Abstract. Shooting methods are employed to obtain solutions of the threepoint boundary value problem for the second order equation, y = f (x, y, y ), y(x 1 ) = y 1 , y(x 3 ) − y(x 2 ) = y 2 , where f : (a, b) × R 2 → R is continuous, a < x 1 < x 2 < x 3 < b, and y 1 , y 2 ∈ R, and conditions are imposed implying that solutions of such problems are unique, when they exist.
We will expand the scope of application of a fixed point theorem due to Krasnosel'skiȋ and Zabreiko to the family of second-order dynamic equations described by u ΔΔ (t) = f (u σ (t)), t ∈ [0,1] ∩ T, with multipoint boundary conditions u(0) = 0, u(σ 2 (1)) = n i=1 α i u(η i), and n i=1 α i ≤ 1 for the purpose of establishing existence results. We will determine sufficient conditions on our function f such that the assumptions of the fixed point theorem are satisfied, which in return gives us the existence of solutions.
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