Limit properties of solutions of singular second-order differential equations
This interesting paper deals with the differential equation \[ u''(t)={\frac{a}{t}}u'(t)+f(t,u(t),u'(t)),\quad\text{ a.e. on }(0,T]\tag{\(*\)} \] where \(a\neq 0\) and the function \(f\) might be unbounded for some values of \(t\). Solutions of the equation \((*)\) are functions in \(AC^1[0,T]\) satisfying \((*)\) almost everywhere on \((0,T].\) In Section 3 of the paper, the authors show that the solution \(u\) of \[ u''(t)={\frac{a}{t}}u'(t)+h(t),\quad\text{ a.e. on }(0,T],\tag{\(**\)} \] with \(h\in L_p\) and \(a>0,\) satisfies \(\lim_{0+}u(t)\in{\mathbb{R}}\) and \(\lim_{0+}u'(t)=0.\) Also, if \({\mathcal{U}}(t):=(u(t),u'(t))^T,\) then they show (in Theorem 3.4) that given \(a>0\), \(p>1\) \(h\in L_p\) and \({\beta}\in{{\mathbb{R}}^2}\) there is a unique solution of \((**)\) satisfying the boundary conditions \(B_0{\mathcal{U}}(0)+B_1{\mathcal{U}}(T)=\beta,\) if-f the following matrix \(B_0\begin{pmatrix} 1&0\\0&0\end{pmatrix}+ B_1\begin{pmatrix} 1&T^{a+1} \\ 0 & (a+1)T^a\end{pmatrix}\) is nonsingular. Also in Theorem 3.5 they show the following: Let \(u\) be a solution of \((**)\). If \(-1<a<0\) then only one of the following properties holds: (i) \(\lim_{t\to+0}u(t)\in\mathbb{R}\) and \(\lim_{t\to+0}u'(t)=0,\) (ii) \(\lim_{t\to+0}u(t)\in\mathbb{R}\) and \(\lim_{t\to+0}u'(t)=\pm\infty.\) If \(a\leq-1,\) then only one of the following properties holds: (i) \(\lim_{t\to+0}u(t)\in\mathbb{R}\) and \(\lim_{t\to+0}u'(t)=0,\) (ii) \(\lim_{t\to+0}u(t)=\mp\infty\) and \(\lim_{t\to+0}u'(t)=\pm\infty.\) In the same section they give the following result: Theorem 3.6: Let \(a<0\) and \(p>1.\) Then for any \(h\in L_p\) and reals \(b_2, \beta\) there exists a unique solution of equation \((**)\) satisfying \(u'(0)=0\) and \(b_0u(0)+b_1u(T)+b_2u'(T)=\beta,\) if-f \(b_0+b_1\neq 0.\) Section 4 is devoted to some properties of the solutions of equation \((*).\) Section 5 contains existence results and (boundedness) properties of solutions of equation \((*)\) satisfying the conditions \(u(0)=u(T)\) and \(u'(0)=u'(T)\). A number of examples illustrate the results.
scientific article; zbMATH DE number 5626176
scientific article