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The purpose of this paper is to present some new contraction mappings via control functions. In addition, some fixed point results for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi mathvariant="normal">Θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi mathvariant="normal">Ψ</mi></mfenced></semantics></math></inline-formula> contraction, rational <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi mathvariant="normal">Θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi mathvariant="normal">Ψ</mi></mfenced></semantics></math></inline-formula> contraction and almost <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="(" close=")"><mi mathvariant="normal">Θ</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi mathvariant="normal">Ψ</mi></mfenced></semantics></math></inline-formula> contraction mappings are obtained. Moreover, under contraction mappings of types (I), (II), and (III) of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><mi mathvariant="normal">Θ</mi><mo>,</mo><mi>θ</mi><mo>,</mo><mi mathvariant="normal">Ψ</mi></mfenced><msub><mi>υ</mi><mn>0</mn></msub></msub></semantics></math></inline-formula>, several fixed circle solutions are provided in the setting of a <i>G</i>-Metric space. Our results extend, unify, and generalize many previously published papers in this direction. In addition, some examples to show the reliability of our results are presented. Finally, a supporting application that discusses the possibility of a solution to a nonlinear integral equation is incorporated.