<p>In this paper, we solve an open problem concerning the connection between <i>F</i>-contractions (Wardowski type contractions) and Meir-Keeler contractions. We prove that for a nondecreasing function <i>F</i> that has a point of discontinuity on the right, there exists a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(F-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>contraction that is not a Meir-Keeler contraction. Moreover, we give a condition on nonlinear <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\varphi ,F)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-contractions to be Meir-Keeler contractions. In the final part of the paper, we give a class of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((E,F)-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>contractions (in the sense of the paper arXiv:2009.13157 [math.FA] 28 Sep 2020), that are Meir-Keeler contractions.</p>

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On the connections between F-contractions and Meir-Keeler contractions

  • Paşc Găvruţa,
  • Laura Manolescu

摘要

In this paper, we solve an open problem concerning the connection between F-contractions (Wardowski type contractions) and Meir-Keeler contractions. We prove that for a nondecreasing function F that has a point of discontinuity on the right, there exists a \(F-\) F - contraction that is not a Meir-Keeler contraction. Moreover, we give a condition on nonlinear \((\varphi ,F)\) ( φ , F ) -contractions to be Meir-Keeler contractions. In the final part of the paper, we give a class of \((E,F)-\) ( E , F ) - contractions (in the sense of the paper arXiv:2009.13157 [math.FA] 28 Sep 2020), that are Meir-Keeler contractions.