scholarly journals On the Existence of a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 136
Author(s):  
Mikhail Kamenskii ◽  
Valeri Obukhovskii ◽  
Garik Petrosyan ◽  
Jen-Chih Yao
Keyword(s):  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Vector Fields ◽  
A Priori Estimates ◽  
A Priori ◽  
Linear Part ◽  
Monotonicity Condition ◽  
Diffusion Type ◽  
Estimates Of Solutions ◽  
Linear Differential

We obtained results on the existence and uniqueness of a mild solution for a fractional-order semi-linear differential inclusion in a Hilbert space whose right-hand side contains an unbounded linear monotone operator and a Carathéodory-type multivalued nonlinearity satisfying some monotonicity condition in the phase variables. We used the Yosida approximations of the linear part of the inclusion, the method of a priori estimates of solutions, and the topological degree method for condensing vector fields. As an example, we considered the existence and uniqueness of a solution to the Cauchy problem for a system governed by a perturbed subdifferential inclusion of a fractional diffusion type.

scholarly journals Existence and Uniqueness of Weak Solutions for Novel Anisotropic Nonlinear Diffusion Equations Related to Image Analysis

2021 ◽  
Vol 2021 ◽  
pp. 1-18 ◽  
Author(s):  
Anas Tiarimti Alaoui ◽  
Mostafa Jourhmane
Keyword(s):  
Weak Solutions ◽  
Nonlinear Diffusion ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Monotonicity Condition ◽  
Restoration Technique

This paper establishes the existence and uniqueness of weak solutions for the initial-boundary value problem of anisotropic nonlinear diffusion partial differential equations related to image processing and analysis. An implicit iterative method combined with a variational approach has been applied to construct approximate solutions for this problem. Then, under some a priori estimates and a monotonicity condition, the existence of unique weak solutions for this problem has been proven. This work has been complemented by a consistent and stable approximation scheme showing its great significance as an image restoration technique.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

Existence of bistable waves for a nonlocal and nonmonotone reaction-diffusion equation

2019 ◽  
Vol 150 (2) ◽  
pp. 721-739
Author(s):  
Sergei Trofimchuk ◽  
Vitaly Volpert
Keyword(s):  
Diffusion Equation ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlocal Nonlinearity ◽  
Estimates Of Solutions ◽  
Priori Estimates

AbstractReaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

scholarly journals The Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition

2020 ◽  
Vol 11 (4) ◽  
pp. 1991-2022
Author(s):  
Annamaria Barbagallo ◽  
Vincenzo Esposito
Keyword(s):  
Sobolev Spaces ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Existence And Uniqueness Results ◽  
Mixed Problems ◽  
Uniqueness Results ◽  
Hyperbolic Operators ◽  
Priori Estimates

Abstract The mixed Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition is investigated. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, existence and uniqueness results for the mixed problems are obtained.

A priori estimates of solutions of nonlinear boundary value problems for singular in a phase variable second order differential inequalities

2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Ivan Kiguradze
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Second Order ◽  
Phase Variable ◽  
Differential Inequalities ◽  
Nonlinear Boundary Value Problems ◽  
Estimates Of Solutions ◽  
Priori Estimates

Abstract.For singular in a phase variable second order differential inequalities, a priori estimates of positive solutions, satisfying nonlinear nonlocal boundary conditions, are established.

scholarly journals Estimates of Solutions of Elliptic Differential-Difference Equations with Degeneration

2018 ◽  
Vol 64 (1) ◽  
pp. 131-147
Author(s):  
V A Popov
Keyword(s):  
Difference Equations ◽  
A Priori Estimates ◽  
Difference Operator ◽  
A Priori ◽  
Difference Operators ◽  
Elliptic Differential Operator ◽  
Friedrichs Extension ◽  
Estimates Of Solutions ◽  
Priori Estimates ◽  
Differential Difference Equation

We consider a second-order differential-difference equation in a bounded domain Q ⊂ Rn. We assume that the differential-difference operator contains some difference operators with degeneration corresponding to differentiation operators. Moreover, the differential-difference operator under consideration cannot be expressed as a composition of a difference operator and a strongly elliptic differential operator. Degenerated difference operators do not allow us to obtain the G˚arding inequality. We prove a priori estimates from which it follows that the differential-difference operator under consideration is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space W 1(Q). However, using the obtained estimates, we can prove some smoothness of solutions, though not in the whole domain Q, but inside some subdomains Qr generated by the shifts of the boundary, where U Qr = Q.

A difference scheme for equations of ocean dynamics on unstructured grids

2014 ◽  
Vol 29 (3) ◽  
Author(s):  
Alexey V. Drutsa
Keyword(s):  
Difference Scheme ◽  
Existence And Uniqueness ◽  
Large Scale ◽  
A Priori Estimates ◽  
Unstructured Grids ◽  
A Priori ◽  
Ocean Dynamics ◽  
Priori Estimates ◽  
Uniqueness Of The Solution ◽  
Grid Operators

AbstractA difference scheme on unstructured grids is constructed for the system of equations of large scale ocean dynamics. The properties of the grid problem and grid operators are studied, in particular, a series of a priori estimates and the theorem on existence and uniqueness of the solution are proved.

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