Dirichlet boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses

Abstract

Abstract The paper deals with the boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses of the form ϕ ( z ′ ( t ) ) ′ = f ( t , z ( t ) , z ′ ( t ) ) for a.e. t ∈ [ 0 , T ] ⊂ R , Δ z ′ ( t ) = M ( z ( t ) , z ′ ( t − ) ) , t = γ ( z ( t ) ) , z ( 0 ) = z ( T ) = 0. $$\begin{array}{} \left(\phi(z'(t))\right)' = f(t,z(t),z'(t))\qquad \text{ for a.e. } t\in [0,T]\subset\mathbb R,\\ \Delta z'(t) = M(z(t),z'(t-)),\qquad t=\gamma (z(t)),\\ z(0) = z(T) = 0. \end{array} $$ Here, T > 0, ϕ : ℝ → ℝ is an increasing homeomorphism, ϕ(ℝ) = ℝ, ϕ(0) = 0, f : [0, T] × ℝ2 → ℝ satisfies Carathéodory conditions, M : ℝ → ℝ is continuous and γ : ℝ → (0, T) is continuous, Δ z′(t) = z′(t+) − z′(t−). Sufficient conditions for the existence of at least one solution to this problem having no pulsation behaviour are provided.