Nonlinear ordinary boundary value problems under a combined effect of periodic and attractive nonlinearities
The author studies the following boundary value problems: \[ -u''(t)- u(t)+ f(u(t))+ g(u(t))= h(t),\quad t\in [0,\pi],\quad u(0)= u(\pi)= 0,\tag{1} \] and \[ -u''(t)- au(t)+ (a- 1)v(t)+ f(v(t))= p(t),\tag{2} \] \[ -v''(t)- av(t)+ (a- 1)u(t)+ g(u(t))= q(t),\quad t\in [0,\pi), \] \[ u(0)= u(\pi)= v(0)= v(\pi)= 0, \] where \(f: \mathbb{R}\to\mathbb{R}\) is a continuous \(T\)-periodic function and \(\int^T_0 f(s) ds= 0\); \(g: \mathbb{R}\to\mathbb{R}\) is a continuous bounded function and \(g(u)u\leq 0\), \(\forall u\in\mathbb{R}\) or \(g(u)u\geq 0\), \(\forall u\in\mathbb{R}\); \(h\in C[0,\pi]\); \(a\) is a real constant; \(p(t)\), \(q(t)\) are functions for which there exist the functions \(u,v\in C^2[0,\pi]\) so that \[ -u''(t)- au(t)+ (a- 1)v(t)= p(t), \] \[ -v''(t)- av(t)+ (a- 1)u(t)= q(t),\quad t\in [0,\pi], \] \[ u(0)= u(\pi)= v(0)= v(\pi)= 0. \] It is proved the existence of solutions to these problems, using the alternative method.
nonlinear ordinary boundary value problems
scientific article; zbMATH DE number 1429700
scientific article