The Eigenvalue Problem for Caputo Type Fractional Differential Equation with Riemann-Stieltjes Integral Boundary Conditions

<jats:p>In this paper, we investigate the eigenvalue problem for Caputo fractional differential equation with Riemann-Stieltjes integral boundary conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msubsup><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mprescripts /><mml:none /><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:mmultiscripts></mml:mrow><mml:mrow><mml:mn mathvariant="normal">0</mml:mn><mml:mo>+</mml:mo></mml:mrow><mml:mrow><mml:mi>θ</mml:mi></mml:mrow></mml:msubsup><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>μ</mml:mi><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn mathvariant="normal">0,1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="normal">′</mml:mi><mml:mi mathvariant="normal">′</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:msubsup><mml:mo stretchy="false">∫</mml:mo><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow><mml:mrow><mml:mn mathvariant="normal">1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>d</mml:mi><mml:mi>A</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:msubsup><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mprescripts /><mml:none /><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:mmultiscripts></mml:mrow><mml:mrow><mml:mn mathvariant="normal">0</mml:mn><mml:mo>+</mml:mo></mml:mrow><mml:mrow><mml:mi>θ</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math> is Caputo fractional derivative, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>θ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">2,3</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn mathvariant="normal">0,1</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:mo>×</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mi mathvariant="normal">∞</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>→</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn mathvariant="normal">0</mml:mn><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mi mathvariant="normal">∞</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> is continuous. By using the Guo-Krasnoselskii’s fixed point theorem on cone and the properties of the Green’s function, some new results on the existence and nonexistence of positive solutions for the fractional differential equation are obtained.</jats:p>

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