scholarly journals A VARIATIONAL APPROACH FOR A PROBLEM INVOLVING A ψ-HILFER FRACTIONAL OPERATOR

2021 ◽  
Vol 11 (3) ◽  
pp. 1610-1630
Author(s):  
J. Vanterler da C. Sousa ◽  
◽  
Leandro S. Tavares ◽  
César E. Torres Ledesma ◽  
◽  
...  
Keyword(s):  
Variational Approach ◽  
Fractional Operator

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities

2020 ◽  
Vol 23 (3) ◽  
pp. 723-752 ◽  
Author(s):  
Alessio Fiscella ◽  
Patrizia Pucci
Keyword(s):  
Scalar Case ◽  
Fractional Operator ◽  
Nontrivial Solutions ◽  
Fractional Systems ◽  
Lack Of Compactness ◽  
Critical Nonlinearities ◽  
Kirchhoff Model

AbstractThis paper deals with the existence of nontrivial solutions for critical possibly degenerate Kirchhoff fractional (p, q) systems. For clarity, the results are first presented in the scalar case, and then extended into the vectorial framework. The main features and novelty of the paper are the (p, q) growth of the fractional operator, the double lack of compactness as well as the fact that the systems can be degenerate. As far as we know the results are new even in the scalar case and when the Kirchhoff model considered is non–degenerate.

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 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 new variational approach for the thermodynamic topology optimization of hyperelastic structures

2020 ◽  
Author(s):  
Philipp Junker ◽  
Daniel Balzani
Keyword(s):  
Topology Optimization ◽  
Variational Approach ◽  
Large Deformations ◽  
Mathematical Structure ◽  
Detailed Model ◽  
Novel Approach ◽  
Extremal Principles ◽  
Model Derivation ◽  
Numerical Discretization ◽  
Total Structure

AbstractWe present a novel approach to topology optimization based on thermodynamic extremal principles. This approach comprises three advantages: (1) it is valid for arbitrary hyperelastic material formulations while avoiding artificial procedures that were necessary in our previous approaches for topology optimization based on thermodynamic principles; (2) the important constraints of bounded relative density and total structure volume are fulfilled analytically which simplifies the numerical implementation significantly; (3) it possesses a mathematical structure that allows for a variety of numerical procedures to solve the problem of topology optimization without distinct optimization routines. We present a detailed model derivation including the chosen numerical discretization and show the validity of the approach by simulating two boundary value problems with large deformations.

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