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Existence of a weak solution for fractional Euler-Lagrange equations. (English) Zbl 1393.35267

Summary: We derive sufficient conditions ensuring the existence of a weak solution \(u\) for fractional Euler-Lagrange equations of the type: \[ \frac{\partial L}{\partial x}(u,D^{\alpha}_-u,t)+D^{\alpha}_+\left(\frac{\partial L} {\partial y}(u,D^{\alpha}_-u,t)\right)=0, \] on a real interval \([a,b]\) and where \(D^{\alpha}_-\) and \(D^{\alpha}_+\) are the fractional derivatives of Riemann-Liouville of order \(0<\alpha<1\).

MSC:

35R11 Fractional partial differential equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35D30 Weak solutions to PDEs
49K21 Optimality conditions for problems involving relations other than differential equations
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