Computation of solution of integral equations via fixed point results

Manar A. Alqudah; Chanchal Garodia; Izhar Uddin; Juan J. Nieto

doi:10.1515/dema-2022-0172

Article Open Access

Computation of solution of integral equations via fixed point results

Manar A. Alqudah , Chanchal Garodia , Izhar Uddin and Juan J. Nieto

Published/Copyright: November 11, 2022

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

The motive of this article is to study a modified iteration scheme for monotone nonexpansive mappings in the class of uniformly convex Banach space and establish some convergence results. We obtain weak and strong convergence results. In addition, we present a nontrivial numerical example to show the convergence of our iteration scheme. To demonstrate the utility of our scheme, we discuss the solution of nonlinear integral equations as an application, which is again supported by a nontrivial example.

Keywords: monotone nonexpansive mappings; fixed point; iteration process; strong and weak convergence; nonlinear integral equation

MSC 2010: 47H09; 47H10

1 Introduction

Fixed point theory is one of the fast growing topics in nonlinear functional analysis. It has got many applications in finding out the solutions of ordinary differential equations, partial differential equations, variational inequalities, and zeros of monotone operators. The iterative methods are employed to obtain solutions to initial and boundary value problems, split feasibility problem, image recovery problems, optimization problems, image restoration problems, image processing problems, etc. Various engineering problems can be transformed into integral equations under suitable conditions like the beam equation and the hanging code equation [1,2, 3,4]. The Banach contraction principle is one of the prime results of fixed point theory. The early findings in fixed point theory revolve around the generalization of the Banach contraction principle [5]. Credit for obtaining the first basic existence results for nonexpansive mappings goes to Browder [6], Göhde [7], and Kirk [8]. Over the last few years, fixed point results in partially ordered sets are the new area of interest among the mathematics community, and numerous results have been obtained in this direction. Ran and Reurings [9] gave the generalization of the Banach contraction principle [5] to ordered metric spaces in 2004. In 2005, Nieto and Rodriguez-Löpez [10] further extended a few more results of fixed points in partially ordered metric spaces by using the same approach and studied the existence of solutions to differential equations.

Recently, Bachar and Khamsi [11], Abdullatif et al. [12], and Song et al. [13] presented some more convergence and existence results for the class of monotone nonexpansive mappings. The weak convergence of the Mann iteration for a monotone nonexpansive mapping was shown by Dehaish and Khamsi [14]. Following this, in 2016, Song et al. [13] used some mild different conditions to prove the weak convergence of the Mann iteration scheme for a monotone nonexpansive mapping in Banach space. Furthermore, in 2018, Uddin et al. [15] proved some convergence results for monotone nonexpansive mappings for the class of CAT(0) spaces using the Mann iteration scheme and utilized the same to study the solution of an integral equation.

Owing to the importance of iteration processes, many new iteration schemes have been obtained in the last few years, and the prime focus of researchers is to obtain iteration schemes that converge at a faster rate than the existing schemes. Some of the most utilized and basic iteration processes are Mann iteration [16], Ishikawa iteration [17], and Halpern iteration [18]. For more iteration, one can see [19,20,21, 22,23].

Recently, Thakur et al. [24] presented a new modified iteration process that can be used for finding the fixed point of nonexpansive mappings. Let J be a nonempty closed convex subset of a uniformly convex Banach space P , then the sequence { c t } is generated iteratively by c 1 ∈ J and

(1.1) a t = ( 1 − μ t ) c t + μ t G c t , b t = ( 1 − δ t ) a t + δ t G a t , c t + 1 = ( 1 − κ t ) G a t + κ t G b t , t ∈ N ,

where { μ t } , { δ t } , and { κ t } are real sequences in ( 0 , 1 ) . Furthermore, they showed that the new iteration process is faster than the above-mentioned iteration processes.

The aim of this article includes the study of the convergence behavior of the Thakur iteration process (1.1) for monotone nonexpansive mappings in uniformly convex Banach space and extending the class of mapping. Also, to support our main results, we provide a nontrivial example and as an application, we study the solution of nonlinear integral equations.

2 Preliminaries

In this section, first, we list a few basic results and definitions that will be used in obtaining our main results.

Recall that a Banach space P satisfies Opial’s condition if for any sequence { c t } in P with c t ⇀ c ( ⇀ denotes weak convergence), we have limsup t → ∞ ‖ c t − c ‖ < limsup t → ∞ ‖ c t − b ‖ for all b ∈ P with b ≠ c .

Hilbert spaces and all l p spaces ( 1 < p < ∞ ) are Banach spaces satisfying Opial’s condition, whereas L p [ 0 , 2 π ] with 1 < p ≠ 2 fail to satisfy Opial’s property.

A mapping G : J → P is demiclosed at b ∈ P if for each sequence { c t } in J and each c ∈ P , c t ⇀ c and G c t → b imply that c ∈ J and G c = b .

Let J be a nonempty closed convex subset of a Banach space P , and let { c t } be a bounded sequence in P . For c ∈ P write:

r ( c , { c t } ) = limsup t → ∞ ‖ c − c t ‖ .

The asymptotic radius of { c t } relative to J is given as follows:

r ( J , { c t } ) = inf { r ( c , { c t } ) : c ∈ J } ,

and the asymptotic center A ( J , { c t } ) of { c t } is defined as follows:

A ( J , { c t } ) = { c ∈ J : r ( c , { c t } ) = r ( J , { c t } ) } .

It is well known that A ( J , { c t } ) consists of exactly one point in a uniformly convex Banach space.

Next, we list the following lemma, which will be useful in furnishing our results.

Lemma 2.3

[25] Let P be a uniformly convex Banach space and { m t } be any sequence such that 0 < p ≤ m t ≤ q < 1 for some p , q ∈ R and for all t ≥ 1 . Let { c t } and { b t } be any two sequences of P such that limsup t → ∞ ‖ c t ‖ ≤ k , limsup t → ∞ ‖ c t ‖ ≤ k , and limsup t → ∞ ‖ m t c t + ( 1 − m t ) b t ‖ = k for some k ≥ 0 . Then, lim t → ∞ ‖ c t − b t ‖ = 0 .

Next, we have the following basic definition.

Let P be a Banach space endowed with the partial order “ ≼ .” Also, we will assume that the order intervals are closed and convex. An order interval is any of the subsets [ z , → ) = { c ∈ P ; z ≼ c } or ( ← , z ] = { c ∈ P : c ≼ z } for any z ∈ P . So, an order interval [ c , b ] for all c , b ∈ P is given by [ c , b ] = { a ∈ P : c ≼ b ≼ a } . It is easy to see that the order interval [ c , b ] is closed and convex.

Definition 2.4

Let J be a nonempty subset of a Banach space P . A mapping G : J → J is known as:

monotone if G c ≼ G b for all c , b ∈ J with c ≼ b
monotone nonexpansive if G is monotone and
∣ ∣ G c − G b ∣ ∣ ≤ ∣ ∣ c − b ∣ ∣
for all c , b ∈ J with c ≼ b .

3 Some convergence results

Throughout this section, J will denote a nonempty, closed, and convex subset of a uniformly convex Banach space ( P , ≼ ) ; G : J → J is a monotone nonexpansive mapping; { c t } is a sequence defined by (1.1), and fix c 1 ∈ J with c 1 ≼ G c 1 . Here, first, we have the following crucial lemma.

Lemma 3.1

We have the following results:

c t ≼ a t ≼ G c t ≼ G a t and a t ≼ b t ≼ G a t ≼ c t + 1 ≼ G b t for any t ≥ 1 ,
c t ≼ e , for all t ≥ 1 provided { c t } weakly converges to a point e ∈ J .

Proof

(i) We will use induction on t to prove the result.

If x 1 , x 2 ∈ J such that x 1 ≼ x 2 , then x 1 ≼ μ x 1 + ( 1 − μ ) x 2 ≼ x 2 holds for any μ ∈ [ 0 , 1 ] . Since we have assumed that order intervals are convex, this is true.

Thus, we are left to obtain that c t ≼ G c t for any t ≥ 1 .

We have already assumed that c 1 ≼ G c 1 and hence inequality holds good for t = 1 . Assume that c t ≼ G c t for t ≥ 2 .

From (1.1), we have

(3.1) c t ≼ ( 1 − μ t ) c t + μ t G c t = a t ≼ ( 1 − μ t ) G c t + μ t G c t = G c t .

Since G is monotone, we obtain c t ≼ a t ≼ G c t ≼ G a t . Again, from ( 1.1 ) , we have

(3.2) a t ≼ ( 1 − δ t ) a t + δ t G a t = b t ≼ ( 1 − δ t ) G a t + δ t G a t = G a t ,

which gives that a t ≼ b t ≼ G a t ≼ G b t . Similarly, we have

(3.3) G a t ≼ ( 1 − κ t ) G a t + κ t G b t = c t + 1 ≼ ( 1 − κ t ) G b t + κ t G b t = G b t .

Thus, we have a t ≼ b t ≼ G a t ≼ c t + 1 ≼ G b t and using the monotonicity of G , we obtain c t + 1 ≼ G c t + 1 . So, the inequality is true for any t ≥ 1 by induction.

(ii) This can be directly obtained from the proof of Lemma 3.1 in [14].□

Lemma 3.2

If F ( G ) ≠ ∅ with e ∈ F ( G ) and e ≼ c 1 , then the following results hold:

lim t → ∞ ∣ ∣ c t − e ∣ ∣ exists.
lim t → ∞ ∣ ∣ G c t − c t ∣ ∣ = 0 .

Proof

It is given that e ≼ c 1 . By using part (i) of Lemma 3.1 for t = 1 , we have c 1 ≼ a 1 ≼ G c 1 ≼ G a 1 and a 1 ≼ b 1 ≼ G a 1 ≼ c 2 ≼ G b 1 . Clearly, we have e ≼ c 1 ≼ c 2 , which yields e ≼ c 2 . Now, using mathematical induction on t , we can easily show that e ≼ c t for any t ≥ 1 .

Again on using Lemma 3.1(i), we have

(3.4) e ≼ c t ≼ a t ≼ G c t ≼ G a t and e ≼ a t ≼ b t ≼ G a t ≼ c t + 1 ≼ G b t ,

for any t ≥ 1 . Now, using (3.4) and monotonicity of G , we have

(3.5) ∣ ∣ a t − e ∣ ∣ = ∣ ∣ ( 1 − μ t ) c t + μ t G c t − e ∣ ∣ ≤ ( 1 − μ t ) ∣ ∣ c t − e ∣ ∣ + μ t ∣ ∣ G c t − e ∣ ∣ ≤ ( 1 − μ t ) ∣ ∣ c t − e ∣ ∣ + μ t ∣ ∣ c t − e ∣ ∣ ≤ ∣ ∣ c t − e ∣ ∣

and

(3.6) ∣ ∣ b t − e ∣ ∣ = ∣ ∣ ( 1 − δ t ) a t + δ t G a t − e ∣ ∣ ≤ ( 1 − δ t ) ∣ ∣ a t − e ∣ ∣ + δ t ∣ ∣ G a t − e ∣ ∣ ≤ ( 1 − δ t ) ∣ ∣ a t − e ∣ ∣ + δ t ∣ ∣ a t − e ∣ ∣ ≤ ∣ ∣ a t − e ∣ ∣ ≤ ∣ ∣ c t − e ∣ ∣ .

On using (3.5) and (3.6), we obtain

(3.7) ∣ ∣ c t + 1 − e ∣ ∣ = ∣ ∣ ( 1 − κ t ) G a t + κ t G b t − e ∣ ∣ ≤ ( 1 − κ t ) ∣ ∣ G a t − e ∣ ∣ + κ t ∣ ∣ G b t − e ∣ ∣ ≤ ( 1 − κ t ) ∣ ∣ a t − e ∣ ∣ + κ t ∣ ∣ b t − e ∣ ∣ ≤ ∣ ∣ c t − e ∣ ∣ .

Thus, we have ∣ ∣ c t + 1 − e ∣ ∣ ≤ ∣ ∣ c t − e ∣ ∣ , which holds for any t ≥ 1 . Therefore, { ∣ ∣ c t − e ∣ ∣ } is a nonincreasing and bounded below sequence of real numbers. Hence, lim t → ∞ ∣ ∣ c t − e ∣ ∣ exists.

Next, we prove ( i i ) . Let

(3.8) lim t → ∞ ∣ ∣ c t − e ∣ ∣ = x .

Then, from (3.5) and (3.6), we have

(3.9) limsup t → ∞ ∣ ∣ a t − e ∣ ∣ ≤ x

and

(3.10) limsup t → ∞ ∣ ∣ b t − e ∣ ∣ ≤ x .

Also, on using (3.4) and monotonicity of G , we have

(3.11) limsup t → ∞ ∣ ∣ G a t − e ∣ ∣ ≤ limsup t → ∞ ∣ ∣ a t − e ∣ ∣ ≤ x ,

(3.12) limsup t → ∞ ∣ ∣ G b t − e ∣ ∣ ≤ limsup t → ∞ ∣ ∣ b t − e ∣ ∣ ≤ x ,

and

(3.13) limsup t → ∞ ∣ ∣ G c t − e ∣ ∣ ≤ limsup t → ∞ ∣ ∣ c t − e ∣ ∣ ≤ x .

Now, we have

x = lim t → ∞ ∣ ∣ c t + 1 − e ∣ ∣ = lim t → ∞ ∣ ∣ ( 1 − κ t ) G a t + κ t G b t − e ∣ ∣ ,

which on using (3.11), (3.12), and Lemma 2.3 gives

(3.14) lim t → ∞ ∣ ∣ G a t − G b t ∣ ∣ = 0 .

Consider

∣ ∣ c t + 1 − e ∣ ∣ = ∣ ∣ ( 1 − κ t ) G a t + κ t G b t − e ∣ ∣ ≤ ( 1 − κ t ) ∣ ∣ G a t − e ∣ ∣ + κ t ∣ ∣ G b t − e ∣ ∣ ≤ ( 1 − κ t ) ∣ ∣ G a t − e ∣ ∣ + κ t ∣ ∣ G b t − G a t ∣ ∣ + κ t ∣ ∣ G a t − e ∣ ∣ = ∣ ∣ G a t − e ∣ ∣ + κ t ∣ ∣ G b t − G a t ∣ ∣ ,

which on using (3.8) and (3.14) yields

(3.15) x ≤ liminf t → ∞ ∣ ∣ G a t − e ∣ ∣ .

From (3.11) and (3.15), we obtain

x ≤ liminf t → ∞ ∣ ∣ G a t − e ∣ ∣ ≤ limsup t → ∞ ∣ ∣ G a t − e ∣ ∣ ≤ limsup t → ∞ ∣ ∣ a t − e ∣ ∣ ≤ x ,

which gives

(3.16) lim t → ∞ ∣ ∣ a t − e ∣ ∣ = x .

Now, (3.8), (3.13), (3.16), and Lemma 2.3 yield that lim t → ∞ ∣ ∣ G c t − c t ∣ ∣ = 0 .□

Lemma 3.3

The conditions c t ⇀ c and lim t → ∞ ∣ ∣ G c t − c t ∣ ∣ = 0 imply that c is a fixed point of G and c ∈ J . In other words, we can say that, I − G is demiclosed at zero.

Proof

It is given that c t ⇀ c , so by using Lemma 3.1 we obtain c t ≼ c for any t ≥ 1 . Then, it follows from the nonexpansiveness of G and lim t → ∞ ∣ ∣ G c t − c t ∣ ∣ = 0 that

∣ ∣ G c − c t ∣ ∣ ≤ ∣ ∣ G c − G c t ∣ ∣ + ∣ ∣ G c t − c t ∣ ∣ limsup t → ∞ ∣ ∣ G c − c t ∣ ∣ ≤ limsup t → ∞ [ ∣ ∣ G c − G c t ∣ ∣ + ∣ ∣ G c t − c t ∣ ∣ ] ≤ limsup t → ∞ ∣ ∣ c − c t ∣ ∣ = r ( c , c t ) .

Thus, we obtain G c = c by uniqueness of asymptotic center.□

Next, we obtain the weak convergence result.

Theorem 3.4

If P satisfies Opial’s condition and F ( G ) ≠ ∅ with e ∈ F ( T ) such that e ≼ c 1 , then { c t } converges weakly to a fixed point of G .

Proof

Since e ∈ F ( G ) such that e ≼ c 1 . Then, from Lemma 3.2, lim t → ∞ ‖ c t − e ‖ exists. We will show that { c t } has a unique weak subsequential limit in F ( G ) , which we will prove the weak convergence of the iteration process (1.1) to a fixed point of G . Consider { c t j } and { c t k } be any two subsequences of { c t } converging weakly to u and v , respectively. From Lemma 3.2, we obtain lim t → ∞ ‖ G c t − c t ‖ = 0 , whereas Lemma 3.3 gives that I − G is demiclosed at zero. Thus, u , v ∈ F ( G ) .

Now, we obtain the uniqueness. lim t → ∞ ‖ c t − u ‖ and lim t → ∞ ‖ c t − v ‖ exist as u , v ∈ F ( G ) . Let if possible u ≠ v . Then, from Opial’s condition, we have

lim t → ∞ ‖ c t − u ‖ = lim j → ∞ ‖ c t j − u ‖ < lim j → ∞ ‖ c t j − v ‖ = lim t → ∞ ‖ c t − v ‖ = lim k → ∞ ‖ c t k − v ‖ < lim k → ∞ ‖ c t k − u ‖ = lim t → ∞ ‖ c t − u ‖ ,

which gives us a contradiction, so u = v . Hence, { c t } converges weakly to a fixed point of G .□

Now, we obtain some strong convergence results.

Theorem 3.5

If F ( G ) ≠ ∅ , then { c t } converges strongly to a fixed point of G if and only if liminf t → ∞ d ( c t , F ( G ) ) = 0 .

Proof

It is trivial that liminf t → ∞ d ( c t , F ( G ) ) = 0 if the sequence { c t } converges to a point c ∈ F ( G ) .

For converse part, let liminf t → ∞ d ( c t , F ( G ) ) = 0 . On using Lemma 3.2(i), we obtain

∣ ∣ c t + 1 − e ∣ ∣ ≤ ∣ ∣ c t − e ∣ ∣ for any e ∈ F ( T ) such that e ≼ c 1

so we have

(3.17) d ( c t + 1 , F ( G ) ) ≤ d ( c t , F ( G ) ) .

Thus, { d ( c t , F ( G ) ) } is a decreasing sequence and bounded below by zero, therefore, lim t → ∞ d ( c t , F ( G ) ) exists. Since, liminf t → ∞ d ( c t , F ( G ) ) = 0 , we obtain lim t → ∞ d ( c t , F ( G ) ) = 0 .

Now, we prove that { c t } is a Cauchy sequence in J . Let α > 0 be arbitrarily chosen. As liminf t → ∞ d ( c t , F ( G ) ) = 0 , there exists t 0 such that for all t ≥ t 0 , we obtain

d ( c t , F ( T ) ) < α 4 .

In particular,

inf { ‖ c t 0 − e ‖ : e ∈ F ( G ) } < α 4 ,

so there must exist a q ∈ F ( G ) such that

‖ c t 0 − q ‖ < α 2 .

Thus, for m , t ≥ t 0 , we have

‖ c t + m − c t ‖ ≤ ‖ c t + m − q ‖ + ‖ c t − q ‖ < 2 ‖ c t 0 − q ‖ < 2 α 2 = α ,

which shows that { c t } is a Cauchy sequence. Since J is a closed subset of a Banach space P , { c t } must converge in J . So, lim t → ∞ c t = q for some q ∈ J .

Now, by using Lemma 3.2(ii) and the fact that G is a monotone nonexpansive mapping, we obtain lim t → ∞ ∣ ∣ G c t − c t ∣ ∣ = 0 . Also, it follows from the proof of Lemma 3.1 in [14] that c t ≼ q for any t ≥ 1 . Thus, we have

∣ ∣ q − G q ∣ ∣ ≤ ∣ ∣ q − c t ∣ ∣ + ∣ ∣ c t − G c t ∣ ∣ + ∣ ∣ G c t − G q ∣ ∣ ≤ ∣ ∣ q − c t ∣ ∣ + ∣ ∣ c t − G ( c t ) ∣ ∣ + ∣ ∣ c t − q ∣ ∣ → 0 as t → ∞

and hence q = G q . Thus, q ∈ F ( G ) . This proves our result.

A mapping G : J → J satisfies Condition (A) ([26]) if there exists a nondecreasing function ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) such that ϕ ( 0 ) = 0 and ϕ ( r ) > 0 for all r ∈ ( 0 , ∞ ) with ‖ c − G c ‖ ≥ ϕ ( d ( c , F ( G ) ) ) for all c ∈ J , where d ( c , F ( G ) ) = inf { ‖ c − e ‖ : e ∈ F ( G ) } .□

Now, with the help of Condition (A), we furnish the following strong convergence result.

Theorem 3.6

If G satisfies Condition (A) and F ( G ) ≠ ∅ , then { c t } converges strongly to a fixed point of G.

Proof

It follows from 3.17 that lim t → ∞ d ( c t , F ( G ) ) exists.

Also, on using Lemma 3.2, we obtain lim t → ∞ ‖ c t − G c t ‖ = 0 .

It results from Condition (A) that

lim t → ∞ ϕ ( d ( c t , F ( G ) ) ) ≤ lim t → ∞ ‖ c t − G c t ‖ = 0 ,

which gives lim t → ∞ ϕ ( d ( c t , F ( G ) ) ) = 0 . As ϕ is a nondecreasing function satisfying ϕ ( 0 ) = 0 and ϕ ( r ) > 0 for all r ∈ ( 0 , ∞ ) , thus lim t → ∞ d ( c t , F ( G ) ) = 0 .□

By using Theorem 3.5, we obtain the desired result that the sequence { c t } converges strongly to a point of F ( G ) .

4 Numerical example

In this section, first we are going to present an example of a monotone nonexpansive mapping that is not a nonexpansive mapping, and then we are going to show the convergence of the iteration scheme to the fixed point.

Let P = [ 0 , + ∞ ) with the usual norm

∣ ∣ c − b ∣ ∣ = ∣ c − b ∣ , b , c ∈ P

and J = [ 0 , 1 ] . Now, consider the order relation c ≼ b as

c ≤ b if c , b ∈ 0 , 1 6 or c ≤ b if c , b ∈ 1 6 , + ∞ .

Let G : J → J be defined by

G ( c ) = c + 4 6 c ∈ 0 , 1 6 c + 5 6 c ∈ 1 6 , 1

for all c ∈ J .

Clearly, G is not continuous at c = 1 6 , so it is not a nonexpansive mapping and c = 1 is the fixed point of G .

Also, it can be easily verified that under the above ordering, G is a monotone nonexpansive mapping.

Now, we will show the convergence of iteration scheme and examine the influence of parameters μ t , δ t , κ t , and initial value. We have used Mathematica for writing all the codes. We will consider the three cases.

Case (I). In this case, we will show the convergence of our iteration scheme for three different sets of parameters with the same initial value. We take the following set of parameters:

μ t = t t + 1 , δ t = t t + 5 , κ t = 1 2 t + 1 for all t ∈ N and c 1 = 0.0025 .
μ t = 2 t 5 t + 2 , δ t = 1 t + 5 , κ t = t t + 9 for all t ∈ N and c 1 = 0.0025 .
μ t = 1 t + 5 , δ t = 1 3 t + 7 , κ t = t + 1 5 t + 1 for all t ∈ N and c 1 = 0.0025 .

We obtain Table 1 and Figure 1 for the initial value 0.0025.

Table 1

Values of iteration for different sets of parameters

Iteration number	Parameter set 1	Parameter set 2	Parameter set 3
1	0.0025	0.0025	0.0025
3	0.99326099	0.98438014	0.98730177
5	0.99998191	0.99979802	0.99982873
7	0.99999997	0.99999749	0.99999747
9	1.00000000	0.99999997	0.99999996
10	1.00000000	1.00000000	0.99999999
11	1.00000000	1.00000000	1.00000000

Figure 1

Graph corresponding to Table 1.

Case (II). In this case, we will show the convergence for three different initial values with the same set of parameters. Set μ t = t t + 1 , δ t = 2 t 5 t + 2 , and κ t = t + 67 t + 68 for all t ∈ N . We obtain Table 2 and Figure 2.

Table 2

Values of iteration for different initial values

Iteration number	When c 1 = 0.12	When c 1 = 0.52	When c 1 = 0.75
1	0.12	0.52	0.75
2	0.93042424	0.96436364	0.98143939
3	0.99626758	0.99808827	0.99900431
4	0.99983485	0.99991541	0.99995594
5	0.99999359	0.99999671	0.99999829
6	0.99999977	0.99999988	0.99999994
7	0.99999999	1.00000000	1.00000000
8	1.00000000	1.00000000	1.00000000

Figure 2

Graph corresponding to Table 2.

Case (III). In this case, we will consider three different sets of parameters with different initial values. We take the following set of parameters:

μ t = t t + 1 , δ t = t t + 5 , κ t = 1 2 t + 1 for all t ∈ N and c 1 = 0.1 .
μ t = 2 t 5 t + 2 , δ t = 1 t + 5 , κ t = t t + 9 for all t ∈ N and c 1 = 0.2 .
μ t = 1 t + 5 , δ t = 1 3 t + 7 , κ t = t + 1 5 t + 1 for all t ∈ N and c 1 = 0.5 .

We obtain Table 3 and Figure 3.

Table 3

Values of iterations in respect of different set of parameters

Iteration number	Parameter set 1	Parameter set 2	Parameter set 3
1	0.1	0.2	0.5
2	0.91312882	0.89982363	0.9476626
3	0.99425028	0.98820274	0.99422459
4	0.99968309	0.99864823	0.99933871
5	0.99998457	0.99984745	0.9999221
6	0.99999931	0.99998295	0.99999061
7	0.99999997	0.99999811	0.99999885
8	1.00000000	0.99999979	0.99999986
9	1.00000000	0.99999998	0.99999998
10	1.00000000	1.00000000	1.00000000

Figure 3

Graph corresponding to Table 3.

From the above three cases, it is clear that our iteration process (1.1) not only converges to the fixed point of G but also shows that the iteration process is stable with respect to the parameters μ t , δ t , and κ t .

5 Application

Integral equations are of paramount importance in many areas of mathematics, engineering, and economics since numerous problems in these subjects can be converted into integral equations. In the last few years, numerous fixed point results have been obtained to arrive at the solution of different classes of integral equations [27,28, 29,30,31, 32,33,34, 35,36].

This section is devoted to the approximation of solutions of nonlinear integral equations with the help of our results. In this manner, this section depicts the utility of our theoretical results.

Consider the following nonlinear integral equation:

(5.1) u ( q ) = h ( q ) + λ ∫ 0 1 k ( q , s ) f ( s , u ( s ) ) d s ,

where q ∈ I = [ 0 , 1 ] and λ ≥ 0 .

Theorem 5.1

Let I = [ 0 , 1 ] and P = C [ 0 , 1 ] be the space of continuous functions equipped with supremum norm defined as

‖ u − v ‖ = sup q ∈ I ∣ u ( q ) − v ( q ) ∣

for u , v ∈ P . Define ordered relation “ ≼ ” as, for all u , v ∈ P , u ≼ v iff u ( q ) ≤ v ( q ) for all q ∈ [ 0 , 1 ] . Let J be a compact subset of P and assume that the following conditions are true:

h : I → R is continuous.
f : I × J → J is continuous and there exists a constant L ≥ 0 such that for all u , v ∈ J
∣ f ( q , u ) − f ( q , v ) ∣ ≤ L ∣ u ( q ) − v ( q ) ∣ .
k : [ 0 , 1 ] × J → R is continuous such that k ( q , u ) ≥ 0 and ∫ 0 1 k ( q , s ) d s ≤ F for all ( q , u ) ∈ I × J .
λ F L ≤ 1 .
G : J → J is a mapping defined by
G u ( q ) = h ( q ) + λ ∫ 0 1 k ( q , s ) f ( s , u ( s ) ) d s ,
where q ∈ I and λ ≥ 0 .

If integral equation ( 5.1 ) has a solution, then the sequence generated by ( 1.1 ) will converge to that solution.

Proof

Consider

∣ G u ( q ) − G v ( q ) ∣ = ∣ h ( q ) + λ ∫ 0 1 k ( q , s ) f ( s , u ( s ) ) d s − h ( q ) − λ ∫ 0 1 k ( q , s ) f ( s , v ( s ) ) d s ∣ = λ ∣ ∫ 0 1 k ( q , s ) ( f ( s , u ( s ) ) − f ( s , u ( s ) ) ) d s ∣

≤ λ ∫ 0 1 k ( q , s ) ∣ f ( s , u ( s ) ) − f ( s , u ( s ) ) ∣ d s ≤ λ ∫ 0 1 k ( q , s ) L ∣ u ( s ) − v ( s ) ∣ d s .

Taking the supremum norm on both sides, we have

∣ ∣ G u ( q ) − G v ( q ) ∣ ∣ ≤ λ F L ∣ ∣ u − v ∣ ∣ ≤ ∣ ∣ u − v ∣ ∣ .

Thus, G is a monotone nonexpansive mapping. Therefore, all the assumptions of Theorem 3.4 have been satisfied, and the sequence generated by (1.1) will converge to the solution of integral equation (5.1).□

Now, we present the following example.

Example 1

Consider the following integral equation:

(5.2) u ( q ) = 2 ( 1 − 2 q 2 ) + 1 2 ∫ 0 1 q s u ( s ) d s , q ∈ [ 0 , 1 ] .

Now, this example is a linear Fredholm equation, which is a special case of ( 5.1 ) with h ( q ) = 2 ( 1 − 2 q 2 ) ; k ( q , s ) = q s , and f ( q , u ) = u ( q ) . Now, for q ∈ [ 0 , 1 ] .

∫ 0 1 k ( q , s ) d s = ∫ 0 1 q s d s = q 2 ≤ 1 2 .

Also, for arbitrary u , v ∈ C ( I , R ) with u ( q ) ≼ v ( q ) , we have

∣ f ( q , u ) − f ( q , v ) ∣ = ∣ u ( q ) − v ( q ) ∣ .

So, here λ = 1 2 , F = 1 2 , and L = 1 , and all the assumptions of Theorem 5.1 are satisfied. The exact solution of integral equation (5.2) is u ( q ) = 2 ( 1 − 2 q 2 ) , where q ∈ [ 0 , 1 ] .

Let the operator G : C [ 0 , 1 ] → C [ 0 , 1 ] be defined by

(5.3) G u ( q ) = 2 ( 1 − 2 q 2 ) + 1 2 ∫ 0 1 q s u ( s ) d s ,

where q ∈ [ 0 , 1 ] . For two different initial guess u 1 ( q ) = 1 + q and u 1 ( q ) = 1 , q ∈ [ 0 , 1 ] and control sequences μ t = t t + 5 , δ t = 1 t + 1 , and κ t = 2 t 5 t + 2 , we obtain the following approximated values of u t ( q ) (Table 4).

Table 4

Approximated values of u t ( q )

Iteration number	When u 1 ( q ) = 1 + q	When u 1 ( q ) = 1
1	1 + q	1
3	2 ( 1 − 2 q 2 ) + 0.0364210232 q	2 ( 1 − 2 q 2 ) + 0.02185261393 q
5	2 ( 1 − 2 q 2 ) + 0.0003811420736 q	2 ( 1 − 2 q 2 ) + 0.0002286852442 q
7	2 ( 1 − 2 q 2 ) + 3.052742423 × 1 0 − 6 q	2 ( 1 − 2 q 2 ) + 1.831645454 × 1 0 − 6 q
9	2 ( 1 − 2 q 2 ) + 1.967370775 × 1 0 − 8 q	2 ( 1 − 2 q 2 ) + 1.180422465 × 1 0 − 8 q

In Table 5, we obtain numerical errors for Example 5.1 for two different initial guesses and the exact solution. The error functions when u 1 ( q ) = 1 + q and u 1 ( q ) = 1 are plotted in Figure 4, which shows that the iterative scheme ( 1.1 ) converges to the exact solution very fast as the error function for both the functions is approaching zero.

Table 5

Error for different approximated values and the exact solution

Value of q	Error when u 1 ( q ) = 1 + q	Error when u 1 ( q ) = 1
0	0	0
0.1	1.4737833 × 1 0 − 10	8.8427043 × 1 0 − 11
0.2	2.9475689 × 1 0 − 10	1.7685409 × 1 0 − 10
0.3	4.4213522 × 1 0 − 10	2.6528113 × 1 0 − 10
0.4	5.8951377 × 1 0 − 10	3.5370817 × 1 0 − 10
0.5	7.3689188 × 1 0 − 10	4.4213522 × 1 0 − 10
0.6	8.8427021 × 1 0 − 10	5.3056226 × 1 0 − 10
0.7	1.0316485 × 1 0 − 9	6.189893 × 1 0 − 10
0.8	1.1790271 × 1 0 − 9	7.0741635 × 1 0 − 10
0.9	1.3264052 × 1 0 − 9	7.9584339 × 1 0 − 10
1.0	1.4737838 × 1 0 − 9	8.8427043 × 1 0 − 10

Figure 4

Graph showing errors for different initial points.

6 Conclusion

We have obtained some convergence results for monotone nonexpansive mappings. We have proved our results analytically as well as numerically by using a nontrivial example. Furthermore, to illustrate the utility of our results, we have shown that the algorithm converges to the solution of an integral equation using two different initial approximations, one of which is a linear function and the other one is constant. Also, our parameters are not constants.

Acknowledgments

We are very much thankful to the reviewers for their constructive comments and suggestions, which have been useful for the improvement of this article.

Funding information: This work has been partially supported by the Agencia Estatal de Investigacion (AEI) of Spain under Grant PID2020-113275GB-I00, cofinanced by the European Community fund FEDER, as well as Xunta de Galicia grant ED431C 2019/02 for Competitive Reference Research Groups (2019-22).
Author contributions: IU suggested the problem to CG. CG analyzed the results in the literature and made the original draft preparation. IU and CG carried out the formal analysis, writing review and editing. MAA critically read the manuscript and helped in example. JJ gone through the revision critically. All the authors read and approved the final manuscript.
Conflict of interest: The authors declare that they have no conflict of interest.
Data availability statement: The data used to support the findings of this study are available from the corresponding author upon request.
Ethical approval: The conducted research is not related to either human or animal use.

References

[1] A. Sahin, Z. Kalkan, and H. Arisoy, On the solution of a nonlinear Volterra integral equation with delay, Sakarya Univ. J. Sci. 21 (2017), no. 6, 1367–13676. Search in Google Scholar

[2] S. A. Khuri and A. Sayfy, An iterative method for boundary value problems, Nonlinear Sci. Lett. A 8 (2017), no. 2, 178–186. Search in Google Scholar

[3] S. A. Khuri and A. Sayfy, Numerical solution of functional differential equations: a Green’s function based iterative approach, Int. J. Comput. Math. 95 (2018), no. 10, 1937–1949. 10.1080/00207160.2017.1344230Search in Google Scholar

[4] Y. Mudasir, D. Singh, and A. Goyal, A novel approach of graphical rectangular b-metric spaces with an application to the vibrations of a vertical heavy hanging cable, J. Fixed Point Theory Appl. 21 (2019), no. 1, Article number 33. 10.1007/s11784-019-0673-3Search in Google Scholar

[5] S. Banach, Sur les operations dans les ensembles abstraits et leurs applications, Fundam. Math. 3 (1922), 133–181. 10.4064/fm-3-1-133-181Search in Google Scholar

[6] F. P. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. 10.1073/pnas.54.4.1041Search in Google Scholar PubMed PubMed Central

[7] D. Göhde, Zum Prinzip der kontraktiven abbildung, Math Nachr. 30 (1965), 251–258. 10.1002/mana.19650300312Search in Google Scholar

[8] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. 10.2307/2313345Search in Google Scholar

[9] A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc. 132 (2004), 1435–1443. 10.1090/S0002-9939-03-07220-4Search in Google Scholar

[10] J. J. Nieto and R. Rodriguez-Löpez, Contractive mapping theorems in partially ordered sets and application to ordinary differential equations, Order. 22 (2005), 223–239. 10.1007/s11083-005-9018-5Search in Google Scholar

[11] M. Bachar and M. A. Khamsi, On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces, Fixed Point Theory Appl. 2015 (2015), Article ID 160. 10.1186/s13663-015-0405-3Search in Google Scholar

[12] B. Abdullatif, B. Dehaish, and M. A. Khamsi, Browder and Göhde fixed point theorem for monotone nonexpansive mappings, Fixed Point Theory Appl. 2016 (2016), Article ID 20. 10.1186/s13663-016-0505-8Search in Google Scholar

[13] Y. Song, P. Kumam, and Y. J. Cho, Fixed point theorems and iterative approximations for monotone nonexpansive mappings in ordered Banach spaces, Fixed Point Theory Appl. 2016 (2016), Article ID 73. 10.1186/s13663-016-0563-ySearch in Google Scholar

[14] B. A. B. Dehaish, M. A. Khamsi, Mann iteration process for monotone nonexpansive mappings, Fixed Point Theory Appl, (2015), Article ID 177. 10.1186/s13663-015-0416-0Search in Google Scholar

[15] I. Uddin, C. Garodia, and J. J. Nieto, Mann iteration for monotone nonexpansive mappings in ordered CAT(0) space with an application to integral equation, J Inequalities Appl. 2018 (2018), Article ID 339. 10.1186/s13660-018-1925-2Search in Google Scholar PubMed PubMed Central

[16] W. R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953), 506–510. 10.1090/S0002-9939-1953-0054846-3Search in Google Scholar

[17] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150. 10.1090/S0002-9939-1974-0336469-5Search in Google Scholar

[18] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957–961. 10.1090/S0002-9904-1967-11864-0Search in Google Scholar

[19] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), no. 1, 217–229. 10.1006/jmaa.2000.7042Search in Google Scholar

[20] C. Garodia and I. Uddin, A new fixed point algorithm for finding the solution of a delay differential equation, Aims Math. 5 (2020), no. 4, 3182–3200. 10.3934/math.2020205Search in Google Scholar

[21] C. Garodia and I. Uddin, A new iterative method for solving split feasibility problem, J. Appl. Anal. Comput. 10 (2020), no. 3, 986–1004. 10.11948/20190179Search in Google Scholar

[22] C. Garodia, I. Uddin, and S. H. Khan, Approximating common fixed points by a new faster iteration process, Filomat 34 (2020), no. 6, 2047–2060. 10.2298/FIL2006047GSearch in Google Scholar

[23] C. Garodia, I. Uddin, and D Baleanu, On constrained minimization, variational inequality and split feasibility problem via new iteration scheme in Banach spaces, Bulletin Iranian Math. Soc. 48 (2022), no. 4, 1493–1512. 10.1007/s41980-021-00596-6Search in Google Scholar

[24] B. S. Thakur, D. Thakur, and M. Postolache, A new iteration scheme for approximating fixed points of nonexpansive mappings, Filomat. 30 (2016), no. 10, 2711–2720. 10.2298/FIL1610711TSearch in Google Scholar

[25] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bulletin Australian Math. Soc. 43 (1991), 153–159. 10.1017/S0004972700028884Search in Google Scholar

[26] H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Am. Math. Soc. 44 (1974), no. 2, 375–380. 10.1090/S0002-9939-1974-0346608-8Search in Google Scholar

[27] A. Salim, S. Abbas, M. Benchohra, and E. Karapinar, Global stability results for Volterra-Hadamard random partial fractional integral equations, Rendiconti del Circolo MaT. di Palermo Series 2 (2022), 1–13. 10.1007/s12215-022-00770-7Search in Google Scholar

[28] S. Abbas, M. Benchohra, J. Henderson, and J. E. Lazreg, Weak solutions for a coupled system of partial Pettis Hadamard fractional integral equations, Adv. Theory Nonlinear Anal. Appl. 1 (2017), no. 2, 136–146. 10.31197/atnaa.379096Search in Google Scholar

[29] I. A. Rus, Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle, Adv. Theory Nonlinear Anal. Appl. 3 (2019), no. 3, 111–120. 10.31197/atnaa.604962Search in Google Scholar

[30] S. K. Panda, E. Karapınar, and A. Atangana, A numerical schemes and comparisons for fixed point results with applications to the solutions of Volterra integral equations in dislocated extended b-metric space, Alexandria Eng. J. 59 (2020), no. 2, 815–827. 10.1016/j.aej.2020.02.007Search in Google Scholar

[31] B. Alqahtani, H. Aydi, E. Karapınar, and V. Rakocevic, A solution for Volterra fractional integral equations by hybrid contractions, Mathematics 7 (2019), no. 8, 694. 10.3390/math7080694Search in Google Scholar

[32] H. Aydi, E. Karapinar, and H. Yazidi, Modified F-Contractions via alpha-admissible mappings and application to integral equations, Filomat. 31 (2017), no. 5, 1141–1148. 10.2298/FIL1705141ASearch in Google Scholar

[33] E. Karapinar, A. Fulga, N. Shahzad, and A. F. Roldán López de Hierro, Solving integral equations by means of fixed point theory, J. Function Spaces 2022 (2022), 1–16, Article ID 7667499. 10.1155/2022/7667499. Search in Google Scholar

[34] L. Wangwe and S. Kumar, Fixed point theorems for multi-valued α−F-contractions in Partial metric spaces with an application, Results in Nonlinear Anal. 4 (2021), no. 3, 130–148. Search in Google Scholar

[35] E. Karapınar, A. Atangana, and A. Fulga, Pata type contractions involving rational expressions with an application to integral equations, Discrete Continuous Dynam. Syst.-S. 14 (2021), no. 10, 3629. 10.3934/dcdss.2020420Search in Google Scholar

[36] R. P. Agarwal, Ü. Aksoy, E. Karapınar, and İ.M. Erhan, F-contraction mappings on metric-like spaces in connection with integral equations on time scales, RACSAM. 114 (2020), no. 3, 1–12. 10.1007/s13398-020-00877-5Search in Google Scholar

Received: 2022-05-01

Revised: 2022-08-26

Accepted: 2022-10-10

Published Online: 2022-11-11

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0172

Keywords for this article

monotone nonexpansive mappings; fixed point; iteration process; strong and weak convergence; nonlinear integral equation

Creative Commons

BY 4.0