Nonexistence and parameter range estimates for convolution differential equations

Abstract

We consider nonlocal differential equations with convolution coefficients of the form − M ( ( a ∗ u q ) ( 1 ) ) u ( t ) = λ f ( t , u ( t ) ) , t ∈ ( 0 , 1 ) , \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 λ \lambda for which this problem, subject to given boundary data, will not admit a nontrivial positive solution; if a ≡ 1 a\equiv 1 , then the model case − M ( ‖ u ‖ L q ( 0 , 1 ) q ) u ( t ) = λ f ( t , u ( t ) ) , t ∈ ( 0 , 1 ) \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 λ \lambda is calculable in terms of initial data, and our results allow for a variety of kernels, a 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.