On first-order ordinary differential equations with nonnegative right-hand sides
Some new results are given in this paper, concerning the existence of extremal solutions to the scalar initial value problem \[ x'(t)=f \bigl(t,x(t) \bigr),\;t\geq t_0,\;x(t_0)= x_0,\tag{1} \] with \(f:[t_0,T]\times [x_0,R] \to\mathbb{R}\), \(T>t_0\), and \(R>x_0\). Firstly, some preliminary notions are defined. We denote by \(AC([a,b])\) the set of all real functions which are absolutely continuous on the interval \([a,b]\). For a subset \(Y\subset AC([t_0, T])\), \(t_0<T_0\leq T\), we say that \(x_*\in Y\) is a minimal solution to (1) in \(Y\) if \(x_*\) is a solution to (1) and \(x_*(t)\leq x(t)\) for all \(t\in[t_0,T_0]\) and for any other solution \(x\in Y\). We define the maximal solution to (1) in \(Y\) by reversing the inequalities. When both the minimal and the maximal solutions to (1) exist, we call them extremal solutions in \(Y\). We say that \(x_-\) is a subfunction on \([t_0,T_0] \subset[t_0,T]\) for problem (1) if \(x_-\in AC([t_0, T_0])\), \(x_-(t_0)=x_0\) and \((x_-)'(t)\leq f(t,x_-(t))\) for a.e. \(t\in [t_0,T_0]\). If \(T_0=T\) we say that \(x_-\) is a subfunction. Analogously, we say that \(x_+\) is a superfunction on \([t_0,T_0] \subset [t_0,T]\) for (1) if \(x_+\in AC([t_0, T_0])\), \(x_+(t_0)=x_0\), and \((x_+)'(t)\geq f(t,x_+(t))\) for a.e. \(t\in[t_0,T_0]\). If \(T_0=T\) we say that \(x_+\) is a superfunction. There are assumed that for the given right-hand side \(f:[t_0,T] \times [x_0,R]\to\mathbb{R}\) there exists a null measure set \(NC[x_0,R]\) such that the following conditions hold: (f1) for all \(t\in [t_0,T]\), \(f(t,\bullet)\) is measurable on \([x_0,R]\); (f2) for all \(x\in[x_0,R] \setminus N\), \[ \liminf_{s\to t^-} f(s,x)\geq f(t,x)\text{ for all } t\in(t_0,T], \text{ and} \] \[ f(t,x)\geq\limsup_{s\to t^+}f(s,x)\text{ for all }t \in[t_0,T); \] (f3) there exists \(M\in L^1(x_0,R)\) such that for all \(x\in [x_0, R] \setminus N\) and all \(t\in[t_0,T]\), \(0<{1\over M(x)}\leq f(t,x)\). The main result on the local existence states that there exists \(T_0\in(t_0,T]\) such that (1) has extremal solutions on the set \(Y_0=\{x\in AC([t_0,T_0]):x'(t)>0\) for a.a. \(t\in[t_0,T_0]\}\). Moreover, if \(x_*\) and \(x^*\) stand, respectively, for the minimal and the maximum solution in \(Y_0\), then for all \(t\in[t_0,T_0]\) we have \(x^*(t)=\max\{x_-(t)\mid x_-\in Y_0\) is a subfunction on \([t_0,T_0]\}\) and \(x_*(t)=\min\{x_+(t)\mid x_+\in Y_0\) is a superfunction on \([t_0,T_0]\} \). Concerning the global existence, if in addition \(f\) satisfies: (f4) there exists \(m\in L^1(x_0,R)\) such that (f4.1) for all \(x\in[x_0,R] \setminus N\) and all \(t\in[t_0,T]\), \(f(t,x)\leq{1\over m(x)}\), (f4.2) \(\int^R_{x_0}m(s)ds\geq T-t_0\), then problem (1) has extremal solutions in \(Y=\{x\in A\subset ([t_0,T])\), \(x'(t)>0\) for a.a. \(t\in [t_0,T]\}\). Moreover, if \(x_*\) and \(x^*\) stand, respectively, for the minimal and the maximal solution in \(Y\), then for all \(t\in[t_0,T]\) we have \(x^*(t)= \max \{x_-(t) \mid x_-\in Y\) is a subfunction\} and \(x_*(t)= \min\{x_+(t) \mid x_+\in Y\) is a superfunction\}. A generalization of the previous results is presented for the problem \[ x'(t)=\ell(t) f\bigl(t,x(t) \bigr),\;t\geq t_0,\;x (t_0) =x_0, \] where \(\ell:[t_0,T]\to\mathbb{R}\), and \(f:[t_0,T] \times[x_0,R] \to \mathbb{R}\), with \(T>t_0\) and \(R>x_0\). Some examples and remarks are given.
scientific article; zbMATH DE number 1891308