In this paper, we prove an existence theorem for a mixed-type fractional integrodifferential equation of the form: <Equation ID="Equ8"> <EquationSource Format="TEX">$ {\phantom {a}}_T^C \varDelta ^\alpha x(t) =f(t,x(t),(Hx)(t),(Kx)(t)), $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msubsup> <mphantom> <mi>a</mi> </mphantom> <mi>T</mi> <mi>C</mi> </msubsup> <msup> <mi>Δ</mi> <mi>α</mi> </msup> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>with the initial conditions <Equation ID="Equ9"> <EquationSource Format="TEX">$ x(0)=x_0, x_0\in E, t\in I_a=[0,a]\cap T,a>0, \alpha \in (0,1]. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>∈</mo> <mi>E</mi> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <msub> <mi>I</mi> <mi>a</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo stretchy="false">]</mo> </mrow> <mo>∩</mo> <mi>T</mi> <mo>,</mo> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi>α</mi> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Here, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(Hx)(t)=\int _0^t k_1 (t,s)g(s,x(s))\varDelta s,(Kx)(t)= \int _0^a k_2 (t,s)h(s,x(s))\varDelta s$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>t</mi> </msubsup> <msub> <mi>k</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>Δ</mi> <mi>s</mi> <mo>,</mo> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>a</mi> </msubsup> <msub> <mi>k</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>Δ</mi> <mi>s</mi> </mrow> </math></EquationSource> </InlineEquation> , where T denotes a time scale (a nonempty closed subset of the real numbers R), <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$I_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> is a time scale interval. The functions f, g, h, x are assumed to be weakly-weakly sequentially continuous. All integrals are understood in the sense of the Henstock-Kurzweil-Pettis delta integral. Moreover, the functions f, g, h satisfy suitable boundary conditions formulated in terms of measures of weak noncompactness.

Fractional Integrodifferential Equations on Time Scales with Henstock-Kurzweil-Pettis integrals

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发表日期：

2026年2月19日

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Aneta Sikorska-Nowak

摘要

In this paper, we prove an existence theorem for a mixed-type fractional integrodifferential equation of the form: $ {\phantom {a}}_T^C \varDelta ^\alpha x(t) =f(t,x(t),(Hx)(t),(Kx)(t)), $ $_{T}^{C} Δ^{α} x (t) = f (t, x (t), (H x) (t), (K x) (t)),$ with the initial conditions $ x(0)=x_0, x_0\in E, t\in I_a=[0,a]\cap T,a>0, \alpha \in (0,1]. $ $x (0) = x_{0}, x_{0} \in E, t \in I_{a} = [0, a] \cap T, a > 0, α \in (0, 1] .$ Here, $(Hx)(t)=\int _0^t k_1 (t,s)g(s,x(s))\varDelta s,(Kx)(t)= \int _0^a k_2 (t,s)h(s,x(s))\varDelta s$ $(H x) (t) = \int_{0}^{t} k_{1} (t, s) g (s, x (s)) Δ s, (K x) (t) = \int_{0}^{a} k_{2} (t, s) h (s, x (s)) Δ s$ , where T denotes a time scale (a nonempty closed subset of the real numbers R), $I_a$ $I_{a}$ is a time scale interval. The functions f, g, h, x are assumed to be weakly-weakly sequentially continuous. All integrals are understood in the sense of the Henstock-Kurzweil-Pettis delta integral. Moreover, the functions f, g, h satisfy suitable boundary conditions formulated in terms of measures of weak noncompactness.