A faster iterative method for solving nonlinear third-order BVPs based on Green’s function

In this article, we propose an iterative method, called the GA iterative method, to approximate the ﬁxed points of generalized α -nonexpansive mappings in uniformly convex Banach spaces. Further, we obtain some convergence results of the new iterative method. Also, we provide a nontrivial example of a generalized α -nonexpansive mapping and with the example, we carry out a numeral experiment to show that our new iterative algorithm is more eﬃcient than some existing iterative methods. Again, we present an interesting strategy based on the GA iterative method to solve nonlinear third-order boundary value problems (BVPs). For this, we derive a sequence named the GA–Green iterative method and show that the sequence converges strongly to the ﬁxed point of an integral operator. Finally, the approximation of the solution for a nonlinear integrodiﬀerential equation via our new iterative method is considered. We present some illustrative examples to validate our main results in the application sections of this article. Our results are a generalization and an extension of several prominent results of many well-known authors in the literature.


Introduction and main results
In this paper, N denotes the set of all natural numbers and R the set of real numbers.We assume that U is a nonempty subset of a Banach space V.A fixed point of a mapping G : U → U is an element f ∈ U such that Gf = f .The fixed-point set of G is denoted by F(G) = {f ∈ U : Gf = f }.The mapping G is called a contraction if there exists a constant β ∈ [0, 1) such that Gf -Gh ≤ β fh , for all f , h ∈ U and it is said to be nonexpansive if Gf -Gh ≤ fh , for all f , h ∈ U.It is said to be quasinonexpansive if F(G) = ∅ and Gf -≤ f -, for all f ∈ U and ∈ F(G).It is known that if U is a closed, bounded, convex subset of a uniformly convex Banach space V, then F(G) = ∅ for a nonexpansive mapping [5,9].The class of nonexpansive map-pings has been studied deeply because of their diverse applications.In recent years, several authors have considered the extensions and generalizations of nonexpansive mappings.
In [33], Suzuki considered a fascinating generalization of nonexpansive mappings and studied some convergence and the existence results of fixed points for such mappings.Definition 1.1 A mapping G : U → U is said to satisfy the condition (C) if for all f , h ∈ U, In 2011, Aoyama and Kohsaka [4] introduced the class of α-nonexpansive mappings in Banach spaces and studied the fixed-point theory of such mappings.Definition 1.2 A mapping G : U → U is said to be α-nonexpansive if there exists an α ∈ [0, 1) such that for all f , h ∈ U, (1.2) With α = 0, then it is not difficult to see that every nonexpansive mapping is an αnonexpansive mapping.
In [28], Pant and Shukla introduced an important class of mappings that properly contains the class of mappings satisfying condition (C) and further proved some fixed-point theorems.Such mappings are known as generalized α-nonexpansive mappings.Definition 1. 3 A mapping G : U → U is said to be generalized α-nonexpansive if there exists an α ∈ [0, 1) such that for each f , h ∈ U, This class of mappings has been studied by several authors in recent years, see [8,24,36,37].
The theory of fixed points is one of the most powerful and fascinating tools of modern mathematical analysis and has many applications in diverse fields such as optimization theory, physics, biology, economics, mathematical engineering, game theory, chemistry, approximation theory, and many others.
The Mann iterative method [22] is constructed as follows: where {p m } is a sequence in (0, 1).
In 1974, Ishikawa [11] constructed a two-step iterative method for approximating fixed points of nonexpansive mappings as follows: where {q m } and {p m } are sequences in (0, 1).
In 2000, Noor [23] proposed a three-step iterative method that is an extension of the Mann and Ishikawa iterative methods as follows: where {q m }, {p m }, and {y m } are sequences in (0, 1).In 2007, Agarwal et al. [2] modified the Ishikawa iteration method, called the S iteration as follows: where {q m } and {p m } are sequences in (0, 1).The authors showed that this iterative algorithm converges faster than the Mann iterative method for contraction mappings.
In 2013, Khan [15] introduced the Picard-Mann iterative method as follows: where {p m } is a sequence in (0, 1).Abbas and Nazir [1], in 2014 defined an iterative method that converges faster than the S iterative method as follows: where {q m }, {p m }, and {y m } are sequences in (0, 1).
In 2016, Thakur et el.[34] developed a new iterative method as follows: where {q m } and {p m } are sequences in (0, 1).They proved that (1.10) converges faster than all of Mann, Ishikawa, Noor, S, and Abbas and Nazir iterative methods.
In this study, we propose the following iterative method, called the GA iterative method: where {q m } and {p m } are sequences in (0, 1).We prove both weak and strong convergence results concerning the convergence of the GA iterative scheme (1.12) to fixed points of generalized α-nonexpansive mappings.Furthermore, we show numerically via a nontrivial example that the GA iterative method (1.12) enjoys a better rate of convergence than the iterative methods (1.4)- (1.11) for generalized α-nonexpansive mappings.Again, we present an interesting pattern based on the GA iterative method to solve nonlinear third-oder BVPs.In view of this, we derive a sequence named GA-Green iteration and show that the sequence converges strongly to the fixed point of an integral operator.Furthermore, we approximate the solution of a nonlinear integrodifferential equation via our new iterative method.We present some illustrative examples to validate our main results in the application sections.

Lemmas and definition
The following definitions and lemmas will be very useful in proving our main results.Definition 2.1 Opial's condition is said to be satisfied by a Banach space V, if for any sequence {f m } in V that is weakly convergent to f ∈ V implies Let U stand for a nonempty, closed, and convex subset of a Banach space V. Let {f m } be a bounded sequence in V.For f ∈ V, we put The asymptotic radius of {f m } relative to U is defined by The asymptotic center of {f m } relative to U is given as It is well known that in a uniformly convex Banach space, A(U , {f m }) consists of exactly one point.
Let U stand for a nonempty, closed, and convex subset of a Banach space V.A mapping G : U → U is said to be demiclosed with respect to f ∈ V, if for each sequence {f m } that converges weakly to f ∈ U and {Gf m } converges strongly to h implies that Gf = h.Lemma 2.3 ([38]) Let {θ m } be a nonnegative real sequence satisfying the following inequality: where σ m ∈ (0, 1) for all m ∈ N and ∞ m=0 σ m = ∞, then lim m→∞ θ m = 0.

Proof of the main results
In this section, we will establish and prove our main results.

Weak and strong convergence theorems
In this section, we establish and prove some convergence theorems of the GA iterative method (1.12) for generalized α-nonexpansive mappings.Further, we present a nontrivial example of generalized α-nonexpansive mappings to analyze the convergence rate of the GA iterative method (1.12) with some existing iterative methods.
Theorem 3.1 Let G : U → U be a generalized α-nonexpansive mapping defined on a nonempty, closed, and convex subset U of a Banach space V. Let {f m } be a sequence defined by the GA iterative method (1.12), then lim m→∞ f mexists for all ∈ F(G).
Theorem 3.3 Let G, U, V, and {f m } be defined as in Theorem 3.1 such that F(G) = ∅.Suppose that V satisfies Opial's property, then {f m } converges weakly to a member of F(G).
Proof Since G is a generalized α-nonexpansive mapping with F(G) = ∅, then by Theorem 3.1 and Theorem 3.2, we have that lim m→∞ f m -exists and lim m→∞ Gf mf m = 0. Next, we will prove that {f m } have exactly one weakly subsequential limit in F(G).Let k and be two weak subsequential limits of {f m j } and {f m k }, respectively.By Theorem 3.2, the demiclosedness of (I -G) at 0 implies that (I -G)k.Hence, Gk = k and similarly, G = .Now, we have to show the uniqueness.Suppose k = , then from Opial's property, one has which is a contradiction, so k = .Hence, {f m } converges weakly to a fixed point of G. Theorem 3.4 Let G, U, V, and {f m } be defined as in Theorem 3.
Proof The necessity is trivial and will be neglected.Next, we will prove the convex case.If for any ∈ F(G), lim inf m→∞ d(f m , F(G)) = 0.By Theorem 3.1, we have that lim m→∞ f mexists for each ∈ F(G) and it follows that Hence, inf{ f m 0 -: ∈ F(G)} < ℘ 2 .Therefore, there exists ∈ F(G) such that For c, m ≥ m 0 , we have This implies that {f m } is a Cauchy sequence in U. From the completeness of U, we have lim m→∞ f m = q for some q ∈ U. Also, lim m→∞ d(f m , F(G)) = 0 shows that q ∈ F(G).
Theorem 3.5 Let G, U, V, and {f m } be defined as in Theorem 3.
On taking j → ∞, G = , i.e., ∈ F(G).From Theorem 3.1, lim m→∞ f m -exists for every ∈ F(G) and so the sequence {f m } converges strongly to .(3.17) By condition (I) and (5.11), we have i.e., lim m→∞ g(d(f m , F(G))) = 0. Since g is a nondecreasing function such g(0) = 0, g(s) > 0, for each s ∈ (0, ∞), then we obtain By Theorem 3.4, we know that the sequence {f m } converges strongly to a point of F(G).Now, we perform a numerical experiment using a generalized α-nonexpansive mapping that is not a Suzuki generalized nonexpansive mapping.
Example 3.7 Let V = R and U = [3,6].Let the mapping G : U → U be defined by Now, we show that G is a generalized α-nonexpansive mapping for α = 1 3 .To show that, the following cases are considered: From all the above-illustrated cases, we know that G is a generalized α-nonexpansive mapping for α = 1 3 .Now, we show that G does not fulfill condition (C).If we take f = 39 10 and h = 29  7 , then On the other hand, Thus, G does not satisfy condition (C).
Next, using the above example and taking p m = q m = y m = n+2 n+3 for m ∈ N and f 0 = 4, then we obtain Tables 1 and 2, and Figs. 1 and 2.
Clearly, from the above tables and figures, all the iterative methods converge to = 3 and the GA iterative method converges faster to = 3 than all the others.

An application to nonlinear third-order BVPs
A BVP for a given differential equation involves finding a solution of the given differential equation subject to a given set of boundary conditions.BVPs emanate in many branches of physics as any physical differential equation will have them.Problems involving wave equations are in most cases expressed as BVPs.Several important BVPs include Sturm-Liouville problems.
The third-order differential equations occur in diverse areas of physics and applied mathematics, for example, a three-layer beam, in the deflection of a curved beam having a constant or varying cross section, gravity-driven flows or electromagnetic waves.Thirdorder BVPs were discussed in many papers in recent years, for instance, see [6,14,21,35] and references therein.
Solutions of BVPs can sufficiently be approximated by some efficient numerical methods.Some of these are the finite-difference method, the standard 5-point formula, the standard analytic method, and the iterative method.
In recent years, several iterative methods based on Green's function have been developed for solving second-and third-order nonlinear BVPs.Some of these are Picard-Green, Krasnosel'skii-Green, Mann-Green, Ishikawa-Green, and Khan-Green iterative methods, e.g., see [12,13,16,18] and the references therein.
Very recently, Khuri and Louhichi [17] presented a fascinating approach that is based on embedding Green's function into the Ishikawa iterative method for solutions of nonlinear third-order BVPs.Motivated by the above results, in this section, we present a strategy based on the GA iterative method (1.12) to solve a nonlinear third-order BVP in the form of a Green's function.

Brief demonstration of Green's functions
Consider the following general linear third-order BVP, where a ≤ ψ ≤ b and subject to the boundary conditions: where c = a or b.The Green's function is defined to be the solution for the following equation where δ is the Kronecker Delta that is subject to It is worth noting that for operators that are not self-adjoint, the right-hand side of (4.3) will be replaced by -δ(ψs).For ψ = s, we solve Li[G(ψ, s)] = 0 and obtain where η 1 , η 2 , η 3 are linearly independent solutions of Li[η] = 0 and the constants are derived through the following properties: (V 1 ) G satisfies the associated homogeneous boundary conditions (V 4 ) At ψ = s, G has a jump discontinuity:

GA-Green iterative method
Applying the Green's function to the GA iterative method (1.12), the following differential equation will be considered: where Li[ρ] and No[ρ] are linear and nonlinear operators in ρ, respectively, and ϕ(ψ, ρ) is a function in ρ that could be linear or nonlinear.
We now define the following linear integral operator in terms of Green's function as follows: where G is the Green's function that is corresponding to the linear differential operator Li [ρ].Observe that has a fixed point if and only if ρ is a solution of (4.8).
From (4.9), we have the following: Now, by applying the GA iterative method (1.11), we obtain where {q m } and {p m } are sequences in [0, 1].Then, for all m ∈ N, this leads to (4.12) Remark 4.1 Our new iterative method (4.12) is independent of all Picard-Green, Mann-Green, Ishikawa-Green, and Khan-Green iterative methods that are already existing in the literature.

Convergence result
In this section, we prove the convergence theorem of the proposed iterative method (4.12).Without loss of generality, we will consider the convergence analysis of our method for the following nonlinear BVP: Solving the associated homogeneous equation η = 0 implies (4.14) The unknowns e 1 , e 2 , e 3 , d 1 , d 2 , and d 3 can be obtained using the properties (V 1 )-(V 4 ).
After finding the unknowns, we then have the following Green's function: Thus, the GA-Green iterative method (4.12) now has the form: where the initial iterate f 0 fulfilled the corresponding equation f = 0 and the boundary conditions f 0 (1) = Q, f 0 (1) = P, and f 0 (2) = W . Next, we define the operator ϒ G : Then, (4.15) reduces to the following form On the other hand, by using the method of integration by parts three times to evaluate 2 1 G(ψ, s)f (s) ds in (4.16) and since 2 1 Our next target is to prove that under some mild conditions on the function ϕ, the integral operator ϒ G is a contraction on the Banach space C 2 ([1, 2]) with respect to the norm Theorem 4.2 Suppose that the function ϕ, which appears in the definition of the operator ϒ G , fulfills the following Lipschitz condition: where Θ 1 , Θ 2 , and Θ 3 are positive constants satisfying Then, ϒ G is a contraction on the Banach space (C 2 ([1, 2]), • C 2 ) and the sequence {f m } defined by the GA-Green iterative method (4.17) converges strongly to the fixed point of ).Then, by (4.19), we obtain Thus, ϒ G is a contraction.Next, we prove that the sequence {f m } defined by the GA-Green iterative method (4.17) converges strongly to the fixed point of ϒ G .Since ϒ G is a contraction, then by the Banach contraction principle, we know that ϒ G has a unique fixed point in (C 2 ([1, 2]), • C 2 ), say .We will now show that f m → as m → ∞.Using (4.17), we have , we have Also, using (4.22), we obtain Since 0 < β < 1 and p m , q m ∈ [0, 1], then it follows that Thus, using (4.24), (4.23) becomes By (4.25), we obtain Finally, from (4.26), we obtain By induction, we have Since 0 < β < 1, then we have that {f m } converges strongly to .

An application to nonlinear Volterra delay integrodifferential equations
In this section, we consider the following delay nonlinear Volterra integrodifferential equation: Assume that the following assumptions hold: Apparently, from condition (A 1 ), the IVP (5.1)-(5.2) is equivalent to the following equations: In [20], Kucche and Shikhare studied the existence and uniqueness results concerning the solution of the problem (5.1)-(5.2) as follows.
Theorem 5.1 Suppose the conditions (A 1 )-(A 4 ) are fulfilled, then the problem (5.1)-( 5.2) has a unique solution and the equation (5.1) is generalized Ulam-Hyers-Rassias stable with respect to the function ψ.
Our main aim in this section is to solve the delay nonlinear Volterra integrodifferential equation (5.1)-(5.2) via a new efficient iterative method (1.12).In view of this, we state and prove the following theorem.Proof Note that B = C([-z, ], R) endowed with the Chebyshev norm • C is a Banach space.Let {f m } be the sequence defined by the iterative method (1.12) for the operator T : B → B defined by Let denote the fixed point of T. Next, we show that f m → as m → ∞.It is clear that for s ∈ [-z, 0], f m → as m → ∞.Now, for s ∈ I, we have and and and and Using (5.4), (5.5), (5.6), (5.7), and (5.8), we obtain Using assumption (A 3 ), we obtain Observe that (5.10) takes the form Thus, all the assumptions of Theorem 3.1 are fulfilled.Hence, lim m→∞ f m -= 0.
Next, we give an example to validate our main result obtained in Theorem 5.2 as follows.

Conclusions
In this paper, we have introduced a five-step iterative method (1.12) for approximating the fixed points of generalized α-nonexpansive mappings.Furthermore, we studied the weak and strong convergence theorems of the new iterative method.Again, a numerical experiment that was carried out showed that our new iterative method has a better speed of convergence than several existing iterative methods.We also utilized our iterative method to find the solutions of nonlinear third-order BVPs based on Green's function and a delay nonlinear Volterra integrodifferential equation.A nontrivial example that authenticates the assumptions used in our main result of Theorem 5.2 is provided.The delay nonlinear Volterra integrodifferential equation (5.1)-(5.2) studied in this manuscript properly includes the nonlinear delay differential equations considered in [3,7,10,[24][25][26][27]36].Hence, our results are generalizations and extensions of the corresponding results in [3,7,10,[24][25][26][27]36] and several others in the existing literature.

Theorem 3 . 2 6 )
Let G, U, V, and {f m } be defined as in Theorem 3.1.Then, F(G) = ∅ if and only if the sequence {f m } is bounded and lim m→∞ Gf mf m = 0. Proof By Theorem 3.1, we have shown that {f m } is bounded and lim m→∞ f m -exists for any ∈ F(G).Since a generalized α-nonexpansive mapping with a nonempty fixed point set F(G) = ∅ is a quasinonexpansive mapping, we obtain lim sup m→∞ Gf m -≤ lim sup m→∞ f m -= j .(3.7)

Theorem 3 . 6
Let G, U, V, and {f m } be defined as in Theorem 3.1 such that F(G) = ∅.If condition (I) is satisfied by G, then {f m } converges strongly to a fixed point of G. Proof By Theorem 3.2, we have that lim m→∞ Gf mf m = 0.

Lemma 2.6 (
[30]) Suppose V is a uniformly convex Banach space and {ι m } is any sequence satisfying 0 < f ≤ ι m ≤ h < 1 for all m ≥ 1. Suppose {f m } and {h m } are any sequences of V we have shown in Theorem 3.2 that lim m→∞ Gf mf m = 0. Due to the compactness of C, one can have a subsequence {f m j } of {f m } such that lim m→∞ f m j → ∈ U.By Proposition 2.7, we have

Table 1
Comparison of the GA iterative method with Mann, S, Picard-Mann and Picard-Ishikawa iterative methods

Table 2
Comparison of speed of convergence of the GA iterative method with Ishikawa, Noor, Abbas, and Thakur iterative methods