Solution and positive solution of a semilinear third-order equation

Abstract Points of intersection of a circle and a torus are used to find a solution to the inverse kinematics problem for a three revolute manipulator. Both geometrical and algebraic solution procedures are discussed. The algebraic procedure begins with a third order equation instead of the usual fourth order equation. Since the procedure is basically geometrical it lends itself to a computer implementation which graphically displays each steps in the solution procedure. The potential of this approach for both design and pedagogy is discussed.

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On the Construction of the Optimal Control for a Linear Third Order Equation with a Quadratic Quality Criterion Using the Method of Moments

Journal of Automation and Information Sciences ◽

10.1615/jautomatinfscien.v48.i8.80 ◽

2016 ◽

Vol 48 (8) ◽

pp. 80-88

Author(s):

Mekhriban Mamed kyzy Yagubova

Keyword(s):

Optimal Control ◽

Method Of Moments ◽

Quality Criterion ◽

Order Equation ◽

Third Order

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Asymptotic formulae for linear equations

Glasgow Mathematical Journal ◽

10.1017/s0017089500001579 ◽

1972 ◽

Vol 13 (2) ◽

pp. 147-152 ◽

Cited By ~ 3

Author(s):

Don B. Hinton

Keyword(s):

Asymptotic Behaviour ◽

Fourth Order ◽

Linear Equations ◽

Order Equation ◽

Third Order ◽

Fourth Order Equation ◽

The Third ◽

Asymptotic Formulae ◽

Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

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Positive Solutions for Resonant and Nonresonant Nonlinear Third-Order Multipoint Boundary Value Problems

Abstract and Applied Analysis ◽

10.1155/2013/519346 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9

Author(s):

Liu Yang ◽

Chunfang Shen ◽

Dapeng Xie

Keyword(s):

Boundary Value Problem ◽

Positive Solution ◽

Positive Solutions ◽

Boundary Value ◽

Type Theorem ◽

Third Order ◽

Multipoint Boundary ◽

Multipoint Boundary Value Problems ◽

First Time ◽

Multipoint Boundary Value Problem

Positive solutions for a kind of third-order multipoint boundary value problem under the nonresonant conditions and the resonant conditions are considered. In the nonresonant case, by using the Leggett-Williams fixed point theorem, the existence of at least three positive solutions is obtained. In the resonant case, by using the Leggett-Williams norm-type theorem due to O’Regan and Zima, the existence result of at least one positive solution is established. It is remarkable to point out that it is the first time that the positive solution is considered for the third-order boundary value problem at resonance. Some examples are given to demonstrate the main results of the paper.

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Strongly Oscillatory and Nonoscillatory Subspaces of Linear Equations

Canadian Journal of Mathematics ◽

10.4153/cjm-1975-012-8 ◽

1975 ◽

Vol 27 (1) ◽

pp. 106-110 ◽

Cited By ~ 1

Author(s):

J. Michael Dolan ◽

Gene A. Klaasen

Keyword(s):

Linear Equation ◽

Linear Space ◽

Nontrivial Solution ◽

Linear Equations ◽

Order Equation ◽

Third Order ◽

Nonoscillatory Solutions ◽

Oscillatory Solutions ◽

The Third ◽

All Solutions

Consider the nth order linear equationand particularly the third order equationA nontrivial solution of (1)n is said to be oscillatory or nonoscillatory depending on whether it has infinitely many or finitely many zeros on [a, ∞). Let denote respectively the set of all solutions, oscillatory solutions, nonoscillatory solutions of (1)n. is an n-dimensional linear space. A subspace is said to be nonoscillatory or strongly oscillatory respectively if every nontrivial solution of is nonoscillatory or oscillatory. If contains both oscillatory and nonoscillatory solutions then is said to be weakly oscillatory.

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