Boundedness of solutions of nonlinear differential delay equations with positive Green functions and the Aizerman-Myshkis problem
Here, the author investigates the scalar equation \[ x^{(n)}(t)+\sum\limits_{k=0}^{n-1}\int_{0}^{\eta}x^{(k)}(t-\tau) dr_k(\tau)= N(D)F(t,x_t)\quad t>0, \] where \(r_k\), \(k=0,\ldots,n-1\), are bounded nondecreasing functions defined on a finite segment \([0,\eta], D\equiv d/dt\), \(N(\lambda)=1+\cdots+b_m\lambda^m\), \(b_k= \text{const}\), \(k=1,\ldots,m<n\), is a polynomial and \(x_t(\theta)=x(t+\theta)\) with \(\theta\in [-\eta,0]\), \(F\in \mathbb{R}_+\times \mathbb{C}[-\eta,0]\). He establishes explicit conditions for solution boundedness and absolute stability of equation (1) with the positive Green function (the fundamental solution, the impulse function).
scientific article; zbMATH DE number 1787984
scientific article