<p>One of the most intensively studied and generalized results in metric fixed point theory is Branciari’s fixed point theorem, which asserts that in a complete metric space <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(M, \rho )$</EquationSource> </InlineEquation>, a mapping <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>T</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> </math></EquationSource> <EquationSource Format="TEX">$T: M \to M$</EquationSource> </InlineEquation> satisfying <Equation ID="Equa"> <EquationSource Format="MATHML"><math> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mi>x</mi> <mo>,</mo> <mi>T</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="0.2em" /> <mi>d</mi> <mi>t</mi> <mo>≤</mo> <mi>c</mi> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="0.2em" /> <mi>d</mi> <mi>t</mi> </math></EquationSource> <EquationSource Format="TEX">$ \int _{0}^{\rho (Tx, Ty)} \omega (t) \, dt \leq c \int _{0}^{\rho (x, y)} \omega (t) \, dt $</EquationSource> </Equation> for some <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>ω</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\omega : [0, \infty ) \to [0, \infty )$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>c</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$c \in (0,1)$</EquationSource> </InlineEquation>, and all <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>M</mi> </math></EquationSource> <EquationSource Format="TEX">$x, y \in M$</EquationSource> </InlineEquation>, admits a unique fixed point <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>∈</mo> <mi>M</mi> </math></EquationSource> <EquationSource Format="TEX">$x^{*} \in M$</EquationSource> </InlineEquation>. This reduces to Banach fixed theorem when <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$\omega (t) = 1$</EquationSource> </InlineEquation>. We extend this to Riemann–Liouville fractional integral contractions, proving the existence of a fixed point for <i>T</i> under <Equation ID="Equb"> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mi>x</mi> <mo>,</mo> <mi>T</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msup> <mrow> <mo maxsize="2.4ex" minsize="2.4ex" stretchy="true">(</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mi>x</mi> <mo>,</mo> <mi>T</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>t</mi> <mo maxsize="2.4ex" minsize="2.4ex" stretchy="true">)</mo> </mrow> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="0.2em" /> <mi>d</mi> <mi>t</mi> <mo>≤</mo> <mi>c</mi> <mo>⋅</mo> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msup> <mrow> <mo maxsize="2.4ex" minsize="2.4ex" stretchy="true">(</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>t</mi> <mo maxsize="2.4ex" minsize="2.4ex" stretchy="true">)</mo> </mrow> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="0.2em" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </math></EquationSource> <EquationSource Format="TEX">$ \frac{1}{\Gamma (\alpha )} \int _{0}^{\rho (Tx, Ty)} \bigl( \rho (Tx, Ty) - t \bigr)^{\alpha -1} \varphi (t) \, dt \leq c \cdot \frac{1}{\Gamma (\alpha )} \int _{0}^{\rho (x, y)} \bigl( \rho (x, y) - t \bigr)^{\alpha -1} \varphi (t) \, dt, $</EquationSource> </Equation> which recovers Branciari’s integral condition for <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mi>α</mi> <mo>=</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$\alpha = 1$</EquationSource> </InlineEquation>.</p>

Banach fixed point theorem for fractional integral contraction

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2025年12月22日

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Irshad Ayoob

摘要

One of the most intensively studied and generalized results in metric fixed point theory is Branciari’s fixed point theorem, which asserts that in a complete metric space $(M, ρ)$ $(M, \rho )$ , a mapping $T : M \to M$ $T: M \to M$ satisfying $\int_{0}^{ρ (T x, T y)} ω (t) d t \leq c \int_{0}^{ρ (x, y)} ω (t) d t$ $ \int _{0}^{\rho (Tx, Ty)} \omega (t) \, dt \leq c \int _{0}^{\rho (x, y)} \omega (t) \, dt $ for some $ω : [0, \infty) \to [0, \infty)$ $\omega : [0, \infty ) \to [0, \infty )$ , $c \in (0, 1)$ $c \in (0,1)$ , and all $x, y \in M$ $x, y \in M$ , admits a unique fixed point $x^{*} \in M$ $x^{*} \in M$ . This reduces to Banach fixed theorem when $ω (t) = 1$ $\omega (t) = 1$ . We extend this to Riemann–Liouville fractional integral contractions, proving the existence of a fixed point for T under $\frac{1}{Γ (α)} \int_{0}^{ρ (T x, T y)} {(ρ (T x, T y) - t)}^{α - 1} φ (t) d t \leq c \cdot \frac{1}{Γ (α)} \int_{0}^{ρ (x, y)} {(ρ (x, y) - t)}^{α - 1} φ (t) d t,$ $ \frac{1}{\Gamma (\alpha )} \int _{0}^{\rho (Tx, Ty)} \bigl( \rho (Tx, Ty) - t \bigr)^{\alpha -1} \varphi (t) \, dt \leq c \cdot \frac{1}{\Gamma (\alpha )} \int _{0}^{\rho (x, y)} \bigl( \rho (x, y) - t \bigr)^{\alpha -1} \varphi (t) \, dt, $ which recovers Branciari’s integral condition for $α = 1$ $\alpha = 1$ .