Abstract
In this paper, we study the solvability of various two-point boundary value problems for x ( 4 ) = f ( t , x , x ′ , x ″ , x ‴ ) , t ∈ ( 0 , 1 ) , where f may be defined and continuous on a suitable bounded subset of its domain. Imposing barrier strips type conditions, we give results guaranteeing not only positive solutions, but also monotonic ones and such with suitable curvature.
Highlights
This paper is devoted to the solvability of boundary value problems (BVPs) for the equation x (4) = f (t, x, x 0, x 00, x 000 ), t ∈ (0, 1), (1)
We provide the reader to formulate variants of Theorems 6 and 7 for the rest BVPs (1),(k), k = 4, 9
It is easy to check in this case that if we choose, for example, F2 = r2 − θ, F1 = r2, L1 = r1, L2 = r1 + θ and σ = θ/2, (H1 ) and (H2 ) hold and so each BVP for (30) with one of the boundary conditions (k), k = 2, 9, has a solution in C4 [0, 1] by Theorem 5
Summary
Each solution x (t) ∈ C4 [0, 1] to a BVP for (1)λ with one of the boundary conditions (k), k = 2, 9, satisfies the bounds Let x (t) ∈ C4 [0, 1] be a solution of (1)λ ,(4); the assertion follows for all the rest families of BVPs. By the mean value theorem, for each t ∈ [0, 1) there is a ξ ∈ (t, 1) such that x 0 (1) − x 0 (t) = x 00 (ξ )(1 − t), from where, using Lemma 3, we get (23).
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