<p>We define a new Caputo fractional differential equation of higher-order on unbounded interval <Equation ID="Equa"> <EquationSource Format="MATHML"><math> <mrow> <mo>{</mo> <mtable> <mtr> <mtd columnalign="left"> <mmultiscripts> <mi>D</mi> <none /> <mi>α</mi> <mprescripts /> <none /> <mi>c</mi> </mmultiscripts> <mi>w</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="0.25em" /> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mspace width="0.25em" /> </mtd> <mtd columnalign="left"> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mi>w</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <msup> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.25em" /> </mtd> <mtd columnalign="left"> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd columnalign="left"> <msup> <mi>w</mi> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>η</mi> </msubsup> <mfrac> <msup> <mrow> <mo stretchy="false">(</mo> <mi>η</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>s</mi> <mo>,</mo> <mspace width="0.25em" /> <msup> <munder> <mo movablelimits="false">lim</mo> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>c</mi> </msup> <msup> <mi>D</mi> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>w</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="0.25em" /> <mn>0</mn> <mo><</mo> <mi>η</mi> <mo><</mo> <mi>ξ</mi> <mo><</mo> <mi>t</mi> <mo>.</mo> </mtd> <mtd columnalign="left"> <mspace width="1em" /> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> <EquationSource Format="TEX">$ \textstyle\begin{cases} ^{c}D^{\alpha }w(t)=f(t),\ 0\leq t< \infty ,\ &\quad \\ w(0)=w'(0)=w^{(4)}(0)=w^{(5)}(0)=\cdots =w^{(n)}(0)=0,\ &\quad \\ w'''(0)=\int _{0}^{\eta } \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}f(s)ds,\ \displaystyle { \lim _{t\rightarrow \infty }}^{c}D^{\alpha -1}w(t)=w(\xi ),\ 0< \eta < \xi < t. &\quad \end{cases} $</EquationSource> </Equation> Also, we introduce a new family of measures of noncompactness in Hölder space <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$C^{0,q}(\Omega )$</EquationSource> </InlineEquation> and we state a version of fixed point theorem of Darbo’s type. Then we study the existence of solutions of new Caputo fractional differential equation in Hölder space by using version of Darbo’s fixed point theorem associated with this new measure of noncompactness. An example is given to demonstrate application of our results.</p>

New Caputo fractional differential equation of higher order on unbounded interval and solvability in the Hölder space $C^{0,q}(\Omega )$

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2026年3月5日

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Hojjatollah Amiri Kayvanloo,
Farzaneh Pouladi Najafabadi,
Ekrem Savaş,
Mohammad Mursaleen

摘要

We define a new Caputo fractional differential equation of higher-order on unbounded interval ${\begin{matrix} {}^{c}D^{α} w (t) = f (t), 0 \leq t < \infty, \\ w (0) = w^{'} (0) = w^{(4)} (0) = w^{(5)} (0) = \dots = w^{(n)} (0) = 0, \\ w^{‴} (0) = \int_{0}^{η} \frac{{(η - s)}^{α - 1}}{Γ (α)} f (s) d s, {lim_{t \to \infty}}^{c} D^{α - 1} w (t) = w (ξ), 0 < η < ξ < t . \end{matrix}$ $ \textstyle\begin{cases} ^{c}D^{\alpha }w(t)=f(t),\ 0\leq t< \infty ,\ &\quad \\ w(0)=w'(0)=w^{(4)}(0)=w^{(5)}(0)=\cdots =w^{(n)}(0)=0,\ &\quad \\ w'''(0)=\int _{0}^{\eta } \frac{(\eta -s)^{\alpha -1}}{\Gamma (\alpha )}f(s)ds,\ \displaystyle { \lim _{t\rightarrow \infty }}^{c}D^{\alpha -1}w(t)=w(\xi ),\ 0< \eta < \xi < t. &\quad \end{cases} $ Also, we introduce a new family of measures of noncompactness in Hölder space $C^{0, q} (Ω)$ $C^{0,q}(\Omega )$ and we state a version of fixed point theorem of Darbo’s type. Then we study the existence of solutions of new Caputo fractional differential equation in Hölder space by using version of Darbo’s fixed point theorem associated with this new measure of noncompactness. An example is given to demonstrate application of our results.

New Caputo fractional differential equation of higher order on unbounded interval and solvability in the Hölder space \(C^{0,q}(\Omega )\)

摘要