Consider the equation

where are linear positive continuous operators and f : Cloc(ℝ;ℝ) → Lloc(ℝ;ℝ) is a continuous operator satisfying the local Carathéodory conditions. Efficient conditions guaranteeing the existence of a global solution, which is bounded and non-negative in the neighbourhood of –∞, to the equation considered are established provided that ℓ0, ℓ1 and f are Volterra-type operators. The existence of a solution that is positive on the whole real line is discussed as well. Furthermore, the asymptotic properties of such solutions are studied in the neighbourhood of –∞. The results are applied to certain models appearing in the natural sciences.

This paper deals with a more general class of singularly perturbed boundary value problem for a differential-difference equations with small shifts. In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter ε and the shifts depend on the small parameter ε has been considered. The fitted-mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layer. The cubic B-spline basis functions with fitted-mesh are considered in the procedure which yield a tridiagonal system which can be solved efficiently by using any well-known algorithm. The stability and parameter-uniform convergence analysis of the proposed method have been discussed. The method has been shown to have almost second-order parameter-uniform convergence. The effect of small parameters on the boundary layer has also been discussed. To demonstrate the performance of the proposed scheme, several numerical experiments have been carried out.

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.

We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process with linear and exponential penalty costs for a detection delay. The optimal times of alarms are found as the first times at which the weighted likelihood ratios hit stochastic boundaries depending on the current observations. The proof is based on the reduction of the initial problems into appropriate three-dimensional optimal stopping problems and the analysis of the associated parabolic-type free-boundary problems. We provide closed-form estimates for the value functions and the boundaries, under certain nontrivial relations between the coefficients of the observable diffusion.

In this paper we investigate the population dynamics of a species with age structure in the case where the diffusion and death rates of the matured population are both age-dependent. We develop a new application of the age-structure technique in terms of an integral equation. For unbounded spatial domains, we study the existence of travelling waves, whilst in bounded domains, we investigate the existence of positive steady-state solutions and their stability.

Nonlinear eigenvalue problems for fluxons in long Josephson junctions with exponentially varying width are treated. Appropriate algorithms are created and realized numerically. The results obtained concern the stability of the fluxons, the centering both magnetic field and current for the magnetic flux quanta in the Josephson junction as well as the ascertaining of the impact of the geometric and physical parameters on these quantities. Each static solution of the nonlinear boundary-value problem is identified as stable or unstable in dependence on the eigenvalues of associated Sturm-Liouville problem. The above compound problem is linearized and solved by using of the reliable Continuous analogue of Newton method.

In this paper we shall discuss nonlinear multipoint boundary value problems for second order differential equations when deviating arguments depend on the unknown solution. Sufficient conditions under which such problems have extremal and quasi-solutions are given. The problem of when a unique solution exists is also investigated. To obtain existence results, a monotone iterative technique is used. Two examples are added to verify theoretical results.

A class of first-order impulsive functional differential equations with forcing terms is considered. It is shown that, under certain assumptions, there exist positive T-periodic solutions, and under some other assumptions, there exists no positive T-periodic solution. Applications and examples are given to illustrate the main results.

Some new Gronwall–Ou-Iang type integral inequalities in two independent variables are established. We also present some of its application to the study of certain classes of integral and differential equations.

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

Second order nonlinear delay differential equations with positive delays are considered, and sufficient conditions are given that guarantee the existence of positive increasing solutions on the half-line with first order derivatives tending to zero at infinity. The approach is elementary and is essentially based on an old idea which appeared in the author's paper Arch. Math. (Basel)36 (1981), 168–178. The application of the result obtained to second order Emden-Fowler type differential equations with constant delays and, especially, to second order linear differential equations with constant delays, is also presented. Moreover, some (general or specific) examples demonstrating the applicability of the main result are given.

Existence principles are given for systems of differential equations with reflection of the argument. These are derived using fixed point analysis, specifically the Nonlinear Alternative. Then existence results are deduced for certain classes of first and second order equations with reflection of the argument.

The concept of essential map and topological transversality due to A. Granas is extended to multi-valued maps in locally convex spaces and it is next applied to prove the solvability of boundary value problems for certain neutral functional differential equations. In order to achieve a required compactness property, the weak topology in a Sobolev space is considered. The topological tool established in the first part of the paper allows to avoid some obstacles which are encountered when trying to use standard degree-theoretical arguments.