Convergence and stability of an iteration process and solution of a fractional

R E S E A R C H Open Access

Convergence and stability of an iteration process and solution of a fractional

differential equation

Mohd Jubair¹, Faeem Ali¹and Javid Ali^1*

*Correspondence:

javid.mm@amu.ac.in

1Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

Abstract

In this paper, we prove that a three-step iteration process is stable for contractive-like mappings. It is also proved analytically and numerically that the considered process converges faster than some remarkable iterative processes for contractive-like mappings. Furthermore, some convergence results are proved for the mappings satisfying Suzuki’s condition (C) in uniformly convex Banach spaces. A couple of nontrivial numerical examples are presented to support the main results and the visualization is showed by Matlab. Finally, by utilizing our main result the solution of a nonlinear fractional differential equation is approximated.

MSC: 47H09; 47H10; 54H25

Keywords: Suzuki’s condition (C); Contractive-like mapping; Iteration processes;

Fixed point; Fractional differential equation; Uniformly convex Banach space

1 Introduction

Throughout this paper,Z⁺denotes the set of all nonnegative integers. We assume thatU is a nonempty subset of a Banach spaceWandF(F) ={t∈U:F:U→UandFt=t}. A mappingF:U→Uis called non-expansive ifFx–Fy ≤ x–y,∀x,y∈U. It is said to be a quasi-non-expansive ifF(F)=∅andFx–t ≤ x–t,∀x∈Uand∀t∈F(F).

Hardy and Rogers [10] introduced generalized non-expansive mapping which is defined as follows:

A self-mapFonUis called generalized non-expansive if for allx,y∈Uthere exist real numbersa,b,c≥0 witha+ 2b+ 2c≤1 such that

Fx–Fy ≤ax–y+b

x–Fx+y–Fy +c

x–Fy+y–Fx

. (1.1) It can be easily verified that ifF(F)=∅, thenFis a quasi-non-expansive mapping but the converse is not true in general.

In 2008, Suzuki [22] defined a condition on the mappings, called condition (C); such mappings are also known as generalized non-expansive mappings.

©The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

A mappingF:U→U is said to satisfy condition (C) if, 1

2x–Fx ≤ x–y ⇒ Fx–Fy ≤ x–y, ∀x,y∈U.

Suzuki [22] proved existence and convergence theorems for such mappings. He also ex- hibited that every non-expansive mapping satisfies condition (C), but the reverse is not true in general. Moreover, ifF(F)=∅and satisfies condition (C) then it is a quasi-non- expansive mapping. Recently, a number of researchers studied the fixed points of Suzuki generalized non-expansive mappings; e.g. see [3,5,8,23,24].

The generalized non-expansive mappings coined by Hardy and Rogers, and Suzuki are generalizations of non-expansive mappings. So, most recently, Ali et al. [2] compared the classes of mappings due to Suzuki, and Hardy and Rogers and showed that the two classes of mappings do not imply each other. They also presented two examples to verify their claim.

In 2003, Imoru and Olantiwo [11] defined the class of contractive-like mappings which is wider than the classes of contractions, Zamfirescu mappings, weak contractions, etc.

They also proved that the Picard and Mann iteration processes are stable with respect to contractive-like mappings. The definition of contractive-like mapping runs as follows.

Definition 1.1([11]) Letϕ: [0,∞)→[0,∞) be a strictly increasing and continuous func- tion withϕ(0) = 0 and a constantδ∈[0, 1). A self-mapFonUis said to be contractive-like if, for allx,y∈U, we have

Fx–Fy ≤δx–y+ϕ

x–Fx .

During approximation of fixed points, the better speed of convergence of iteration pro- cess saves time. Berinde [7] gave the following definitions to compare the rate of conver- gence of iteration processes.

Definition 1.2 Let (W, · ) be a normed space andF:W→Wa mapping. Suppose that the two fixed point iteration processes{τ_n}and{σ_n}converge to the same pointt.

Furthermore, assume that the error estimates τ_n–t ≤α_n,

σ_n–t ≤β_n,

are available (and these estimates are the best ones available), where{α_n}and{β_n}are two sequences of nonnegative real numbers that converge to 0. Now, in order to compare the two fixed point sequences{τ_n}and{σ_n}inW, it suffices to compare the two sequences of real numbers{α_n}and{β_n}converging to 0. For this, one can use the following concept of rate of convergence of two sequences given by Berinde [7].

Definition 1.3 Let{α_n}and{β_n}be two sequences of nonnegative real numbers that con- verge toxandy, respectively. Assume that

= lim

n→∞

|αn–x|

|β_n–y|.

(i) If= 0, then{α_n}converges toxfaster than{β_n}toy.

(ii) If 0 <<∞, then{αn}and{βn}have the same rate of convergence.

We say that the given fixed point iteration process is stable if and only if the approximate sequence{tn}still converges to the fixed point ofF. To approximate fixed points of the mappings, we consider an approximate sequence{tn}instead of the theoretical sequence {τ_n}, because of rounding errors and numerical approximation of functions. In view of this fact, Ostrowski [18] was first to coin the concept of stability for a fixed point iteration process and proved that Picard iteration process is stable for contraction mapping. The definition of stability due to Ostrowski runs as follows.

Definition 1.4([18]) Consider an approximate sequence{t_n}in a subsetU of a Banach spaceW. Then an iteration procedureτ_n+1=f(F,τ_n) is said to beF-stable or stable with respect toFfor some functionf, converging to a fixed pointt, if for n=tn+1–f(F,tn), n∈Z⁺, we havelim_{n→∞ n}= 0⇔lim_n→∞t_n=t.

In the last three decades, the study of fixed point iteration processes has taken an em- inent place in the fixed point theory and applied mathematics. The iteration processes are used to solve initial and boundary value problems, image recovery problems, image restoration problems, image processing problems, variational inequality problems, func- tional equations [19] etc. Thus, several authors introduced and studied remarkable itera- tion processes to approximate the fixed point of different nonlinear mappings. The follow- ing iteration processes have been introduced by Mann [15], Ishikawa [12], Agrawal et al.

(S) [1], Gursoy and Karakaya (Picard-S) [9] and Noor [16], respectively. Here the sequence {τ_n}with an initial guessτ₀∈Ufor the self-mappingFonUis defined as follows:

τ_n+1= (1 –θ_n)τn+θ_nFτ_n, n∈Z⁺, (1.2)

⎧⎨

⎩

τ_n+1= (1 –θ_n)τn+θ_nFσ_n,

σ_n= (1 –μ_n)τn+μ_nFτ_n, n∈Z⁺,

(1.3)

⎧⎨

⎩

τ_n+1= (1 –θ_n)Fτn+θ_nFσn,

σ_n= (1 –μ_n)τn+μ_nFτ_n, n∈Z⁺, (1.4)

⎧⎪

⎪⎨

⎪⎪

⎩

τ_n+1=Fσ_n,

σ_n= (1 –θ_n)Fτ_n+θ_nFξ_n,

ξ_n= (1 –μ_n)τ_n+μ_nFτ_n, n∈Z⁺,

(1.5)

⎧⎪

⎪⎨

⎪⎪

⎩

τ_n+1= (1 –θ_n)τ_n+θ_nFσn, σ_n= (1 –μ_n)τn+μ_nFξ_n,

ξ_n= (1 –γ_n)τ_n+γ_nFτn, n∈Z⁺,

(1.6)

where the sequences{θ_n},{μ_n}and{γ_n}are in (0, 1).

Most recently, Ali et al. [2] introduced a new iteration process, called JF iteration process and approximated the fixed points of Hardy and Rogers generalized non-expansive map- pings in uniformly convex Banach spaces. In this process, the sequence{τ_n}is generated

by an initial guessτ₀∈Uand defined as follows:

⎧⎪

⎪⎨

⎪⎪

⎩

τ_n+1=F((1 –θ_n)σ_n+θ_nFσn), σ_n=Fξ_n,

ξ_n=F((1 –μ_n)τ_n+μ_nFτ_n), n∈Z⁺,

(1.7)

where{θ_n}and{μ_n}are in (0, 1). They claimed numerically that JF iteration process con- verges to the fixed point of Hardy and Rogers mappings faster than some well-known iteration processes. They also approximated the solution of a delay differential equation via JF iteration process.

Motivated by the above, we prove the stability and rate of convergence of the JF itera- tion process for contractive-like mappings. We also prove some convergence results for Suzuki generalized non-expansive mappings via the JF iteration process in uniformly con- vex Banach spaces. In the last section, we estimate the solution of a nonlinear fractional differential equation via the JF iteration process. A couple of illustrative numerical exam- ples are presented to validate the results. The results of this paper are remarkable from the point of view of the results of Ali et al. [2] and extend several relevant results in the literature.

2 Preliminaries

This section contains some lemmas, propositions and definitions that will be used in the main results.

Lemma 2.1([6]) Let{ n}and{u_n}be sequences of positive real numbers satisfying u_n+1≤ δun+ n,n∈Z⁺,whereδ∈[0, 1).Iflim_{n→∞ n}= 0thenlim_n→∞un= 0.

Lemma 2.2([22]) LetUbe a weakly compact convex subset of a uniformly convex Banach spaceW andF:U →U be a mapping satisfying Suzuki’s condition(C).ThenF has a fixed point.

Lemma 2.3([22]) LetUbe a nonempty closed convex subset of a uniformly convex Banach spaceW andF :U →U a mapping satisfying Suzuki’s condition (C).If{τ_n} converges weakly to t∈U andlim_n→∞τn–Fτn= 0,thenFt=t implies I–F is demiclosed at zero.

Lemma 2.4([20]) LetW be a uniformly convex Banach space and0 <a≤sn≤b< 1 for all n≥1.Let{τ_n} and{σ_n} be two sequences inW such thatlim sup_n→∞τ_n ≤w, lim sup_n→∞σ_n ≤w andlim_n→∞s_nτ_n+ (1 –s_n)σ_n=w holds, for some w≥0.Then lim_n→∞τn–σn= 0.

Proposition 2.5([22]) LetUbe a nonempty subset of a Banach spaceWandF:U→U be a mapping satisfying condition(C).Then

x–Fy ≤3Fx–x+x–y, ∀x,y∈U.

Definition 2.6 A Banach spaceW is said to satisfy Opial’s property [17] if for any se- quence{τ_n} ⊂Wwith{τ_n}x(denotes weak convergence) it implies that

lim inf

n→∞τ_n–x<lim inf

n→∞τ_n–y holds, for ally∈Wwithy=x.

Definition 2.7([21]) A self-mapFonU is said to satisfy condition (I), if there exists a nondecreasing functionψ: [0,∞)→[0,∞) withψ(0) = 0 andψ(z) > 0,∀z> 0 such that d(x,Fx)≥ψ(d(x,F(F))),∀x∈U.

Definition 2.8 LetWbe a Banach space and letUa nonempty, closed and convex subset ofW,{τ_n}a bounded sequence inWand forx∈U,

r x,{τ_n}

=lim sup

n→∞ τ_n–x.

The asymptotic radius of{τ_n}relative toU is defined by r

U,{τ_n}

=inf r

x,{τ_n} :x∈U

.

The asymptotic center of{τ_n}relative toUis defined by A

U,{τ_n}

=

x∈U:r x,{τ_n}

=r

U,{τ_n} .

It is known that ifWis a uniformly convex Banach space, thenA(U,{τ_n}) contains only one point.

3 Rate of convergence and stability results for contractive-like mappings Throughout this section, we presume thatUis a nonempty, closed and convex subset of a Banach spaceWandF:U→Ua contractive-like mapping. The purpose of this section is to prove stability and convergence results for contractive-like mappings via JF iteration process.

Theorem 3.1 Let{τ_n}be an iteration process defined by(1.7).Then iteration process(1.7) isF-stable.

Proof Suppose{tn}is an arbitrary sequence inU andtn+1=f(F,tn) is the sequence gen- erated by (1.7) and _n=t_n+1–f(F,t_n)for alln∈Z⁺. We have to prove thatlim_{n→∞ n}= 0⇐⇒lim_n→∞tn=t.

Supposelim_{n→∞ n}= 0, then by iteration process (1.7), we have tn+1–t ≤tn+1–f(F,tn)+f(F,tn) –t

≤ n+f(F,t_n) –t

≤ n+δ²

1 – (1 –δ)θn

1 – (1 –δ)μn

tn–t. (3.1)

Since 0 < (1 – (1 –δ)θn)≤1 and 0 < (1 – (1 –δ)μn)≤1 and using (3.1), we get tn+1–t ≤ n+δ²tn–t.

Defineun=tn–t, then un+1≤δ²un+ n.

Sincelim_{n→∞ n}= 0, so by Lemma2.1, we havelim_n→∞u_n= 0i.e.,lim_n→∞t_n=t.

Conversely, supposelim_n→∞tn=t, we have

n=tn+1–f(F,tn)

≤ t_n+1–t+f(F,t_n) –t

≤ t_n+1–t+δ²

1 – (1 –δ)θ_n

1 – (1 –δ)μ_n t_n–t

≤ t_n+1–t+δ²t_n–t.

This implies thatlim_{n→∞ n}= 0. Hence iteration process (1.7) isF- stable.

Theorem 3.2 Let F(F)=∅and{τ_n} be a sequence defined by(1.7),then{τ_n}converges faster than the iteration processes(1.2)-(1.6).

Proof From (1.5), for anyt∈F(F), we have ξ_n–t=(1 –μ_n)τ_n+μ_nFτn–t

≤(1 –μ_n)τ_n–t+μ_nδτ_n–t

=

1 – (1 –δ)μ_n

τ_n–t. (3.2)

Using (3.2), we get

σ_n–t=(1 –θ_n)Fτ_n+θ_nFξ_n–t

≤(1 –θ_n)δτ_n–t+θ_nδξ_n–t

≤δ

1 – (1 –δ)θnμ_n

τ_n–t. (3.3)

Using (3.3), we get

τ_n+1–t=Fσn–t ≤δσ_n–t ≤δ²

1 – (1 –δ)θ_nμ_n

τ_n–t. (3.4)

By using the fact 0 < (1 – (1 –δ)θnμ_n)≤1, we get τ_n+1–t ≤δ²τ_n–t.

Inductively, we get

τ_n–t ≤δ²⁽ⁿ⁺¹⁾τ₀–t. (3.5)

Now from (1.7), we have ξn–t=F

(1 –μn)τn+μnFτn

–t

≤δ(1 –μ_n)τ_n+μ_nFτ_n–t

≤δ

(1 –μ_n)τ_n–t+μ_nδτ_n–t

≤δ

1 – (1 –δ)μn

τ_n–t. (3.6)

Using (3.6), we get σ_n–t=Fξ_n–t

≤δξ_n–t

≤δ²

1 – (1 –δ)μ_n

τ_n–t. (3.7)

Using (3.7), we get τ_n+1–t=F

(1 –θ_n)σn+θ_nFσ_n –t

≤δ(1 –θn)σn+θnFσn–t

≤δ³

1 – (1 –δ)θ_n

1 – (1 –δ)μ_n τ_n–t.

By using the fact 0 < (1 – (1 –δ)θn)≤1 and 0 < (1 – (1 –δ)μn)≤1, we have τ_n+1–t ≤δ³τ_n–t.

Inductively, we get

τ_n+1–t ≤δ³⁽ⁿ⁺¹⁾τ₀–t. (3.8)

Letα_n=δ³⁽ⁿ⁺¹⁾τ₀–tandβ_n=δ²⁽ⁿ⁺¹⁾τ₀–t, then

n→∞lim α_n β_n= lim

n→∞

δ³⁽ⁿ⁺¹⁾τ₀–t δ²⁽ⁿ⁺¹⁾τ₀–t = 0.

Hence iteration process (1.7) converges faster than iteration process (1.5).

Similarly we can show that iteration process (1.7) converges faster than (1.2)–(1.4) and (1.6) iteration processes.

In support of Theorem3.2, we construct the following example.

Example3.3 LetW=R²be a Banach space with taxicab norm andU = [0, 6]×[0, 6] a subset ofR². LetF:U→Ube defined by

F(x1,x2) =

⎧⎨

⎩

(^x₃¹,^x₃²), if (x₁,x₂)∈[0, 3)×[0, 3), (^x₆¹,^x₆²), if (x1,x2)∈[3, 6]×[3, 6].

Clearly, (0, 0) is the fixed point ofF. Now, we show thatFis a contractive-like mapping but not a contraction. For this, we define a functionϕ:R⁺→R⁺byϕ(x) =^x₄. Then,ϕis strictly increasing and continuous function. Also,ϕ(0) = 0. We show that

Fx–Fy=δx–y+ϕ

x–Fx

(3.9) for all x,y∈U andδ∈(0, 1). Before going ahead, let us note the following. Whenx= (x1,x2)∈[0, 3)×[0, 3), then

x–Fx=

(x1,x2) – x1

3,x2

3 =

2x1

3 ,2x2

3

and ϕ

x–Fx

=ϕ

2x1

3 ,2x2

3

=

x1

6,x2

6 =

x1

6 +

x2

6

. (3.10)

Similarly, whenx= (x1,x2)∈[3, 6]×[3, 6], then x–Fx=

(x1,x2) – x1

6,x2

6 =

5x1

6 ,5x2

6

and ϕ

x–Fx

=ϕ

5x1

6 ,5x2

6

=

5x1

24,5x2

24

= 5x1

24 +

5x2

24

. (3.11)

Now, we have the following cases:

Case (i): Letx,y∈[0, 3)×[0, 3), using (3.10) we get Fx–Fy=

x1

3,x2

3

– y1

3,y2

3

= x1

3 –y1

3 +

x2

3 –y2

3

= 1

3|x1–y1|+1 3|x2–y2|

= 1

3(x1,x2) – (y1,y2)

≤1

3x–y+ x1

6 +

x2

6

= 1

3x–y+ϕ

x–Fx

. (3.12)

Case (ii): Letx,y∈[3, 6]×[3, 6], using (3.10) we get Fx–Fy=

x₁

6,x₂ 6

–

y₁ 6,y₂

6

= x1

6 –y1

6 +

x2

6 –y2

6

= 1

6|x1–y1|+1 6|x2–y2|

= 1

6(x1,x2) – (y1,y2)

≤1

3x–y+ 5x1

24 +

5x2

24

= 1

3x–y+ϕ

x–Fx

. (3.13)

Case (iii): Letx∈[0, 3) andy∈[3, 6], then using (3.10) we get Fx–Fy=

x1

3,x2

3

– y1

6,y2

6

=

x1

3 –y1

6

, x2

3 –y2

6

=

x₁ 6 +x₁

6 –y₁ 6

, x₂

6 +x₂ 6 –y₂

6

= x1

6 +x1

6 –y1

6 +

x2

6 +x2

6 –y2

6

≤ x1

6 +

x2

6 +

x1

6 –y1

6 +

x2

6 –y2

6

= 1 6

|x1–y1|+|x2–y2| +ϕ

x–Fx

≤1

3(x₁,x2) – (y₁,y2)+ϕ

x–Fx

= 1

3x–y+ϕ

x–Fx

. (3.14)

Case (iv): Letx∈[3, 6] andy∈[0, 3), then using (3.10) we get Fx–Fy=

x₁

6,x₂ 6

–

y₁ 3,y₂

3

=

x1

6 –y1

3

, x2

6 –y2

3

≤

x1

3 –x1

6 –y1

3

, x2

3 –x2

6 –y2

3

= x₁

3 –x₁ 6 –y₁

3 +

x₂ 3 –x₂

6 –y₂ 3

≤ x1

6 +

x2

6 +

x1

3 –y1

3 +

x2

3 –y2

3

= 1 3

|x1–y1|+|x2–y2| +ϕ

x–Fx

= 1

3(x1,x2) – (y1,y2)+ϕ

x–Fx

= 1

3x–y+ϕ

x–Fx .

So, (3.9) is satisfied withδ=¹₃. Thus,Fis a contractive-like mapping.

Table 1 Computational table of different iteration processes

Iter. Mann Ishikawa S

1 (1.000000, 2.500000) (1.000000, 2.500000) (1.000000, 2.500000)

2 (0.666667, 1.666667) (0.622222, 1.555556) (0.288889, 0.722222)

.. .

.. .

.. .

.. .

14 (0.005138, 0.012846) (0.002096, 0.005239) (0.000000, 0.000000)

16 (0.002284, 0.005709) (0.000811, 0.002028) (0.000000, 0.000000)

..

. ... ... ...

33 (0.000002, 0.000006) (0.000000, 0.000001) (0.000000, 0.000000)

34 (0.000002, 0.000004) (0.000000, 0.000000) (0.000000, 0.000000)

.. .

.. .

.. .

.. .

39 (0.000000, 0.000001) (0.000000, 0.000000) (0.000000, 0.000000)

40 (0.000000, 0.000000) (0.000000, 0.000000) (0.000000, 0.000000)

Table 2 Computational table of different iteration processes

Iter. Picard-S Noor JF

1 (1.000000, 2.500000) (1.000000, 2.500000) (1.000000, 2.500000)

2 (0.096296, 0.240741) (0.617778, 1.544444) (0.069136, 0.172840)

.. .

.. .

.. .

.. .

7 (0.000001, 0.000002) (0.055590, 0.138974) (0.000000, 0.000000)

8 (0.000000, 0.000000) (0.034342, 0.085855) (0.000000, 0.000000)

.. .

.. .

.. .

.. .

33 (0.000000, 0.000000) (0.000000, 0.000001) (0.000000, 0.000000)

34 (0.000000, 0.000000) (0.000000, 0.000000) (0.000000, 0.000000)

.. .

.. .

.. .

.. .

40 (0.000000, 0.000000) (0.000000, 0.000000) (0.000000, 0.000000)

Figure 1Convergence behavior of the sequences defined by distinct iteration processes

It can be easily seen in Tables1–2and Fig.1that JF iteration process converges to a fixed point (0, 0) of the mappingFfaster than the leading iteration processes with initial point (1, 2.5) and control sequencesθ_n= 0.5,μ_n= 0.4 andγ_n= 0.3,n∈Z⁺.

4 Convergence results for Suzuki’s generalized non-expansive mappings Throughout this section, we presume thatU is a nonempty closed and convex subset of a uniformly convex Banach spaceWand letF:U→Ube a mapping satisfying Suzuki’s condition (C).

Lemma 4.1 LetUbe a nonempty closed and convex subset of a uniformly convex Banach spaceWandF:U →U a mapping satisfying Suzuki’s condition(C).Suppose F(F)=∅ and{τ_n}is a sequence developed by iteration process(1.7),thenlim_n→∞τ_n–texists for all t∈F(F).

Proof Lett∈F(F) andx∈U. SinceFsatisfies Suzuki’s condition (C), we have Fx–t ≤ x–t, for allx∈Uand for allt∈F(F).

Now from iteration process (1.7), we get ξ_n–t=F(1 –μ_n)τ_n+μ_nFτn–t

≤(1 –μ_n)τn+μ_nFτ_n–t

≤(1 –μ_n)τ_n–t+μ_nτ_n–t

=τn–t (4.1)

and

σ_n–t=Fξ_n–t

≤ ξ_n–t

≤ τ_n–t. (4.2)

Using (4.1) and (4.2), we have τ_n+1–t=F

(1 –θ_n)σ_n+θ_nFσ_n –t

≤(1 –θ_n)σn+θ_nFσ_n–t

≤(1 –θ_n)σ_n–t+θ_nσ_n–t

=σ_n–t

≤ τ_n–t. (4.3)

This shows that the sequence{τ_n–t}is non-increasing and bounded below for allt∈

F(F). Thuslim_n→∞τ_n–texists.

Lemma 4.2 Let{τ_n}be a sequence developed by iteration process(1.7)and sequence{μ_n} satisfying condition0 <a≤μ_n≤b< 1for all n≥1.Then F(F)=∅if and only if{τ_n}is bounded andlim_n→∞τ_n–Fτ_n= 0.

Proof By Lemma4.1, it follows thatlim_n→∞τ_n–texists.

Presume thatlim_n→∞τ_n–t=c.

By the inequalities (4.1) and (4.2), we get lim sup

n→∞ ξ_n–t ≤c (4.4)

and lim sup

n→∞ σn–t ≤c, (4.5)

respectively. SinceFsatisfies Suzuki’s condition (C), we have

Fτn–t ≤ τn–t, Fσn–t ≤ σn–t, Fξn–t ≤ ξn–t. lim sup

n→∞ Fτ_n–t ≤c, (4.6)

lim sup

n→∞ Fσ_n–t ≤c, (4.7)

lim sup

n→∞ Fξ_n–t ≤c. (4.8)

Since

τ_n+1–t=F

(1 –θ_n)σn+θ_nFσ_n –t

≤(1 –θ_n)σ_n+θ_nFσn–t

≤(1 –θ_n)σ_n–t+θ_nσ_n–t

=σ_n–t.

Taking thelim infon both sides, we get c=lim inf

n→∞τ_n+1–t ≤lim inf

n→∞σ_n–t. (4.9)

Thus, (4.5) and (4.9) give

n→∞lim σ_n–t=c.

We have c=lim inf

n→∞σ_n–t=lim inf

n→∞Fξ_n–t

≤lim inf

n→∞ξ_n–t. (4.10)

From (4.4) and (4.10), we have

n→∞lim ξ_n–t=c.

So,

c= lim

n→∞ξ_n–t

= lim

n→∞F

(1 –μn)τn+μnFτn

–t

≤ lim

n→∞(1 –μ_n)τn+μ_nFτ_n–t

≤ lim

n→∞(1 –μ_n)(τn–t) +μ_n(Fτ_n–t)

≤ lim

n→∞

(1 –μ_n)τ_n–t+μ_nτ_n–t

≤ lim

n→∞τ_n–t=c, which implies that

n→∞lim(1 –μ_n)(τ_n–t) +μ_n(Fτn–t)=c. (4.11) From (4.11) and Lemma2.4, we have

n→∞lim τ_n–Fτ_n= 0.

On the contrary, assume that{τn}is bounded andlim_n→∞τn–Fτn= 0. Supposet∈ A(U,{τ_n}), so by Proposition2.5, we have

r

Ft,{τ_n}

=lim sup

n→∞ τ_n–Ft

≤lim sup

n→∞ (3Fτ_n–τ_n)+τ_n–t)

=lim sup

n→∞ τ_n–t

=r t,{τ_n}

=r U,{τ_n}

.

This impliesFt∈A(U,{τ_n}). SinceWis uniformly convex,A(U,{τ_n}) is singleton, hence

we haveFt=t.

Theorem 4.3 Assume thatWsatisfies Opial’s condition,then the sequence{τ_n}developed by iteration process(1.7)converge weakly to a point of F(F).

Proof From Lemma4.1, we see thatlim_n→∞τ_n–texists. In order to show the weak convergence of the iteration process (1.7) to a fixed point ofF, we will prove that{τ_n}has a unique weak subsequential limit inF(F). For this, let{τnj}and{τn_k}be two subsequences of{τ_n}which converges weakly toxandy, respectively. From Lemma4.2,lim_n→∞τ_n– Fτ_n= 0 andI–Fis demiclosed at zero by Lemma2.3. Thus (I–F)x= 0, that is,x=Fx.

Similarlyy=Fy.

Now we show uniqueness. Ifx=y, by Opial’s condition, we have

n→∞lim τ_n–x= lim

nj→∞τ_nj–x

< lim

nj→∞τ_nj–y

= lim

n→∞τ_n–y

= lim

nk→∞τ_nk–y

< lim

n_k→∞τ_nk–x

= lim

n→∞τ_n–x,

which is a contradiction; hencex=y. Consequently, the{τ_n}converge weakly to a point

ofF(F).

Theorem 4.4 LetF,UandWbe defined as in Lemma4.1.Then the sequence{τ_n}devel- oped by iteration process(1.7)converges to a point of F(F)if and only iflim inf_n→∞d(τ_n, F(F)) = 0,where d(τn,F(F)) =inf{τ_n–t:t∈F(F)}.

Proof The first part is trivial. So, we prove the converse part. Presume thatlim inf_n→∞d(τ_n, F(F)) = 0. From Lemma4.1,lim_n→∞τ_n–texists, for allt∈F(F) thereforelim_n→∞d(τn, F(F)) = 0 by hypothesis.

Now our assertion is that{τ_n}is a Cauchy sequence inU. Sincelim_n→∞d(τ_n,F(F)) = 0, and for a givenλ> 0, there existsm0∈Nsuch that for alln≥m0

d

τ_n,F(F)

<λ 2

⇒ inf

τ_n–t:t∈F(F)

<λ 2.

In particular,inf{τ_m0–t:t∈F(F)}<^λ₂. Therefore there existst∈F(F) such that τ_m0–t<λ

2. Now, form,n≥m0,

τ_n+m–τ_n ≤ τ_n+m–t+τ_n–t

≤ τ_m0–t+τ_m0–t

= 2τ_m0–t<λ.

Thus{τ_n}is a Cauchy sequence inU. AsU is closed, then there exists a pointq∈U such thatlim_n→∞τ_n=q. Now,lim_n→∞d(τn,F(F)) = 0 impliesd(q,F(F)) = 0, hence we getq∈

F(F).

Theorem 4.5 LetF:U→U be a mapping satisfying Suzuki’s condition(C),whereU is a nonempty,compact and convex subset of a uniformly convex Banach spaceW.Then the sequence{τ_n}developed by iteration process(1.7)converges strongly to a fixed point ofF.

Proof By Lemma2.2,F(F)=∅, so by Lemma4.2, we havelim_n→∞Fτ_n–τ_n= 0. Since U is compact, there exists a subsequence{τ_nj}of{τ_n}such thatτ_nj→tstrongly for some t∈U. By Proposition2.5, we have

τ_nj–Ft ≤3Fτ_nj–τ_nj+τ_nj–t, ∀j≥1.

Asj→ ∞, we getτ_nj→Ft, impliesFt=t,i.e.t∈F(F). Also,lim_n→∞τ_n–texists by Lemma4.1. Thustis the strong limit of the sequence{τ_n}itself.

Applying condition (I) we now prove a strong convergence result.

Theorem 4.6 LetF,U andW be defined as in Lemma4.1.Assume that the mapping F also satisfies condition(I).Then the sequence{τ_n}developed by iteration process(1.7) converges strongly to a fixed point ofF.

Proof We proved in Lemma4.2that

n→∞lim τ_n–Fτ_n= 0. (4.12)

Applying condition (I) and (4.12), we get 0≤ lim

n→∞ψ d

τ_n,F(F)

≤ lim

n→∞τ_n–Fτ_n= 0

⇒ lim

n→∞ψ d

τ_n,F(F)

= 0.

And hence

n→∞lim d

τn,F(F)

= 0.

So by Theorem4.4, the sequence{τ_n}converge strongly to a fixed point ofF. 5 An illuminate numerical example

The purpose of this section is to present a numerical example to compare the rate of con- vergence for a mapping satisfying Suzuki’s condition (C).

Example5.1 LetF: [0, 2]→[0, 2] be a mapping defined by

F(x) =

⎧⎨

⎩

2 –x, ifx∈[0,¹₉),

x+16

9 , ifx∈[¹₉, 2].

HereFsatisfies Suzuki’s condition (C), butFis not a non-expansive mapping.

Verification Forx=₁₀¹ andy=¹₉, we obtain x–y=

1 10–1

9 = 1

90. We have

Fx–Fy= 2 – 1

10–145 81

= 89 810> 1

90=x–y.

HenceFis not a non-expansive mapping.

We now show thatFsatisfies Suzuki’s condition (C).

We have the following cases:

Case I If eitherx,y∈[0,¹₉) orx,y∈[¹₉, 2], then obviouslyFsatisfies Suzuki’s condition (C).

Case II Letx∈[0,¹₉). Then¹₂x–Fx=¹₂x– (2 –x)=¹₂2x– 2=x– 1 ∈(⁸₉, 1]. For

1

2x–Fx ≤ x–y, we should have 1 –x≤y–ximplyingy≥1 andy∈[1, 2]. Now, Fx–Fy=

y+ 16

9 – 2 +x =

y+ 9x– 2 9

<1 9 and

x–y=|x–y|>

1 –1 9

= 8

9 =8

9 >1 9.

Hence,¹₂x–Fx ≤ x–y ⇒ Fx–Fy ≤ x–y.

Case III Let x∈[¹₉, 2]. Then ¹₂x–Fx= ¹₂^x+16₉ –x=^16–8x₁₈ ∈[0,¹³⁶₁₆₂]. For ¹₂x– Fx ≤ x–y, we should have^16–8x₁₈ ≤ |x–y|, which indicates two possibilities:

(a) Letx<y, then ^16–8x₁₈ ≤y–x,i.e. ^10x+16₁₈ ≤y ⇒ y∈[¹⁵⁴₁₆₂, 2]⊂[¹₉, 2]. So Fx–Fy=

x+ 16

9 –y+ 16 9

=1

9x–y ≤ x–y. Hence,¹₂x–Fx ≤ x–y ⇒ Fx–Fy ≤ x–y.

(b) Letx>y, then^16–8x₁₈ ≤x–y,i.e.y≤^26x–16₁₈ ⇒ y≤^–118₁₆₂ andy≤2, soy∈[0, 2]. Since y∈[0, 2] andy≤^26x–16₁₈ ⇒ ^18y+16₂₆ ≤x. Since the casex∈[¹⁶₂₆, 2] andy∈[¹₉, 2] is already discussed in Case I. Now consider,x∈[¹⁶₂₆, 2] andy∈[0,¹₉). Then

Fx–Fy= x+ 16

9 – 2 +y =

x+ 9y– 2 9

<1 9 and

x–y=|x–y|>

16 26–1

9 =

144 – 26 234

=118 234>1

9.

Hence,¹₂x–Fx ≤ x–y ⇒ Fx–Fy ≤ x–y. ThusFsatisfies Suzuki’s condition (C).

It can be easily seen from Table3and Fig.2that the JF iteration process converges to a fixed pointt= 2 of the mappingFfaster than the leading iteration processes with initial pointτ₀= 0.11 and control sequencesθ_n= 0.75,μ_n= 0.65 andγ_n= 0.55,n∈Z⁺.

6 Application to a nonlinear fractional differential equation

In this section, by using iteration process (1.7) we approximate the solution of a nonlinear fractional differential equation. Consider the following nonlinear fractional differential equation:

⎧⎨

⎩

D^αx(t) +D^βx(t) =f(t,x(t)) (0≤t≤1, 0 <β<α< 1),

x(0) =x(1) = 0, (6.1)

wheref : [0, 1]×R→Ris a continuous function andD^αandD^βdenote the Caputo frac- tional derivatives of orderαandβ, respectively.

Table 3 Numerical comparison of iteration processes

Iter. Mann Ishikawa S Picard-S Noor JF

1 0.110000 0.110000 0.110000 0.110000 0.110000 0.110000

2 1.370000 1.461000 1.881000 1.986778 1.466561 1.999635

3 1.790000 1.846285 1.992507 1.999907 1.849441 2.000000

4 1.930000 1.956163 1.999528 1.999999 1.957506 2.000000

5 1.976667 1.987498 1.999970 2.000000 1.988006 2.000000

6 1.992222 1.996435 1.999998 2.000000 1.996615 2.000000

7 1.997407 1.998983 2.000000 2.000000 1.999045 2.000000

.. .

.. .

.. .

.. .

.. .

.. .

.. .

13 1.999996 1.999999 2.000000 2.000000 2.000000 2.000000

14 1.999999 2.000000 2.000000 2.000000 2.000000 2.000000

15 2.000000 2.000000 2.000000 2.000000 2.000000 2.000000

Figure 2Graphical representation of the convergence of iteration processes

LetW=C[0, 1] be a Banach space of continuous function from [0, 1] intoRendowed with the supremum norm. The Green’s function associated to (6.1) is defined by

G(t) =t^α–1Eα–β,α

–t^α–β , whereE_α–β,α(–t^α–β) =_∞

k=0

(–t^α–β)^k

((α–β)k+α)is the Mittag–Leffler function. Many authors stud- ied the existence of the solution of problem (6.1) [e.g. see [4,13,14]]. Now we approximate the solution of problem (6.1) by utilizing iteration process (1.7) with the following assump- tion:

(C₁) Assume that

f(t,a) –f(t,b)≤c|a–b|

for allt∈[0, 1],a,b∈Randc≤α.

Theorem 6.1 LetW=C[0, 1]be a Banach space with supremum norm.Let{τ_n}be a se- quence defined by JF iteration process(1.7)for the operatorF:W→Wdefined by

F x(t)

= t

0

G(t–w)f w,x(w)

dw, (6.2)

∀t∈[0, 1],∀x∈W.Assume that the condition(C1)is satisfied.Then the sequence defined by JF iteration process(1.7)converges to a solution,say x^∗∈Wof the problem(6.1).

Proof Observe thatx^∗∈Wis a solution of (6.1) if and only ifx^∗is a solution of the integral equation

x(t) = t

0

G(t–w)f w,x(w)

dw.

Now, letx,y∈Wand for allt∈[0, 1]. Using (C1), we get F

x(t) –F

y(t)= _t

0

G(t–w)f w,x(w)

dw– _t

0

G(t–w)f w,y(w)

dw

≤ _t

0

G(t–w)f w,x(w)

–f

w,y(w)dw

≤ _t

0

G(t–w)cx(w) –y(w)dw

≤

sup

t∈[0,1]

t 0

G(t–w)dw

cx–y

≤ c αx–y.

Note that G(t) = t^α–1E_α–β,α(–t^α–β) ≤ t^α–1_1+|–t¹_α–β| ≤ t^α–1 for all t ∈ [0, 1]. Thus, sup_t∈[0,1]t

0G(t–w)dw≤_α¹. Hence, forx,y∈Wand for allt∈[0, 1], we have Fx–Fy ≤ x–y.

ThusF is a Suzuki generalized non-expansive mapping. Hence the JF iteration process

converges to the solution of (6.1).

Now, we present the following example for the validity of Theorem6.1.

Example6.2 Consider the following fractional differential equation:

⎧⎨

⎩

D^0.5x(t) +D^0.25x(t) =t³+ 1 (0≤t≤1),

x(0) =x(1) = 0. (6.3)

The exact solution of problem (6.3) is given by x(t) =

t 0

G(t–w)f w,x(w)

dw.

The operatorF:C[0, 1]→C[0, 1] is defined by Fx(t) =

_t

0

G(t–w)f w,x(w)

dw. (6.4)

For the initial guessτ₀(t) =t(1 –t),t∈[0, 1] and control sequencesθ_n= 0.85,μ_n= 0.65, n∈Z⁺, we observe that JF iteration process converges to the exact solution of problem (6.3) for the operator defined in (6.4) which is shown in Tables4–5and Figs.3–6. Further- more, we consider a Mittag-Leffler series expansion atk= 3,k= 11,k= 31 andk= 500