Multiple periodic solutions for asymptotically linear Duffing equations with resonance
The author investigates multiple periodic solutions of an asymptotically linear Duffing equation of the form \[ x''(t) + f(t,x(t)) = 0, {\text{with}}\,\, x(1) - x(0) = 0 = x'(1) - x'(0) \tag{*} \] where \(f\) satisfies \[ \int^1_0 f(t, +\infty)\,dt > 0 > \int^1_0 f(t, -\infty)\,dt. \tag{\(H_1\)} \] The main result of the paper is: Assume that \(\lambda\) and \(\lambda_0 \in L^\infty[0, 1]\) and \(f\in C^1([0, 1]\times \mathbb R,\mathbb R)\) satisfies the following two conditions \[ 0 \leq \frac{f(t,x)}{x} < \lambda(t)\text{ for } |x| \geq r > 0,\tag{\(H_2\)} \] where \(r\) is a constant, \(i(\lambda) = 1\), \(\nu(\lambda) = 0;\) \[ f(t,0) = 0, \quad \lambda_0(t) =f'(t,0) \text{ and } 1 \not\in [i(\lambda_0) + \nu(\lambda_0)].\tag{\(H_3\)} \] Then (*) has at least one nontrivial solution. Moreover, if we assume \[ \nu(\lambda_0) = 0,\quad i(\lambda_0) = 3,\tag{\(H_4\)} \] then (*) has at least two nontrivial solutions. Index theory and Morse theory are used in the proof of the results.
asymptotically linear Duffing equation with resonance
scientific article; zbMATH DE number 5871222