In this article, the existence of heteroclinic solution of a class of generalized Hamiltonian system with potential $V : {\open R}^{n} \longmapsto {\open R}$ having a finite or infinite number of global minima is studied. Examples include systems involving the p-Laplacian operator, the curvature operator and the relativistic operator. Generalized conservation of energy is established, which leads to the property of equipartition of energy enjoyed by heteroclinic solutions. The existence problem of heteroclinic solution is studied using both variational method and the metric method. The variational approach is classical, while the metric method represents a more geometrical point of view where the existence problem of heteroclinic solution is reduced to that of geodesic in a proper length metric space. Regularities of the heteroclinic solutions are discussed. The results here not only provide alternative solution methods for Φ-Laplacian systems, but also improve existing results for the classical Hamiltonian system. In particular, the conditions imposed upon the potential are very mild and new proof for the compactness is given. Finally in ℝ2, heteroclinic solutions are explicitly written down in closed form by using complex function theory.

We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory. An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown. A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows. The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation. The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms.

It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from.

Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of Möbius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.

The nonlinear Sturm-Liouville equation

$$-{{\left( p{y}' \right)}^{\prime }}\,+\,qy\,=\,\text{ }\!\!\lambda\!\!\text{ }\left( 1-f \right)ry\,on\,[0,\,1]$$

is considered subject to the boundary conditions

$$\left( {{\text{a}}_{j}}\text{ }\lambda \text{ }\text{+}{{\text{b}}_{j}} \right)y\left( j \right)=\left( {{c}_{j}}\text{ }\lambda \text{ }+{{d}_{j}} \right)\left( p{y}' \right)\left( j \right),j=0,1$$

Here ${{\text{a}}_{0}}\,=\,0\,=\,{{c}_{0}}$ and $p,\,r\,>\,0$ and $q$ are functions depending on the independent variable $x$ alone, while $f$ depends on $x,\,y\,and\,{y}'$. Results are given on existence and location of sets of $(\lambda ,\,y)$ bifurcating from the linearized eigenvalues, and for which $y$ has prescribed oscillation count, and on completeness of the $y$ in an appropriate sense.