Green’s Function and Positive Solutions of a Third-Order Equation with Periodic Boundary Conditions

(ird-order differential equations arise in many areas of physics and engineering [1] and describe, for example, deflection of a curved beam having a constant or varying cross section, a three-layered beam, and electromagnetic waves. Boundary value problems for third-order differential equations have been studied by many authors, for example, [2–9] just to name a few. In this paper, we consider a wellknown [10, 11] boundary value problem:


Introduction
ird-order differential equations arise in many areas of physics and engineering [1] and describe, for example, deflection of a curved beam having a constant or varying cross section, a three-layered beam, and electromagnetic waves. Boundary value problems for third-order differential equations have been studied by many authors, for example, [2][3][4][5][6][7][8][9] just to name a few. In this paper, we consider a wellknown [10,11] boundary value problem: We improve the results of [10,11] and obtain positive solutions using the fixed point index. e solutions to (1) and (2) will be sought in the Banach space B � C[0, 2π] endowed with the max-norm. In order to obtain positive solutions, we apply the fixed point theorem of Guo and Lakshmikantham [12] stated in Section 2. Also Denoting is a decreasing function It is a lengthy exercise to verify that x 1 (ρ) is increasing It follows from (8) In fact, g 1 ″ (x 1 ) < 0; thus, By a similar argument, one can show that so that We are in position to state and prove our first lemma.
It is clear that the map T: B ⟶ B defined by is completely continuous. Define Obviously, C is a cone in B. By (34), T: C ⟶ C and u ∈ B is a solution of (1) and (2) We apply the following theorem [12].
Theorem 1. Let B be a Banach space. Assume that T: C r ⟶ C is completely continuous such that Tu ≠ u for u ∈ zC r .
We will restrict our attention to the case. ( If there exists δ > 0 such that f(x) ≥ δx, then for u ∈ C, Similarly, if there exists δ > 0 such that f(x) ≤ cx, then, for u ∈ C, From these inequalities, by eorem 1, we have the following result.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.