A Multiplicity Result for an Elliptic Anisotropic Differential Inclusion Involving Variable Exponents

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

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Blow-up solutions of a time-fractional diffusion equation with variable exponents

Tbilisi Mathematical Journal ◽

10.32513/tbilisi/1578020574 ◽

2019 ◽

Vol 12 (4) ◽

pp. 149-157

Author(s):

J. Manimaran ◽

L. Shangerganesh

Keyword(s):

Diffusion Equation ◽

Blow Up ◽

Fractional Diffusion ◽

Fractional Diffusion Equation ◽

Variable Exponents ◽

Time Fractional Diffusion Equation

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Existence of solutions for a second order boundary value problem with the Clarke subdifferential

Filomat ◽

10.2298/fil1709763a ◽

2017 ◽

Vol 31 (9) ◽

pp. 2763-2771 ◽

Cited By ~ 1

Author(s):

Dalila Azzam-Laouir ◽

Samira Melit

Keyword(s):

Boundary Value Problem ◽

Differential Inclusion ◽

Existence Of Solutions ◽

Boundary Value ◽

Second Order ◽

Clarke Subdifferential ◽

Lipschitzian Function ◽

The Clarke Subdifferential

In this paper, we prove a theorem on the existence of solutions for a second order differential inclusion governed by the Clarke subdifferential of a Lipschitzian function and by a mixed semicontinuous perturbation.

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On Small Perturbations of a Periodic Homogeneous Differential Inclusion with an Asymptotically Stable Set

Automation and Remote Control ◽

10.1134/s0005117919050035 ◽

2019 ◽

Vol 80 (5) ◽

pp. 834-839 ◽

Cited By ~ 2

Author(s):

M. V. Morozov

Keyword(s):

Differential Inclusion ◽

Stable Set ◽

Asymptotically Stable ◽

Small Perturbations

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Positive Solvability for Conjugate Fractional Differential Inclusion of (k,n-k) Type without Continuity and Compactness

Axioms ◽

10.3390/axioms10030170 ◽

2021 ◽

Vol 10 (3) ◽

pp. 170

Author(s):

Ahmed Salem ◽

Aeshah Al-Dosari

Keyword(s):

Differential Inclusion ◽

Existence Of Solutions ◽

Inclusion Type ◽

Fractional Differential Inclusion ◽

Fractional Differential

The monotonicity of multi-valued operators serves as a guideline to prove the existence of the results in this article. This theory focuses on the existence of solutions without continuity and compactness conditions. We study these results for the (k,n−k) conjugate fractional differential inclusion type with λ>0,1≤k≤n−1.

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Variable Anisotropic Hardy Spaces with Variable Exponents

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2020-0124 ◽

2021 ◽

Vol 9 (1) ◽

pp. 65-89

Author(s):

Zhenzhen Yang ◽

Yajuan Yang ◽

Jiawei Sun ◽

Baode Li

Keyword(s):

Hardy Spaces ◽

Variable Exponent ◽

Atomic Decomposition ◽

Integral Operators ◽

Moment Condition ◽

Variable Exponents ◽

Hölder Continuous ◽

Multi Level ◽

The Moment ◽

Variable Anisotropy

Abstract Let p(·) : ℝ n → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of ℝ n introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces Hp (·)(Θ) via the radial grand maximal function and then obtain its atomic decomposition, which generalizes that of Hardy spaces Hp (Θ) on ℝ n with pointwise variable anisotropy of Dekel et al. [16] and variable anisotropic Hardy spaces of Liu et al. [24]. As an application, we establish the boundedness of variable anisotropic singular integral operators from Hp (·)(Θ) to Lp (·)(ℝ n ) in general and from Hp (·)(Θ) to itself under the moment condition, which generalizes the previous work of Bownik et al. [6] on Hp (Θ).

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