Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity

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• 作者：Anna Oleynik^a ; ^{anna.oleynik@nmbu.no" class="auth_mail" title="E-mail the corresponding author} ; Arcady Ponosov^a ; ^{arkadi@nmbu.no" class="auth_mail" title="E-mail the corresponding author} ; Vadim Kostrykin^b ; ^{kostrykin@mathematik.uni-mainz.de" class="auth_mail" title="E-mail the corresponding author} ; Alexander V. Sobolev^c ; ^{a.sobolev@ucl.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：47H30 ; 47N60 ; 47G10 ; 45L05 ; 45J05 ; 92B20
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 November 2016
• 年：2016
• 卷：261
• 期：10
• 页码：5844-5874
• 全文大小：641 K

文摘

We study the existence of fixed points to a parameterized Hammerstein operator class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616302376&_mathId=si1.gif&_user=111111111&_pii=S0022039616302376&_rdoc=1&_issn=00220396&md5=5ca502d44926d8f19ab8ffcbf7be4555" title="Click to view the MathML source">H_βclass="mathContainer hidden">class="mathCode">${\mathcal{H}}_{\beta }$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616302376&_mathId=si2.gif&_user=111111111&_pii=S0022039616302376&_rdoc=1&_issn=00220396&md5=b5568f682d3c268ea6a622039419676b" title="Click to view the MathML source">β∈(0,∞]class="mathContainer hidden">class="mathCode">$\beta \in (0,\infty ]$, with sigmoid type of nonlinearity. The parameter class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616302376&_mathId=si3.gif&_user=111111111&_pii=S0022039616302376&_rdoc=1&_issn=00220396&md5=9297c8fde42dfc4108e4e84aa457d7ef" title="Click to view the MathML source">β<∞class="mathContainer hidden">class="mathCode">$\beta <\infty$ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616302376&_mathId=si184.gif&_user=111111111&_pii=S0022039616302376&_rdoc=1&_issn=00220396&md5=ca05145a194175e41f5191b1451a2572" title="Click to view the MathML source">β=∞class="mathContainer hidden">class="mathCode">$\beta =\infty$ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large β   exist and can be approximated by the fixed points of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616302376&_mathId=si5.gif&_user=111111111&_pii=S0022039616302376&_rdoc=1&_issn=00220396&md5=814ff743238789b607b28caaa07188ef" title="Click to view the MathML source">H_∞class="mathContainer hidden">class="mathCode">${\mathcal{H}}_{\infty }$. These results are of a high importance in biological applications where one often approximates the smooth sigmoid by discontinuous unit step function. Moreover, in order to achieve even better approximation than a solution of the limit problem, we employ the iterative method that has several advantages compared to other existing methods. For example, this method can be used to construct non-isolated homoclinic orbit of a Hamiltonian system of equations. We illustrate the results and advantages of the numerical method for stationary versions of the FitzHugh–Nagumo reaction–diffusion equation and a neural field model.