<p>In this paper, we propose a generalization of the fractional Euler–Lagrange equations of the form: <Equation ID="Equ13"> <EquationSource Format="TEX">$\begin{aligned} \dfrac{\partial F}{\partial y}(t,y,~_{0}D_{t}^{\alpha }y) + ~_{t}D_{T}^{\alpha }\left( \dfrac{\partial F}{\partial ~_{0}D_{t}^{\alpha } y} ( t,y,~_{0}D_{t}^{\alpha }y)\right) = 0, ~\forall t \in [0,T], \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>∂</mi> <mi>F</mi> </mrow> <mrow> <mi>∂</mi> <mi>y</mi> </mrow> </mfrac> </mstyle> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mmultiscripts> <mspace width="3.33333pt" /> <mn>0</mn> <mrow /> </mmultiscripts> <msubsup> <mi>D</mi> <mrow> <mi>t</mi> </mrow> <mi>α</mi> </msubsup> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mmultiscripts> <mspace width="3.33333pt" /> <mi>t</mi> <mrow /> </mmultiscripts> <msubsup> <mi>D</mi> <mrow> <mi>T</mi> </mrow> <mi>α</mi> </msubsup> <mfenced close=")" open="("> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>∂</mi> <mi>F</mi> </mrow> <mrow> <mi>∂</mi> <mmultiscripts> <mspace width="3.33333pt" /> <mn>0</mn> <mrow /> </mmultiscripts> <msubsup> <mi>D</mi> <mrow> <mi>t</mi> </mrow> <mi>α</mi> </msubsup> <mi>y</mi> </mrow> </mfrac> </mstyle> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mmultiscripts> <mspace width="3.33333pt" /> <mn>0</mn> <mrow /> </mmultiscripts> <msubsup> <mi>D</mi> <mrow> <mi>t</mi> </mrow> <mi>α</mi> </msubsup> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mo>∀</mo> <mi>t</mi> <mo>∈</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$_{t}D_{T}^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mi>t</mi> <mrow /> </mmultiscripts> <msubsup> <mi>D</mi> <mrow> <mi>T</mi> </mrow> <mi>α</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$_{0}D_{t}^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>0</mn> <mrow /> </mmultiscripts> <msubsup> <mi>D</mi> <mrow> <mi>t</mi> </mrow> <mi>α</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> are the right and left Riemann-Liouville fractional derivatives of generalization order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n-1< \alpha < n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. Based on the variational methods, the main theorems provide some new results regarding the existence a weak solution which the previous results are a special case of our problem. These equations appear for problems of the calculus of variations with functionals containing fractional derivatives.</p>

Existence of a weak solution for fractional Euler–Lagrange equation by variational method

期刊论文
发表日期：

2025年7月2日

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Saeed Kosari,
Fatemeh Shirmohammadzadeh,
Milad Yadollahzadeh

摘要

In this paper, we propose a generalization of the fractional Euler–Lagrange equations of the form: $\begin{aligned} \dfrac{\partial F}{\partial y}(t,y,~_{0}D_{t}^{\alpha }y) + ~_{t}D_{T}^{\alpha }\left( \dfrac{\partial F}{\partial ~_{0}D_{t}^{\alpha } y} ( t,y,~_{0}D_{t}^{\alpha }y)\right) = 0, ~\forall t \in [0,T], \end{aligned}$ $\begin{matrix} \frac{\partial F}{\partial y} (t, y,_{0} D_{t}^{α} y) +_{t} D_{T}^{α} (\frac{\partial F}{\partial_{0} D_{t}^{α} y}, (t, y,_{0} D_{t}^{α} y)) = 0, \forall t \in [0, T], \end{matrix}$ where $_{t}D_{T}^{\alpha }$ $_{t} D_{T}^{α}$ and $_{0}D_{t}^{\alpha }$ $_{0} D_{t}^{α}$ are the right and left Riemann-Liouville fractional derivatives of generalization order $n-1< \alpha < n$ $n - 1 < α < n$ . Based on the variational methods, the main theorems provide some new results regarding the existence a weak solution which the previous results are a special case of our problem. These equations appear for problems of the calculus of variations with functionals containing fractional derivatives.