scholarly journals Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1665
Author(s):  
Fátima Cruz ◽  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Time Delay ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Fractional Operator ◽  
Different Types ◽  
Generalized Fractional Derivatives ◽  
Necessary And Sufficient ◽  
Distributed Order

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

scholarly journals A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter

2021 ◽  
Vol 5 (1) ◽  
pp. 24
Author(s):  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Variational Problem ◽  
Fractional Derivatives ◽  
Variational Calculus ◽  
Variational Problems ◽  
Sufficient Optimality Conditions ◽  
Caputo Fractional Derivatives ◽  
Independent Variable ◽  
Left And Right ◽  
Necessary And Sufficient ◽  
Fractional Variational Problem

In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results.

scholarly journals Ulam–Hyers–Mittag-Leffler stability for tripled system of weighted fractional operator with TIME delay

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed A. Almalahi ◽  
Satish K. Panchal ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Time Delay ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Fractional Operator ◽  
Caputo Fractional Derivatives ◽  
Picard Operator

AbstractThis study is aimed to investigate the sufficient conditions of the existence of unique solutions and the Ulam–Hyers–Mittag-Leffler (UHML) stability for a tripled system of weighted generalized Caputo fractional derivatives investigated by Jarad et al. (Fractals 28:2040011 2020) in the frame of Chebyshev and Bielecki norms with time delay. The acquired results are obtained by using Banach fixed point theorems and the Picard operator (PO) method. Finally, a pertinent example of the results obtained is demonstrated.

scholarly journals Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Jianke Zhang ◽  
Gaofeng Wang ◽  
Xiaobin Zhi ◽  
Chang Zhou
Keyword(s):  
Fractional Derivative ◽  
Lagrange Function ◽  
Variational Problems ◽  
Sufficient Optimality Conditions ◽  
Lagrange Equations ◽  
Hukuhara Difference ◽  
Generalized Hukuhara Difference ◽  
Fractional Variational Problems ◽  
Fractional Calculus Of Variations ◽  
Necessary And Sufficient

We study in this paper the Atangana-Baleanu fractional derivative of fuzzy functions based on the generalized Hukuhara difference. Under the condition of gH-Atangana-Baleanu fractional differentiability, we prove the generalized necessary and sufficient optimality conditions for problems of the fuzzy fractional calculus of variations with a Lagrange function. The new kernel of gH-Atangana-Baleanu fractional derivative has no singularity and no locality, which was not precisely illustrated in the previous definitions.

scholarly journals Higher-order optimality conditions for a minimax

1996 ◽  
Vol 54 (3) ◽  
pp. 509-516 ◽  
Author(s):  
Do Van Luu ◽  
W. Oettli
Keyword(s):  
Optimality Conditions ◽  
Minimax Problem ◽  
Higher Order ◽  
Regularity Conditions ◽  
Sufficient Optimality Conditions ◽  
Basic Assumptions ◽  
Necessary And Sufficient ◽  
Inequality Type

Higher-order necessary and sufficient optimality conditions for a nonsmooth minimax problem with infinitely many constraints of inequality type are established under suitable basic assumptions and regularity conditions.

scholarly journals Convex sequences of higher order

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4655-4663
Author(s):  
Daniel Sofonea ◽  
Ioan Ţincu ◽  
Ana Acu
Keyword(s):  
Real Number ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Necessary And Sufficient Conditions ◽  
Difference Operators ◽  
Real Sequence ◽  
Convex Sequence ◽  
Different Types ◽  
The Difference ◽  
Necessary And Sufficient

In this paper we study the class of convex sequences of higher order defined using the difference operators and investigate their properties. The notion of the convex sequence of order r ? N will be extended for r a real number. Some necessary and sufficient conditions such that a real sequence belongs to the class of convex sequences of higher order r ? R are introduced. Using different types of means we will investigate the convexity of higher order for real sequences.

scholarly journals Noether currents for higher-order variational problems of Herglotz type with time delay

2018 ◽  
Vol 11 (1) ◽  
pp. 91-102 ◽  
Author(s):  
Simão P. S. Santos ◽  
◽  
Natália Martins ◽  
Delfim F. M. Torres
Keyword(s):  
Time Delay ◽  
Variational Problems ◽  
Higher Order

scholarly journals Higher-Order Weakly Generalized Epiderivatives and Applications to Optimality Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Qilin Wang ◽  
Guolin Yu
Keyword(s):  
Optimality Conditions ◽  
Optimization Problem ◽  
Efficient Solutions ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Contingent Epiderivative ◽  
Constraint Set ◽  
Necessary And Sufficient ◽  
Henig Efficient Solutions ◽  
Generalized Contingent Epiderivative

The notions of higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative for set-valued maps are proposed. By virtue of the higher-order weakly generalized contingent (adjacent) epiderivatives, both necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. The imposed assumptions are relaxed in comparison with those of recent results in the literature. Examples are provided to show some advantages of our notions and results.

scholarly journals New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 592
Author(s):  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Optimality Conditions ◽  
Fractional Derivatives ◽  
Lagrange Function ◽  
Sufficient Conditions ◽  
Necessary Optimality Conditions ◽  
Variational Problems ◽  
Final State ◽  
Fractional Variational Problems ◽  
Generalized Fractional Derivatives ◽  
Arbitrary Kernel

This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided.

Sign in / Sign up

Export Citation Format

Share Document