Existence principle for BVPs with state-dependent impulses

The paper provides an existence principle for the Sturm-Liouville boundary value problem with state-dependent impulses {gather} z''(t) = f(t,z(t),z'(t)) for a.e. t ∈ [0,T] ⊂ , \\ z(0) - az'(0) = c_1, z(T) + bz'(T) = c_2, \\ z({τ}{i}+) - z({τ}{i}) = J_i({τ}{i},z({τ}{i})), z'({τ}{i}+) - z'({τ}{i}-) = _i({τ}{i},z({τ}{i})), {gather} where the points τ1, …, τp depend on z through the equations {equation*} {τ}{i} = γ(z({τ}{i})), i = 1,…,p, \ p ∈ . {equation*} Provided a, b ∈ [0,∞), cⱼ ∈, $j = 1,2$, and the data functions f, Jᵢ, ᵢ, i=1,…,p, are bounded, transversality conditions for barriers γᵢ, i = 1,…,p, which yield the solvability of the problem, are delivered. An application to the problem with unbounded data functions is demonstrated.