Existence and iteration of monotone positive solutions for third-order nonlocal BVPs involving integral conditions

This paper is concerned with the existence of monotone positive solution for the following third-order nonlocal boundary value problem u′′′ (t)+f (t, u (t) , u′ (t)) = 0, 0 < t < 1; u (0) = 0, au′ (0) − bu′′ (0) = α[u], cu′ (1) + du′′ (1) = β[u], where f ∈ C([0, 1] × R+ × R+, R+), α[u] = ∫ 1 0 u(t)dA(t) and β[u] = ∫ 1 0 u(t)dB(t) are linear functionals on C[0, 1] given by Riemann-Stieltjes integrals. By applying monotone iterative techniques, we not only obtain the existence of monotone positive solution but also establish an iterative scheme for approximating the solution. An example is also included to illustrate the main results.


Introduction
Third-order differential equation arises in a variety of different areas of applied mathematics and physics, e.g., 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].
BVPs with Stieltjes integral boundary condition (BC for short) have been considered recently as both multipoint and integral type BCs are treated in a single framework.For more comments on Stieltjes integral BC and its importance, we refer the reader to the papers by Webb and Infante [2,3,4] and their other related works.In recent years, thirdorder nonlocal BVPs have received much attention from many authors, see, for example [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] and the references therein.For fourthorder or higher-order nonlocal BVPs, one can refer to [3,4,21,22].In particular, it should be pointed out that Webb and Infante in [2], gave a unified approach for studying the existence of multiple positive solutions of second-order BVPs subject to various nonlocal BCs.In [3], they extended their method to cover equations of order N with any number up to N of nonlocal BCs in a single theory.
Recently, iterative methods have been successfully employed to prove the existence of positive solutions of nonlinear BVPs for ordinary differential equations, see [23,24,25,26,27] and the references therein.It is worth mentioning that, Sun et al. [24] obtained the existence of monotone positive solutions for third-order three-point BVPs, the main tools used were monotone iterative techniques.Inspired by the above mentioned excellent works, the aim of this paper is to investigate the existence and iteration of monotone positive solution for the following BVP where are linear functionals on C[0, 1] given by Riemann-Stieltjes integrals and a, b, c, d are nonnegative constants with ρ := ac + ad + bc > 0. By a positive solution of BVP (1.1), we understand a solution u(t) which is positive on t ∈ (0, 1) and satisfies BVP (1.1).By applying monotone iterative techniques, we construct a successive iterative scheme whose starting point is a zero function, which is very useful and feasible for computational purpose.An example is also included to illustrate the main results.

Preliminary lemmas
In this section, the ideas and the method we will adopt, which have been widely used, are due to Webb and Infante in [2,3].
In our case, the existence of positive solutions of nonlocal BVP (1.1) with two nonlocal boundary terms α[u], β[u], can be studied, via a perturbed Hammerstein integral equation of the type (2.1) Here γ(t), δ(t) are linearly independent and given by a+2b ; G(t, s) is the Green's function for the corresponding problem with local terms when α[u] and β[u] are identically 0.
We first make the following hypotheses on the Green's function: (H1) The kernel G is measurable, non-negative, and for every τ (H3) A, B are functions of bounded variation, and where In the remainder of this paper, we always assume that 0 As shown in Theorem 2.3 in [3], if u is a fixed point of T in (2.1), then u is a fixed point of S, which is now given by in our case.The kernel G S is the Green's function corresponding to the BVP (1.1).
The proof is same to the Theorem 2.4 in [2], so omitted.
Moreover, we easily know that for a function Φ 2 , i.e., for s ∈ [0, 1], We will use the classical Banach space and let c 0 be same as in Lemma 2.2, then define Then it is to verify that P and K are cones in E. Lemma 2.4 T : K → K and S : P → K.
Lemma 2.5 T and S have the same fixed points (in K).

Main results
Now we apply monotone iterative techniques to seek solution of BVP (1.1) as fixed point of the integral operator S.
and there exists a constant r > 0 such that If we construct an iterative sequence which is a monotone positive solution of the BVP (1.1) and satisfies which together with the condition (3.1) and Lemma 2.2 and (2.8) implies that Hence, we have shown that S : , which is a monotone positive solution of the BVP (1.1) and satisfies In what follows, we prove that {v n } ∞ n=0 is monotone by induction.Firstly, by v 0 = 0 and S : P → K, we easily know v 1 − v 0 ∈ K, which shows that v 0 v 1 .Next, we assume that v k−1 v k .Then, in view of Lemma 2.2 and (3.1), we have At the same time, by Lemma , we also have It follows from (3.2) and (3.3) that v k+1 (t) − v k (t) ∈ K, which shows that v k v k+1 .Thus, we have shown that v n v n+1 , n = 0, 1, 2 . . . .Since {v n } ∞ n=0 is relatively compact and monotone, there exists a v * ∈ K r such that v n − v * → 0 (n → ∞), which together with the continuity of S and the fact that v n+1 = Sv n implies that v * = Sv * .Moreover, in view of f (t, 0, 0) ≡ 0 for t ∈ (0, 1), we know that the zero function is not a solution of BVP (1.1).Thus, v * ∞ > 0. So, it follows from v * ∈ K r that 0 < v * (t) ≤ r for t ∈ (0, 1], 0 ≤ (v * ) ′ (t) ≤ r for t ∈ [0, 1].   and δ(t) = t 2 2 .By simple calculation shows that

An example
Moreover, the iterative scheme is