It is well known that there exists no non-trivial bounded solution of $-y''+y=0$ in $\mathbb R.$ Is this result even true, the problem $$ \bigg(-\frac{d^2}{dx^2}\bigg)^{s} y+y=0 $$ has no bounded solution in $\mathbb R$ where $s\in (0, 1).$

It is well known that there exists no non-trivial bounded solution of $-u''+u=0$ in $\mathbb R.$ Is this result even true, the problem $$ \bigg(-\frac{d^2}{dx^2}\bigg)^{s} u+u=0 $$ has no bounded solution in $\mathbb R$ where $s\in (0, 1).$ Here $$\bigg(-\frac{d^2}{dx^2}\bigg)^{s} u(x)=c\int_{\mathbb R}\frac{u(x)-u(y)}{|x-y|^{1+2s}}dy$$ where $c$ is positive constant depending on $s.$