Conditions of solvability of the nonlocal boundary-value problem for a differential-operator equation with weak nonlinearity

2021 ◽  
Vol 18 (1) ◽  
pp. 37-59
Author(s):  
Volodymyr Il'kiv ◽  
Nataliya Strap ◽  
Iryna Volyanska
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Linear Part ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Iteration Scheme ◽  
Nonlocal Boundary Value Problem ◽  
Small Denominators ◽  
Nonlinear Part ◽  
Complex Coefficients

We study the solvability of a nonlocal boundary-value problem for a differential equation with nonlinearity. The linear part of the equation has complex coefficients which together with the coefficient in nonlocal conditions are considered to be the parameters of the problem. The area of change of each parameter is limited by a complex circle with its center at the origin. The nonlinear part of the equation is given by a smooth function that satisfies, together with its derivatives, some conditions of growth in the Dirichlet--Fourier space scale. The proofs are based on the differentiable Nash--Moser iteration scheme, where the main difficulty is to get estimates of the interpolation type for the inverse linearized operators obtained at each step of the iteration. The estimation is connected with the problem of small denominators which is solved, by using a metric approach on the set of parameters of the problem.

scholarly journals On nonlocal boundary value problem for the equation of motion of a homogeneous elastic beam with pinned-pinned ends

2018 ◽  
Vol 10 (1) ◽  
pp. 105-113
Author(s):  
T.P. Goy ◽  
M. Negrych ◽  
I.Ya. Savka
Keyword(s):  
Boundary Value Problem ◽  
Lebesgue Measure ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Elastic Beam ◽  
Equation Of Motion ◽  
Nonlocal Boundary Value Problem ◽  
Small Denominators ◽  
Local Boundary ◽  
Almost All

In the current paper, in the domain $D=\{(t,x): t\in(0,T), x\in(0,L)\}$ we investigate the boundary value problem for the equation of motion of a homogeneous elastic beam $$ u_{tt}(t,x)+a^{2}u_{xxxx}(t,x)+b u_{xx}(t,x)+c u(t,x)=0, $$ where  $a,b,c \in \mathbb{R}$, $b^2<4a^2c$, with nonlocal two-point conditions $$u(0,x)-u(T, x)=\varphi(x), \quad u_{t}(0, x)-u_{t}(T, x)=\psi(x)$$ and local boundary conditions $$u(t, 0)=u(t, L)=u_{xx}(t, 0)=u_{xx}(t, L)=0.$$ Solvability of this problem is connected with the problem of small denominators, whose estimation from below is based on the application of the metric approach. For almost all (with respect to Lebesgue measure) parameters of the problem, we establish conditions for the  solvability of the problem in the Sobolev space. In particular, if $\varphi\in\mathbf{H}_{q+\rho+2}$ and $\psi \in\mathbf{H}_{q+\rho}$, where $\rho>2$, then for almost all (with respect to Lebesgue measure in $\mathbb{R}$) numbers $a$ exists a unique solution $u\in\mathbf{C}^{\,2}([0,T];\mathbf{H}_{q})$ of the problem considered.

scholarly journals Monotonic Positive Solutions of Nonlocal Boundary Value Problems for a Second-Order Functional Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
E. M. Hamdallah ◽  
Kh. W. El-kadeky
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Functional Differential Equation ◽  
Nonlocal Condition ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Second Order ◽  
Nonlocal Boundary Value Problem ◽  
Functional Differential

We study the existence of at least one monotonic positive solution for the nonlocal boundary value problem of the second-order functional differential equationx′′(t)=f(t,x(ϕ(t))),t∈(0,1), with the nonlocal condition∑k=1makx(τk)=x0,x′(0)+∑j=1nbjx′(ηj)=x1, whereτk∈(a,d)⊂(0,1),ηj∈(c,e)⊂(0,1), andx0,x1>0. As an application the integral and the nonlocal conditions∫adx(t)dt=x0,x′(0)+x(e)-x(c)=x1will be considered.

scholarly journals On second order nonlocal boundary value problem at resonance

2016 ◽  
Vol 56 (1) ◽  
pp. 143-153 ◽  
Author(s):  
Katarzyna Szymańska-Dębowska
Keyword(s):  
Boundary Value Problem ◽  
Bounded Variation ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Second Order ◽  
Function Of Bounded Variation ◽  
Nonlocal Boundary Value Problem ◽  
At Resonance ◽  
Problem At Resonance

Abstract This work is devoted to the existence of solutions for a system of nonlocal resonant boundary value problem $$\matrix{{x'' = f(t,x),} \hfill & {x'(0) = 0,} \hfill & {x'(1) = {\int_0^1 {x(s)dg(s)},} }} $$ where f : [0, 1] × ℝk → ℝk is continuous and g : [0, 1] → ℝk is a function of bounded variation.

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

2020 ◽  
Vol 12 (1) ◽  
pp. 173-188
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M.I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Mixed Boundary Conditions ◽  
Multipoint Problem ◽  
Uniqueness Of The Solution

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

scholarly journals A note on the nonlocal boundary value problem for a third order partial differential equation

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 801-808 ◽  
Author(s):  
Kh. Belakroum ◽  
A. Ashyralyev ◽  
A. Guezane-Lakoud
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Order Partial Differential Equation ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
Partial Differential

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space with a self-adjoint positive definite operator is considered. Applying operator approach, the theorem on stability for solution of this nonlocal boundary value problem is established. In applications, the stability estimates for the solution of three nonlocal boundary value problems for third order partial differential equations are obtained.

Sign in / Sign up

Export Citation Format

Share Document