A continuation theorem for periodic boundary value problems with oscillatory nonlinearities
The authors investigate whether the differential equation \[ x''+ F(t, x, x')= 0,\quad x(0)- x(T)= x'(0)- x'(T)= 0\tag{1} \] has a solution, where \(F: \mathbb{R}^ 3\to \mathbb{R}\) is continuous and \(T\)-periodic in the first variable. There is an extensive literature about this problem, and in this paper, a generalization of previous results is obtained. First, the authors consider a linear (unbounded) Fredholm operator \(L\) of index 0 from a Banach space \(X\) into a Banach space \(Z\) and an \(L\)-completely continuous operator \(N: X\times [0, 1]\to Z\). Then \(\Sigma_ \lambda:= \{x\in \text{dom } L\times [0, 1]: Lx= N(x,\lambda)\}\), \(\lambda\in [0, 1]\}\), is the set of solutions of \(Lx= N(x,\lambda)\), and it is assumed that \(\Sigma_ 0\neq \emptyset\). It is shown that also \(\Sigma_ 1\neq \emptyset\) if certain conditions are satisfied. In the application to (1) one replaces \(F(t,x,x')\) by \(f(t,x,x',\lambda)\), where \(f(t,x,x',0)= f_ 0(x,x')\) and \(f(t,x,x',1)= F(t,x,x')\). For the autonomous problem at \(\lambda=0\), a topological degree argument gives \(\Sigma_ 1\neq \emptyset\). As an application, the Duffing equation is considered, i.e., (2) \(F(t,x,x')= g(x)- p(t,x,x')\), where \(g\) and \(p\) are continuous functions, and \(p\) is additionally bounded. Setting \(G(x):= \int^ x_ 0 g(s) ds\), \(h(c)< 0\) for \(c> 0\) such that \(G(h(c))= G(c)\), and \(\tau_ g(c)= \sqrt 2 \int^ c_{h(c)} {du\over \sqrt{G(c)-G(u)}}\), the following theorem is proved: Assume \(\lim_{| x|\to +\infty} g(x)\text{sign}(x)= +\infty\) and \[ \forall c_ 1> 0\;\exists c_ 2> 0: AB> 0 \& | \sqrt{G(B)}- \sqrt{G(A)}|< c_ 1\Rightarrow | B- A|< c_ 2, \] and suppose that there is a sequence \(M_ n\to +\infty\) such that \(\lim_{n\to+\infty} {T\over \tau_ g(M_ n)}= \rho\in [0,+\infty]\), with \(\rho\) not a positive integer. Then (1), (2) has at least one solution. Finally, some special cases are considered, where the above conditions are replaced by stronger conditions which might be easier to verify.
scientific article; zbMATH DE number 762090
scientific article