>=</mo> <mi>C</mi> <mfenced open="(" close=")"> <mrow> <mfenced open="[" close="]"> <mrow> <mn>0</mn> <mo>,</mo> <mi>η</mi> </mrow> </mfenced> <mo>,</mo> <mi>ℝ</mi> </mrow> </mfenced> </math> is the space of all <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mi>ℝ</mi> </math> -valued continuous functions on the interval <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <mfenced open="[" close="]"> <mrow> <mn>0</mn> <mo>,</mo> <mi>η</mi> </mrow> </mfenced> </math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <mi>Γ</mi> <mo>:</mo> <mfenced open="[" close="]"> <mrow> <mn>0</mn> <mo>,</mo> <mi>η</mi> </mrow> </mfenced> <mo>×</mo> <mfenced open="[" close="]"> <mrow> <mn>0</mn> <mo>,</mo> <mi>η</mi> </mrow> </mfenced> <mo>×</mo> <mi>ℝ</mi> <mo>⟶</mo> <mi>ℝ</mi> </math> .