### AIMS Mathematics

2021, Issue 5: 5351-5369. doi: 10.3934/math.2021315
Research article Special Issues

# Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels

• Received: 22 December 2020 Accepted: 10 March 2021 Published: 12 March 2021
• MSC : 26A33, 49K05

• In this work we study necessary and sufficient optimality conditions for variational problems dealing with a new fractional derivative. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with arbitrary kernels. After proving a fractional integration by parts formula, we obtain the Euler-Lagrange equation and natural boundary conditions for the fundamental variational problem. Also, fractional variational problems with integral and holonomic constraints are considered. We end with some examples to exemplify our results.

Citation: Fátima Cruz, Ricardo Almeida, Natália Martins. Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels[J]. AIMS Mathematics, 2021, 6(5): 5351-5369. doi: 10.3934/math.2021315

### Related Papers:

• In this work we study necessary and sufficient optimality conditions for variational problems dealing with a new fractional derivative. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with arbitrary kernels. After proving a fractional integration by parts formula, we obtain the Euler-Lagrange equation and natural boundary conditions for the fundamental variational problem. Also, fractional variational problems with integral and holonomic constraints are considered. We end with some examples to exemplify our results.

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