Hussain et al. (2013) established new fixed point results in complete metric space. In this paper, we prove fixed point results of α-admissible mappings with respect to η, for modified contractive condition in complete metric space. An example is given to show the validity of our work. Our results generalize/improve several recent and classical results existing in the literature.
1. Preliminaries and Scope
The study of fixed point problems in nonlinear analysis has emerged as a powerful and very important tool in the last 60 years. Particularly, the technique of fixed point theory has been applicable to many diverse fields of sciences such as engineering, chemistry, biology, physics, and game theory. Over the years, fixed point theory has been generlized in many directions by several mathematicians (see [1–36]).
In 1973, Geraghty [12] studied different contractive conditions and established some useful fixed point theorems.
In 2012, Samet et al. [33] introduced a concept of α-ψ-contractive type mappings and established various fixed point theorems for mappings in complete metric spaces. Afterwards Karapinar and Samet [10] refined the notions and obtained various fixed point results. Hussain et al. [17] extended the concept of α-admissible mappings and obtained useful fixed point theorems. Subsequently, Abdeljawad [4] introduced pairs of α-admissible mappings satisfying new sufficient contractive conditions different from those in [17, 33] and proved fixed point and common fixed point theorems. Lately, Salimi et al. [32] modified the concept of α-ψ-contractive mappings and established fixed point results.
We define Ω the family of nondecreasing functions ψ:[0,+∞)→[0,+∞) such that ∑n=1+∞ψn(t)<+∞, and ψ(0)=0 for each t>0 where ψn is the nth term of ψ.
Lemma 1 (see [32]).
If ψ∈Ω, then ψ(t)<t for all t>0.
Definition 2 (see [33]).
Let (X,d) be a metric space and let S:X→X be a given mapping. We say that S is an α-ψ-contractive mapping if there exist two functions α:X×X→[0,+∞) and ψ∈Ω such that
(1)α(x,y)d(Sx,Sy)≤ψ(d(x,y)),
for all x,y∈X.
Definition 3 (see [33]).
Let S:X→X and α:X×X→[0,+∞). One says that S is α-admissible if x,y∈X, α(x,y)≥1⇒α(Sx,Sy)≥1.
Example 4.
Consider X=[0,∞). Define S:X→X and α:X×X→[0,∞) by Sx=2x, for all x,y∈X and
(2)α(x,y)={ey/xifx≥y,x≠00ifx<y.
Then S is α-admissible.
Definition 5 (see [32]).
Let S:X→X and let α,η:X×X→[0,+∞) be two functions. One says that S is α-admissible mapping with respect to η if x,y∈X, α(x,y)≥η(x,y)⇒α(Sx,Sy)≥η(Sx,Sy). Note that if one takes η(x,y)=1, then this definition reduces to definition [33]. Also if we take α(x,y)=1, then one says that S is an η-subadmissible mapping.
2. Main Results
In this section, we prove fixed point theorems for α-admissible mappings with respect to η, satisfying modified (α-η)-contractive condition in complete metric space.
Theorem 6.
Let (X,d) be a complete metric space and let S is α-admissible mappings with respect to η. Assume that there exists a function β:[0,+∞)→[0,1) such that, for any bounded sequence {tn} of positive reals, β(tn)→1 implies tn→0 such that
(3)(d(Sx,Sy)+l)α(x,Sx)α(y,Sy)≤(β(d(x,y))d(x,y)+l)η(x,Sx)η(y,Sy)
for all x,y∈X where l≥1; then suppose that one of the following holds:
Sis continuous;
if {xn} is a sequence in X such that α(xn,xn+1)≥η(xn,xn+1) for all n∈N∪{0} and xn→p∈X as n→+∞, then
(4)α(p,Sp)≥η(p,Sp).
If there exists x0,x1∈X such that α(x0,x1)≥η(x0,x1), then S has a unique fixed point.
Proof.
Let x0∈X and define
(5)xn+1=Sxn,∀n≥0.
We will assume that xn≠xn+1 for each n. Otherwise, there exists an n such that xn=xn+1. Then xn=Sxn and xn is a fixed point of S. Since α(x0,x1)≥η(x0,x1) and S is α-admissible mapping with respect to η, we have
(6)α(x1,x2)=α(Sx0,Sx1)≥η(Sx0,Sx1)=η(x1,x2).
By continuing in this way, we have
(7)α(xn,xn+1)≥η(xn,xn+1)
for all n∈N∪{0}. From (7), we have
(8)α(xn-1,xn)α(xn,xn+1)≥η(xn-1,xn)η(xn,xn+1).
Thus applying the inequality (3), with x=xk-1 and y=xk, we obtain
(9)(d(xk,xk+1)+l)η(xk-1,Sxk-1)η(xk,Sxk)=(d(Sxk-1,Sxk)+l)η(xk-1,Sxk-1)η(xk,Sxk)≤(d(Sxk-1,Sxk)+l)α(xk-1,Sxk-1)α(xk,Sxk)≤(β(d(xk-1,xk))d(xk-1,xk)+l)η(xk-1,Sxk-1)η(xk,Sxk)
which implies that
(10)d(xk,xk+1)≤β(d(xk-1,xk))d(xk-1,xk).
We suppose that
(11)d(xk,xk+1)≤d(xk-1,xk).
Then we prove that d(xk-1,xk)→0. It is clear that {d(xk-1,xk)} is a decreasing sequence. Therefore, there exists some positive number ϱ such that limn→∞d(xk,xk+1)=ϱ. Now we will prove that ϱ=0. From (10), we have
(12)d(xk,xk+1)d(xk-1,xk)≤β(d(xk-1,xk))≤1.
Now by taking limit k→∞, we have
(13)1=dd=limk→∞d(xk,xk+1)limk→∞d(xk-1,xk)≤β(d(xk-1,xk))≤1,limk→∞β(d(xk-1,xk))=1.
By using property of β function, we have limk→∞d(xk-1,xk)=0. Thus
(14)limk→∞d(xk,xk+1)=0.
Now we prove that sequence {xn} is Cauchy sequence. Suppose on contrary that {xn} is not a Cauchy sequence. Then there exists ϵ>0 and sequences {xmk} and {xnk} such that, for all positive integers k, we have nk>mk>k,
(15)d(xmk,xnk)≥ϵ,d(xmk,xnk-1)<ϵ.
By the triangle inequality, we have
(16)ϵ≤d(xmk,xnk)≤d(xmk,xnk-1)+d(xnk-1,xnk)<ϵ+d(xnk-1,xnk)
for all k∈N. Now taking limit as k→+∞ in (16) and using (14), we have
(17)limk→∞d(xmk,xnk)=ϵ.
Again using triangle inequality, we have
(18)d(xmk,xnk)≤d(xmk,xmk+1)+d(xmk+1,xnk+1)+d(xnk+1,xnk),d(xmk+1,xnk+1)≤d(xmk+1,xmk)+d(xmk,xnk)+d(xnk,xnk+1).
Taking limit as k→+∞ and using (14) and (17), we obtain
(19)limk→+∞d(xmk+1,xnk+1)=ϵ.
By using (3), (17), and (19), we have
(20)(d(xmk+1,xnk+1)+l)η(xmk,Sxmk)η(xnk,Sxnk)≤(d(xmk+1,xnk+1)+l)α(xmk,Sxmk)α(xnk,Sxnk)≤(d(Sxmk,Txnk)+l)α(xmk,Sxmk)α(xnk,Sxnk)≤(β(d(xmk,xnk))d(xmk,xnk)+l)η(xmk,Sxmk)η(xnk,Sxnk)
which implies that
(21)d(xmk+1,xnk+1)≤β(d(xmk,xnk))d(xmk,xnk).
Therefore, we have
(22)d(xmk+1,xnk+1)d(xmk,xnk)≤β(d(xmk,xnk))≤1.
Now taking limit as k→+∞ in (22), we get
(23)limn→∞β(d(xmk,xnk))=1.
Hence limk→∞d(xmk,xnk)=0<ϵ, which is a contradiction. Hence {xn} is a Cauchy sequence. Since X is complete so there exists p∈X such that xn→p. Now we prove that p=Sp. Suppose (i) holds; that is, S is continuous, so we get
(24)Sp=Slimn→∞xn=limn→∞Sxn=limn→∞xn+1=p.
Thus p=Sp. Now we suppose that (ii) holds. Since
(25)α(xn,xn+1)≥η(xn,xn+1)
for all n∈N∪{0}. By the hypotheses of (ii), we have
(26)α(p,Sp)α(xk,Sxk)≥η(p,Sp)η(xk,Sxk).
Using the triangle inequality and (3), we have
(27)(d(Sp,xk+1)+l)η(p,Sp)η(xk,Sxk)=(d(Sp,Sxk)+l)η(p,Sp)η(xk,Sxk)≤(d(Sp,Sxk)+l)α(p,Sp)α(xk,Sxk)≤(β(d(p,xk))d(p,xk)+l)η(p,Sp)η(xk,Sxk)
which implies that
(28)d(Sp,xk+1)≤β(d(p,xk))d(p,xk).
Letting k→∞ then we have d(p,Sp)=0. Thus p=Sp. Let there exists q to be another fixed point of Sq∈X, s.t q=Sq;
(29)(d(p,q)+l)η(p,Sp)η(q,Sq)=(d(Sp,Sq)+l)η(p,Sp)η(q,Sq)≤(d(Sp,Sq)+l)α(p,Sp)α(q,Sq)≤(β(d(p,q))d(p,q)+l)η(p,Sp)η(q,Sq)
which implies that
(30)d(p,q)+l≤β(d(p,q))d(p,q)+l.
By the property of β function, β(d(p,q))=1, implies d(p,q)=0; then we have p=q. Hence S has a unique fixed point.
If η(x,y)=1 in Theorem 6, we get the following corollary.
Corollary 7 (see [17]).
Let (X,d) be a complete metric space and let S be α-admissible mapping. Assume that there exists a function β:[0,+∞)→[0,1) such that, for any bounded sequence {tn} of positive reals, β(tn)→1 implies tn→0 such that
(31)(d(Sx,Sy)+l)α(x,Sx)α(y,Sy)≤β(d(x,y))d(x,y)+l,
for all x,y∈X, where l≥1. Suppose that either
S is continuous, or
if {xn} is a sequence in X such that α(xn,xn+1)≥1 for all n∈N∪{0} and xn→p∈X as n→+∞, then
(32)α(p,Sp)≥1.
If there exists x0,x1∈X such that α(x0,x1)≥1; then S has a fixed point.
If α(x,y)=1 in Theorem 6, we get the following corollary.
Corollary 8.
Let (X,d) be a complete metric space and let S be η-subadmissible mapping. Assume that there exists a function β:[0,+∞)→[0,1) such that, for any bounded sequence {tn} of positive reals, β(tn)→1 implies tn→0 such that
(33)(d(Sx,Sy)+l)≤(β(d(x,y))d(x,y)+l)η(x,Sx)η(y,Sy)
for all x,y∈X where l≥1; then suppose that one of the following holds:
S is continuous;
if {xn} is a sequence in X such that η(xn,xn+1)≤1 for all n∈N∪{0} and xn→p∈X as n→+∞, then
(34)η(p,Sp)≤1.
If there exists x0,x1∈X such that η(x0,x1)≤1, then S has a fixed point.
Example 9.
Let X=[0,∞) with usual metric d(x,y)=|x-y| for all x,y∈X and S:X→X, α:X×X→[0,∞) and β:[0,+∞)→[0,1] for all x,y∈X be defined by
(35)Sx={0ifx∈[0,1]xifx∈(1,5],α(x,y)={1ifx≥y0ifx<y,β(t)=1t,β(0)∈[0,1].
We prove that Corollary 7 can be applied to S. Let x,y∈X; clearly Sx≤x and Sy≤y, then S of α-admissible mapping α(x,y)≥1, and α(x,Sx)≥1, α(y,Sy)≥1, and α(x,Sx)α(y,Sy)≥1 imply that
(36)(d(Sx,Sy)+l)α(x,Sx)α(y,Sy)=Sx-Sy+l=x-y+l≤x-yx+y+l≤2(x-y)3x-y+l=β(d(x,y))(d(x,y))+l.
If α(x,Sx)α(y,Sy)=0, then we have
(37)(d(Sx,Sy)+l)α(x,Sx)α(y,Sy)=1≤β(d(x,y))(d(x,y))+l.
Let x=5 and y=2; then
(38)d(S5,S2)α(5,S5)α(3,S3)=0.8218≤β(d(5,3))(d(5,3))=1.4142.
Theorem 10.
Let (X,d) be a complete metric space and let S be α-admissible mappings with respect to η. Assume that there exists a function β:[0,+∞)→[0,1) such that, for any bounded sequence {tn} of positive reals, β(tn)→1 implies tn→0 such that
(39)α(x,Sx)α(y,Sy)d(Sx,Sy)≤η(x,Sx)η(y,Sy)β(d(x,y))d(x,y)
for all x,y∈X; then suppose that one of the following holds:
S is continuous;
if {xn} is a sequence in X such that α(xn,xn+1)≥η(xn,xn+1) for all n∈N∪{0} and xn→p∈X as n→+∞, then
(40)α(p,Sp)≥η(p,Sp).
If there exists x0,x1∈X such that α(x0,x1)≥η(x0,x1), then S has a fixed point.
Proof.
Let x0∈X and define
(41)xn+1=Sxn,∀n≥0.
We will assume that xn≠xn+1 for each n. Otherwise, there exists an n such that xn=xn+1. Then xn=Sxn and xn is a fixed point of S. Since α(x0,x1)≥η(x0,x1) and S is α-admissible mapping with respect to η, we have
(42)α(x1,x2)=α(Sx0,Sx1)≥η(Sx0,Sx1)=η(x1,x2).
By continuing in this way, we have
(43)α(xn,xn+1)≥η(xn,xn+1)
for all n∈N∪{0}. From (43), we have
(44)α(xn-1,xn)α(xn,xn+1)≥η(xn-1,xn)η(xn,xn+1).
Thus applying the inequality (39), with x=xk-1 and y=xk, we obtain
(45)η(xk-1,Sxk-1)η(xk,Sxk)d(xk,xk+1)=η(xk-1,Sxk-1)η(xk,Sxk)d(Sxk-1,Sxk)≤α(xk-1,Sxk-1)α(xk,Sxk)d(Sxk-1,Sxk)≤η(xk-1,Sxk-1)η(xk,Sxk)β(d(xk-1,xk))×d(xk-1,xk)
which implies that
(46)d(xk,xk+1)≤β(d(xk-1,xk))d(xk-1,xk).
We suppose that
(47)d(xk,xk+1)≤d(xk-1,xk).
Then we prove that d(xk-1,xk)→0. It is clear that {d(xk-1,xk)} is a decreasing sequence. Therefore, there exists some positive number ϱ such that limn→∞d(xk,xk+1)=ϱ. Now we will prove that ϱ=0. From (47), we have
(48)d(xk,xk+1)d(xk-1,xk)≤β(d(xk-1,xk))≤1.
Now by taking limit k→∞, we have
(49)1=dd=limk→∞d(xk,xk+1)limk→∞d(xk-1,xk)≤β(d(xk-1,xk))≤1,limk→∞β(d(xk-1,xk))=1.
By using property of β function, we have limk→∞d(xk-1,xk)=0. Thus
(50)limk→∞d(xk,xk+1)=0.
Now we prove that sequence {xn} is Cauchy sequence. Suppose on contrary that {xn} is not a Cauchy sequence. Then there exists ϵ>0 and sequences {mk} and {nk} such that, for all positive integers k, we have nk>mk>k,
(51)d(xmk,xnk)≥ϵ,d(xmk,xnk-1)<ϵ.
By the triangle inequality, we have
(52)ϵ≤d(xmk,xnk)≤d(xmk,xnk-1)+d(xnk-1,xnk)<ϵ+d(xnk-1,xnk)
for all k∈N. Now taking limit as k→+∞ in (52) and using (50), we have
(53)limk→∞d(xmk,xnk)=ϵ.
Again using triangle inequality, we have
(54)d(xmk,xnk)≤d(xmk,xmk+1)+d(xmk+1,xnk+1)+d(xnk+1,xnk),d(xmk+1,xnk+1)≤d(xmk+1,xmk)+d(xmk,xnk)+d(xnk,xnk+1).
Taking limit as k→+∞ and using (50) and (53), we obtain
(55)limk→+∞d(xmk+1,xnk+1)=ϵ.
By using (39), (53), and (55), we have
(56)η(xmk,Sxmk)η(xnk,Sxnk)d(xmk+1,xnk+1)≤α(xmk,Sxmk)α(xnk,Sxnk)d(xmk+1,xnk+1)≤α(xmk,Sxmk)α(xnk,Sxnk)d(Sxmk,Txnk)≤η(xmk,Sxmk)η(xnk,Sxnk)β(d(xmk,xnk))×d(xmk,xnk)
which implies that
(57)d(xmk+1,xnk+1)≤β(d(xmk,xnk))d(xmk,xnk).
Therfore, we have
(58)d(xmk+1,xnk+1)d(xmk,xnk)≤β(d(xmk,xnk))≤1.
Now taking limit as k→+∞ in (58), we get
(59)limn→∞β(d(xmk,xnk))=1.
Hence limk→∞d(xmk,xnk)=0<ϵ, which is a contradiction. Hence {xn} is a Cauchy sequence. Since X is complete so there exists p∈X such that xn→p. Now we prove that p=Sp. Suppose (i) holds; that is, S is continuous, so we get
(60)Sp=Slimn→∞xn=limn→∞Sxn=limn→∞xn+1=p.
Thus p=Sp. Now we suppose that (ii) holds. Since
(61)α(xn,xn+1)≥η(xn,xn+1)
for all n∈N∪{0}. By the hypotheses of (ii), we have
(62)α(p,Sp)α(xk,Sxk)≥η(p,Sp)η(xk,Sxk).
Using the triangle inequality and (39), we have
(63)η(p,Sp)η(xk,Sxk)d(Sp,xk+1)=η(p,Sp)η(xk,Sxk)d(Sp,Sxk)≤α(p,Sp)α(xk,Sxk)d(Sp,Sxk)≤η(p,Sp)η(xk,Sxk)β(d(p,xk))d(p,xk),
which implies that
(64)d(Sp,xk+1)≤β(d(p,xk))d(p,xk).
Letting k→∞, we have d(p,Sp)=0. Thus p=Sp. Let there exists q to be another fixed point of Sq∈X, s.t q=Sq;
(65)η(p,Sp)η(q,Sq)d(Sp,Sq)≤α(p,Sp)α(q,Sq)d(Sp,Sq)≤η(p,Sp)η(q,Sq)β(d(p,q))d(p,q),
implies
(66)d(Sp,Sq)≤β(d(p,q))d(p,q).
By the property of β function, β(d(p,q))=1 implies d(p,q)=0; then we have p=q. Hence S has a unique fixed point.
If η(x,y)=1 in Theorem 10, we get the following corollary.
Corollary 11 (see [17]).
Let (X,d) be a complete metric space and let S be α-admissible mapping. Assume that there exists a function β:[0,+∞)→[0,1) such that, for any bounded sequence {tn} of positive reals, β(tn)→1 implies tn→0 such that
(67)α(x,Sx)α(y,Sy)d(Sx,Sy)≤β(d(x,y))d(x,y)
for all x,y∈X. Suppose that either
S is continuous, or
if {xn} is a sequence in X such that α(xn,xn+1)≥1 for all n∈N∪{0} and xn→p∈X as n→+∞, then
(68)α(p,Sp)≥1.
If there exists x0,x1∈X such that α(x0,x1)≥1, then S has a fixed point. Our results are more general than those in [17, 32, 33] and improve several results existing in the literature.
Conflict of Interests
The authors declare that they have no competing interests.
Authors’ Contribution
All authors contributed equally and significantly to writing this paper. All authors read and approved the final paper.
Acknowledgments
Marwan Amin Kutbi gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The authors thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper.
AbbasM.RhoadesB. E.Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type20072007954101MR234633410.1155/2007/54101AbbasM.RhoadesB. E.Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition2008132295301MR2488678ZBL1175.470502-s2.0-76149129010AbbasM.KhanA. R.Common fixed points of generalized contractive hybrid pairs in symmetric spaces2009200911869407MR256577310.1155/2009/869407AbdeljawadT.Meir-Keeler α-contractive fixed and common fixed point theorems20132013, article 1910.1186/1687-1812-2013-19MR3022841AlioucheA.A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type2006322279680210.1016/j.jmaa.2005.09.068MR2250617ZBL1111.470462-s2.0-33747207562ArshadM.Some fixed point results for α*-ψ-contractive multi-valued mapping in partial metric spacesJournal of Advanced Research in Applied Mathematics. In pressArshadM.AzamA.VetroP.Some common fixed point results in cone metric spaces2009200911493965MR250148910.1155/2009/493965BanachS.Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales19223133181BranciariA.A fixed point theorem for mappings satisfying a general contractive condition of integral type200229953153610.1155/S0161171202007524MR1900344ZBL0993.540402-s2.0-17844395696KarapinarE.SametB.Generalized α-ψ contractive type mappings and related fixed point theorems with applications201220121779348610.1155/2012/793486GairolaU. C.RawatA. S.A fixed point theorem for integral type inequality2008213–16709712MR2482672GeraghtyM. A.On contractive mappings19734060460810.1090/S0002-9939-1973-0334176-5MR0334176ZBL0245.54027GuF.YeH.Common fixed point theorems of Altman integral type mappings in G-metric spaces2012201213630457MR299496310.1155/2012/630457GuptaV.ManiN.A common fixed point theorem for two weakly compatible mappings satisfying a new contractive condition of integral type201111HaghiR. H.RezapourS.ShahzadN.Some fixed point generalizations are not real generalizations20117451799180310.1016/j.na.2010.10.052MR27643802-s2.0-78651357943HussainN.ArshadM.ShoaibA.Shoaib and Fahimuddin, Common fixed point results for α-ψ-contractions on a metric space endowed with graph20142014, article 136HussainN.KarapınarE.SalimiP.AkbarF.α-admissible mappings and related fixed point theorems20132013, article 1141110.1186/1029-242X-2013-114MR3047105HussainN.AbbasM.Common fixed point results for two new classes of hybrid pairs in symmetric spaces2011218254254710.1016/j.amc.2011.05.098MR28205162-s2.0-79960848489HussainN.ChoY. J.Weak contractions, common fixed points, and invariant approximations200920091039063410.1155/2009/390634MR25007402-s2.0-76649144207JungckG.RhoadesB. E.Fixed points for set valued functions without continuity1998293227238MR16179192-s2.0-0032383032JungckG.HussainN.Compatible maps and invariant approximations200732521003101210.1016/j.jmaa.2006.02.058MR2270066ZBL1110.540242-s2.0-33750327353JungckG.RhoadesB. E.Fixed point theorems for occasionally weakly compatible mappings200672287296MR2284600JungckG.RhoadesB. E.Erratum: “Fixed point theorems for occasionally weakly compatible mappings” [Fixed Point Theory, vol. 7 (2006), no. 2, 287–296]200891383384KannanR.Some results on fixed points1968607176MR0257837ZBL0209.27104MoradiS.OmidM.A fixed point theorem for integral type inequality depending on another function2010232392442NadlerS. B.Multi-valued contraction mappings19693047548810.2140/pjm.1969.30.475MR0254828OjhaD. B.MishraM. K.KatochU.A common fixed point theorem satisfying integral type for occasionally weakly compatible maps2010232392442-s2.0-79953063345PathakH. K.TiwariR.KhanM. S.A common fixed point theorem satisfying integral type implicit relations20077222228MR2346046RhoadesB. E.A comparison of various definitions of contractive mappings197722625729010.1090/S0002-9947-1977-0433430-4MR0433430RhoadesB. E.Two fixed-point theorems for mappings satisfying a general contractive condition of integral type2003634007401310.1155/S0161171203208024MR2030391ZBL1052.470522-s2.0-17844377836ShrivastavaP. K.BawaN. P. S.NigamS. K.Fixed point theorems for hybrid contractions200222275281MR1982957SalimiP.LatifA.HussainN.Modified α-ψ-contractive mappings with applications201320131910.1186/1687-1812-2013-151MR3074018SametB.VetroC.VetroP.Fixed point theorems for α-ψ-contractive type mappings20127542154216510.1016/j.na.2011.10.014MR28709072-s2.0-84655168090VijayarajuP.RhoadesB. E.MohanrajR.A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type2005152359236410.1155/IJMMS.2005.2359MR2184475ZBL1113.540272-s2.0-29144502330LiY.GuF.Common fixed point theorem of altman integral type mappings200924214218MR2562261ZhangX.Common fixed point theorems for some new generalized contractive type mappings2007333278078610.1016/j.jmaa.2006.11.028MR2331693ZBL1133.540282-s2.0-34248546144