Existence of Positive Solution for a Third-Order BVP with Advanced Arguments and Stieltjes Integral Boundary Conditions

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves or gravity-driven flows, and so on [1]. Recently, third-order boundary value problems (BVPs for short) with integral boundary conditions, which cover thirdorder multipoint BVPs as special cases, have attracted much attention from many authors; see [2–6] and the references therein. In particular, in 2012, by using a fixed point theorem due to Avery and Peterson [7], Jankowski [4] established the existence of at least three nonnegative solutions to the following BVP:


Introduction
Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves or gravity-driven flows, and so on [1].
The situation with a signed measure Λ was first discussed in [8,9] for second-order differential equations; it was also discussed in [10,11] for second-order impulsive differential equations.For some other related results, one can refer to [12][13][14].
Throughout, we assume that the following conditions are fulfilled: For convenience, we denote Obviously, ,   ≥ 0. In the remainder of this paper, we always assume that  < Δ.
which together with implies that If  ∈ (, 1), then So, which together with (23) and (28) implies that It follows from ( 25), (30), and (34) that min we have This shows that  :  → .Similarly, we can prove that  :  → .Lemma 5.The operators  and  have the same fixed points in .
Proof.Suppose that  ∈  is a fixed point of .Then, which shows that So, which indicates that  is a fixed point of .Suppose that  ∈  is a fixed point of .Then, which shows that So, which indicates that  is a fixed point of .Lemma 6. ,  :  →  is completely continuous.
Next, we show that  is compact.Let  ⊂  be a bounded set.Then, there exists  1 > 0 such that ‖‖ ≤  1 for any  ∈ .Since Λ is a function of bounded variation, there exists Then, for any  ∈ , which shows that () is uniformly bounded.On the other hand, for any  > 0, since (, ) is uniformly continuous on [0, 1] × [0, 1], there exists  1 () > 0 such that for any which indicates that  is continuous.Therefore,  :  →  is completely continuous.Similarly, we can prove that  :  →  is also completely continuous.
Then, for any  ∈  ∩ Ω 2 , which together with (57) and (58) implies that This shows that On the other hand, since  1  ∞ < 1, there exists  4 > 0 so that  1 ( ∞ +  4 ) < 1. (67) By the definition of  ∞ , we may choose  4 > 0 such that which implies that Therefore, it follows from Theorem 7 that the BVP (74) has at least one positive solution.