Classes of functions on common fixed points for two mappings of integral type with semi-implicit contractive conditions in metric spaces

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Adv. Fixed Point Theory, 6 (2016), No. 4, 486-497 ISSN: 1927-6303

CLASSES OF FUNCTIONS ON COMMON FIXED POINTS FOR TWO MAPPINGS OF INTEGRAL TYPE WITH SEMI-IMPLICIT CONTRACTIVE CONDITIONS IN

METRIC SPACES

ARSLAN HOJAT ANSARI1, YONGJIE PIAO2,∗, NAWAB HUSSAIN3 1_{Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran} 2_{Department of Mathematics, College of Science, Yanbian University, Yanji, China}

3_{Department of Mathematics, King Abdulaziz University P.O. Box 80203, Jeddah 21589, Saudi Arabia} Copyright c2016 Ansari, Piao, and Hussain. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. In this paper, by using the classesC andΦuof functions to introduce a generalization of the known

classΨof 5-dimensional functions, we discuss the existence problems of common fixed points for two mappings

of integral type with semi-implicit contractive conditions and give more general results and some particular forms.

Keywords:classC; classΨ∗; classΦu; semi-implicit; sub-additive; common fixed point.

2010 AMS Subject Classification:47H05, 47H10, 54E40, 54H25.

1. Introduction and Preliminaries

Throughout this paper, we assume that_R+= [0,+∞)and

Φ={φ :φ:_R+→_R+satisfying that φ is Lebesgue integral, summable on each compact

subset of_R+and Rε

0 φ(t)dt >0 for eachε >0}

The famous Banach’s contraction principle is as follows:

∗_{Corresponding author}

E-mail address: [email protected]

Received July 25, 2016

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Theorem 1.1.[1]Let f be a self mapping on a complete metric space(X,d)satisfying

d(f x,f y)≤cd(x,y), ∀x,y∈X, (1.1)

wherec∈[0,1)is a constant. Then f has a unique fixed point ˆx∈X such that limn→∞fnx=xˆ

for eachx∈X.

It is known that the Banach contraction principle has a lot of generalizations and various applications in many directions, see, for examples, [2-12] and the references cited therein. Es-pecially, in 1962, Rakotch[13] extended the Banach contraction principle with replacing the contraction constantcin (1.1) by a contraction functionγ and obtained the next theorem:

Theorem 1.2.[13]Let f be a self-mapping on a complete metric space(X,d)satisfying

d(f x,f y)≤γ(d(x,y))d(x,y), ∀x,y∈X, (1.2)

whereγ:_R+→[0,1)is a monotonically decreasing function. Then f has a unique fixed point

ˆ

x∈X such that limn→∞fnx=xˆfor eachx∈X.

In 2002, Branciari[14] gave an integral version of Theorem 1.1 as follows:

Z d(f x,f y)

0

φ(t)dt ≤c

Z d(x,y)

0

φ(t)dt, ∀x,y∈X, (1.3)

where c∈(0,1) is a constant and φ ∈Φ. Then f has a unique fixed point ˆx∈X such that

limn→∞fnx=xˆfor eachx∈X.

In 2011, Liu and Li[15] modified the method of Rakotch to generalize the Branciari’s fixed point theorem with replacing the contraction constantcin (1.3) by contraction functionsα and β and established the following fixed point theorem:

Z d(f x,f y)

0

φ(t)dt≤ α(d(x,y))

Z d(x,f x)

0

φ(t)dt+β(d(x,y))

Z d(y,f y)

0

φ(t)dt,∀x,y∈X, (1.4)

whereφ ∈Φandα,β :_R+→[0,1)are two functions with

α(t) +β(t)<1,∀t∈_R+; lim sup

s→0+

β(s)<1; lim sup

s→t+

α(s)

1−β(s) <1, ∀

t>0.

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In [16], Jin and Piao discussed the existence problems of unique common fixed points for two mappings of integral type with variable coefficient in metric spaces and obtained the more general results, the main results generalize and improve Theorem 1.4. Also they introduce the following two definitions and obtain a unique common fixed point theorem for two mappings of integral type with semi-implicit contractive conditions:

The functionφ ∈Φis called to be sub-additive if and only if for alla,b∈_R+,

Z a+b

0

φ(t)dt ≤ Z a

0

φ(t)dt+ Z b

0

φ(t)dt.

Letψ ∈Ψ[16]if and only ifψ:_R+5→_R+is a continuous and nondecreasing function about

the 4th and 5th variables and the following conditions hold:

(i) there existsh1∈(0,1)such thatu≤ψ(v,u,v,0,u+v)impliesu≤h1v; (ii) there existsh₂∈(0,1)such thatu≤ψ(v,v,u,u+v,0)impliesu≤h2v; (iii)ψ(t,0,0,t,t)<t,ψ(0,t,0,0,t)<t andψ(0,0,t,t,0)<tfor allt>0.

Theorem 1.5.[16] Let(X,d) be a complete metric space, f,g:X →X two mappings. If for eachx,y∈X,

Z d(f x,gy)

0

φ(t)dt≤ψ

Z d(x,y)

0

φ(t)dt,

Z d(x,f x)

0

φ(t)dt,

Z d(y,gy)

0

φ(t)dt,

Z d(x,gy)

0

φ(t)dt,

Z d(y,f x)

0

φ(t)dt

,

(1.5)

whereφ ∈Φis sub-additive andψ ∈Ψ. Then f andghave a unique common fixed point.

The aim of this paper is to use two classesC and Φu of real functions to introduce a real

generalization of the above classΨ and to discuss the unique existence problems of common

fixed points for two self-mappings of integral type with a new semi-implicit limitation in a non-complete metric space. The obtained results further generalize and improve the corresponding conclusions in the literature.

To do this, we introduce the definitions ofC andΦuand give a well-known lemma.

The following concept of classC of functions was introduced by A.H.Ansari[17].

Definition 1.6.[17]A mappingF:[0,∞)2→_Ris calledC-classfunction if it is continuous and

satisfies following axioms: (1)F(s,t)≤s;

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Note for someF we have thatF(0,0) =0. We denoteC-class functions asC.

Example 1.7.[17]The following functionsF :[0,∞)2→Rare elements ofC, for all s,t ∈[0,∞):

(1)F(s,t) =s−t,F(s,t) =s⇒t =0;

(2)F(s,t) =ms, 0<m<1,F(s,t) =s⇒s=0;

(3)F(s,t) = ₍₁₊s_t₎r;r∈(0,∞),F(s,t) =s⇒s=0 ort =0;

(4)F(s,t) =log(t+as)/(1+t),a>1,F(s,t) =s⇒s=0 ort=0; (5)F(s,t) =ln(1+as)/2,a>e,F(s,1) =s⇒s=0;

(6)F(s,t) = (s+l)(1/(1+t)r)−l,l>1,r∈(0,∞),F(s,t) =s⇒t =0;

(7)F(s,t) =slog_t₊_aa,a>1,F(s,t) =s⇒s=0 ort=0; (8)F(s,t) =s−(1₂+₊s_s)(₁₊t_t),F(s,t) =s⇒t =0;

(9)F(s,t) =sβ(s),β :[0,∞)→[0,1)is continuous,F(s,t) =s⇒s=0;

(10)F(s,t) =s−ϕ(s),F(s,t) =s⇒s=0,hereϕ:[0,∞)→[0,∞)is a continuous function

such thatϕ(t) =0⇔t=0;

(11) F(s,t) = sh(s,t),F(s,t) = s⇒s =0,here h:[0,∞)×[0,∞)→[0,∞)is a continuous

function such thath(t,s)<1 for allt,s>0; (12)F(s,t) =s−(₁2₊+t_t)t,F(s,t) =s⇒t=0; (13)F(s,t) =pn _ln₍₁₊_sn₎_,_F₍_s_,_t_{) =}_s_⇒_s₌_0;

(14) F(s,t) =φ(s),F(s,t) =s⇒s=0,here φ :[0,∞)→[0,∞) is a upper semicontinuous

function such thatφ(0) =0,andφ(t)<t fort>0,

(15)F(s,t) = ₍₁₊s_s₎r;r∈(0,∞),F(s,t) =s⇒s=0 ;

(16)F(s,t) =ϑ(s);ϑ :_R+×_R+→_Ris a generalized Mizoguchi-Takahashi type function

,F(s,t) =s⇒s=0; (17)F(s,t) = s

Γ(1/2) R∞

0 e

−x

√

x+tdx, whereΓis the Euler Gamma function.

Definition 1.8.LetΦube a set of all functionsϕ:R+→R+satisfying the following conditions: (1)ϕ is continuous;

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Lemma 1.9.[18] Suppose (X,d) is a metric space. Let {x_n} be a sequence in X such that d(x_n,x_n₊₁)→0 as n→∞. If {xn} is not a Cauchy sequence, then there exist an ε >0 and

sequences of positive integers{m(k)}and {n(k)} with m(k)>n(k)>k for eachk∈_Nsuch thatd(x_m₍_k₎,x_n₍_k₎)≥ε,d(x_m₍_k₎₋₁,x_n₍_k₎)<ε and the following result holds

lim

k→∞

d(x_m₍_k₎₋₁,x_n₍_k₎₊₁) = lim

k→∞

d(x_m₍_k₎,x_n₍_k₎) = lim

k→∞

d(x_m₍_k₎₋₁,x_n₍_k₎) =ε.

Remark 1.10. Under the conditions of Lemma 1.9, We easily obtian the following result: lim

k→∞

d(x_m₍_k₎,x_n₍_k₎₊₁) = lim

k→∞

d(x_m₍_k₎₊₁,x_n₍_k₎) =lim

k→∞

d(x_m₍_k₎₊₁,x_n₍_k₎₊₁) =lim

k→∞

d(x_m₍_k₎,x_n₍_k₎₋₁) =ε.

2. Common fixed points

Lemma 2.1.[15]Letφ ∈Φand{rn}n∈Nbe a nonnegative sequence with limn→∞rn=a. Then

lim

n→∞

Z r_n

0

φ(t)dt = Z a

0

φ(t)dt.

Lemma 2.2.[15]Letφ ∈Φand{rn}n∈Nbe a nonnegative sequence. Then

lim

n→∞

Z r_n

0

φ(t)dt =0⇐⇒ lim

n→∞rn=0.

Now, we give a new and real generalization of the classΨin [16] as follows.

Letψ ∈Ψ∗ if and only ifψ :R+5→R+ is a continuous and nondecreasing function about the 4th and 5th variables and the following conditions hold:

(i) there existsF1∈C,ϕ1∈Φu,such thatu≤ψ(v,u,v,0,u+v)impliesu≤F1(v,ϕ1(v)); (ii) there existsF₂∈C,ϕ2∈Φu,such thatu≤ψ(v,v,u,u+v,0)impliesu≤F2(v,ϕ2(v)); (iii)ψ(t,0,0,t,t)<t,ψ(0,t,0,0,t)<t andψ(0,0,t,t,0)<tfor allt>0.

Example 2.1.Defineψ :_R+5→_R+ as follows

ψ(x1,x2,x3,x4,x5) =a1x1+a2x2+a3x3+a4x4+a5x5,∀x1,x2,x3,x4,x5∈R+,

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We also know that the definition ofψ∈Ψ∗is a proper generalization ofψ ∈Ψ.

The functionφ ∈Φis called to be strictly increasing about integral type if for anyx,y∈[0,∞)

withx<y,

Z x

0

φ(t)dt < Z y

0

φ(t)dt.

Example 2.2.Letφ :_R+ →_R+,φ(t) =₁₊1_t for eacht∈_R+.Then obviouslyφ ∈Φand for all

a,b∈_R+_,

Z a+b

0

φ(t)dt =ln(1+a+b)≤ln(1+a+b+ab)=ln(1+a)(1+b)= Z a

0

φ(t)dt+ Z b

0

φ(t)dt.

And for 0≤x<y,

Z x

0 1

1+tdt =ln

(1+x)_<

ln(1+y)= Z y

0 1 1+tdt.

Henceφ(t) = ₁1₊_t is a sub-additive and strictly increasing function about integral type.

The following results is the main common fixed point theorem.

Theorem 2.3. Let(X,d)be a metric space, f,g:X →X two mappings. Suppose that for each x,y∈X,

Z d(f x,gy)

0

φ(t)dt≤ψ

Z d(x,y)

0

φ(t)dt,

Z d(x,f x)

0

φ(t)dt,

Z d(y,gy)

0

φ(t)dt,

Z d(x,gy)

0

φ(t)dt,

Z d(y,f x)

0

φ(t)dt

,

(2.1)

whereφ ∈Φis sub-additive and strictly increasing about the integral type andψ ∈Ψ∗. If f X

orgX is complete, then f andghave a unique common fixed point.

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Since

Z d₂_n

0

φ(t)dt=

Z d(f x₂n,gx₂_n−₁)

0

φ(t)dt

≤ψ

Z d(x₂_n,x₂_n−₁)

0

φ(t)dt,

Z d(x₂_n,f x₂_n)

0

φ(t)dt,

Z d(x₂_n−₁,gx₂_n−₁)

0

φ(t)dt,

Z d(x₂_n,gx₂_n−₁)

0

φ(t)dt,

Z d(x₂_n−₁,f x₂_n)

0

φ(t)dt

=ψ

Z d₂_n−₁

0

φ(t)dt, Z d₂_n

0

φ(t)dt, Z d₂_n−₁

0

φ(t)dt,0,

Z d(x₂_n−₁,x₂_n+₁)

0

φ(t)dt

≤ψ

Z d2n−1

0

φ(t)dt, Z d₂_n

0

φ(t)dt,0,

Z d₂_n−₁+d₂_n+₁

0

φ(t)dt

≤ψ

Z d₂_n−₁

0

φ(t)dt, Z d₂_n

0

φ(t)dt,0, Z d₂_n−₁

0

φ(t)dt+ Z d₂_n

0

φ(t)dt

.

So by (i),

Z d₂_n

0

φ(t)dt ≤F1

Z d₂_n−₁

0

φ(t)dt,ϕ1(

Z d₂_n−₁

0

φ(t)dt)

. (2.2)

Similarly,

Z d₂_n+₁

0

φ(t)dt =

Z d(x₂_n+₁,x₂_n+₂)

0

φ(t)dt =

Z d(f x₂_n,gx₂_n+₁)

0

φ(t)dt

≤ψ

Z d(x2n,x2n+1)

0

φ(t)dt,

Z d(x₂_n,f x₂_n)

0

φ(t)dt,

Z d(x₂_n+₁,gx₂_n+₁)

0

φ(t)dt,

Z d(x₂_n,gx₂_n+₁)

0

φ(t)dt,

Z d(x₂_n+₁,f x₂_n)

0

φ(t)dt

=ψ

Z d₂_n

0

φ(t)dt, Z d₂_n

0

φ(t)dt, Z d₂_n+₁

0

φ(t)dt,

Z d(x₂_n,x₂_n+₂)

0

φ(t)dt,0

≤ψ Z d₂_n

0

φ(t)dt, Z d₂_n

0

φ(t)dt,

Z d₂_n+d₂_n+₁

0

φ(t)dt,0

≤ψ Z d₂_n

0

φ(t)dt, Z d₂_n

0

φ(t)dt, Z d₂_n

0

φ(t)dt+ Z d₂_n+₁

0

φ(t)dt,0

.

So by (ii),

Z d₂_n+₁

0

φ(t)dt ≤F2

Z d2n

0

φ(t)dt,ϕ2(

Z d₂_n

0

φ(t)dt)

. (2.3)

Combining (2.3) and (2.4) and using the property ofF1andF2, we have

Z d_n+₁

0

φ(t)dt ≤ Z d_n

0

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If there existsn₀∈_Nsuch thatd_n₀+1>dn0, then by the strictly increasing property ofφ about integral type, we obtain

Z d_n

0+1

0

φ(t)dt> Z d_n

0

φ(t)dt,

which is a contradiction with (2.4), hence we have

d_n₊₁≤d_n,∀n=0,1,2,· · ·. (2.5)

So there isu∈_R+ such that limn→∞dn=u. Ifu>0, then by Lemma 2.1 and (2.2),

Z u

0

φ(t)dt=lim

n→∞

Z d₂_n

0

φ(t)dt≤F1(lim

n→∞

Z d₂_n−₁

0

φ(t)dt,lim

n→∞ϕ1(

Z d₂_n−₁

0

φ(t)dt))

=F1(

Z u

0

φ(t)dt,ϕ1(

Z u

0

φ(t)dt)).

So Ru

0 φ(t)dt =0 orϕ1(

Ru

0 φ(t)dt) =0 by the property ofF1, thus

Ru

0φ(t)dt =0, which is a contradiction with the condition ofφ. Therefore,u=0, i.e., limn→∞dn=0.

We claim that{x_n}is Cauchy. Otherwise, by Lemma 1.9 and Remark 1.10, there existsε>0

such that for each k∈_N, there exist m(k),n(k)∈_N with m(k)>n(k)such that the parity of m(k)andn(k)is different and the following result holds

lim

k→∞

d(x_m₍_k₎,x_n₍_k₎) = lim

k→∞

d(x_m₍_k₎₊₁,x_n₍_k₎) =lim

k→∞

d(x_m₍_k₎₊₁,x_n₍_k₎₊₁) = lim

k→∞

d(x_m₍_k₎,x_n₍_k₎₊₁) =ε.

(2.6)

Ifm(k)is even andn(k)is odd, then by Lemma 2.1, (2.1), (2.6) and (iii),

0<

Z ε

0

φ(t)dt= lim

k→∞

Z d(x_m(k)+₁,x_n(k)+₁)

0

φ(t)dt= lim

k→∞

Z d(f x_m(k),gx_n(k))

0

φ(t)dt

≤lim

k→∞

ψ

Z d(x_m(k),x_n(k))

0

φ(t)dt,

Z d(x_m(k),f x_m(k))

0

φ(t)dt,

Z d(x_n(k),gx_n(k))

0

φ(t)dt,

Z d(x_m(k),gx_n(k))

0

φ(t)dt,

Z d(f x_m(k),x_n(k))

0

φ(t)dt

=ψ

Z ε

0

φ(t)dt,0,0, Z ε

0

φ(t)dt, Z ε

0

φ(t)dt

<

Z ε

0

φ(t)dt.

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If f X is complete, then{x_n} is a Cauchy sequence and x₂_n+1∈ f X(for all n=0,1,2,· · ·) implies there existsx∗∈ f X such thatx₂_n₊₁→x∗.Hence

d(x₂_n₊₂,x∗)≤d(x₂_n₊₂,x₂_n₊₁) +d(x₂_n₊₁,x∗)

impliesx₂_n₊₂→x∗, thereforex_n→x∗ asn→∞.Similarly we havexn→y∗ for somey∗∈gX

for the case thatgX is complete. Hence we can assume thatx_n→x∗∈ f X∪gX asn→∞for

any case.

If f x∗6=x∗, thend(f x∗,x∗)>0, hence by Lemma 2.1 and (2.1) and (iii),

0<

Z d(f x∗,x∗)

0

φ(t)dt = lim

n→∞

Z d(f x∗,gx₂_n+₁)

0

φ(t)dt

≤limψ

Z d(x∗,x2n+1)

0

φ(t)dt,

Z d(x∗,f x∗)

0

φ(t)dt,

Z d(x₂_n+₁,gx₂_n+₁)

0

φ(t)dt,

Z d(x∗,gx₂_n+₁)

0

φ(t)dt,

Z d(f x∗,x₂_n+₁)

0

φ(t)dt

=ψ0,

Z d(f x∗,x∗)

0

φ(t)dt,0,0,

Z d(f x∗,x∗)

0

φ(t)dt

<

Z d(f x∗,x∗)

0

φ(t)dt.

This is a contradiction, hence f x∗ =x∗. Similarly, we obtain gx∗ =x∗. Therefore, x∗ is a common fixed point of f andg.

Ify∗is another common fixed point of f andg, thend(x∗,y∗)>0, hence by (2.1) and (iii),

0<

Z d(x∗,y∗)

0

φ(t)dt =

Z d(f x∗,gy∗)

0

φ(t)dt

≤ψ

Z d(x∗,y∗)

0

φ(t)dt,

Z d(x∗,f x∗)

0

φ(t)dt,

Z d(y∗,gy∗)

0

φ(t)dt,

Z d(x∗,gy∗)

0

φ(t)dt,

Z d(f x∗,y∗)

0

φ(t)dt

=ψ

Z d(x∗,y∗)

0

φ(t)dt,0,0,

Z d(x∗,y∗)

0

φ(t)dt,

Z d(x∗,y∗)

0

φ(t)dt

<

Z d(x∗,y∗)

0

φ(t)dt.

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Theorem 2.4. Let(X,d)be a metric space, f,g:X →X two mappings. Suppose that for all x,y∈X,

ln(1+d(f x,gy))≤ψ

ln(1+d(x,y)),ln(1+d(x,f x)),ln(1+d(y,gy)),ln(1+d(x,gy)),ln(1+d(y,f x))

,

whereψ ∈Ψ∗. If f X orgX is complete, then f andghave a unique common fixed point.

Combining Theorem 2.4 and Example 2.1, we obtain the following result.

Theorem 2.5. Let (X,d) be a metric space, f,g:X →X two mappings. Suppose that for eachx,y∈X,

1+d(f x,gy)≤(1+d(x,y))a1₍₁₊_d₍_x_,_{f x}₎₎a2₍₁₊_d₍_y_,_gy₎₎a3₍₁₊_d₍_x_,_gy₎₎a4₍₁₊_d₍_y_,_{f x}₎₎a5_, wherea_i≥0 for alli=1,2,3,4,5 anda₁+a₂+a₃+2a4+2a5<1. If f X orgX is complete, then f andghave a unique common fixed pointu.

From Theorem 2.3, we obtain the next more general common fixed point theorem.

Theorem 2.6. Let(X,d)be a metric space,m,n∈_Nand f,g:X →X two mappings. Suppose that for eachx,y∈X,

Z d(fmx,gny)

0

φ(t)dt

≤ψ

Z d(x,y)

0

φ(t)dt,

Z d(x,fmx)

0

φ(t)dt,

Z d(y,gny)

0

φ(t)dt,

Z d(x,gny)

0

φ(t)dt,

Z d(y,fmx)

0

φ(t)dt

,

(2.7)

whereφ ∈Φis sub-additive and strictly increasing about integral type andψ ∈Ψ∗. If fmX or

gnX is complete, then f andghave a unique common fixed point.

Proof.LetF= fmandG=gn, thenF andGsatisfy all of the conditions of Theorem 2.3, hence there exists an unique elementu∈X such that fmu=Fu=u=Gu=gnu.If f u6=u, then by (2.7) and (iii),

0<

Z d(f u,u)

0

φ(t)dt=

Z d(fmf u,gnu)

0

φ(t)dt

≤ψ

Z d(f u,u)

0

φ(t)dt,

Z d(f u,fmf u)

0

φ(t)dt,

Z d(u,gnu)

0

φ(t)dt,

Z d(f u,gnu)

0

φ(t)dt,

Z d(u,fmf u)

0

φ(t)dt

=ψ

Z d(f u,u)

0

φ(t)dt,0,0,

Z d(f u,u)

0

φ(t)dt,

Z d(u,f u)

0

φ(t)dt

<

Z d(f u,u)

0

(11)

which is a contradiction. Hence f u=u. Similarly, we can obtaingu=u. Souis the common fixed point of f andg. The uniqueness is obvious.

Conflict of Interests

The authors declare that there is no conflict of interests.

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