Nonexistence and parameter range estimates for convolution differential equations

Goodrich, Christopher

doi:10.1090/bproc/130

Nonexistence and parameter range estimates for convolution differential equations
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by Christopher S. Goodrich HTML | PDF

Proc. Amer. Math. Soc. Ser. B 9 (2022), 254-265

Abstract:

We consider nonlocal differential equations with convolution coefficients of the form \begin{equation} -M\Big (\big (a*u^q\big )(1)\Big )u''(t)=\lambda f\big (t,u(t)\big ),t\in (0,1),\notag \end{equation} and we demonstrate an explicit range of $\lambda$ for which this problem, subject to given boundary data, will not admit a nontrivial positive solution; if $a\equiv 1$, then the model case \begin{equation} -M\Big (\Vert u\Vert _{L^q(0,1)}^{q}\Big )u''(t)=\lambda f\big (t,u(t)\big ),t\in (0,1)\notag \end{equation} is obtained. The range of $\lambda$ is calculable in terms of initial data, and our results allow for a variety of kernels, $a$, to be utilized, including, for example, those leading to a fractional integral coefficient of Riemann-Liouville type. Two examples are provided in order to illustrate the application of the result.

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