Positive Periodic Solutions of Second-Order Differential Equations with Delays

The existence results of positive ω-periodic solutions are obtained for the second-order differential equation with delays −u″+a(t)=f(t,u(t−τ1),...,u(t−τn)), where a∈C(ℝ,(0,∞)) is a ω-periodic function, f:ℝ×[0,∞)n→[0,∞) is a continuous function, which is ω-periodic in t, and τ1,τ2,...,τn are positive constants. Our discussion is based on the fixed point index theory in cones

The Existence of solutions to nonlinear second order periodic boundary value problems

We consider existence of periodic boundary value problems of nonlinear second order ordinary differential equations. Under certain half Lipschitzian type conditions several existence results are obtained. As applications positive periodic solutions of some $\phi$-Laplacian type equations and Duffing type equations are investigated.Comment: 17 pages, to appear in Nonlinear Analysis Series A: Theory, Methods & Application

Positive periodic solutions of singular systems

The existence and multiplicity of positive periodic solutions for second order non-autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. Our results provide a unified treatment for the problem and significantly improve several results in the literature. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.Comment: Journal of Differential Equations, 201

Positive periodic and subharmonic solutions of second order singular differential equations with impulsive effects

In this paper, we study the existence and multiplicity of positive periodic and subharmonic solutions of second order singulardifferential equations with impulsive effects. The proof is based on a generalized version of the Poincar\u27{e}-Birkhoff twist theorem due to Ding and some phase plane analysis methods

On positive solutions for singular boundary value problems of differential and difference equations / Noor Halimatus Sa'diah Ismail

This thesis is concerned with the existence and multiplicity of positive solutions to singular boundary value problems (BVPs) of differential and difference equations. By using the Krasnoselskii fixed point theorem on compression and expectation in cone, sufficient conditions for the existence of positive solutions are established for a singular system of first-order differential equations and singular second-order BVPs of difference equations. Our results give an almost complete structure of the existence of positive solutions for the problems studied with an appropriately chosen parameter. By choosing appropriate cone, the singularity of the equations is essentially removed and the associated positive operator becomes well defined for certain ranges of functions even when et is negative. By employing the Krasnoselskii fixed point theorem in cone, the existence and multiplicity of positive periodic solutions for a singular system of first-order ordinary differential equations is established. As an extension, the discrete analogue of singular differential problems of second-order BVPs with a parameter is derived. The existence of positive solutions is obtained by applying the Krasnoselskii fixed point theorem in cone. The result is then extended to a singular discrete system of second-order two point BVPs. Also the existence of positive solutions is investigated for a singular discrete system of second-order multi-point BVPs

Nonlinear differential equations having non-sign-definite weights

In the present PhD thesis we deal with the study of the existence, multiplicity and complex behaviors of solutions for some classes of boundary value problems associated with second order nonlinear ordinary differential equations of the form $u''+f(u)u'+g(t,u)=s,$ or $u''+g(t,u)=0,$ $t\in I$, where $I$ is a bounded interval, $f\colon\mathbb{R}\to\mathbb{R}$ is continuous, $s\in\mathbb{R}$ and $g: I\times \mathbb{R}\to\mathbb{R}$ is a perturbation term characterizing the problems. The results carried out in this dissertation are mainly based on dynamical and topological approaches. The issues we address have arisen in the field of partial differential equations. For this reason, we do not treat only the case of ordinary differential equations, but also we take advantage of some results achieved in the one dimensional setting to give applications to nonlinear boundary value problems associated with partial differential equations. In the first part of the thesis, we are interested on a problem suggested by Antonio Ambrosetti in ``Observations on global inversion theorems'' (2011). In more detail, we deal with a periodic boundary value problem associated with the first differential equation where the perturbation term is given by $g(t,u):=a(t)\phi(u)-p(t)$. We assume that $a,$ $p\in L^{\infty}(I)$ and $\phi\colon\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying $\lim_{|\xi|\to\infty}\phi(\xi)=+\infty$. In this context, if the weight term $a(t)$ is such that $a(t)\geq 0$ for a.e. $t\in I$ and $\int_{I}a(t)\,dt>0$, we generalize the result of multiplicity of solutions given by Fabry, Mawhin and Nakashama in ``A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations'' (1986). We extend this kind of improvement also to more general nonlinear terms under local coercivity conditions. In this framework, we also treat in the same spirit Neumann problems associated with second order ordinary differential equations and periodic problems associated with first order ones. Furthermore, we face the classical case of a periodic Ambrosetti-Prodi problem with a weight term $a(t)$ which is constant and positive. Here, considering in the second differential equation a nonlinearity $g(t,u):=\phi(u)-h(t)$, we provide several conditions on the nonlinearity and the perturbative term that ensure the presence of complex behaviors for the solutions of the associated $T$-periodic problem. We also compare these outcomes with the result of stability carried out by Ortega in ``Stability of a periodic problem of Ambrosetti-Prodi type'' (1990). The case with damping term is discussed as well. In the second part of this work, we solve a conjecture by Yuan Lou and Thomas Nagylaki stated in ``A semilinear parabolic system for migration and selection in population genetics'' (2002). The problem refers to the number of positive solutions for Neumann boundary value problems associated with the second differential equation when the perturbation term is given by $g(t,u):=\lambda w(t)\psi(u)$ with $\lambda>0$, $w\in L^{\infty}(I)$ a sign-changing weight term such that $\int_{I}w(t)\,dt<0$ and $\psi\colon[0,1]\to[0,\infty[$ a non-concave continuous function satisfying $\psi(0)=0=\psi(1)$ and such that the map $\xi\mapsto \psi(\xi)/\xi$ is monotone decreasing. In addition to this outcome, other new results of multiplicity of positive solutions are presented as well, for both Neumann or Dirichlet boundary value problems, by means of a particular choice of indefinite weight terms $w(t)$ and different positive nonlinear terms $\psi(u)$ defined on the interval $[0,1]$ or on the positive real semi-axis $[0,+\infty[$

Positive periodic solutions of second-order nonlinear neutral differential equations with variable coefficients

In this paper, we use Krasnoselskii's fixed point theorem to establish the existence of positive periodic solutions of second-order nonlinear neutral differential equations. Our techniques can be used and applied to study other classes of problems and extension some results

Applications of Schauder’s Fixed Point Theorem to Semipositone Singular Differential Equations

We study the existence of positive periodic solutions of second-order singular differential equations. The proof relies on Schauder’s fixed point theorem. Our results generalized and extended those results contained in the studies by Chu and Torres (2007) and Torres (2007) . In some suitable weak singularities, the existence of periodic solutions may help

Periodic Sturm-Liouville problems related to two Riccati equations of constant coefficients

"We consider two closely related Riccati equations of constant parameters whose particular solutions are used to construct the corresponding class of supersymmetrically-coupled second-order differential equations. We solve an-alytically these parametric periodic problems along the positive real axis. Next, the analytically solved model is used as a case study for a powerful numerical approach that is employed here for thefirst time in the investigation of the en-ergy band structure of periodic not necessarily regular potentials. The approach is based on the well-known self-matching procedure of James (1949) and imple-ments the spectral parameter power series solutions introduced by Kravchenko (2008). We obtain additionally an efficient series representation of the Hill dis-criminant based on on Kravchenko´s series.