<p>This study investigates a class of fractional differential equations of the form <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3331_Article_Equa.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="380" /> </MediaObject> <EquationSource Format="MATHML"><math> <msubsup> <mi>D</mi> <msup> <mn>0</mn> <mo>+</mo> </msup> <mrow> <mi>μ</mi> <mo>,</mo> <mi>ϕ</mi> </mrow> </msubsup> <mi>q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mn>0</mn> <mo><</mo> <mi>n</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mn>1</mn> <mo><</mo> <mi>μ</mi> <mo>≤</mo> <mn>2</mn> <mo>,</mo> </math></EquationSource> <EquationSource Format="TEX">$ D^{\mu ,\phi}_{0^{+}} q(n) + f(n, q(n)) = 0, \quad 0 < n < 1, \quad 1 < \mu \leq 2, $</EquationSource> </Equation> where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3331_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msubsup> <mi>D</mi> <msup> <mn>0</mn> <mo>+</mo> </msup> <mrow> <mi>μ</mi> <mo>,</mo> <mi>ϕ</mi> </mrow> </msubsup> </math></EquationSource> <EquationSource Format="TEX">$D^{\mu ,\phi}_{0^{+}} $</EquationSource> </InlineEquation> represents the Caputo fractional derivative. The nonlinear function <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3331_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="193" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$f: [0,1] \times [0,\infty ) \rightarrow [0,\infty ) $</EquationSource> </InlineEquation> is assumed to be continuous, and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3331_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> <mo>∈</mo> <msup> <mi>C</mi> <mn>2</mn> </msup> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </math></EquationSource> <EquationSource Format="TEX">$\phi \in C^{2}[0,1] $</EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3331_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\phi '(n) > 0 $</EquationSource> </InlineEquation>. The problem is supplemented with nonlocal boundary conditions: <Equation ID="Equb"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3331_Article_Equb.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="381" /> </MediaObject> <EquationSource Format="MATHML"><math> <mi>q</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>q</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ρ</mi> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mn>1</mn> </msubsup> <mi>p</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msup> <mi>ϕ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mspace width="0.2em" /> <mi>d</mi> <mi>r</mi> <mo>,</mo> <mspace width="1em" /> <mn>0</mn> <mo><</mo> <mi>ρ</mi> <mo><</mo> <mn>1</mn> <mo>.</mo> </math></EquationSource> <EquationSource Format="TEX">$ q(0) = 0, \quad q(1) = \rho \int _{0}^{1} p(r) q(r) \phi '(r) \, dr, \quad 0 < \rho < 1. $</EquationSource> </Equation></p><p>By constructing an equivalent integral representation using Green’s function, the existence of nontrivial positive solutions is established through the application of fixed point theorems. The analysis provides new insights into the solvability and properties of solutions for this class of fractional boundary value problems.</p>

Existence and properties of nontrivial solutions for novel fractional differential equations with ϕ-Hilfer operators and boundary conditions

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2025年7月1日

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Zahra Salemi,
Hojjat Afshari,
Asghar Ahmadkhanlu

摘要

This study investigates a class of fractional differential equations of the form $D_{0^{+}}^{μ, ϕ} q (n) + f (n, q (n)) = 0, 0 < n < 1, 1 < μ \leq 2,$ $ D^{\mu ,\phi}_{0^{+}} q(n) + f(n, q(n)) = 0, \quad 0 < n < 1, \quad 1 < \mu \leq 2, $ where $D_{0^{+}}^{μ, ϕ}$ $D^{\mu ,\phi}_{0^{+}} $ represents the Caputo fractional derivative. The nonlinear function $f : [0, 1] \times [0, \infty) \to [0, \infty)$ $f: [0,1] \times [0,\infty ) \rightarrow [0,\infty ) $ is assumed to be continuous, and $ϕ \in C^{2} [0, 1]$ $\phi \in C^{2}[0,1] $ satisfies $ϕ^{'} (n) > 0$ $\phi '(n) > 0 $ . The problem is supplemented with nonlocal boundary conditions: $q (0) = 0, q (1) = ρ \int_{0}^{1} p (r) q (r) ϕ^{'} (r) d r, 0 < ρ < 1 .$ $ q(0) = 0, \quad q(1) = \rho \int _{0}^{1} p(r) q(r) \phi '(r) \, dr, \quad 0 < \rho < 1. $

By constructing an equivalent integral representation using Green’s function, the existence of nontrivial positive solutions is established through the application of fixed point theorems. The analysis provides new insights into the solvability and properties of solutions for this class of fractional boundary value problems.