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

Abstract

Abstract We are concerned with Dirichlet problems of impulsive differential equations − u ″ ( x ) − λ u ( x ) + g ( x , u ( x ) ) + ∑ j = 1 p I j ( u ( x ) ) δ ( x − y j ) = 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 ∈ L 2 ( 0 , π ) f\in {L}^{2}\left(0,\pi ) , I j ∈ C ( R , R ) {I}_{j}\in C\left({\mathbb{R}},{\mathbb{R}}) , j = 1 , 2 , … , p j=1,2,\ldots ,p , p ∈ N p\in {\mathbb{N}} , the nonlinearity g : [ 0 , π ] × R → R g:\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 < y 1 < y 2 < ⋯ < y p < π 0\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.