Multiple positive periodic solutions for delay differential system
The authors study the following general non-autonomous delay differential systems \[ \dot y_i(t)=-a_i(t)y_i(t)+f_i(t,\Upsilon_1(t), \Upsilon_2(t),\dots,\Upsilon_m(t)),\quad i=1,2,\dots,m\tag{1} \] and \[ \dot y_i(t)=a_i(t)y_i(t)-f_i(t,\Upsilon_1(t), \Upsilon_2(t),\dots,\Upsilon_m(t)),\quad i=1,2,\dots,m, \tag{2} \] where \[ \Upsilon_i(t):=(y_i(t-\tau_1(t)), y_i(t-\tau_2(t)), \dots, y_i(t-\tau_n(t))), \;t\in\mathbb R, \;i\in \Lambda_1=\{1,2,\dots,m\} \] and \(a_i, f_i\) \((i\in \Lambda_1)\) and \(\tau_j\) \((j\in \Lambda_2=\{1,2,\dots,n\})\) are given functions, here \(a_i\in C(\mathbb R,[0,\infty)),\) \(a_i(t)\not\equiv 0\), \(a_i(t)=a_i(t+T)\), \(t\in\mathbb R\), \(i\in \Lambda_1\), \(T>0\) is a fixed constant; \(f_i\in C(\mathbb R\times[0,\infty)^{m\times n},[0,\infty))\), \(i\in \Lambda_1\); \(f_i (i\in \Lambda_1)\) are \(T\)-periodic functions \(t\); \(\tau_j\in C(R,[0,\infty))\), \(\tau_j(t+T)=\tau_j(t)\), \(t\in\mathbb R\), \(j\in \Lambda_2\). By using multiple fixed-point theorems, the existence of multiple positive periodic solutions of the systems (1) and (2) is derived. Some examples are given to illustrate the main results.
scientific article; zbMATH DE number 5701457
scientific article