Abstract
By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for nonlinear higher-order differential equation boundary value problems with sign-changing Green’s function. The theorems obtained are very general and complement previous known results.
Highlights
Boundary value problems (BVPs for short) for nonlinear differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory
The study of multipoint BVPs for second-order differential equations was initiated by Bicadze and Samarskiı [1] and later continued by II’in and Moiseev [2, 3] and Gupta [4]
Many results on the existence of solutions for multipoint BVPs have been obtained; the methods used therein mainly depend on the fixed point theorems, degree theory, upper and lower techniques, and monotone iteration
Summary
Boundary value problems (BVPs for short) for nonlinear differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory. By applying the fixed point theorems on cones, the authors of papers [5–7] established the existence and multiplicity of positive solutions for the nth-order three-point BVP: u(n) (t) + a (t) f (t, u (t)) = 0, t ∈ (0, 1) , u (0) = u (0) = ⋅ ⋅ ⋅ = u(n−2) (0) = 0, u (1) = αu (η) , (1). By using the cone theory and the Banach contraction mapping principle, the author [26] established the existence and uniqueness for singular third-order three-point boundary value problems. The purpose of this paper is to investigate the existence and uniqueness of solution of the following higher-order differential equation boundary value problem: u(n) (t) + f (t, u (t) , u (t) , . The methods used in this paper are motivated by [26], and the arguments are based upon the cone theory and the Banach contraction mapping principle
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