Existence Theorems for Second Order Multi-Point Boundary Value Problems

We are interested in the existence of nontrivial solutions for the second order nonlinear differential equation (E): y′′(t) = f ( t, y(t) ) = 0, 0 < t < 1 subject to multipoint boundary conditions at t = 1 and either Dirichlet or Neumann conditions at t = 0. Assume that f(t, y) satisfies |f(t, y)| ≤ k(t)|y| + h(t) for non-negative functions k, h ∈ L(0, 1) for all (t, y) ∈ (0, 1) × R and f(t, 0) 6≡ 0 for t ∈ (0, 1). We show without any additional assumption on h(t) that if ‖k‖1 is sufficiently small where ‖ · ‖1 denotes the norm of L(0, 1) then there exists at least one non-trivial solution for such boundary value problems. Our results reduce to that of Sun and Liu [11] and Sun [10] for the three point problem with Neumann boundary condition at t = 0.

In [10], Sun considered a similar boundary value problem also with Neumann boundary condition at t = 0, i.e. equation (1.1) subject to and proved Theorem B (Sun [10]) Suppose that f (t, y) satisfies the same assumptions as in then the boundary value problem (1.1), (1.5) has a nontrivial solution.
We need to show that the set is equicontinuous, we observe This proves that A ′ j s are completely continuous for j = 1, 2, 3.

Remark 2
The boundary conditions involving the derivative of a solution at some interior points in general give rise to kernels associated with the operators A j in (2.1) which are discontinuous in two variables t, s.However, they are shown above to be completely continuous operators.
In [10], [11], the authors used the more customary integral operator I(t) defined by instead of the Green's operator G j [y](t) given in (2.1).
Writing I(t) = I[y](t), G j (t) = G j [y](t) for short, we can relate G j (t) with I(t) as follows: Using (2.10), (2.11), (2.12), we can rewrite the operator equations in (2.1) as follows: Results in [10], [11] can then be proved using the operator equation ( 2 We now prove a result generalizing both Theorems A and B for the boundary value problem (BVP1).
Theorem 2 Under the same assumptions as in Theorem 1, if k(t) satisfies for then the (BVP1) has at least one non-trivial solution where I[k](t) and I ′ [k](t) are defined like (2.9) by Proof.We use the integral representation (2.13) for the operator A 1 .Since f (t, 0) ≡ 0 in [0, 1], we also have Γ 1 (h) > 0 by (1.2).Using (3.5) we define the positive constant r 1 > 0 by To apply the Schauder Fixed Point Theorem, we suppose that there exists y ∈ ∂Ω r 1 = y ∈ Ω r 1 : y = r 1 such that A 1 y = λy for some λ > 1.Now apply (1.2), (3.5) to the integral representation given by (2.13), and obtain by (3.7) which contradicts the assumption that λ > 1.Now Schauder ′ s Fixed point theroem shows that there exists y ∈ Ω r 1 such A 1 y = y.Since f (t, 0) ≡ 0, so y cannot be the identically zero solution.This complets of the proof.
Theorem 4 Under the same assumptions as in Theorem 3, if k(t) satisfies as defined by (2.9) and can prove the following results for (BVP2), (BVP3).
Theorem 5 Under the same assumptions of Theorem 3, if k(t) satisfies then the boundary value problem (BVP2) has at least one non-trivial solution.
Theorem 6 Under the same assumptions of Theorem 5, if k(t) satisfies then the boundary value problem (BVP3) has at least one non-trivial solution.
The proofs of Theorems 3, 4, 5, 6 are similar to those given for Theorem 1 and 2 and we shall not repeat them here.

Discussion
We illustrate our results with examples in three point boundary value problems and begin with two examples discussed in [10], [11].