<p>In this paper, we present new fixed point theorems for sets that are endowed with a quasi-metric, which is a generalization of a metric space, where the triangle inequality is modified into a less restrictive form known as the relaxed triangle inequality: <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathfrak {D}_{q}(x,y) \le s[\mathfrak {D}_{q}(x,z) + \mathfrak {D}_{q}(z,y)]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="fraktur">D</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi>s</mi> <mrow> <mo stretchy="false">[</mo> <msub> <mi mathvariant="fraktur">D</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi mathvariant="fraktur">D</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s \ge 1.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Furthermore, we apply our results to iterated function system theory to generate fractals, showcasing their usefulness in fractal construction. At the end, we discuss how sensitivity on maps carry over to their products and same for iterated function systems in the framework of quasi-metric spaces.</p>

Fixed points and fractal construction via cyclic IFS in quasi-metric spaces

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发表日期：

2025年12月29日

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H. Baranwal,
A. K. B. Chand,
A. Petruşel,
J.-C. Yao

摘要

In this paper, we present new fixed point theorems for sets that are endowed with a quasi-metric, which is a generalization of a metric space, where the triangle inequality is modified into a less restrictive form known as the relaxed triangle inequality: $\mathfrak {D}_{q}(x,y) \le s[\mathfrak {D}_{q}(x,z) + \mathfrak {D}_{q}(z,y)]$ $D_{q} (x, y) \leq s [D_{q} (x, z) + D_{q} (z, y)]$ , $s \ge 1.$ $s \geq 1 .$ Furthermore, we apply our results to iterated function system theory to generate fractals, showcasing their usefulness in fractal construction. At the end, we discuss how sensitivity on maps carry over to their products and same for iterated function systems in the framework of quasi-metric spaces.