scholarly journals The nonlocal boundary problem with perturbations of antiperiodicity conditions for the eliptic equation with constant coefficients

2018 ◽  
Vol 10 (2) ◽  
pp. 215-234
Author(s):  
Ya.O. Baranetskij ◽  
I.Ya. Ivasiuk ◽  
P.I. Kalenyuk ◽  
A.V. Solomko
Keyword(s):  
Riesz Basis ◽  
Nonlocal Boundary ◽  
Nonlocal Problem ◽  
Fourier Method ◽  
Sufficient Conditions ◽  
Nonlocal Boundary Conditions ◽  
Transformation Operator ◽  
Constant Coefficients ◽  
Adjoint Functions ◽  
Nonlocal Boundary Problem

In this article, we investigate a problem with nonlocal boundary conditions which are perturbations of antiperiodical conditions in bounded $m$-dimensional parallelepiped using Fourier method. We describe properties of a transformation operator $R:L_2(G) \to L_2(G),$ which gives us a connection between selfadjoint operator $L_0$ of the problem with antiperiodical conditions and operator $L$ of perturbation of the nonlocal problem $RL_0=LR.$ Also we construct a commutative group of transformation operators $\Gamma(L_0).$ We show that some abstract nonlocal problem corresponds to any transformation operator $R \in \Gamma(L_0):L_2(G) \to L_2(G)$ and vice versa. We construct a system $V(L)$ of root functions of operator $L,$ which consists of infinite number of adjoint functions. Also we define conditions under which the system $V(L)$ is total and minimal in the space $L_{2}(G),$ and conditions under which it is a Riesz basis in the space $L_{2}(G)$. In case if $V(L)$ is a Riesz basis in the space $L_{2}(G),$ we obtain sufficient conditions under which the nonlocal problem has a unique solution in the form of Fourier series by system $V(L).$

scholarly journals The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. I

2019 ◽  
Vol 11 (2) ◽  
pp. 228-239 ◽  
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M.I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Boundary Conditions ◽  
Riesz Basis ◽  
Mixed Boundary ◽  
Nonlocal Boundary ◽  
Nonlocal Problem ◽  
Fourier Method ◽  
Sufficient Conditions ◽  
Mixed Boundary Conditions ◽  
Nonlocal Boundary Conditions ◽  
Constant Coefficients

In this article we investigate a problem with nonlocal boundary conditions which are multipoint perturbations of mixed boundary conditions in the unit square $G$ using the Fourier method. The properties of a generalized transformation operator $R: L_2(G) \to L_2(G)$ that reflects normalized eigenfunctions of the operator $L_0$ of the problem with mixed boundary conditions in the eigenfunctions of the operator $L$ for nonlocal problem with perturbations, are studied. We construct a system $V(L)$ of eigenfunctions of operator $L.$ Also, we define conditions under which the system $V(L)$ is total and minimal in the space $L_{2}(G),$ and conditions under which it is a Riesz basis in the space $L_{2}(G).$ In the case if $V(L)$ is a Riesz basis in $L_{2}(G),$ we obtain sufficient conditions under which nonlocal problem has a unique solution in form of Fourier series by system $V(L).$

ON POSITIVE EIGENFUNCTIONS OF STURM-LIOUVILLE PROBLEM WITH NONLOCAL TWO‐POINT BOUNDARY CONDITION

2007 ◽  
Vol 12 (2) ◽  
pp. 215-226 ◽  
Author(s):  
Sigita Pečiulytė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Liouville Problem ◽  
Point Boundary ◽  
Existence Domain ◽  
Classical Boundary ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Nonlocal Boundary Problem

Positive eigenvalues and corresponding eigenfunctions of the linear Sturm‐Liouville problem with one classical boundary condition and another nonlocal two‐point boundary condition are considered in this paper. Four cases of nonlocal two‐point boundary conditions are analysed. We get positive eigenfunctions existence domain for each case of these problems. This domain depends on the parameters of the nonlocal boundary problem and it gives necessary and sufficient conditions for existing positive eigenvalues with positive eigenfunctions.

scholarly journals Blowup Properties for a Semilinear Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Dengming Liu ◽  
Chunlai Mu
Keyword(s):  
Boundary Condition ◽  
Global Existence ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Nonlocal Boundary Conditions ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Subsolution And Supersolution

We investigate the blowup properties of the positive solutions for a semilinear reaction-diffusion system with nonlinear nonlocal boundary condition. We obtain some sufficient conditions for global existence and blowup by utilizing the method of subsolution and supersolution.

scholarly journals The Langevin Equation in Terms of Generalized Liouville–Caputo Derivatives with Nonlocal Boundary Conditions Involving a Generalized Fractional Integral

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 533 ◽  
Author(s):  
Bashir Ahmad ◽  
Madeaha Alghanmi ◽  
Ahmed Alsaedi ◽  
Hari M. Srivastava ◽  
Sotiris K. Ntouyas
Keyword(s):  
Boundary Conditions ◽  
Langevin Equation ◽  
Differential Operators ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Nonlocal Boundary Conditions ◽  
Generalized Fractional Integral ◽  
Fractional Differential ◽  
Nonlinear Langevin Equation ◽  
Fractional Differential Operators

In this paper, we establish sufficient conditions for the existence of solutions for a nonlinear Langevin equation based on Liouville-Caputo-type generalized fractional differential operators of different orders, supplemented with nonlocal boundary conditions involving a generalized integral operator. The modern techniques of functional analysis are employed to obtain the desired results. The paper concludes with illustrative examples.

scholarly journals On the solvability of spatial nonlocal boundary value problemsfor one-dimensional pseudoparabolic and pseudohyperbolic equations

2017 ◽  
Vol 21 (3) ◽  
pp. 29-43
Author(s):  
N.S. Popov
Keyword(s):  
Integral Equations ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Integral Conditions ◽  
Regular Solutions ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Systems Of Integral Equations ◽  
Constant Coefficients

In the present work we study the solvability of spatial nonlocal boundary value problems for linear one-dimensional pseudoparabolic and pseudohyperbolic equations with constant coefficients, but with general nonlocal boundary conditions by A.A. Samarsky and integral conditions with variables coefficients. The proof of the theorems of existence and uniqueness of regular solutions is carried out by the method of Fourier. The study of solvability in the classes of regular solutions leads to the study of a system of integral equations of Volterra of the second kind. In particular cases nongeneracy conditions of the obtained systems of integral equations in explicit form are given.

scholarly journals Sign-constancy of Green’s functions for impulsive nonlocal boundary value problems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
A. Domoshnitsky ◽  
Iu. Mizgireva
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Impulsive Differential Equation ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Nonlocal Boundary Conditions ◽  
Nonlocal Boundary Value Problems ◽  
T Tau

Abstract We consider the following second order impulsive differential equation with delays: $$ \textstyle\begin{cases} (Lx)(t)\equiv x''(t)+\sum_{j=1}^{p} a_{j}(t) x'(t-\tau _{j}(t)) + \sum_{j=1}^{p} b_{j}(t) x(t-\theta _{j}(t)) = f(t), \quad t \in [0, \omega ], \\ x(t_{k})=\gamma _{k} x(t_{k}-0), \quad\quad x'(t_{k})=\delta _{k} x'(t_{k}-0), \quad k=1,2,\ldots,r. \end{cases} $${(Lx)(t)≡x″(t)+∑j=1paj(t)x′(t−τj(t))+∑j=1pbj(t)x(t−θj(t))=f(t),t∈[0,ω],x(tk)=γkx(tk−0),x′(tk)=δkx′(tk−0),k=1,2,…,r. In this paper we consider sufficient conditions of nonpositivity of Green’s function for impulsive differential equation with nonlocal boundary conditions.

scholarly journals Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations of Orderα∈(2,3]with Nonlocal Boundary Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Lihong Zhang ◽  
Guotao Wang ◽  
Guangxing Song
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Nonlocal Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential

We investigate the existence and uniqueness of solutions to the nonlocal boundary value problem for nonlinear impulsive fractional differential equations of orderα∈(2,3]. By using some well-known fixed point theorems, sufficient conditions for the existence of solutions are established. Some examples are presented to illustrate the main results.

scholarly journals Existence and Numerical Simulation of Solutions for Fractional Equations Involving Two Fractional Orders with Nonlocal Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Jing Zhao ◽  
Peifen Lu ◽  
Yiliang Liu
Keyword(s):  
Numerical Simulation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Nonlocal Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Fractional Equations ◽  
Fractional Langevin Equation ◽  
Fractional Orders

We study a boundary value problem for fractional equations involving two fractional orders. By means of a fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. In addition, we describe the dynamic behaviors of the fractional Langevin equation by using theG2algorithm.

scholarly journals Positive solutions to advanced fractional differential equations with nonlocal boundary conditions

2015 ◽  
Vol 9 (2) ◽  
pp. 209-220 ◽  
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Nonlocal Boundary Conditions ◽  
Multiple Positive Solutions ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
The Fixed Point Theorem

In this paper, we study the existence of positive solutions for a class of higher order fractional differential equations with advanced arguments and boundary value problems involving Stieltjes integral conditions. The fixed point theorem due to Avery-Peterson is used to obtain sufficient conditions for the existence of multiple positive solutions. Certain of our results improve on recent work in the literature.

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