On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations

Naito, Yūki; Tanaka, Satoshi

doi:10.1016/j.na.2003.10.020

Cited by 60 publications

(34 citation statements)

References 20 publications

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“…They are thankful to one of the referees who pointed out that the shooting method has also been used by Naito and Tanaka [33] and by Kong [34] for the study of two-point boundary value problems.…”

Section: Acknowledgmentsmentioning

confidence: 99%

The Shooting Method and Nonhomogeneous Multipoint BVPs of Second-Order ODE

Kwong

Wong

2007

Boundary Value Problems

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Section: Acknowledgmentsmentioning

confidence: 99%

The Shooting Method and Nonhomogeneous Multipoint BVPs of Second-Order ODE

Kwong

Wong

2007

Boundary Value Problems

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“…In recent years, progress has also been made to the study of nodal solutions, i.e., solutions with a specific zero-counting property in (a, b), for nonlinear BVPs consisting of Eq. (1.1) and two-point separated BCs, see [10,11,19,21,22,23]. As regard to nonlocal BVPs, results on the existence of nodal solutions have been obtained only for the special BVP consisting of the equation 4) and the multi-point BC…”

Section: −(P(t)ymentioning

confidence: 99%

Nodal Solutions of Nonlocal Boundary Value Problems

Chamberlain

Kong

2009

Mathematical Modelling and Analysis

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Abstract. We study the nonlinear boundary value problem consisting of the second order differential equation on [a, b] and a boundary condition involving a RiemannStieltjes integral. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes of the existence of different types of nodal solutions as the problem changes.

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“…In [15], Erbe, Hu and Wang showed that there were at least two positive solutions in the case of superlinearity at one end (zero or infinity) and sublinearity at the other end. Naito and Tanaka [16] and Ma and Thompson [17] established a precise condition concerning the behavior of the ratio f (s)/s for the existence and nonexistence of solutions. Their main result was that the problem had at least k solutions if the ratio f (s)/s crossed the k eigenvalues of the associated eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%

Bifurcations of optimal solutions for coordinated robotic systems: Numerical and homotopy methods

Deng

Valenzuela

2010

2010 IEEE International Conference on Robotics and Automation

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Abstract-This paper presents the relatively rich and interesting bifurcation structure that is present in the nature of optimal solutions to a multi-robot formation control problem. The problem considered is a two point nonlinear boundaryvalue problem that can only be solved numerically. Since common numerical solution techniques such as the shooting method are local in nature and hence are difficult to use to find multiple solutions, an alternative formulation of the problem is presented that can be solved through homotopy methods for polynomial systems. These methods are guaranteed to find all solutions within the resolution of the system description's discretization. Specifically, this paper studies a group of unicyclelike autonomous mobile robots operating in a 2-dimensional obstacle-free environment. Each robot has a predefined initial state and final state and the problem is to find the optimal path between two states for every robot. The path is optimized with respect to the control effort and the deviation from a desired formation. The bifurcation parameter is the relative weight given to penalizing the deviation from the desired formation versus control effort. It is shown that as this number varies, bifurcations of solutions are obtained. Considering the common use of optimization methods in robotic navigation and coordination problems, understanding the existence and structure of bifurcating and multiple solutions is of great importance in robotics.

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On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations

Cited by 60 publications

References 20 publications

The Shooting Method and Nonhomogeneous Multipoint BVPs of Second-Order ODE

The Shooting Method and Nonhomogeneous Multipoint BVPs of Second-Order ODE

Nodal Solutions of Nonlocal Boundary Value Problems

Bifurcations of optimal solutions for coordinated robotic systems: Numerical and homotopy methods

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