Ordinary Differential Equations

Hartman, Philip

doi:10.1137/1.9780898719222

Cited by 2,215 publications

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“…The technique used for the existence and asymptotic behaviour of solutions of (1) is based on the well-known Schauder's fixed point theorem and Wazewski's topological method for ordinary differential equations (see [5] …”

Section: The Main Resultsmentioning

confidence: 99%

On an Initial Value Problem for Singular Integro-Differential Equations

Šmarda¹

2002

Demonstratio Mathematica

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Abstract. The purpose of this paper is to study the existence and asymptotic behaviour of solutions of a nonlinear singular integro-differential equation. IntroductionIn the past two decades, several papers have been devoted to the study of singular initial value problems for differential and integro-differential equations under various conditions on the nonlinearity and the kernel (see e.g. [7]). Integro-differential equations have different properties from ordinary differential equations even in the simplest cases (see [2]). Therefore known qualitative methods of investigation of ordinary differential equations, e.g. Wazewki's topological method, cannot be applied to integrodifferential equations. The fundamental tools used in the existence proofs of all above mentioned works are essentially Schauder-Tychonoff's fixed point theorem and Banach contraction principle.In this paper we deal with the following problem t (1) g(t)y'(t) = a(t)y(t){l + f{t,y(t),\K(t,s,y(s))ds)), y(

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Section: The Main Resultsmentioning

confidence: 99%

On an Initial Value Problem for Singular Integro-Differential Equations

Šmarda¹

2002

Demonstratio Mathematica

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“…We use linear analysis for this. In a small neighborhood of a hyperbolic equilibrium (v,û) = (0, 0) for (3), (4), flow of the nonlinear system is topologically equivalent with that of its linearization if (0, 0) is a hyperbolic steady state (Hartman, 1964). The linearized equations of Eqs.…”

Section: Standard Functional Responsesmentioning

confidence: 99%

Continuous Traveling Waves for Prey-Taxis

2008

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Spatially moving predators are often considered for biological control of invasive species. The question arises as to whether introduced predators are able to stop an advancing pest or foreign population. In recent studies of reaction-diffusion models, it has been shown that the prey invasion can only be stopped if the prey dynamics observes an Allee effect.In this paper, we include prey-taxis into the model. Prey-taxis describe the active movement of predators to regions of high prey density. This effect leads to the observation that predators are drawn away from the leading edge of a prey invasion where its density is low. This leads to counterintuitive result that prey-taxis can actually reduce the likelihood of effective biocontrol.

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“…This is the effect of the assumptions (3) and (4). For more information on this point the reader is referred to Section X.3 of the book [16]. After classifying the points in Γ we are ready to complete the proof.…”

Section: Theoremmentioning

confidence: 99%

Retracts, fixed point index and differential equations

Ortega

2008

Rev. R. Acad. Cien. Serie A. Mat.

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Abstract. Some problems in differential equations evolve towards Topology from an analytical origin. Two such problems will be discussed: the existence of solutions asymptotic to the equilibrium and the stability of closed orbits of Hamiltonian systems. The theory of retracts and the fixed point index have become useful tools in the study of these questions. Retractos,índice de punto fijo y ecuaciones diferencialesResumen. Algunos problemas de las ecuaciones diferenciales evolucionan hacia la Topología desde un origen analítico. Se discutirán dos problemas de este tipo: la existencia de soluciones asintóticas al equilibrio y la estabilidad de lasórbitas cerradas de los sistemas Hamiltonianos. La teoría de retractos y elíndice de punto fijo se han convertido en herramientas muyútiles para el estudio de estas cuestiones.

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Ordinary Differential Equations

Cited by 2,215 publications

References 0 publications

On an Initial Value Problem for Singular Integro-Differential Equations

On an Initial Value Problem for Singular Integro-Differential Equations

Continuous Traveling Waves for Prey-Taxis

Retracts, fixed point index and differential equations

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