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We investigate the uniqueness of solutions to an initial value problem associated with a nonlinear fractional differential equation of order α∈(0,1). The differential operator is of Caputo type whereas the nonlinearity cannot be expressed as a Lipschitz function. Instead, the Riemann–Liouville derivative of this nonlinearity verifies a special inequality.

We improve several recent results in the asymptotic integration theory of nonlinear ordinary differential equations via a variant of the method devised by J. K. Hale and N. Onuchic The results are used for investigating the existence of positive solutions to certain reaction-diffusion equations.

We present a class of second order linear differential equations that are integrable and also possess a vanishing-at-infinity oscillatory solution. This feature shows that integrability might play a significant role in the theory of perturbed oscillations.

We establish a flexible oscillation criterion based on an averaging technique that improves upon a result due to C. G. Philos.

We give a simple proof of existence for the smooth traveling waves with a single crest profile of maximum amplitude of a recently derived integrable two-component shallow water system.

We establish that the elliptic equationdefined in an exterior domain of ℝn,n≥3, has a positive solution which decays to 0 as$\vert x\vert \rightarrow +\infty $under quite general assumptions uponfandg.

With a change of perspective in Hille’s paper on oscillation [Trans. Amer. Math. Soc. 64 (1948), 234–252], we provide optimal integral conditions for an application of the method of sub-solutions and super-solutions to investigating the existence of solutions for the semi-linear elliptic equation \(\Delta U + F(x,U) = 0\) in the euclidean space \({\mathbb{R}}^{n}\), n ≥ 3, that decay to zero as \(|x| \rightarrow +\infty\).

We investigate the existence of positive solutions vanishing as x ->+infinity to the semilinear elliptic equation Delta u + f (x, u) + g(x)x . del u = 0, x > A > 0, in R-n, n >= 3, under mild assumptions on the functions f, g. Conditions for these solutions to be radial are also given.

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation $${\frac{{\rm d}^{2}x}{{\rm d}t^{2}}=a(t)x^{\lambda}}$$ , t ≥ t 0 > 0, with solutions x such that $${x(t)\sim c\cdot t^{2}}$$ as $${t\rightarrow +\infty}$$ , $${c\not= 0}$$ , and the equation $${\frac{{\rm d}^{2}y}{{\rm d}u^{2}}=b(u)e^{y}}$$ , u > 0, with solutions y such that $${\lim sup_{u\searrow k}\frac{y(u)}{\ln (u-k)}=c_{1} < 0}$$ for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation $${\Delta U=K(\left| x\right|)e^U}$$ , $${\left| x\right| > x_0 > 0}$$ , that blow up as $${\left| x\right|\searrow x_0}$$ in the two dimensional case.

A new approach to the analysis of wave-breaking solutions to the Dullin-Gottwald-Holm equation is presented in this paper. Introduction of a set of variables allows for solving the singularities. A continuous semigroup of solutions is also built. The solutions have constant $H^{1}$-energy for almost every time with respect to the Lebesgue measure.

Using a novel Wintner-type formulation of the classical Peano's existence theorem (Math. Ann. 37 (1890), 182-228), we enhance Wazewski's result on invertibility of maps defined on closed balls (Ann. Soc. Pol. Math. 20 (1947), 81-125) securing the size of the domain of invertibility that agrees with the bounds derived by John (Comm. Pure Appl. Math. 21 (1968), 77-110) and Sotomayor (Z. Angew. Math. Phys. 41 (1990), 306-310).

Via a special integral transformation, asymptotic integration results for ordinary differential equations are used to establish accurate asymptotic developments for radial solutions of the elliptic equation Δu + K(|x|)eu = 0, |x| > x0 > 0, in the bidimensional case.

The local well-posedness of a generalized Camassa-Holm equation is established by means of Kato's theory for quasilinear evolution equations and two types of results for the blow-up of solutions with smooth initial data are given.

We establish the local existence and uniqueness of solutions for the Dullin-Gottwald-Holm equation with continuously differentiable, periodic initial data. The regularity conditions needed for the Cauchy problem via the semigroup approach of quasilinear hyperbolic equations of evolution or the viscosity method are significantly lowered.

Analyzing the asymptotic properties of solutions to a class of second-order differential equations, we give criteria for the existence of edge waves for variable seabed profiles in the presence of longshore currents.

For a wide class of higher order nonlinear differential equations, global existence of solutions which behave at infinity like polynomials is established by combining two fixed point results.

We give constructive proof of the existence of vanishing at infinity oscillatory solutions for a second-order perturbed nonlinear differential equation. In contrast to most results reported in the literature, we do not require oscillatory character of the associated unperturbed equation, monotonicity of nonlinearity, and we establish global existence of oscillatory solutions without assuming it a priori. Furthermore, as our example demonstrates, existence of bounded oscillatory solutions does not exclude existence of unbounded nonoscillatory solutions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We are concerned with the nonexistence of L 2 -solutions of a nonlinear differential equation x ″= a ( t ) x + f ( t , x ). By applying technique similar to that exploited by Hallam [SIAM J. Appl. Math. 19 (1970) 430–439] for the study of asymptotic behavior of solutions of this equation, we establish nonexistence of solutions from the class L 2 ( t 0 ,∞) under milder conditions on the function a ( t ) which, as the examples show, can be even square integrable. Therefore, the equation under consideration can be classified as of limit-point type at infinity in the sense of the definition introduced by Graef and Spikes [Nonlinear Anal. 7 (1983) 851–871]. We compare our results to those reported in the literature and show how they can be extended to third order nonlinear differential equations.

The paper addresses two open problems related to global existence of solutions with a "linear-like" behavior at infinity. For a class of second-order nonlinear differential equations, we establish global existence of solutions under milder assumption on the rate of decay of the coefficient. Furthermore, as opposed to results reported in the literature, we prove for another class of second-order nonlinear differential equations that the region of the initial data for the solutions with desired asymptotic behavior is unbounded and proper.

where the nonlinearity f satis9es certain monotonicity conditions speci9ed in the sequel. We always assume without mentioning that the function f(t; u; v) is continuous in the domain D= {(t; u; v): t ∈ [t0;+∞); u; v∈R}, where t0 ? 1. It is supposed also that all equations and inequalities involving t hold for t? t0 unless otherwise speci9ed, the symbols “o” and “O” have their standard meanings as t → ∞; and we use the notation R+ def =(0;+∞). Eq. (1) is often used for mathematical modelling of various physical, chemical and biological systems and attracts constant interest of researchers. A great deal of papers published during the last three decades are concerned with local and global existence of solutions of Eq. (1) and its numerous particular cases, uniqueness of solutions