Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses

Abstract

We are concerned with Dirichlet problems of impulsive differential equations −u″(x)−λu(x)+g(x,u(x))+∑j=1pIj(u(x))δ(x−yj)=f(x)for a.e.x∈(0,π),u(0)=u(π)=0,\left\{\begin{array}{l}-{u}^{^{\prime\prime} }\left(x)-\lambda u\left(x)+g\left(x,u\left(x))+\mathop{\displaystyle \sum }\limits_{j=1}^{p}{I}_{j}\left(u\left(x))\delta \left(x-{y}_{j})=f\left(x)\hspace{1em}\hspace{0.1em}\text{for a.e.}\hspace{0.1em}\hspace{0.33em}x\in \left(0,\pi ),\\ u\left(0)=u\left(\pi )=0,\\ \end{array}\right. where λ\lambda is a parameter and runs near 1, f∈L2(0,π)f\in {L}^{2}\left(0,\pi ), Ij∈C(R,R){I}_{j}\in C\left({\mathbb{R}},{\mathbb{R}}), j=1,2,…,pj=1,2,\ldots ,p, p∈Np\in {\mathbb{N}}, the nonlinearity g:[0,π]×R→Rg:\left[0,\pi ]\times {\mathbb{R}}\to {\mathbb{R}} satisfies the Carathéodory condition, δ=δ(x)\delta =\delta \left(x) denote the Dirac delta impulses concentrated at 0, which are applied at given points 0<y1<y2<⋯<yp<π0\lt {y}_{1}\lt {y}_{2}\hspace{0.33em}\lt \cdots \lt {y}_{p}\lt \pi . We show the existence and multiplicity of solutions to the aforementioned problem for λ\lambda in a neighborhood of 1 by using degree theory and bifurcation theory.

Keywords

• near resonance
• impulsive differential equations
• a priori bounds
• bifurcation from infinity
• 34b15
• 34b37
• 34c23