Recommended by Kanishka Perera In a recent paper, Sun et al. (2007) studied the existence of positive solutions of nonhomo-geneous multipoint boundary value problems for a second-order differential equation. It is the purpose of this paper to show that the shooting method approach proposed in the recent paper by the first author can be extended to treat this more general problem. This is an ope...

In this paper, we investigate the following quasilinear and singular problem : { −∆pu = λ uδ + u q in Ω u|∂Ω = 0 , u > 0 in Ω (1) where Ω is an open bounded domain with smooth boundary, 1 < p, p− 1 < q and λ, δ > 0. We first prove that there exist weak solutions for λ > 0 small in W 1,p 0 (Ω)∩C(Ω̄) if and only if δ < 2+ 1 p−1 . Investigating the radial symmetric case (Ω = BR(0)), we prove by a s...

For the real-time computation of closed-loop controls, guidance methods have been recently developed by the authors which are based on the theory of neighboring extremals and/or the multiple shooting algorithm. Because of their close relationship, these methods have a comparable domain of controllability, i.e., the set of all deviations from the nominal optimal trajectory which can be compensat...

This article deals with the problem of computing energy-minimal trajectories between the invariant manifolds in the neighborhood of the equilibrium point L1 of the restricted 3-body problem. Initializing a simple shooting method with solutions of the corresponding linear optimal control problem, we numerically compute energy-minimal extremals from the Pontryagin’s Maximum principle, whose optim...

Uniqueness of and numerical techniques for the inverse Sturm-Liouville problem with eigenparameter dependent boundary conditions will be discussed. We will use a Gel’fand-Levitan technique to show that the potential q in u00 þ qu 1⁄4 u, 0 < x < 1 uð0Þ 1⁄4 0, ða þ bÞuð1Þ 1⁄4 ðc þ d Þu0ð1Þ can be uniquely determined using spectral data. In the presence of finite spectral data, q can be reconstruc...

The shoot2.0 package implements an indirect shooting method for optimal control problems. It is speci cally designed to handle control discontinuities, with an automatic switching detection that requires no assumptions concerning the number of switchings. Special care is also devoted to the computation of the Jacobian matrix of the shooting function, using the variational system instead of clas...

where 0 < αi < βi < 1, 0 < ξi < ηi < 1, γi > 0, ρi > 0, i = 1, 2, . . . ,n, λ > 0, μ > 0. J = [0, 1], J∗ = J\{t1, t2, . . . , tp}, 0 = t0 < t1 < t2 < · · · < tp < tp+1 = 1, u(tk) = u(t+ k ) – u(t– k ) denotes the jump of u(t) at t = tk , u(t+ k ) and u(t – k ) represent the right and left limits of u(t) at t = tk , respectively, u′(tk) has a similar meaning for u′(t). In fact, multi-strip condi...

A new adaptive L₁/₂ shooting regularization method for variable selection based on the Cox's proportional hazards mode being proposed. This adaptive L₁/₂ shooting algorithm can be easily obtained by the optimization of a reweighed iterative series of L₁ penalties and a shooting strategy of L₁/₂ penalty. Simulation results based on high dimensional artificial data show that the adaptive L₁/₂ sho...

* Correspondence: [email protected] Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, PR China Abstract In this article, we consider the uniqueness of positive radial solutions to the Dirichlet boundary value problem u + f (|x|, u) + g(|x|)x · ∇u = 0, x ∈ , u = 0, x ∈ ∂ , where Ω denotes an annulus in R (n ≥ 3). The uniqueness criterion is established by applyi...

We study both existence and the exact number of positive solutions of the problem . All righ . Iturria ðPkÞ k ju0j u0 0 þ f ðuÞ 1⁄4 0 in ð0;1Þ; uð0Þ 1⁄4 uð1Þ 1⁄4 0; <: where k is a positive parameter, p > 1, the nonlinearity f is positive in (0,1), and f ð0Þ 1⁄4 f ð1Þ 1⁄4 0. Assuming that f satisfies the condition lims!1 f ðsÞ ð1 sÞ 1⁄4 x > 0 where h 2 ð0; p 1Þ, we study its behavior near zero,...