scholarly journals Nonpotentiality of a diffusion system and the construction of a semi-bounded functional

2021 ◽  
Vol 104 (4) ◽  
pp. 118-129
Author(s):  
V.M. Savchin ◽  
◽  
L.T. Huyen
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Variational Methods ◽  
Variational Principles ◽  
Boundary Value ◽  
Diffusion System ◽  
System Of Equations ◽  
The Given ◽  
Variational Formulations

The wide prevalence and the systematic variational principles are used in mathematics and applications due to a series of remarkable consequences among which the possibility to establish the existence of the solutions of the initial equations, and the determination of stable approximations of the solutions of the considered equations by the so-called variational methods. In this connection, it is natural for a given system of equations to investigate the problem of the existence of its variational formulations. It can be considered as the inverse problem of the calculus of variations. The main goal of this work is to study this problem for a diffusion system of partial differential equations. A key object is the criterion of potentiality. On its ground, the nonpotentiality of the operator of the given boundary value problem with respect to the classical bilinear form is proved. This system does not admit a matrix variational multiplier of the given form. Thus, the diffusion system cannot be deduced from the classical Hamilton’s principle. We posed the question that whether there exists a functional semi-bounded on solutions to the boundary value problem. We have done the algorithm of the constructive determination of such a functional. The main value of constructed functional action will be in applications of direct variational methods.

The M-matrix inverse problem for the Sturm—Liouville equation on graphs

2009 ◽  
Vol 139 (4) ◽  
pp. 775-796 ◽  
Author(s):  
Sonja Currie ◽  
Bruce A. Watson
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Inverse Spectral Problem ◽  
Boundary Value ◽  
Unitary Equivalence ◽  
Matrix Inverse ◽  
Normal Boundary ◽  
Sturm Liouville ◽  
The Given

We consider an inverse spectral problem for Sturm–Liouville boundary-value problems on a graph with formally self-adjoint boundary conditions at the nodes, where the given information is the M-matrix. Based on the authors' previous results, using Green's function, we prove that the poles of the M-matrix are at the eigenvalues of the associated boundary-value problem and are simple, located on the real axis, and that the residue at a pole is a negative semi-definite matrix with rank equal to the multiplicity of the eigenvalue. We define the so-called norming constants and relate them to the spectral measure and the M-matrix. This enables us to recover, from the M-matrix, the boundary conditions and the potential, up to a unitary equivalence for co-normal boundary conditions.

scholarly journals An Analysis on the Positive Solutions for a Fractional Configuration of the Caputo Multiterm Semilinear Differential Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Sh. Rezapour ◽  
B. Azzaoui ◽  
B. Tellab ◽  
S. Etemad ◽  
H. P. Masiha
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Caputo Fractional Derivatives ◽  
Control Functions ◽  
Existence Of Positive Solutions ◽  
Semilinear Differential Equation ◽  
The Given ◽  
Existence Criteria

In this paper, we consider a multiterm semilinear fractional boundary value problem involving Caputo fractional derivatives and investigate the existence of positive solutions by terms of different given conditions. To do this, we first study the properties of Green’s function, and then by defining two lower and upper control functions and using the wellknown Schauder’s fixed-point theorem, we obtain the desired existence criteria. At the end of the paper, we provide a numerical example based on the given boundary value problem and obtain its upper and lower solutions, and finally, we compare these positive solutions with exact solution graphically.

Existence of Two Solutions for a Second-Order Discrete Boundary Value Problem

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Pasquale Candito ◽  
Giovanni Molica Bisci
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Second Order ◽  
Nontrivial Solutions ◽  
Discrete Boundary Value Problem

AbstractThe existence of two nontrivial solutions for a class of nonlinear second-order discrete boundary value problems is established. The approach adopted is based on variational methods.

EXISTENCE AND UNIQUENESS OF WEAK AND CLASSICAL SOLUTIONS FOR A FOURTH-ORDER SEMILINEAR BOUNDARY VALUE PROBLEM

2019 ◽  
Vol 61 (3) ◽  
pp. 305-319
Author(s):  
CRISTIAN-PAUL DANET
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Fourth Order ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Classical Solutions ◽  
Maximum Principles ◽  
Uniqueness Results

This paper is concerned with the problem of existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem. The existence and uniqueness for weak solutions follows from standard variational methods, while similar uniqueness results for classical solutions are derived using maximum principles.

scholarly journals On a class of non-local discrete boundary value problem

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Anass Ourraoui ◽  
Abdesslem Ayoujil
Keyword(s):  
Boundary Value Problem ◽  
Variational Methods ◽  
Critical Points ◽  
Boundary Value ◽  
Discrete Problem ◽  
Three Solutions ◽  
Content Type ◽  
Three Critical Points Theorem ◽  
Discrete Boundary Value Problem ◽  
Non Local

PurposeIn this article, the authors discuss the existence and multiplicity of solutions for an anisotropic discrete boundary value problem in T-dimensional Hilbert space. The approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.Design/methodology/approachThe approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.FindingsThe authors study the existence of results for a discrete problem, with two boundary conditions type. Accurately, the authors have proved the existence of at least three solutions.Originality/valueAn other feature is that problem is with non-local term, which makes some difficulties in the proof of our results.

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