Uniqueness of solutions to an Abel type nonlinear integral equation on the half line
The author studies the integral equation \[ u(x)=\int_{-\infty}^{x}(x-s)^{\alpha-1}g(u(s))\;ds\text{, }-\infty<x\text{, }0<\alpha, \] and obtains that any two nontrivial solutions of this equation are translations of each other; note that in this paper it is assumed that the constant \(\alpha\) appearing above satisfies the inequality \(1<\alpha<4\). More precisely, the author proves that if \(u_1\) and \(u_2\) are nontrivial solutions of the above integral equation, then it holds that \(u_2(x)=u_1(x+c)\), for some constant \(c\in\mathbb{R}\) and each \(x>-\infty\). In order to prove this result, the author assumes that the function \(g\;: \;[0,+\infty)\rightarrow[0,+\infty)\) satisfies \(g(0)=0\), \(g(u)>0\) for all \(u>0\), and \(g\) is nondecreasing; furthermore, it is supposed that \(g\) is continuous on \([0,+\infty)\). It is also worth noting that in addition to the aforementioned assumptions imposed on \(g\) it is also supposed that \(g\) satisfies the inequality \[ \int_0^{\delta}\left(\frac{s}{g(s)}\right)^{\frac{1}{\alpha}}\frac{ds}{s}=+\infty \] for some \(\delta>0\). With all these assumptions in hand, the previously mentioned uniqueness-type result is obtained. This paper is clearly written and could be of interest to researchers interested in integral equations generally and, more specifically, in both the existence and uniqueness of solution to integral equations.
scientific article; zbMATH DE number 6137111
scientific article