Abstract
General nonlocal boundary value problems are considered for systems of impulsive equations with finite and fixed points of impulses. Sufficient conditions are established for the solvability and unique solvability of these problems, among them effective spectral conditions.
Highlights
General nonlocal boundary value problems are considered for systems of impulsive equations with finite and fixed points of impulses
H(x) – (x) ≤ (x) + x s for x ∈ Cs [a, b], Rn; τ, . . . , τm hold, where : Cs([a, b], Rn; τ, . . . , τm ) → Rn and : Cs([a, b], Rn; τ, . . . , τm ) → Rn+ are, respectively, linear continuous and positive homogeneous continuous operators, the pair (P, {Jl}ml= ) satisfies the Opial condition with respect to the pair (, ); α ∈ Car([a, b] × R+, R+) is a function nondecreasing in the second variable, and βl ∈ C(R+, R+) (l =, . . . , m ) and ∈ C(R+, Rn+) are nondecreasing, respectively, functions and vector functions such that lim ρ→+∞ ρ b m
Hold, where P ∈ Car ([a, b] × Rn, Rn×n), Pi ∈ L([a, b], Rn×n) (i =, ), Jil ∈ Rn×n (i =, ; l =, . . . , m ), : Cs([a, b], Rn; τ, . . . , τm ) → Rn and : Cs([a, b], Rn; τ, . . . , τm ) → Rn+ are, respectively, linear continuous and positive homogeneous continuous operators; α ∈ Car([a, b] × R+, R+) is a function nondecreasing in the second variable, and βl ∈ C([a, b], R+) (l =, . . . , m ) and ∈ C(R+, Rn+) are nondecreasing, respectively, functions and vector function such that the condition ( . ) holds
Summary
General nonlocal boundary value problems are considered for systems of impulsive equations with finite and fixed points of impulses. M ), satisfy the Opial condition with respect to the pair ( , ) if: (a) there exist a matrix function ∈ L([a, b], R+n×n) and constant matrices l ∈ Rn×n Has only the trivial solution for every matrix function A ∈ L([a, b], Rn×n) and constant matrices Gl
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