Existence and approximation of solutions to nonlinear hybrid ordinary differential equations

In this note, we consider the initial value problem for first order nonlinear hybrid ordinary differential equations and we discuss the existence and approximation of the solutions. The main results are related to a recent work by Dhage et al. (2014) through the restructuring of some of the hypotheses imposed and the extension of some of the results there. In addition, we provide an example to illustrate the applicability of the abstract results developed.     

On global continuability of solutions of infinite-dimensional ordinary differential equations

We prove necessary and sufficient conditions for global in time existence of solutions of ordinary differential equations on infinite dimensional Banach spaces and manifolds under some natural additional hypotheses, in particular, for equations with right-hand sides, given on everywhere dense subsets of phase spaces.     

On oscillatory solutions of fourth order ordinary differential equations

The paper deals with the oscillation of a differential equation L ₄ y + P(t)L ₂ y + Q(t)y 0 as well as with the structure of its fundamental system of solutions.     

On convexity of the functional for inverse problems of hyperbolic equations

In this paper, a strategy to design a functional for inverse problems of hyperbolic equations is proposed. For an inverse source problem, it is shown that the designed functional is globally strictly convex. For an inverse coefficient problem, we can only prove that it is strictly convex near true solution. This strategy can be generalized to other inverse problems, as long as Lipschitz stability is given.     

On improper integrals and differential equations in ordered Banach spaces

In this paper we study the existence of improper integrals of vector-valued mappings. The so obtained results combined with fixed point results in partially ordered functions spaces are then applied to derive existence and comparison results for least and greatest solutions of initial- and boundary-value problems in ordered Banach spaces. The considered problems can be singular, functional, nonlocal, implicit and discontinuous. Concrete examples are also solved.     

Some remarks on a paper by samimi on nonuniqueness criteria for ordinary differential equations

In [4] M. SAMIMI gives nonuniqueness theorems for initial value problems x′= f(t,x), x(0) =x₀. where f is not defined at t = 0. But the result for the scalar case only holds with the additional assumption that one solution exists. The applicability of the revised theorem is shown. The result for the n-dimensional case is generalized.     

Some existence results for differential equations of second order

Existence results for initial value problem and Dirichlet boundary value problem for nonlinear differential equations of second order are obtained.     

On elliptic solutions of nonlinear ordinary differential equations

The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented.     

On periodic solutions of even-order ordinary differential equations

Optimal in a certain sense sufficient conditions are given for the existence and uniqueness of ω-periodic solutions of the nonautonomous ordinary differential equation u ^(2m) =f(t,u,...,u ^(m-1) ), where the function f:ℝ×ℝ ^m →ℝ is periodic with respect to the first argument with period ω. Received: December 21, 1999; in final form: August 12, 2000?Published online: October 2, 2001     

A survey of numerical techniques for solving singularly perturbed ordinary differential equations

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory. Starting from Prandtl's work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984–2000. The work done in this area during the period 1905–1984 has already been surveyed by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223] for details. Due to the space constraints we have covered only singularly perturbed one-dimensional problems.     

Approximating solutions of nonlinear hybrid differential equations

We prove the existence as well as approximations of the solutions of initial value problems of first order ordinary nonlinear hybrid differential equations. We rely our results on a recent hybrid fixed point theorem of Dhage (2014) in partially ordered normed linear spaces. A realization is also indicated to illustrate the abstract theory developed in the paper.