Limits of solutions of a perturbed linear differential equation (Q2781087)

The authors give two theorems concerning the following second-order boundary value problem NEWLINE\[NEWLINE\text{(A)}\quad \ddot{x}+ 2 f(t) \dot{x} + x + g(t,x) = 0, \qquad\text{(B)}\quad x(+\infty)=\dot{x}(+\infty)=0,NEWLINE\]NEWLINE where \(f:[0,+\infty)\rightarrow (0,+\infty)\) and \(g:[0,+\infty)\times\mathbb{R}\rightarrow\mathbb{R}\) are continuous functions. NEWLINENEWLINENEWLINEThe first theorem is the following (Theorem 2.1 in the paper): there exists an \(a>0\) such that every solution \(x\) to (A) with \(|x(0)|<a\) is defined on \([0,+\infty)\) and satisfies (B), under the hypotheses: NEWLINENEWLINENEWLINE(i) \(f\in C^1([0,+\infty))\), \(f(+\infty)=0\) and \(\int_0^{+\infty} f(s) ds=+\infty\); NEWLINENEWLINENEWLINE(ii) there exists \(K\in (0,1)\) such that \(|f'(t)+f^2(t)|\leq K f(t)\) for all \(t\in [0,+\infty)\); NEWLINENEWLINENEWLINE(iii) there exist \(M>0\) and \(\alpha >1\) such that \(|g(t,x)|\leq M f(t) |x|^\alpha\) for all \(t\in [0,+\infty)\). NEWLINENEWLINENEWLINEAlthough not mentioned by the authors in the statement, the existence of continuous solutions is also proved, by means of the Schauder-Tikhonov fixed-point theorem. NEWLINENEWLINENEWLINEThe second theorem is the following (Theorem 3.1 in the paper): suppose that (i), (ii) and (iii) are fulfilled, and in addition, for every \(n\in \mathbb{N}^\ast\) there exists \(L_n\in [0,1)\) such that for every \(x_1,x_2\in [0,+\infty)\) with \(|x_i|\leq \rho\) (\(\rho<a\)), \(|g(t,x_1)-g(t,x_2)|\leq L_n |x_1 - x_2|\) (we have corrected a missprint in the paper) for all \(t\in [0,n]\). Then the problem admits a unique solution which is convergent to zero. NEWLINENEWLINENEWLINEThe Banach fixed-point theorem for Frechet spaces is employed in the proof of this result. Not mentioned by the authors in the statement is the fact that the solution is found in the space of continuous functions.