The concept of tangential for single-valued
mappings is extended to multivalued mappings and used to prove
the existence of a common fixed point theorem of Gregus type for
four mappings satisfying a strict general contractive condition of
integral type. Consequently, several known fixed point results generalized
and improved the corresponding recent result of Pathak and
Shahzad (2009) and many authors.

1. Introduction

The first important result on fixed points for contractive-type mappings was the well-known Banach contraction principle, published for the first time in 1922 in [1] (see also [2]). Banach contraction principle has been extended in many different directions, see [3–5], and so forth. Many authors in [3, 5–12] established fixed point theorems involving more general contractive conditions. In 1969, Nadler [13] combines the ideas of set-valued mapping and Lipschitz mapping and prove some fixed point theorems about multivalued contraction mappings. Afterward, the study of fixed points for multivalued contractions using the Hausdorff metric was initiated by Markin [14]. Later, an interesting and rich fixed point theory for such maps was developed (see [15–18]). The theory of multivalued maps has applications in optimization problems, control theory, differential equations, and economics.

Sessa [19] introduced the concept of weakly commuting maps. Jungck [20] defined the notion of compatible maps in order to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible but the converse is not true. This concept was further improved by Jungck and Rhoades [21] with the notion of weakly compatible mappings. In 2002, Aamri and Moutawakil [22] defined property (E.A). This concept was frequently used to prove existence theorems in common fixed point theory. Three years later, Liu et al. [23] introduced common property (E.A). The class of (E.A) maps contains the class of noncompatible maps. Branciari [3] studied contractive conditions of integral type, giving an integral version of the Banach contraction principle, that could be extended to more general contractive conditions. Recently, Pathak and Shahzad [24] introduced the new concept of weak tangent point and tangential property for single-valued mappings and established common fixed point theorems. Very recently, Vetro [25] obtained an interesting theorem for mappings satisfying a contractive condition of integral type which is a generalization of Branciari [3, Theorem 2].

The aim of this paper is to define a tangential property for multivalued mappings which generalize the concept of tangential property for single-valued mappings of Pathak and Shahzad [24] and prove a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type.

2. Preliminary

Throughout this paper, (X,d) denotes a metric space. We denote by CB(X), the class of all nonempty bounded closed subsets of X. The Hausdorff metric induced by d on CB(X) is given by H(A,B)=max{supa∈Ad(a,B),supb∈Bd(b,A)},
for every A,B∈CB(X), where d(a,B)=d(B,a)=inf{d(a,b):b∈B} is the distance from a to B⊆X. Let f:X→X and T:X→CB(X). A point x∈X is a fixed point of f (resp. T) if fx=x (resp. x∈Tx). The set of all fixed points of f (resp. T) is denoted by F(f) (resp. F(T)). A point x∈X is a coincidence point of f and T if fx∈Tx. The set of all coincidence points of f and T is denoted by C(f,T). A point x∈X is a common fixed point of f and T if x=fx∈Tx. The set of all common fixed points of f and T is denoted by F(f,T).

Definition 2.1.

The maps f:X→X and g:X→X are said to be commuting if fgx=gfx, for all x∈X.

Definition 2.2 (see [<xref ref-type="bibr" rid="B25">19</xref>]).

The maps f:X→X and g:X→X are said to be weakly commuting if d(fgx,gfx)≤d(fx,gx), for all x∈X.

Definition 2.3 (see [<xref ref-type="bibr" rid="B13">20</xref>]).

The maps f:X→X and g:X→X are said to be compatible if limn→∞d(fgxn,gfxn)=0 whenever {xn} is a sequence in X such that limn→∞fxn=limn→∞gxn=z, for some z∈X.

Definition 2.4 (see [<xref ref-type="bibr" rid="B14">26</xref>]).

The maps f:X→X and g:X→X are said to be weakly compatible fgx=gfx, for all x∈C(f,g).

Definition 2.5 (see [<xref ref-type="bibr" rid="B1">22</xref>]).

Let f:X→X and g:X→X. The pair (f,g) satisfies property (E.A) if there exist the sequence {xn} in X such that
limn→∞fxn=limn→∞gxn=z∈X.

See example of property (E.A) in Kamran [27, 28] and Sintunavarat and Kumam [11].

Definition 2.6 (see [<xref ref-type="bibr" rid="B19">23</xref>]).

Let f,g,A,B:X→X. The pair (f,g) and (A,B) satisfy a common property (E.A) if there exist sequences {xn} and {yn} in X such that
limn→∞fxn=limn→∞gxn=limn→∞Ayn=limn→∞Byn=z∈X.

Remark 2.7.

If A=f, B=g, and {xn}={yn} in (2.3), then we get the definition of property (E.A).

Definition 2.8 (see [<xref ref-type="bibr" rid="B22">24</xref>]).

Let f,g:X→X. A point z∈X is said to be a weak tangent point to (f,g) if there exist sequences {xn} and {yn} in X such that
limn→∞fxn=limn→∞gyn=z∈X.

Remark 2.9.

If {xn}={yn} in (2.4), we get the definition of property (E.A).

Definition 2.10 (see [<xref ref-type="bibr" rid="B22">24</xref>]).

Let f,g,A,B:X→X. The pair (f,g) is called tangential with respect to the pair (A,B) if there exist sequences {xn} and {yn} in X such that
limn→∞fxn=limn→∞gyn=limn→∞Axn=limn→∞Byn=z∈X.

3. Main Results

In this section, we first introduce the notion of tangential property for two single-valued and two multivalued mappings. Throughout this section, ℝ+ denotes the set of nonnegative real numbers.

Definition 3.1.

Let f,g:X→X and A,B:X→CB(X). The pair (f,g) is called tangential with respect to the pair (A,B) if
limn→∞Axn=limn→∞Byn=D∈CB(X),
whenever sequences {xn} and {yn} in X such that
limn→∞fxn=limn→∞gyn=z∈D,
for some z∈X.

Example 3.2.

Let (ℝ+,d) be a metric space with usual metric d. Let f,g:ℝ+→ℝ+ and A,B:ℝ+→CB(ℝ+) be mappings defined by fx=x+1, gx=x+2, Ax={x2/2+1}, and Bx={x2+2}, for all x∈ℝ+. Clearly, there exists two sequences {xn=2+1/n} and {yn=1+1/n} such that
limn→∞Axn=limn→∞Byn={3}∈CB(R+)
whenever
limn→∞fxn=limn→∞gyn=3∈R+.
So, the pair (f,g) is tangential with respect to the pair (A,B).

Definition 3.3.

Let f:X→X and A:X→CB(X). The mapping f is called tangential with respect to the mapping A if
limn→∞Axn=limn→∞Ayn=D∈CB(X),
whenever sequences {xn} and {yn} in X such that
limn→∞fxn=limn→∞fyn=z∈D,
for some z∈X.

Example 3.4.

Let (ℝ+,d) be a metric space with usual metric d. Let f:ℝ+→ℝ+ and A:ℝ+→CB(ℝ+) be mappings defined by
fx=x+1,Ax={x2+1}.
Clearly, there exist two sequences {xn=1+1/n} and {yn=1-1/n} such that
limn→∞Axn=limn→∞Ayn={2}∈CB(R+)
whenever
limn→∞fxn=limn→∞fyn=2∈R+.
So, the mapping f is tangential with respect to the mapping A.

Now, we state and prove our main result.

Theorem 3.5.

Let f,g:X→X and A,B:X→CB(X) satisfy
(1+α∫0d(fx,gy)ψ(t)dt)∫0H(Ax,By)ψ(t)dt<α(∫0d(Ax,fx)ψ(t)dt∫0d(By,gy)ψ(t)dt+∫0d(Ax,gy)ψ(t)dt∫0d(fx,By)ψ(t)dt)+a∫0d(fx,gy)ψ(t)dt+(1-a)max{(∫0d(Ax,gy)ψ(t)dt)1/2∫0d(Ax,fx)ψ(t)dt,∫0d(By,gy)ψ(t)dt,(∫0d(Ax,fx)ψ(t)dt)1/2(∫0d(Ax,gy)ψ(t)dt)1/2,(∫0d(fx,By)ψ(t)dt)1/2(∫0d(Ax,gy)ψ(t)dt)1/2},
for all x,y∈X for which the right-hand side of (3.10) is positive, where 0<a<1, α≥0 and ψ:ℝ+→ℝ+ is a Lebesgue integrable mapping which is a summable nonnegative and such that
∫0ϵψ(t)dt>0,
for each ϵ>0. If the following conditions (a)–(d) hold:

there exists a point z∈f(X)∩g(X) which is a weak tangent point to (f,g),

(f,g) is tangential with respect to (A,B),

ffa=fa, ggb=gb, and Afa=Bgb for a∈C(f,A) and b∈C(g,B),

the pairs (f,A) and (g,B) are weakly compatible.

Then, f, g, A, and B have a common fixed point in X. Proof.

Since z∈f(X)∩g(X), z=fu=gv for some u,v∈X. It follows from a point z which is a weak tangent point to (f,g) that there exist sequences {xn} and {yn} in X such that
limn→∞fxn=limn→∞gyn=z.
Because the pair (f,g) is tangential with respect to the pair (A,B), we get
limn→∞Axn=limn→∞Byn=D,
for some D∈CB(X). Since z=fu=gv and (3.12) and (3.13) are true, we have
z=fu=gv=limn→∞fxn=limn→∞gyn∈limn→∞Axn=limn→∞Byn=D.

We claim that z∈Bv. If not, then condition (3.10) implies(1+α∫0d(fxn,gv)ψ(t)dt)∫0H(Axn,Bv)ψ(t)dt<α(∫0d(Axn,fxn)ψ(t)dt∫0d(Bv,gv)ψ(t)dt+∫0d(Axn,gv)ψ(t)dt∫0d(fxn,Bv)ψ(t)dt)+a∫0d(fxn,gv)ψ(t)dt+(1-a)max{(∫0d(Ax,gy)ψ(t)dt)1/2∫0d(Axn,fxn)ψ(t)dt,∫0d(Bv,gv)ψ(t)dt,(∫0d(Axn,fxn)ψ(t)dt)1/2(∫0d(Axn,gv)ψ(t)dt)1/2,(∫0d(fxn,Bv)ψ(t)dt)1/2(∫0d(Axn,gv)ψ(t)dt)1/2}.
Letting n→∞, we get
∫0H(D,Bv)ψ(t)dt≤(1-a)∫0d(Bv,z)ψ(t)dt.
Since
∫0d(z,Bv)ψ(t)dt<∫0H(D,Bv)ψ(t)dt≤(1-a)∫0d(Bv,z)ψ(t)dt<∫0d(z,Bv)ψ(t)dt,
which is a contradiction, then z∈Bv.

Again, we claim that z∈Au. If not, then condition (3.10) implies(1+α∫0d(fu,gyn)ψ(t)dt)∫0H(Au,Byn)ψ(t)dt<α(∫0d(Au,fu)ψ(t)dt∫0d(Byn,gyn)ψ(t)dt+∫0d(Au,gyn)ψ(t)dt∫0d(fu,Byn)ψ(t)dt)+a∫0d(fu,gyn)ψ(t)dt+(1-a)max{(∫0d(Ax,gy)ψ(t)dt)1/2∫0d(Au,fu)ψ(t)dt,∫0d(Byn,gyn)ψ(t)dt,(∫0d(Au,fu)ψ(t)dt)1/2(∫0d(Au,gyn)ψ(t)dt)1/2,(∫0d(fu,Byn)ψ(t)dt)1/2(∫0d(Au,gyn)ψ(t)dt)1/2}.
Letting n→∞, we get
∫0H(Au,D)ψ(t)dt≤(1-a)∫0d(Au,z)ψ(t)dt.
Since
∫0d(z,Au)ψ(t)dt<∫0H(Au,D)ψ(t)dt≤(1-a)∫0d(Au,z)ψ(t)dt<∫0d(z,Au)ψ(t)dt,
which is a contradiction, then z∈Au.

Now, we conclude z=gv∈Bv and z=fu∈Au. It follows from v∈C(g,B), u∈C(f,A) that ggv=gv, ffu=fu, and Afu=Bgv. Hence, gz=z, fz=z and Az=Bz.

Since the pair (g,B) is weakly compatible, gBv=Bgv. Thus gz∈gBv=Bgv=Bz. Similarly, we can prove that fz∈Az. Consequently, z=fz=gz∈Az=Bz. Therefore the maps f,g,A, and B have a common fixed point.

If α=0 in Theorem 3.5, we get the following corollary.

Corollary 3.6.

Let f,g:X→X and A,B:X→CB(X) satisfy
∫0H(Ax,By)ψ(t)dt<a∫0d(fx,gy)ψ(t)dt+(1-a)max{(∫0d(Ax,gy)ψ(t)dt)1/2∫0d(Ax,fx)ψ(t)dt,∫0d(By,gy)ψ(t)dt,(∫0d(Ax,fx)ψ(t)dt)1/2(∫0d(Ax,gy)ψ(t)dt)1/2,(∫0d(fx,By)ψ(t)dt)1/2(∫0d(Ax,gy)ψ(t)dt)1/2},
for all x,y∈X for which the right-hand side of (3.21) is positive, where 0<a<1 and ψ:ℝ+→ℝ+ is a Lebesgue integrable mapping which is a summable nonnegative and such that
∫0ϵψ(t)dt>0,
for each ϵ>0. If the following conditions (a)–(d) hold:

there exists a point z∈f(X)∩g(X) which is a weak tangent point to (f,g),

(f,g) is tangential with respect to (A,B),

ffa=fa, ggb=gb and Afa=Bgb for a∈C(f,A) and b∈C(g,B),

the pairs (f,A) and (g,B) are weakly compatible.

Then, f, g, A, and B have a common fixed point in X.

If α=0, g=f, and B=A in Theorem 3.5, we get the following corollary.

Corollary 3.7.

Let f:X→X and A:X→CB(X) satisfy
∫0H(Ax,Ay)ψ(t)dt<a∫0d(fx,fy)ψ(t)dt+(1-a)max{(∫0d(Ax,gy)ψ(t)dt)1/2∫0d(Ax,fx)ψ(t)dt,∫0d(Ay,fy)ψ(t)dt,(∫0d(Ax,fx)ψ(t)dt)1/2(∫0d(Ax,fy)ψ(t)dt)1/2,(∫0d(fx,Ay)ψ(t)dt)1/2(∫0d(Ax,fy)ψ(t)dt)1/2},
for all x,y∈X for which the right-hand side of (3.23) is positive, where 0<a<1 and ψ:ℝ+→ℝ+ is a Lebesgue integrable mapping which is a summable nonnegative and such that
∫0ϵψ(t)dt>0
for each ϵ>0. If the following conditions (a)–(d) hold:

there exists sequence {xn} in X such that limn→∞fxn∈X,

f is tangential with respect to A,

ffa=fa for a∈C(f,A),

the pair (f,A) is weakly compatible.

Then, f and A have a common fixed point in X.

If ψ(t)=1 in Theorem 3.5, we get the following corollary.

Corollary 3.8.

Let f,g:X→X and A,B:X→CB(X) satisfy
(1+αd(fx,gy))H(Ax,By)<α(d(Ax,fx)d(By,gy)+d(Ax,gy)d(fx,By))+ad(fx,gy)+(1-a)max{d(Ax,fx),d(By,gy),(d(Ax,fx))1/2(d(Ax,gy))1/2,(d(fx,By))1/2(d(Ax,gy))1/2}
for all x,y∈X for which the right-hand side of (3.25) is positive, where 0<a<1 and α≥0. If the following conditions (a)–(d) holds:

there exists a point z∈f(X)∩g(X) which is a weak tangent point to (f,g),

(f,g) is tangential with respect to (A,B),

ffa=fa, ggb=gb and Afa=Bgb for a∈C(f,A) and b∈C(g,B),

the pairs (f,A) and (g,B) are weakly compatible.

Then, f, g, A, and B have a common fixed point in X.

If ψ(t)=1 and α=0 in Theorem 3.5, we get the following corollary.

Corollary 3.9.

Let f,g:X→X and A,B:X→CB(X) satisfy
H(Ax,By)<ad(fx,gy)+(1-a)max{d(Ax,fx),d(By,gy),(d(Ax,fx))1/2(d(Ax,gy))1/2,(d(fx,By))1/2(d(Ax,gy))1/2},
for all x,y∈X for which the right-hand side of (3.26) is positive, where 0<a<1. If the following conditions (a)–(d) hold:

there exists a point z∈f(X)∩g(X) which is a weak tangent point to (f,g),

(f,g) is tangential with respect to (A,B),

ffa=fa, ggb=gb and Afa=Bgb for a∈C(f,A) and b∈C(g,B),

the pairs (f,A) and (g,B) are weakly compatible.

Then, f, g, A, and B have a common fixed point in X.

If ψ(t)=1, α=0, g=f and B=A in Theorem 3.5, we get the following corollary.

Corollary 3.10.

Let f:X→X and A:X→CB(X) satisfy
H(Ax,Ay)<ad(fx,fy)+(1-a)max{d(Ax,fx),d(Ay,fy),(d(Ax,fx))1/2(d(Ax,fy))1/2,(d(fx,Ay))1/2(d(Ax,fy))1/2}
for all x,y∈X for which the right-hand side of (3.27) is positive, where 0<a<1. If the following conditions (a)–(d) holds:

there exists sequence {xn} in X such that limn→∞fxn∈X,

f is tangential with respect to A,

ffa=fa for a∈C(f,A),

the pairs (f,A) is weakly compatible.

Then, f and A have a common fixed point in X.Acknowledgments

W. Sintunavarat would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the Faculty of Science, KMUTT for financial support during the preparation of this manuscript for Ph.D. Program at KMUTT. Moreover, the authors also would like to thank the National Research University Project of Thailand's Office of the Higher Education Commission for financial support (under the CSEC project no. 54000267). Finally, the authors would like to thank Professor Frank Werner for your help and encouragement. Special thanking are also due to the reviewers, who have made a number of valuable comments and suggestions which have improved the manuscript greatly.

BanachS. Sur les opérations dans les ensembles abstraits et leurs applications
aux équations intéralesCaccioppoliR.Un teorema generale sull esistenza di elementi uniti in una trasformazione funzionaleBranciariA.A fixed point theorem for mappings satisfying a general contractive condition of integral typeShahzadN.Invariant approximations and R-subweakly commuting mapsVijayarajuP.RhoadesB. E.MohanrajR.A fixed point theorem for a pair of maps satisfying a general contractive condition of integral typeAgarwalR. P.O'ReganD.ShahzadN.Fixed point theory for generalized contractive maps of Meir-Keeler typeAltunI.TürkoğluD.RhoadesB. E.Fixed points of weakly compatible maps satisfying a general contractive condition of integral typeAltunI.TurkogluD.Some fixed point theorems for weakly compatible multivalued mappings satisfying some general contractive conditions of integral typeAzamA.ArshadM.Common fixed points of generalized contractive maps in cone metric spacesRazaniA.MoradiR.Common fixed point theorems of integral type in modular spacesSintunavaratW.KumamP.Coincidence and common fixed points for hybrid strict contractions without the weakly commuting conditionSintunavaratW.poomteun@hotmail.comKumamP.poom.kum@kmutt.ac.thWeak condition for generalized multi-valued (f,α,β)-weak contraction mappingsNadlerS. B.Jr.Multi-valued contraction mappingsMarkinJ. T.Continuous dependence of fixed point setsAubinJ.-P.SiegelJ.Fixed points and stationary points of dissipative multivalued mapsCovitzH.NadlerS. B.Jr.Multi-valued contraction mappings in generalized metric spacesSintunavaratW.KumamP.PatthanangkoorP.Common random fixed points for multivalued random operators without S- and T-weakly commuting randomWangT. X.Fixed-point theorems and fixed-point stability for multivalued mappings on metric spacesSessaS.On a weak commutativity condition of mappings in fixed point considerationsJungckG.Compatible mappings and common fixed pointsJungckG.RhoadesB. E.Fixed point theorems for occasionally weakly compatible mappingsAamriM.El MoutawakilD.Some new common fixed point theorems under strict contractive conditionsLiuY.WuJ.LiZ.Common fixed points of single-valued and multivalued mapsPathakH. K.ShahzadN.Gregus type fixed point results for tangential mappings satisfying contractive conditions of integral typeVetroC.On Branciari's theorem for weakly compatible mappingsJungckG.Common fixed points for noncontinuous nonself maps on nonmetric spacesKamranT.Coincidence and fixed points for hybrid strict contractionsKamranT.Multivalued f-weakly Picard mappings