Stokes phenomena with respect to a large parameter are investigated for Shrödinger-type ordinary differential equations having a Stokes curve of loop-type. For this purpose, we employ a Bessel-type equation as a canonical form and compute the Voros coefficient of the equation. Combining the formula describing the Stokes automorphism for the Voros coefficient and the formal coordinate transformation connecting the Shrödinger-type equation and the Bessel-type equation, we have some formulas describing the action of alien derivatives and Stokes automorphism for WKB solutions of the Shrödinger-type equation.
We show that solutions to a classical Fokker-Plank equation can be approximated by solutions to nonlocal evolution problems when a rescaling parameter that controls the size of the nonlocality goes to zero.
In this paper, we show the existence of a classical solution to a class of fractional logistic equations in an open bounded subset with smooth boundary. We use the method of sub- and super-solutions with variational arguments to establish the existence of a unique positive solution. We also establish the stability and nondegeneracy of the positive solution.
Okaie et al.  utilized the Keller-Segel model for mobile bionanosensor networks for target tracking. They introduced a mathematical formulation and described numerical results. In this paper, we would like to study analytically their model. We first construct a unique local solution for model equations. Second, we establish a priori estimates for local solutions to obtain a global solution. Finally, after constructing a non-autonomous dynamical system, we will show existence of exponential attractors.
In the theory of linear autonomous neutral functional differential equations with infinite delay, the spectrum distribution of the infinitesimal generator of its solution operators is studied under a certain phase space. Thereafter, we prove the representation theorem of the solution operators, which is later employed to obtain exponential dichotomy properties in terms of semigroup theory. Formal adjoint theory for linear autonomous NFDEs with infinite delay is established including such topics as formal adjoint equations, the relationship between the formal adjoint and true adjoint, and decomposing the phase space with formal adjoint equation. Finally, the algorithm for calculating the Hopf bifurcation properties for nonlinear NFDEs with infinite delay is presented based on the theory of linear equations.
Lotka-Volterra systems associated with skew-symmetric interaction between species are studied. We pick up some form of this model provided with conserved quantities, which makes all the solutions to be periodic-in-time except for the equilibrium. This class is explicitly given by a set of algebraic conditions on coefficients. If the system takes N components, we have 2N-3 and 2N-1 degrees of freedom without and with linear terms, respectively.