Sharp conditions for rapid nonlinear oscillations (Q1961259)

The authors consider the equation \[ \ddot x + g(x)=f(t,x,\dot x) \tag{1} \] where \(g:{\mathbb{R}}\to {\mathbb{R}}\) and \(f:[a,b]\times {\mathbb{R}}\times {\mathbb{R}}\to{\mathbb{R}}\) are \(C^1\)-functions, \(xg(x)>0\) for \(|x|\) large. \(g(x)\) is superlinear and associated boundary value problems are referred to as superlinear also. One says that equation (1) possesses the \(S\)-property in a finite interval \((a,b)\) if: 1. all solutions extend to \([a,b]\)~; 2. for any \(N\) there exists \(\Delta\) such that the number of zeros in \([a,b]\) of any solution \(x(t)\) to the initial value problem (1) and \(x(a)=x_0\)~, \(\dot x(a)=\dot x_0\), \(\|x_0\|=|x_0|+|\dot x_0|\to \infty\) is greater than \(N\), if \(\|x_0\|>\Delta\). The interest in the \(S\)-property is that equation (1) possessing the \(S\)-property has infinite number of solutions satisfying the Sturm-Liouville boundary conditions \[ a_1x(0) + a_2\dot x(0)=A, \quad b_1x(T) + b_2\dot x(T)=B, \quad a_1^2+a_2^2>0, \quad b_1^2+b_2^2>0. \] Main results concern the study of interdependence of the growth rate of \(g(x)\) and the growth rate of \(f(t,x,\dot x)\) with respect to \(\dot x\) at infinity with the \(S\)-property. Two theorems are proved: Theorem 1: Suppose that all solutions to equation (1) extend to the interval \([a,b]\) and there exists a continuous function \(\omega:(0,+\infty)\to (0,+\infty)\) and numbers \(\Delta >0\), \(k,c>0\) such that 1. \(|g(x)|\geq \omega(x)+k^2|x|^\gamma\) for \(|x|>\Delta\); 2. \(|f(t,x,\dot x)|\leq \omega(x)+cy^{2\gamma/(\gamma+1)}\) for \(t\in [a,b]\) and \(x^2+y^2>\Delta^2\); 3. \(k^2>c_*\), where \(c_*={{c}\over{1+\gamma}} [{{2\gamma c}\over{(\gamma+1)^2}})^\gamma.\) Then equation (1) possesses the \(S\)-property. Theorem 2: Suppose that all the solutions to equation (1) extend to the interval \([a,b]\), and the following conditions are fulfilled: 1. \(xg(x)>0\) for \(x\) large; 2. \(|g(x)|\leq \omega(x)+>k^2|x|^\gamma\) for large \(|x|\), \(k\) is a number; 3. \(f(t,x,y)\geq\omega(x)+cy^{2\gamma/\gamma+1)}\) for \(t\in[a,b]\) and large \(x^2+y^2\), \(x>0\), or\(f(t,x,y)\leq-\omega(x)-cy^{2\gamma/\gamma+1)}\) for \(t\in[a,b]\) and large \(x^2+y^2\)~, \(x<0\), \(c>0\); 4. \(k^2\leq c_*= {c\over{\gamma+1}} \left[ {{2\gamma c}\over{(\gamma+1)^2}}\right]^\gamma\). Then the \(S\)-property is not fulfilled for equation (1).