Eigenvalues of a Class of Singular Boundary Value Problems of Impulsive Differential Equations in Banach Spaces

Abstract EN

This paper is devoted to investigating the eigenvalue problems of a class of nonlinear impulsive singular boundary value problem in Banach spaces: μ x ′′ + f ( t , x ) = 0 , t ∈ ( 0,1 ) , t ≠ t i ; Δ x | t = t i = α i x ( t i - 0 ) , i = 1,2 , … , k ; a x ( 0 ) - b x ′ ( 0 ) = θ ; c x ( 1 ) + d x ′ ( 1 ) = θ , where θ denotes the zero element of Banach space, Δ x | t = t i = x ( t i + 0 ) - x ( t i - 0 ) , α i > - 1 , a , b , c , d ∈ R + , γ = a c + a d + b c > 0 , μ is a parameter, and f ( t , x ) may be singular at t = 0 , 1 and x = θ .

The arguments are mainly based upon the theory of fixed point index, measure of noncompactness, and the special cone, which is constructed to overcome the singularity.