The conditions of solvability and the structure of the set of solutions of the equation x(n) = f(t,x,x′, ..., x(n-1)), on a given interval [a,b] (-∞(i)(a) = Ai (i = 0, ..., k), x(j)(b) = Bj (j = 0, ..., l) and addit...
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The conditions of solvability and the structure of the set of solutions of the equation x(n) = f(t,x,x′, ..., x(n-1)), on a given interval [a,b] (-∞boundary conditions x(i)(a) = Ai (i = 0, ..., k), x(j)(b) = Bj (j = 0, ..., l) and additional restriction α(t)≤x(t)≤β(t) (a≤t≤b) were investigated. It is assumed that n≥4;the function f:[a,b]×Rn→R satisfies the Caratheodory conditions;k, l≥0;(k+1)+(l+1)≤n;α and β (α<βi.e.α(t)<β(t), for every t∈[a,b]) are solutions of x(n) = f(t,x,x′, ..., x(n-1)). The main statements are obtained as the consequence of special suppositions of the uniqueness for solutions of auxiliary boundaryvalueproblems.
An existence and uniqueness result for the boundaryvalue Sigma(k=1)(m) A(k)x((2k-1)) + (-1)(m-1) f(t,x) = e(t),x((j))(0) =x((j))(pi) = 0,j = 0,1,..., (C) 2002 Published by Elsevier Science Inc.
An existence and uniqueness result for the boundaryvalue Sigma(k=1)(m) A(k)x((2k-1)) + (-1)(m-1) f(t,x) = e(t),x((j))(0) =x((j))(pi) = 0,j = 0,1,..., (C) 2002 Published by Elsevier Science Inc.
A two-point boundary value problem for Volterra-Fredholm integro-differential equations is considered. To solve this problem, Dzhumabaev parametrization method is used. Conditions for the existence and uniqueness of t...
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A two-point boundary value problem for Volterra-Fredholm integro-differential equations is considered. To solve this problem, Dzhumabaev parametrization method is used. Conditions for the existence and uniqueness of two-point boundary value problem for Volterra-Fredholm integro-differential equations are established. An algorithm for finding solution of this problem is proposed. Then, numerical approach for analysis of two-point boundary value problem for Volterra-Fredholm integro-differential equations is offered. Results are illustrated by numerical example.
A two-point boundary value problem with a positive parameter Q arising in the study of surface-tension-induced flows of a liquid metal or semiconductor is studied. On the basis of the upper-lower solution method and S...
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A two-point boundary value problem with a positive parameter Q arising in the study of surface-tension-induced flows of a liquid metal or semiconductor is studied. On the basis of the upper-lower solution method and Schauder's fixed-point theorem, it is proved that when 0 less than or equal to Q less than or equal to 13.213, the problem admits a solution. This improves a recent result where 0 less than or equal to Q less than or equal to 1.
The asymptotic output tracking problem is studied for a class of non-minimum phase nonlinear systems without requiring the a priori knowledge, or even the existence, of a finite-dimensional exosystem that generates th...
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The asymptotic output tracking problem is studied for a class of non-minimum phase nonlinear systems without requiring the a priori knowledge, or even the existence, of a finite-dimensional exosystem that generates the prescribed periodic reference signal. The design strategy is illustrated in two steps. First, it is shown that the knowledge of a solution to a certain two-point boundary value problem involving the underlying zero-dynamics of the plant is instrumental and sufficient to construct a state feedback regulator that achieves boundedness of the trajectories and (exact) asymptotic tracking, hence completely circumventing the need for solving partial differential equations. Then, since the computation of the latter solution may be affected by numerical errors that are particularly detrimental in the presence of unstable zero-dynamics, the above architecture is robustified by means of an additional hybrid feedback loop whose trajectories converge to a solution of the two-point boundary value problem. Once the latter scheme has been established and discussed, the extension to the case of output feedback is presented, firstly in the specially structured case in which the input vector field depends only on the measured output and then extended to the generic case. The theory is then corroborated by means of a physically motivated numerical example involving an inverted pendulum on a cart. Interestingly, it is also shown that the solution provided by the hybrid scheme above coincides with the limiting solution of a suitably defined cheap optimal control problem. (C) 2021 European Control Association. Published by Elsevier Ltd. All rights reserved.
In this paper, two-point boundary value problems have been solved by the well-known variational iteration method. Considering the situation in which the nonlinear part is a polynomial (unction with degree of >= 2, ...
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In this paper, two-point boundary value problems have been solved by the well-known variational iteration method. Considering the situation in which the nonlinear part is a polynomial (unction with degree of >= 2, the steady temperature distribution in a rod has been computed. The strongly nonlinear differential equation has been become a reduced differential equation by the aid of a proper transformation and variational iteration method has been applied to the boundaryvalueproblem. (C) 2009 Elsevier Ltd. All rights reserved.
We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold (M, g) from an enumerative point of view. We prove a finiteness result for solutions joining twopoints p...
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We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold (M, g) from an enumerative point of view. We prove a finiteness result for solutions joining twopoints p, q is an element of M that are non-conjugate in a suitable sense, under the assumption that (M, g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies. (C) 2003 Elsevier Science B.V. All rights reserved.
The aim of this note is to establish the existence of three solutions for a perturbed two-point boundary value problem depending of two real parameters. The approach is based on variational methods. (C) 2010 Elsevier ...
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The aim of this note is to establish the existence of three solutions for a perturbed two-point boundary value problem depending of two real parameters. The approach is based on variational methods. (C) 2010 Elsevier Ltd. All rights reserved.
In this paper, we investigate a quasilinear second-order differential equation with Dirichlet boundary conditions. The existence of an open interval of parameters which ensures this problem admits at least three solut...
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In this paper, we investigate a quasilinear second-order differential equation with Dirichlet boundary conditions. The existence of an open interval of parameters which ensures this problem admits at least three solutions is determined by using the critical point theory. (C) 2003 Elsevier Science Ltd. All rights reserved.
In this paper, we develop quintic nonpolynomial spline methods for the numerical solution of fourth order two-point boundary value problems. Using this spline function a few consistency relations are derived for compu...
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In this paper, we develop quintic nonpolynomial spline methods for the numerical solution of fourth order two-point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. The present approach gives better approximations and generalizes all the existing polynomial spline methods up to order four. This approach has less computational cost. Convergence analysis of these methods is discussed. two numerical examples are included to illustrate the practical usefulness of our methods. (c) 2008 Elsevier B.V. All rights reserved.
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