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Subject to given boundary data, nonexistence of solution to the one-dimensional Kirchhoff-like equation − M ( ( a ∗ | u | q ) ( 1 ) ) u ( t ) = λ f ( t , u ( t ) ) , 0 > t > 1 \begin{equation*} -M\Big (\big (a*|u|^q\big )(1)\Big )u(t)=\lambda f\big (t,u(t)\big ),\ 0>t>1 \end{equation*} is considered. In particular, a condition is provided on the parameter λ \lambda such that for each λ > λ 0 \lambda >\lambda _0 , where λ 0 \lambda _0 is defined in terms of initial data, the boundary value problem has no nontrivial positive solution.
We consider nonlocal differential equations with convolution coefficients of the form − M ( a * | u | q ) ( 1 ) μ ( t ) u ″ ( t ) = λ f t , u ( t ) , t ∈ ( 0,1 ) , $$-M\left(\left(a {\ast} \vert u{\vert }^{q}\right)\left(1\right)\mu \left(t\right)\right){u}^{{\prime\prime}}\left(t\right)=\lambda f\left(t,u\left(t\right)\right)\text{,\,}t\in \left(0,1\right),$$ where q > 0, subject to given boundary data. The function μ ∈ C [ 0,1 ] $\mu \in \mathcal{C}\left(\left[0,1\right]\right)$ modulates the strength of the nonlocal element. We demonstrate that the nonlocality has a strong deregularising effect in the specific sense that nonexistence theorems for this problem are directly affected by the magnitude of the function μ. A specific example illustrates the application of the nonexistence results presented herein.
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