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 [<xref ref-type="bibr" rid="B31">32</xref>]).

If ψ∈Ω, then ψ(t)<t for all t>0.

Definition 2 (see [<xref ref-type="bibr" rid="B32">33</xref>]).

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 [<xref ref-type="bibr" rid="B32">33</xref>]).

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 [<xref ref-type="bibr" rid="B31">32</xref>]).

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 [<xref ref-type="bibr" rid="B17">17</xref>]).

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 [<xref ref-type="bibr" rid="B17">17</xref>]).

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 typeAbbasM.RhoadesB. E.Common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive conditionAbbasM.KhanA. R.Common fixed points of generalized contractive hybrid pairs in symmetric spacesAbdeljawadT.Meir-Keeler α-contractive fixed and common fixed point theoremsAlioucheA.A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral typeArshadM.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 spacesBanachS.Sur les opérations dans les ensembles abstraits et leur application aux équations intégralesBranciariA.A fixed point theorem for mappings satisfying a general contractive condition of integral typeKarapinarE.SametB.Generalized α-ψ contractive type mappings and related fixed point theorems with applicationsGairolaU. C.RawatA. S.A fixed point theorem for integral type inequalityGeraghtyM. A.On contractive mappingsGuF.YeH.Common fixed point theorems of Altman integral type mappings in G-metric spacesGuptaV.ManiN.A common fixed point theorem for two weakly compatible mappings satisfying a new contractive condition of integral typeHaghiR. H.RezapourS.ShahzadN.Some fixed point generalizations are not real generalizationsHussainN.ArshadM.ShoaibA.Shoaib and Fahimuddin, Common fixed point results for α-ψ-contractions on a metric space endowed with graphHussainN.KarapınarE.SalimiP.AkbarF.α-admissible mappings and related fixed point theoremsHussainN.AbbasM.Common fixed point results for two new classes of hybrid pairs in symmetric spacesHussainN.ChoY. J.Weak contractions, common fixed points, and invariant approximationsJungckG.RhoadesB. E.Fixed points for set valued functions without continuityJungckG.HussainN.Compatible maps and invariant approximationsJungckG.RhoadesB. E.Fixed point theorems for occasionally weakly compatible mappingsJungckG.RhoadesB. E.Erratum: “Fixed point theorems for occasionally weakly compatible mappings” [Fixed Point Theory, vol. 7 (2006), no. 2, 287–296]KannanR.Some results on fixed pointsMoradiS.OmidM.A fixed point theorem for integral type inequality depending on another functionNadlerS. B.Multi-valued contraction mappingsOjhaD. B.MishraM. K.KatochU.A common fixed point theorem satisfying integral type for occasionally weakly compatible mapsPathakH. K.TiwariR.KhanM. S.A common fixed point theorem satisfying integral type implicit relationsRhoadesB. E.A comparison of various definitions of contractive mappingsRhoadesB. E.Two fixed-point theorems for mappings satisfying a general contractive condition of integral typeShrivastavaP. K.BawaN. P. S.NigamS. K.Fixed point theorems for hybrid contractionsSalimiP.LatifA.HussainN.Modified α-ψ-contractive mappings with applicationsSametB.VetroC.VetroP.Fixed point theorems for α-ψ-contractive type mappingsVijayarajuP.RhoadesB. E.MohanrajR.A fixed point theorem for a pair of maps satisfying a general contractive condition of integral typeLiY.GuF.Common fixed point theorem of altman integral type mappingsZhangX.Common fixed point theorems for some new generalized contractive type mappings