(2008) pp. 43–59
http://www.ektf.hu/ami
Common fixed point theorems for pairs of single and multivalued D -maps satisfying
an integral type
H. Bouhadjera, A. Djoudi
Laboratoire de Mathématiques Appliquées Université Badji Mokhtar, Annaba, Algérie
Submitted 4 March 2008; Accepted 28 June 2008
Abstract
This contribution is a continuation of [1, 3, 14]. The concept of subcom- patibility between single maps and between single and multivalued maps is used as a tool for proving existence and uniqueness of common fixed points on complete metric and symmetric spaces. Extensions of known results, in particularly results given by Djoudi and Aliouche, Elamrani and Mehdaoui, Pathak et al. are thereby obtained.
Keywords: Commuting and weakly commuting maps, compatible and com- patible maps of type (A), (B), (C) and (P), weakly compatible maps, δ- compatible maps, subcompatible maps,D-maps, integral type, common fixed point theorems, metric space.
MSC:47H10, 54H25
1. Introduction and preliminaries
Let(X, d)be a metric space and letB(X)be the class of all nonempty bounded subsets ofX. For allA, B in B(X), define
δ(A, B) = sup{d(a, b) :a∈A, b∈B}.
IfA={a}, we writeδ(A, B) =δ(a, B). Also, ifB ={b}, it yields thatδ(A, B) = d(a, b).
From the definition ofδ(A, B), for allA, B, C inB(X)it follows that δ(A, B) =δ(B, A)>0,
δ(A, B)6δ(A, C) +δ(C, B), 43
δ(A, A) =diamA,
δ(A, B) = 0 iff A=B ={a}.
In his paper [15], Sessa introduced the notion of weak commutativity which generalized the notion of commutativity.
Later on, Jungck [6] gave a generalization of weak commutativity by introducing the concept of compatibility.
Again, to generalize weakly commuting maps, the same author with Murthy and Cho [8] introduced the concept of compatible maps of type(A).
Extending type(A), Pathak and Khan [13] made the notion of compatible maps of type(B).
In [11], the concept of compatible maps of type (P) was introduced and com- pared with compatible and compatible maps of type(A).
In 1998, Pathak, Cho, Kang and Madharia [12] defined the notion of compatible maps of type(C)as another extension of compatible maps of type (A).
In his paper [7], Jungck generalized all the concepts of compatibility by giving the notion of weak compatibility (subcompatibility).
The authors of [9] extended the concept of compatible maps to the setting of single and multivalued maps by giving the notion ofδ-compatible maps.
Also, the same authors [10] extended the definition of weak compatibility to the setting of single and multivalued maps by introducing the concept of subcompatible maps.
In their paper [2], Djoudi and Khemis introduced the notion ofD-maps which is a generalization ofδ-compatible maps.
Definition 1.1 ([4]). A sequence {An} of nonempty subsets of X is said to be convergent to a subsetA ofX if:
(i) each pointa∈Ais the limit of a convergent sequence{an}, wherean ∈An forn∈N,
(ii) for arbitraryǫ >0, there exists an integermsuch thatAn⊆Aǫforn > m, whereAǫ denotes the set of all pointsxinX for which there exists a pointainA, depending onx, such thatd(x, a)< ǫ.
Lemma 1.2 ([4, 5]). If {An} and {Bn} are sequences in B(X) converging to A andB inB(X), respectively, then the sequence{δ(An, Bn)} converges toδ(A, B).
Lemma 1.3 ([5]). Let {An} be a sequence in B(X) and y be a point in X such that δ(An, y)→0. Then the sequence{An} converges to the set{y} inB(X).
Definition 1.4 ([15]). The self-maps f and g of a metric spaceX are said to be weakly commuting ifd(f gx, gf x)6d(gx, f x)for allx∈ X.
Definition 1.5 ([6, 8, 13, 12, 11]). The self-mapsf andgof a metric spaceX are said to be
(1) compatible if
n→∞lim d(f gxn, gf xn) = 0,
(2) compatible of type(A)if
n→∞lim d(f gxn, g2xn) = 0and lim
n→∞d(gf xn, f2xn) = 0, (3) compatible of type(B)if
n→∞lim d(f gxn, g2xn)61 2
h lim
n→∞d(f gxn, f t) + lim
n→∞d(f t, f2xn)i ,
n→∞lim d(gf xn, f2xn)61 2
h lim
n→∞d(gf xn, gt) + lim
n→∞d(gt, g2xn)i ,
(4) compatible of type(C)if
n→∞lim d(f gxn, g2xn)6 1 3 h
n→∞lim d(f gxn, f t) + lim
n→∞d(f t, f2xn) + lim
n→∞d(f t, g2xn)i ,
n→∞lim d(gf xn, f2xn)6 1 3
h lim
n→∞d(gf xn, gt) + lim
n→∞d(gt, g2xn) + lim
n→∞d(gt, f2xn)i ,
(5) compatible of type(P)if
n→∞limd(f2xn, g2xn) = 0 whenever {xn} is a sequence in X such that lim
n→∞f xn = lim
n→∞gxn = t for some t∈ X.
Definition 1.6 ([7]). The self-mapsf andgof a metric spaceX are called weakly compatible iff x=gx,x∈ X impliesf gx=gf x.
Definition 1.7 ([9]). The maps f:X → X andF: X →B(X) areδ-compatible if
n→∞limδ(F f xn, f F xn) = 0
whenever{xn}is a sequence inXsuch thatf F xn∈B(X),f xn →tandF xn → {t}
for some t∈ X.
Definition 1.8 ([10]). Mapsf:X → X andF:X →B(X)are subcompatible if they commute at coincidence points; i.e., for each point u∈ X such that F u = {f u}, we haveF f u=f F u.
Definition 1.9 ([2]). The maps f: X → X and F: X → B(X) are said to be D-maps iff there exists a sequence{xn}in X such that for somet∈ X
n→∞limf xn=t and lim
n→∞F xn ={t}.
Recently in 2007, Pathak et al. [14] established a general common fixed point theorem for two pairs of weakly compatible maps satisfying integral type implicit relations. The first main object of this paper is to prove a common fixed point theorem for a quadruple of maps satisfying certain integral type implicit relations.
Our result extended the result of [14] to the setting of single and multivalued maps.
For this consideration we need the following:
Let Φ = {ϕ:R+→Ris a Lebesgue-integrable map which is summable} and letF be the set of all continuous functions F: R6+ →R+ satisfying the following conditions:
(Fa)RF(u,0,0,u,u,0)
0 ϕ(t)dt60impliesu= 0;
(Fb)RF(u,0,u,0,0,u)
0 ϕ(t)dt60 impliesu= 0.
The function F satisfies the condition (F1) if RF(u,u,0,0,u,u)
0 ϕ(t)dt > 0 for all u >0.
2. Main results
Theorem 2.1. Let f, g be self-maps of a metric space (X, d) and let F, G: X → B(X)be two multivalued maps such that
(1)FX ⊆gX andGX ⊆fX, (2)
Z F(δ(F x,Gy),d(f x,gy),δ(f x,F x),δ(gy,Gy),δ(f x,Gy),δ(gy,F x)) 0
ϕ(t)dt60 for allx, y inX, whereF ∈ F andϕ∈Φ. If either
(3)f andF are subcompatible D-maps; g andG are subcompatible andFX is closed, or
(3′)g andGare subcompatibleD-maps; f andF are subcompatible andGX is closed.
Then,f, g, F andGhave a unique common fixed point t∈ X such that
F t=Gt={f t}={gt}={t}.
Proof. Suppose that f and F are D-maps, then, there exists a sequence {xn} in X such that f xn → t and F xn → {t} for some t ∈ X. Since FX is closed and FX ⊆ gX, then, there is a point u ∈ X such that gu = t. We show that Gu={gu}={t}. Using inequality(2), we have
Z F(δ(F xn,Gu),d(f xn,gu),δ(f xn,F xn),δ(gu,Gu),δ(f xn,Gu),δ(gu,F xn)) 0
ϕ(t)dt60.
SinceF is continuous, we get at infinity
Z F(δ(gu,Gu),0,0,δ(gu,Gu),δ(gu,Gu),0) 0
ϕ(t)dt60
which implies, by using condition (Fa), δ(gu, Gu) = 0; i.e., Gu = {gu} = {t}.
Since the pair(g, G)is subcompatible, it follows thatGgu=gGu; i.e., Gt={gt}.
Ift6=gt, using(2)we have
Z F(δ(F xn,Gt),d(f xn,gt),δ(f xn,F xn),δ(gt,Gt),δ(f xn,Gt),δ(gt,F xn)) 0
ϕ(t)dt60.
Taking limit asn→ ∞, we get
Z F(d(t,gt),d(t,gt),0,0,d(t,gt),d(gt,t)) 0
ϕ(t)dt60,
which contradicts(F1). Hence, Gt={gt}={t}. SinceGX ⊆fX, there isv ∈ X such that{t}=Gt={f v}. IfF v6={t}, using(2)again, we have
Z F(δ(F v,Gt),d(f v,gt),δ(f v,F v),δ(gt,Gt),δ(f v,Gt),δ(gt,F v)) 0
ϕ(t)dt
=
Z F(δ(F v,t),0,δ(t,F v),0,0,δ(t,F v)) 0
ϕ(t)dt60,
which implies by using condition (Fb)thatδ(F v, t) = 0, hence,F v={t}={f v}.
Since F and f are subcompatible, it follows thatF f v =f F v; i.e., F t={f t}. If t6=f t, using(2)we have
Z F(δ(F t,Gt),d(f t,gt),δ(f t,F t),δ(gt,Gt),δ(f t,Gt),δ(gt,F t)) 0
ϕ(t)dt
=
Z F(d(f t,t),d(f t,t),0,0,d(f t,t),d(t,f t)) 0
ϕ(t)dt60, which contradicts(F1). Thus,{f t}={t}=F t.
We get the same conclusion if we use(3′)instead of(3).
The uniqueness of the common fixed point follows easily from conditions (2)
and(F1).
Corollary 2.2. Letf be a map from a metric space (X, d)into itself and letF be a map fromX intoB(X). If
(i)FX ⊆fX,
(ii)f andF are subcompatible D-maps, (iii)
Z F(δ(F x,F y),d(f x,f y),δ(f x,F x),δ(f y,F y),δ(f x,F y),δ(f y,F x)) 0
ϕ(t)dt60
for all x, y in X, where ϕ∈Φ and F is continuous and satisfies conditions (Fa) and(F1)or(Fb)and(F1). IfFX is closed, then, f andF have a unique common fixed point in X.
The next Theorem is a generalization of Theorem 2.1.
Theorem 2.3. Let f, g be self-maps of a metric space (X, d) and let Fn:X → B(X), wheren= 1,2, . . . be multivalued maps such that
(i)FnX ⊆gX andFn+1X ⊆fX, (ii)
Z F(δ(Fnx,Fn+1y),d(f x,gy),δ(f x,Fnx),δ(gy,Fn+1y),δ(f x,Fn+1y),δ(gy,Fnx)) 0
ϕ(t)dt60 for allx, y inX, whereF ∈ F andϕ∈Φ. If either
(iii) f and Fn are subcompatible D-maps; g and Fn+1 are subcompatible and FnX is closed, or
(iii)′ g andFn+1 are subcompatible D-maps; f and Fn are subcompatible and Fn+1X is closed.
Then,f, g andFn have a unique common fixed pointt∈ X such that
Fnt={f t}={gt}={t}.
Now, let Ψ be the set of all maps ψ: R+ → R+ such that ψ is a Lebesgue- integrable which is summable, nonnegative and satisfies Rǫ
0ψ(t)dt > 0 for each ǫ >0.
In [3], a common fixed point theorem for a pair of generalized contraction self- maps and a pair of multivalued maps in a complete metric space was obtained.
Our second main subject is to complement and improve the result of [3] by relax- ing the notion ofδ-compatibility to subcompatibility, removing the assumption of continuity imposed on at least one of the four maps and deleting some conditions required on the functionsΦ,a,bandcby using an integral type in a metric space instead of a complete metric space.
Theorem 2.4. Let f, gbe self-maps of a metric space(X, d)and letF, Gbe maps from X intoB(X) satisfying the following conditions
(1′)f andg are surjective, (2′)
Z ̥(δ(F x,Gy)) 0
ψ(t)dt6a(d(f x, gy))
Z ̥(d(f x,gy)) 0
ψ(t)dt +b(d(f x, gy))
Z ̥(δ(f x,F x))+̥(δ(gy,Gy)) 0
ψ(t)dt +c(d(f x, gy))
Z min{̥(δ(f x,Gy)),̥(δ(gy,F x))}
0
ψ(t)dt for all x, y inX, where̥: [0,∞)→[0,∞)is an upper semi-continuous map such that ̥(t) = 0ifft= 0;a, b, c: [0,∞)→[0,1)are upper semi-continuous such that a(t) +c(t)<1 for everyt >0 andψ∈Ψ. If either
(3′)f andF are subcompatibleD-maps; g andGare subcompatible, or
(3′′)g andGare subcompatibleD-maps; f andF are subcompatible.
Then,f, g, F andGhave a unique common fixed point t∈ X such that
F t=Gt={f t}={gt}={t}.
Proof. Suppose that f and F are D-maps, then, there is a sequence{xn} in X such that lim
n→∞f xn =tand lim
n→∞F xn={t}for somet∈ X. By condition(1′), there exist pointsu, vinX such thatt=f u=gv. First, we show thatGv={gv}={t}.
Using inequality (2′)we get Z ̥(δ(F xn,Gv))
0
ψ(t)dt 6a(d(f xn, gv))
Z ̥(d(f xn,gv)) 0
ψ(t)dt +b(d(f xn, gv))
Z ̥(δ(f xn,F xn))+̥(δ(gv,Gv)) 0
ψ(t)dt +c(d(f xn, gv))
Z min{̥(δ(f xn,Gv)),̥(δ(gv,F xn))}
0
ψ(t)dt.
Taking the limit asn→ ∞, one obtains Z ̥(δ(gv,Gv))
0
ψ(t)dt6b(0)
Z ̥(δ(gv,Gv)) 0
ψ(t)dt <
Z ̥(δ(gv,Gv)) 0
ψ(t)dt this contradiction implies that Gv = {gv} = {t}. Since the pair (g, G) is sub- compatible, then, Ggv =gGv; i.e., Gt ={gt}. We claim that Gt= {gt} ={t}.
Suppose not, then, by condition(2′)we have Z ̥(δ(F xn,Gt))
0
ψ(t)dt6a(d(f xn, gt))
Z ̥(d(f xn,gt)) 0
ψ(t)dt +b(d(f xn, gt))
Z ̥(δ(f xn,F xn))+̥(δ(gt,Gt)) 0
ψ(t)dt +c(d(f xn, gt))
Z min{̥(δ(f xn,Gt)),̥(δ(gt,F xn))}
0
ψ(t)dt.
Whenn→ ∞we obtain Z ̥(δ(t,Gt))
0
ψ(t)dt=
Z ̥(d(t,gt)) 0
ψ(t)dt
6[a(d(t, gt)) +c(d(t, gt))]
Z ̥(d(t,gt)) 0
ψ(t)dt
<
Z ̥(d(t,gt)) 0
ψ(t)dt
which is a contradiction. Hence, {gt} = {t} = Gt. Next, we claim that F u = {f u}={t}. If not, then, by(2′)we get
Z ̥(δ(F u,f u)) 0
ψ(t)dt=
Z ̥(δ(F u,Gt)) 0
ψ(t)dt 6a(d(f u, gt))
Z ̥(d(f u,gt)) 0
ψ(t)dt +b(d(f u, gt))
Z ̥(δ(f u,F u))+̥(δ(gt,Gt)) 0
ψ(t)dt +c(d(f u, gt))
Z min{̥(δ(f u,Gt)),̥(δ(gt,F u))}
0
ψ(t)dt
=b(0)
Z ̥(δ(f u,F u)) 0
ψ(t)dt <
Z ̥(δ(f u,F u)) 0
ψ(t)dt which is a contradiction. Thus,F u={f u}={t}. SinceF andf are subcompati- ble, then, F f u=f F u; i.e.,F t={f t}. Suppose thatf t6=t. Then, the use of(2′) gives
Z ̥(d(f t,t)) 0
ψ(t)dt=
Z ̥(δ(F t,Gt)) 0
ψ(t)dt 6a(d(f t, gt))
Z ̥(d(f t,gt)) 0
ψ(t)dt +b(d(f t, gt))
Z ̥(δ(f t,F t))+̥(δ(gt,Gt)) 0
ψ(t)dt +c(d(f t, gt))
Z min{̥(δ(f t,Gt)),̥(δ(gt,F t))}
0
ψ(t)dt
= [a(d(f t, t)) +c(d(f t, t))]
Z ̥(d(f t,t)) 0
ψ(t)dt
<
Z ̥(d(f t,t)) 0
ψ(t)dt
this contradiction implies thatf t=tand henceF t={f t}={t}. Thereforet is a common fixed point of bothf, g, F andG.
The uniqueness of the common fixed point follows easily from condition(2′).
We get the same conclusion if we consider(3′′)in lieu of(3′).
Remark 2.5. Theorem 3.1 of [3] becomes a special case of Theorem 2.4 with ψ(x) = 1.
If we putf =gin Theorem 2.4, we get the next corollary.
Corollary 2.6. Let(X, d)be a metric space and letf:X → X;F, G:X →B(X) be maps. Suppose that
(i)f is surjective, (ii)
Z ̥(δ(F x,Gy)) 0
ψ(t)dt6a(d(f x, f y))
Z ̥(d(f x,f y)) 0
ψ(t)dt +b(d(f x, f y))
Z ̥(δ(f x,F x))+̥(δ(f y,Gy)) 0
ψ(t)dt +c(d(f x, f y))
Z min{̥(δ(f x,Gy)),̥(δ(f y,F x))}
0
ψ(t)dt
for allx, y inX, where̥, ψ, a, b, care as in Theorem 2.4. If either
(iii)f andF are subcompatible D-maps; f andGare subcompatible, or (iii)′ f andGare subcompatibleD-maps; f andF are subcompatible.
Then,f, F andGhave a unique common fixed pointt∈ X such that
F t=Gt={f t}={t}.
For a single map f: X → X (resp. a multivalued map F: X → B(X)), Ff
(resp.FF) will denote the set of fixed point off (resp.F).
Theorem 2.7. Let F, G:X →B(X)be multivalued maps and let f, g:X → X be single maps on the metric space X. If inequality (2′) holds for allx, y inX, then,
(Ff∩ Fg)∩ FF = (Ff∩ Fg)∩ FG.
Proof. We can check the above equality by using inequality (2′).
Theorems 2.4 and 2.7 imply the next one.
Theorem 2.8. Let f, g be self-maps of a metric space (X, d) and let Fn, where n= 1,2, . . . be maps fromX intoB(X)such that
(i)f andg are surjective, (ii)
Z ̥(δ(Fnx,Fn+1y)) 0
ψ(t)dt 6a(d(f x, gy))
Z ̥(d(f x,gy)) 0
ψ(t)dt +b(d(f x, gy))
Z ̥(δ(f x,Fnx))+̥(δ(gy,Fn+1y)) 0
ψ(t)dt +c(d(f x, gy))
Z min{̥(δ(f x,Fn+1y)),̥(δ(gy,Fnx))}
0
ψ(t)dt for allx, y inX, where̥, ψ, a, b, care as in Theorem 2.4. If either
(iii)f andF1 are subcompatible D-maps; g andF2 are subcompatible, or
(iii)′ g andF2 are subcompatibleD-maps; f andF1 are subcompatible.
Then,f, g andFn have a unique common fixed pointt∈ X such that Fnt={f t}={gt}={t} for n= 1,2, . . . .
Let Ω be the family of all maps ω: R+ → R+ such that ω is upper semi- continuous andω(t)< tfor eacht >0.
In [1], Djoudi and Aliouche proved a common fixed point theorem of Greguš type for four maps satisfying a contractive condition of integral type in a metric space using the concept of weak compatibility. Our aim henceforth is to extend this result to multivalued maps by using the concept ofD-maps.
Theorem 2.9. Let (X, d)be a metric space and let f, g:X → X;Fk:X →B(X) be single and multivalued maps, respectively. Suppose that
(i)FkX ⊆gX andFk+1X ⊆fX, (ii)
Z δ(Fkx,Fk+1y) 0
ψ(t)dt
!p
6ω a
Z d(f x,gy) 0
ψ(t)dt
!p
+ (1−a) max (
α
Z δ(f x,Fkx) 0
ψ(t)dt
!p
,
β
Z δ(gy,Fk+1y) 0
ψ(t)dt
!p
,
Z δ(f x,Fkx) 0
ψ(t)dt
!p2 Z δ(gy,Fkx) 0
ψ(t)dt
!p2
,
Z δ(gy,Fkx) 0
ψ(t)dt
!p2 Z δ(f x,Fk+1y) 0
ψ(t)dt
!p2
,
1 2
Z δ(f x,Fkx) 0
ψ(t)dt
!p
+
Z δ(gy,Fk+1y) 0
ψ(t)dt
!p!)!
for allx, yinX, wherek∈N∗={1,2, . . .},ω∈Ω,ψ∈Ψ,0< a <1,0< α, β61 andpis an integer such that p>1. If either
(iii) f and Fk are subcompatible D-maps; g and Fk+1 are subcompatible and FkX is closed, or
(iii)′ g and Fk+1 are subcompatible D-maps; f andFk are subcompatible and Fk+1X is closed.
Then,f, g andFk have a unique common fixed point t∈ X such that
Fkt={f t}={gt}={t}.
Proof. Suppose that f and Fk are D-maps, then, there exists a sequence{xn} in X such that lim
n→∞f xn =t and lim
n→∞Fkxn ={t} for some t ∈ X. SinceFkX is closed and FkX ⊆gX, then, there is u∈ X such that gu=t. IfFk+1u6={gu},
using inequality (ii)we get Z δ(Fkxn,Fk+1u)
0
ψ(t)dt
!p
6ω a
Z d(f xn,gu) 0
ψ(t)dt
!p
+ (1−a) max (
α
Z δ(f xn,Fkxn) 0
ψ(t)dt
!p
, β
Z δ(gu,Fk+1u) 0
ψ(t)dt
!p
,
Z δ(f xn,Fkxn) 0
ψ(t)dt
!p2 Z δ(gu,Fkxn) 0
ψ(t)dt
!p2
,
Z δ(gu,Fkxn) 0
ψ(t)dt
!p2 Z δ(f xn,Fk+1u) 0
ψ(t)dt
!p2
,
1 2
Z δ(f xn,Fkxn) 0
ψ(t)dt
!p
+
Z δ(gu,Fk+1u) 0
ψ(t)dt
!p!)!
. Letting n→ ∞we obtain
Z δ(gu,Fk+1u) 0
ψ(t)dt
!p
6ω (1−a) max
β,1 2
Z δ(gu,Fk+1u) 0
ψ(t)dt
!p!
<(1−a) max
β,1 2
Z δ(gu,Fk+1u) 0
ψ(t)dt
!p
<
Z δ(gu,Fk+1u) 0
ψ(t)dt
!p
which is a contradiction. Then Fk+1u={gu} ={t}. Since the pair (g, Fk+1) is subcompatible, we have Fk+1gu = gFk+1u; i.e., Fk+1t = {gt}. If t 6= gt, using inequality (ii)we obtain
Z δ(Fkxn,Fk+1t) 0
ψ(t)dt
!p
6ω a
Z d(f xn,gt) 0
ψ(t)dt
!p
+ (1−a) max (
α
Z δ(f xn,Fkxn) 0
ψ(t)dt
!p
, β
Z δ(gt,Fk+1t) 0
ψ(t)dt
!p
,
Z δ(f xn,Fkxn) 0
ψ(t)dt
!p2
Z δ(gt,Fkxn) 0
ψ(t)dt
!p2 ,
Z δ(gt,Fkxn) 0
ψ(t)dt
!p2
Z δ(f xn,Fk+1t) 0
ψ(t)dt
!p2 ,
1 2
Z δ(f xn,Fkxn) 0
ψ(t)dt
!p
+
Z δ(gt,Fk+1t) 0
ψ(t)dt
!p!)!
. At infinity we get
Z d(t,gt) 0
ψ(t)dt
!p
6ω
Z d(t,gt) 0
ψ(t)dt
!p!
<
Z d(t,gt) 0
ψ(t)dt
!p
which is a contradiction. ThereforeFk+1t={gt}={t}. SinceFk+1X ⊆fX, there exists v∈ X such thatFk+1t={t}={f v}. We claim that Fkv ={f v}, suppose not, then by condition(ii)we have
Z δ(Fkv,Fk+1t) 0
ψ(t)dt
!p
6ω a
Z d(f v,gt) 0
ψ(t)dt
!p
+ (1−a) max (
α
Z δ(f v,Fkv) 0
ψ(t)dt
!p
,
β
Z δ(gt,Fk+1t) 0
ψ(t)dt
!p
,
Z δ(f v,Fkv) 0
ψ(t)dt
!p2
Z δ(gt,Fkv) 0
ψ(t)dt
!p2 ,
Z δ(gt,Fkv) 0
ψ(t)dt
!p2 Z δ(f v,Fk+1t) 0
ψ(t)dt
!p2
,
1 2
Z δ(f v,Fkv) 0
ψ(t)dt
!p
+
Z δ(gt,Fk+1t) 0
ψ(t)dt
!p!)!
, that is,
Z δ(Fkv,f v) 0
ψ(t)dt
!p
6ω (1−a)
Z δ(Fkv,f v) 0
ψ(t)dt
!p!
<(1−a)
Z δ(Fkv,f v) 0
ψ(t)dt
!p
<
Z δ(Fkv,f v) 0
ψ(t)dt
!p
which is a contradiction. Hence Fkv = {f v} = {t}. Since the pair (f, Fk) is subcompatible, then,Fkf v=f Fkv; i.e.,Fkt={f t}. The use of(ii)gives
Z δ(Fkt,Fk+1t) 0
ψ(t)dt
!p
6ω a
Z d(f t,gt) 0
ψ(t)dt
!p
+ (1−a) max (
α
Z δ(f t,Fkt) 0
ψ(t)dt
!p
,
β
Z δ(gt,Fk+1t) 0
ψ(t)dt
!p
,
Z δ(f t,Fkt) 0
ψ(t)dt
!p2 Z δ(gt,Fkt) 0
ψ(t)dt
!p2
,
Z δ(gt,Fkt) 0
ψ(t)dt
!p2 Z δ(f t,Fk+1t) 0
ψ(t)dt
!p2
,
1 2
Z δ(f t,Fkt) 0
ψ(t)dt
!p
+
Z δ(gt,Fk+1t) 0
ψ(t)dt
!p!)!
, i.e.,
Z d(f t,t) 0
ψ(t)dt
!p
6ω
Z d(f t,t) 0
ψ(t)dt
!p!
<
Z d(f t,t) 0
ψ(t)dt
!p
this contradiction implies that{f t}={t}=Fkt. Thus,t is a common fixed point off, g andFk.
The uniqueness of the common fixed point follows from inequality(ii).
If one uses condition(iii)′ instead of(iii), one gets the same conclusion.
Theorem 2.10. Let(X, d)be a metric space and letf, g:X → X;Fn:X →B(X) be single and multivalued maps such that
(i)FnX ⊆gX andFn+1X ⊆fX, (ii)
Z δ(Fnx,Fn+1y) 0
ψ(t)dt
!p
6ω a
Z d(f x,gy) 0
ψ(t)dt
!p
+ (1−a) max
(Z δ(f x,Fnx) 0
ψ(t)dt,
Z δ(gy,Fn+1y) 0
ψ(t)dt,
Z δ(f x,Fnx) 0
ψ(t)dt
!12
Z δ(gy,Fnx) 0
ψ(t)dt
!12 ,
Z δ(gy,Fnx) 0
ψ(t)dt
!12
Z δ(f x,Fn+1y) 0
ψ(t)dt
!12
p
for all x, y in X, where ω ∈Ω, ψ ∈Ψ, 0 < a < 1 and p is an integer such that p>1. If either
(iii) f and Fn are subcompatible D-maps; g and Fn+1 are subcompatible and FnX is closed, or
(iii)′ g andFn+1 are subcompatible D-maps; f and Fn are subcompatible and Fn+1X is closed.
Then,f, g andFn have a unique common fixed pointt∈ X such that Fnt={f t}={gt}={t} for n= 1,2, . . . .
Proof. It is similar to the proof of Theorem 2.9.
Now, we prove a unique common fixed point theorem of Greguš type by using a strict contractive condition of integral type for two pairs of single and multivalued maps in a metric space.
Theorem 2.11. Let f andg be self-maps of a metric space(X, d) and let {Fn}, n= 1,2, . . . be multivalued maps fromX intoB(X)such that
(1′′)f andg are surjective, (2′′)
Z δ(F1x,Fky) 0
ψ(t)dt
< α
Z d(f x,gy) 0
ψ(t)dt+ (1−α) max (
a
Z δ(f x,F1x) 0
ψ(t)dt,
b
Z δ(gy,Fky) 0
ψ(t)dt, c
Z δ(f x,F1x) 0
ψ(t)dt
!12 Z δ(gy,F1x) 0
ψ(t)dt
!12
,
d
Z δ(gy,F1x) 0
ψ(t)dt
!12 Z δ(f x,Fky) 0
ψ(t)dt
!12
for all x, y in X and some k >1 for which the right hand side is positive, where ψ∈Ψ,0< α, a, b, c, d <1 andα+d(1−α)<1. If either
(3′′)f andF1 are subcompatibleD-maps; g andFk are subcompatible, or (3′′′)g andFk are subcompatibleD-maps; f andF1 are subcompatible.
Then,f, g and{Fn} have a unique common fixed pointt∈ X such that Fnt={f t}={gt}={t}, for n= 1,2, . . . .
Proof. Suppose that condition (3′′) holds, then, there is a sequence {xn} in X such that f xn → t and F1xn → {t} as n → ∞ for some t ∈ X. By condition (1′′), there are two elementsuand v in X such thatt =f u=gv. We show that {t}=Fkv. Indeed, using inequality(2′′)we get
Z δ(F1xn,Fkv) 0
ψ(t)dt
< α
Z d(f xn,gv) 0
ψ(t)dt+ (1−α) max (
a
Z δ(f xn,F1xn) 0
ψ(t)dt,
b
Z δ(gv,Fkv) 0
ψ(t)dt, c
Z δ(f xn,F1xn) 0
ψ(t)dt
!12
Z δ(gv,F1xn) 0
ψ(t)dt
!12 ,
d
Z δ(gv,F1xn) 0
ψ(t)dt
!12 Z δ(f xn,Fkv) 0
ψ(t)dt
!12
. Taking limit asn→ ∞, we obtain
Z δ(t,Fkv) 0
ψ(t)dt6b(1−α)
Z δ(t,Fkv) 0
ψ(t)dt <
Z δ(t,Fkv) 0
ψ(t)dt
thus, we have Fkv ={t}={gv} and since g andFk are subcompatible, we have Fkgv=gFkv; that is,Fkt={gt}. Again, by(2′′)we obtain
Z δ(F1xn,Fkt) 0
ψ(t)dt
< α
Z d(f xn,gt) 0
ψ(t)dt+ (1−α) max (
a
Z δ(f xn,F1xn) 0
ψ(t)dt,
b
Z δ(gt,Fkt) 0
ψ(t)dt, c
Z δ(f xn,F1xn) 0
ψ(t)dt
!12 Z δ(gt,F1xn) 0
ψ(t)dt
!12
,
d
Z δ(gt,F1xn) 0
ψ(t)dt
!12 Z δ(f xn,Fkt) 0
ψ(t)dt
!12
. Whenn→ ∞, we get
Z d(t,gt) 0
ψ(t)dt6[α+d(1−α)]
Z d(t,gt) 0
ψ(t)dt <
Z d(t,gt) 0
ψ(t)dt
this contradiction implies that{t}={gt}=Fkt={f u}. We claim thatF1u={t}.
By condition(2′′)we have Z δ(F1u,t)
0
ψ(t)dt=
Z δ(F1u,Fkt) 0
ψ(t)dt
< α
Z d(f u,gt) 0
ψ(t)dt+ (1−α) max (
a
Z δ(f u,F1u) 0
ψ(t)dt,
b
Z δ(gt,Fkt) 0
ψ(t)dt, c
Z δ(f u,F1u) 0
ψ(t)dt
!12
Z δ(gt,F1u) 0
ψ(t)dt
!12 ,
d
Z δ(gt,F1u) 0
ψ(t)dt
!12
Z δ(f u,Fkt) 0
ψ(t)dt
!12
= (1−α) max{a, c}
Z δ(F1u,t) 0
ψ(t)dt <
Z δ(F1u,t) 0
ψ(t)dt
this contradiction demands that F1u={t}={f u}. Since f andF1 are subcom- patible, then, F1f u = f F1u; that is, F1t = {f t}. Moreover, by (2′′) one may get
Z d(f t,t) 0
ψ(t)dt=
Z δ(F1t,Fkt) 0
ψ(t)dt
< α
Z d(f t,gt) 0
ψ(t)dt+ (1−α) max (
a
Z δ(f t,F1t) 0
ψ(t)dt,
b
Z δ(gt,Fkt) 0
ψ(t)dt, c
Z δ(f t,F1t) 0
ψ(t)dt
!12 Z δ(gt,F1t) 0
ψ(t)dt
!12
,
d
Z δ(gt,F1t) 0
ψ(t)dt
!12 Z δ(f t,Fkt) 0
ψ(t)dt
!21
= [α+d(1−α)]
Z d(f t,t) 0
ψ(t)dt <
Z d(f t,t) 0
ψ(t)dt
which is a contradiction. Thus,{f t}={t}=F1t. Therefore, F1t=Fkt={f t}= {gt}={t}.
Uniqueness follows easily from condition(2′′). The proof is thus completed.
Important remark. Every contractive or strict contractive condition of integral type automatically includes a corresponding contractive or strict contractive con- dition, not involving integrals, by setting ϕ(t) = 1 (resp. ψ(t) = 1) overR+. So, our results extend, generalize and complement several various results existing in the literature.
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H. Bouhadjera A. Djoudi
Laboratoire de Mathématiques Appliquées Université Badji Mokhtar
B. P. 12, 23000, Annaba Algérie
e-mail: b_hakima2000@yahoo.fr
