scholarly journals Singularly Perturbed Linear Neumann Problem with the Characteristic Roots on the Imaginary Axis

2010 ◽  
Vol 18 (28) ◽  
pp. 163-168
Author(s):  
Ľudmila Vaculíková ◽  
Vladimír Liška
Keyword(s):  
Integral Equation ◽  
Asymptotic Behavior ◽  
Neumann Problem ◽  
Imaginary Axis ◽  
Singularly Perturbed ◽  
Asymptotic Behavior Of Solutions ◽  
Characteristic Roots

Singularly Perturbed Linear Neumann Problem with the Characteristic Roots on the Imaginary Axis We investigate the problem of existence and asymptotic behavior of solutions for the singularly perturbed linear Neumann problem <img src="/fulltext-image.asp?format=htmlnonpaginated&src=C6551P41673P4147_html\Journal10186_Volume18_Issue28_20_paper.gif" alt=""/> Our approach relies on the analysis of integral equation equivalent to the problem above.

scholarly journals Asymptotic Behavior of Solutions of Integral Equations with Homogeneous Kernels

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 180
Author(s):  
Oleg Avsyankin
Keyword(s):  
Integral Equation ◽  
Asymptotic Behavior ◽  
Integral Equations ◽  
Spherical Harmonics ◽  
Special Class ◽  
Continuous Functions ◽  
Asymptotic Behavior Of Solutions ◽  
Free Term ◽  
Research Technique ◽  
Multidimensional Integral

The multidimensional integral equation of second kind with a homogeneous of degree (−n) kernel is considered. The special class of continuous functions with a given asymptotic behavior in the neighborhood of zero is defined. It is proved that, if the free term of the integral equation belongs to this class and the equation itself is solvable, then its solution also belongs to this class. To solve this problem, a special research technique is used. The above-mentioned technique is based on the decomposition of both the solution and the free term in spherical harmonics.

scholarly journals Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Cristóbal González ◽  
Antonio Jiménez-Melado
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Asymptotic Behavior ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Abstract Differential Equation ◽  
Deviating Arguments ◽  
Vector Integral ◽  
Similar Equation

In this paper, we propose the study of an integral equation, with deviating arguments, of the typey(t)=ω(t)-∫0∞‍f(t,s,y(γ1(s)),…,y(γN(s)))ds,t≥0,in the context of Banach spaces, with the intention of giving sufficient conditions that ensure the existence of solutions with the same asymptotic behavior at∞asω(t). A similar equation, but requiring a little less restrictive hypotheses, isy(t)=ω(t)-∫0∞‍q(t,s)F(s,y(γ1(s)),…,y(γN(s)))ds,t≥0.In the case ofq(t,s)=(t-s)+, its solutions with asymptotic behavior given byω(t)yield solutions of the second order nonlinear abstract differential equationy''(t)-ω''(t)+F(t,y(γ1(t)),…,y(γN(t)))=0,with the same asymptotic behavior at∞asω(t).

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