Well Ordered Monotone Iterative Technique for Nonlinear Second Order Four Point Dirichlet BVPs

. In this article, we develop a monotone iterative technique (MI-technique) with lower and upper (L-U) solutions for a class of four-point Dirichlet nonlinear boundary value problems (NLBVPs), deﬁned as, where 0 < c 0 < 1, c 1 > 0, 0 < η 0 ≤ η 1 < 1, ψ ( x ) ∈ C 2 [0 , 1], the non linear term F ( x, ψ, ψ (cid:48) ) is continuous function in x , one sided Lipschitz in ψ and Lipschitz in ψ (cid:48) . To show the existence result, we construct Green’s function and iterative sequences for the corresponding linear problem. We use quasilinearization to construct these iterative schemes. We prove maximum principle and establish monotonicity of sequences of lower solution ( l m ( x )) m and upper solution ( u m ( x )) m such that l m ( x ) ≤ u m ( x ), ∀ m ∈ N . Then under certain suﬃcient conditions we prove that these sequences converge uniformly to the solution ψ ( x ) in a speciﬁc region where ∂F∂ψ (cid:54) = 0.


Introduction
In real-life, there are many applications of multi-point (m-point) boundary value problems (BVPs), e.g., suspension bridge. Two-point BVPs conduct small size bridges, in order to conduct large size bridges we need m-point boundary conditions (BCs) [15]. Lazer and McKenna [21] have discussed the existence and multiplicity of periodic solutions of possible mathematical models for the nonlinear behavior of suspension bridge.
The study of m-point BVPs have become a broad area of research due to their wide applications. The m-point linear second-order BVPs was first studied by II'in and Moiseev [18]. In this article, we have focused on the existence of solution of second-order four-point Dirichlet NLBVPs. To study the existence, various methods are introduced such as fixed point (FP) index theory, MI-technique, shooting method, etc., we refer the reader to [7,8,19,20]. There are various techniques to study the following class of four-point Dirichlet NLBVPs, ψ (x) + βq(x)F (x, ψ, ψ ) = 0, 0 < x < 1, (1.1) Verma et al. [28] established MI-technique with L-U solution to study the existence of solution, where β = q(x) = 1 and c 0 = 0. Also article [30], deals with the existence and multiplicity of positive solutions of above NLBVPs (1.1)-(1.2), here ψ is involved in F explicitly and β = q(x) = 1. For this, they have used Krasnoselskii FP theorem and triple FP theorem. By using FP theorem Liu et al. [22], studied the existence of positive solutions for second order problem ψ (x) + a(x)f (ψ(x)) = 0 with BCs (1.2).
MI-technique was first introduced by E. Picard in 1890 [25]. By using MItechnique we can obtain existence results for a large class of BVPs. We study this technique to ensure the existence and approximation of solutions lying in a pair of ordered functions called L-U solutions. This technique also gives a constructive way to find the maximal and minimal solutions corresponding to the L-U solutions [9,10,11,12]. To know more about history and development related to MI-technique one can refer [14,27].
In this article, to study the existence of a solution we have explored an iterative process for a class of four-point Dirichlet NLBVPs with nonlinear source term, defined as follows, 1], the non linear term F (x, ψ, ψ ) is continuous function in x, one sided Lipschitz in ψ and Lipschitz in ψ . The iterative schemes we have constructed in this article are defined in (4.2)-(4.3), where λ is non zero real number.
In [23], by using the FP of strict-set-contractions, Liu et. al. studied the existence of at least one or two positive solutions to the four-point BVPs (1.1). Various NLBVPs have been studied with multi-point BCs by using MI-technique. In 2019 [29], we have discussed the existence of solution for four-point NLBVPs (1.1) with BCs ψ (0) = 0 and ψ(1) = c 0 ψ(η 0 ) + c 1 ψ(η 1 ). We also have discussed the existence of solution of the above NLBVPs (1.1) in [26], for the same BCs where F is independent of ψ . For the existence of solutions on three-point NLBVPs with MI-technique, one can refer [28]. For higher order BVP with four point BCs one can refer the work in [1,2,3].
The work of this paper generalises our earlier work [28] and complement a recent paper [29]. Cherpion in his article [13], stated that the iterative scheme (4.2)-(4.3) do not work for constant λ. Also they stated that due to lack of uniform antimaximum principle it is impossible to develop MI-technique for the case l m (x) ≥ u m (x). This statement is true for our case also. The work in this paper is novel in the sense that even with constant λ we are able to generate monotone sequences. Also, our results are based on simple assumptions, hence it can deal with larger class of nonlinear four point BVPs, e.g., we don't require sign restrictions [27, p. 27]. All these further approves the fact that L-U solution technique related to MI-method is most powerful technique to solve class of nonlinear BVPs [31]. This paper is divided into five sections. In the second section, we discuss some preliminaries. In the third section, we describe Green's function, solution of the corresponding linear problem, and maximum principle which is used to construct monotone L-U solutions. In the fourth section, L-U solutions are defined, some assumptions are considered on the source term F , and we establish the main result on the existence of a solution for λ = 0. The final section verifies the theoretical results numerically.
Proof. For the proof of this lemma we refer [28].

Green's function and maximum principle
In this section, we state two lemmas where we obtain Green's function for the BVPs (2.1), and then we show that under some assumptions the Green's function is non-positive. We also obtain the solution for the BVPs (2.1). Further we prove some important inequalities and establish the maximum principle which are used to construct monotone sequences of L-U solutions. Assume that the following conditions hold; Under the assumed sign restriction on B 1 , B 2 and B 3 , it is easy to deduce that D λ − > 0.
With the help of article [28,29] we can construct the Green's function for the BVPs (3.1), and fix its sign. Proof.
For the proof of this lemma we refer [29, lemma 3.2].
Lemma 5. If ψ(x) be a function given in Lemma 4, δ 2 : [0, 1] → [0, ∞) such that δ 2 (0) = 0, then we have the following: is the derivative of ξ + (x, t) with respect to x, given by Proof. (a.) The result can be easily obtained by using (3.2) and (3.6), Lemma 1, and (P 0 ). Let us first prove that, Consider the case 0 ≤ x ≤ t ≤ η 0 , from (3.2) and (3.6), we symbolize Then, putting in (3.7), we get are non-positive under (P 0 ). By applying Lemma 1(a) it can be proved that, Similarly, for rest of the cases, we can prove the result easily by using (P 0 ) and Lemma 1. Proof of (b) follows similarly.

The nonlinear problem
Here we define lower solution l(x), upper solution u(x), construct iterative sequences with initial iterates of L-U solutions, and prove the existence result. Nagumo conditions are used to show the bound on the derivative of the solution. We have also assumed some conditions on nonlinear term F (x, ψ, ψ ) based on l(x) and u(x). Here the Lipschitz constant δ 2 (x) ≥ 0, with respect to ψ , is a function of x, if we do not assume this then it is not easy to generate the monotonic sequences. Let (l m (x)) m , (u m (x)) m ∈ C 2 ([0, 1]), m ∈ N, are two sequences defined as follows: If m = 0, define l 0 = l and u 0 = u, where l and u are L-U solutions of Equation (1.1), respectively. If λ ∈ I 0 ∪ (−∞, 0) then assume that the following hypotheses hold.

The existence theorem
In this subsection, we prove monotonic behavior of L-U solution in well ordered case and existence result. This section is divided into two subsections, in first we see the results for λ ∈ I 0 and in other we see the results for λ ∈ (−∞, 0). such that ψ ∈ [l(x), u(x)], where l(x) and u(x) are L-U solutions of (1.1), respectively.
where l(x) and u(x) are L-U solution of (1.1), respectively.

Proof.
We can prove this result similarly as in Lemma 6.
Proof. Using Propositions 2 and 3, we obtain that the sequences (l m (x)) m and (u m (x)) m such that, From (4.7), we can conclude that (l m (x)) m is monotonically increasing and (u m (x)) m is monotonically decreasing. Also (l m (x)) m and (u m (x)) m are bounded. Therefore by monotone convergence theorem, (l m (x)) m converges to its supremum, say l * (x); and (u m (x)) m converges to its infimum, say u * (x). Hence we can write,

λ ∈ (−∞, 0)
We skip the proof of this section, as all the proofs are similar to the proof of the above subsection, for λ ∈ I 0 .

Conclusions
The work of this paper generalises and complement our earlier work [28,29].
Since assumptions are simple we can deal with larger class of nonlinear four point boundary value problems (see [27, p. 27]) and since method is simple iterative, it is user friendly and can be used to develop software package to compute solutions of multi point BVPs.