# General Common Fixed Point Theorems and Applications

Authors: Shyam Lal Singh; Swami Nath Mishra; Renu Chugh; Raj Kamal
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The main result is a common fixed point theorem for a pair of multivalued maps on a complete metric space extending a recent result ofĐorić and  Lazović (2011) for a multivalued map on a metric space satisfying Ćirić-Suzuki-type-generalized contraction. Further, as a special case, we obtain a generalization of an important common fixed point theorem of Ćirić (1974). Existence of a common solution for a class of functional equations arising in dynamic programming is also discussed.


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## Body

## 1. Introduction

Consistent with Nadler [1, page 620], (X,d) will denote a metric space and CL(X), the collection of all nonempty closed subsets of X. For A,B∈CL(X) and ε>0,(1.1)N(ε,A)={x∈X:d(x,a)<εforsomea∈A},EA,B={ε>0:A⊆N(ε,B),B⊆N(ε,A)},H(A,B)={inf⁡⁡EA,B,ifEA,B≠∅+∞,ifEA,B=∅.
The hyperspace (CL(X),H) is called the generalized Hausdorff metric space induced by the metric d on X.For nonempty subsetsA,B of X,d(A,B) denotes the gap between the subsets A and B, while (1.2)ρ(A,B)=sup⁡⁡{d(a,b):a∈A,b∈B},BN(X)={A:∅≠A⊆XandthediameterofAisfinite}.
As usual, we write d(x,B) (resp. ρ(x,B)) for d(A,B) (resp. ρ(A,B)) when A={x}.LetS,T:X→CL(X). Then u∈X is a fixed point of S  if and only if u∈Su and a common fixed point of S and T if and only if u∈Su∩Tu.LetS and T be maps to be defined specifically in a particular context, while x and y are the elements of a metric space (X,d):
(1.3)M(Sx,Ty)=max⁡{d(x,y),d(x,Sx),d(y,Ty),d(x,Ty)+d(y,Sx)2}.Recently Suzuki [2] and Kikkawa and Suzuki [3] obtained interesting generalizations of the Banach’s classical fixed point theorem and other fixed point results by Nadler [4], Jungck [5], and Meir and Keeler [6]. These results have important outcomes (see, e.g., [7–14]). The following result, due to Đorić and Lazović [9], extends and generalizes fixed point theorems from Ćirić [15], Kikkawa and Suzuki [3], Nadler [4], Reich [16], Rus [17], and others.Theorem 1.1.
Define a nonincreasing functionφ from [0,1) onto (0,1] by
(1.4)φ(r)={1if0≤r<121-rif12≤r<1.
Let X be a complete metric space and T:X→CL(X). Assume there exists r∈[0,1) such that for every x,y∈X,
(1.5)φ(r)d(x,Tx)≤d(x,y)impliesH(Tx,Ty)≤rM(Tx,Ty).
Then there exists z∈X such that z∈Tz.We remark that, for everyx,y∈X, the generalized contraction H(Tx,Ty)≤rM(Tx,Ty), 0≤r<1, was first studied by Ćirić [15]. The following important common fixed point theorem is due to Ćirić [18].Theorem 1.2.
LetX be a complete metric space and S,T:X→X. Assume there exists r∈[0,1) such that for every x,y∈X,
(1.6)d(Sx,Ty)≤rM(Sx,Ty).
Then S and T have a unique common fixed point.For an excellent discussion on several special cases and variants of Theorem1.2, one may refer to Rus [17]. However, the generality of Theorem 1.2 may be appreciated from the fact that (1.6) in Theorem 1.2 cannot be replaced by
(1.7)d(Sx,Ty)≤rmax⁡⁡{d(x,y),d(x,Sx),d(y,Ty),d(x,Ty),d(y,Sx)}.
Indeed, Sastry and Naidu [19, Example  5] have shown that maps S and T satisfying (1.7) need not have a common fixed point on a complete metric space. Notice that the condition (1.7) with S=T is the quasicontraction due to Ćirić [20].The main result of this paper (cf. Theorem2.2) generalizes Theorems 1.1 and 1.2. Further, a corollary of Theorem 2.2 is used to obtain a unique common fixed point theorem for multivalued maps on a metric space with values in BN(X). As another application, we deduce the existence of a common solution for a general class of functional equations under much weaker conditions than those in [12, 14, 21–24].


## 2. Main Results

We shall need the following result essentially due to Nadler [4] (see also [15, 25], [26, page 4], [27], [17, page 76]).Lemma 2.1.
IfA,B∈CL(X) and a∈A, then for each ε>0, there exists b∈B such that d(a,b)≤H(A,B)+ε.Theorem 2.2.
LetX be a complete metric space and S,T:X→CL(X). Assume there exists r∈[0,1) such that for every x,y∈X,
(2.1)φ(r)min⁡⁡{d(x,Sx),d(y,Ty)}≤d(x,y)impliesH(Sx,Ty)≤rM(Sx,Ty).
Then there exists an element u∈X such that u∈Su∩Tu.Proof.
ObviouslyM(Sx,Ty)=0 iff x=y is a common fixed point of S and T. So, we may take without any loss of generality that M(Sx,Ty)>0 for distinct x,y∈X. Let ε>0 be such that β=r+ε<1. Let u0∈X and u1∈Tu0. Then by Lemma 2.1, their exists u2∈Su1 such that
(2.2)d(u2,u1)≤H(Su1,Tu0)+εM(Su1,Tu0).
Similarly, their exists u3∈Tu2 such that
(2.3)d(u3,u2)≤H(Tu2,Su1)+εM(Tu2,Su1).
Continuing in this manner, we find a sequence {un} in X such that
(2.4)u2n+1∈Tu2n,u2n+2∈Su2n+1suchthatd(u2n+1,u2n)≤H(Tu2n,Su2n-1)+εM(Tu2n,Su2n-1),d(u2n+2,u2n+1)≤H(Su2n+1,Tu2n)+εM(Su2n+1,Tu2n).
Now, we consider two cases and show that for any n∈N,
(2.5)d(u2n+1,u2n)≤βd(u2n-1,u2n).Case 1.
Ifd(u2n-1,Su2n-1)≥d(u2n,Tu2n), then
(2.6)φ(r)min⁡⁡{d(u2n-1,Su2n-1),d(u2n,Tu2n)}≤d(u2n-1,u2n).
Therefore by the assumption,
(2.7)H(Su2n-1,Tu2n)≤rM(Su2n-1,Tu2n).Case 2.
Ifd(u2n,Tu2n)≥d(u2n-1,Su2n-1), then
(2.8)φ(r)min⁡⁡{d(u2n-1,Su2n-1),d(u2n,Tu2n)}≤d(u2n-1,u2n).
So by the assumption,
(2.9)H(Su2n-1,Tu2n)≤rM(Su2n-1,Tu2n).
Hence in either case we obtain by (2.7) and (2.9),
(2.10)d(u2n,u2n+1)≤H(Su2n-1,Tu2n)+εM(Su2n-1,Tu2n)≤rM(Su2n-1,Tu2n)+εM(Su2n-1,Tu2n)=βM(Su2n-1,Tu2n)=βmax⁡⁡{d(u2n-1,u2n),d(u2n-1,Su2n-1),d(u2n,Tu2n),d(u2n-1,Tu2n)+d(u2n,Su2n-1)2}≤βmax⁡⁡{d(u2n-1,u2n),d(u2n,u2n+1)}.
This yields (2.5). Analogously, we obtain d(u2n+2,u2n+1)≤βd(u2n+1,u2n), and conclude that for any n∈N,
(2.11)d(un+1,un)≤βd(un,un-1).
Therefore{un} is a Cauchy sequence and has a limit in X. Call it u.
Now we show that for  anyy∈X-{u},
(2.12)d(u,Ty)≤rmax⁡⁡{d(u,y),d(y,Ty)},(2.13)d(u,Sy)≤rmax⁡⁡{d(u,y),d(y,Sy)}.
Since un→u, there exists n0∈N (natural numbers) such that
(2.14)d(u,un)≤13d(u,y)fory≠uandalln≥n0.
Then as in [2, page 1862],
(2.15)φ(r)d(u2n-1,Su2n-1)≤d(u2n-1,Su2n-1)≤d(u2n-1,u2n)≤d(u2n-1,u)+d(u,u2n)≤23d(y,u)=d(y,u)-13d(y,u)≤d(y,u)-d(u2n-1,u)≤d(u2n-1,y).
Therefore
(2.16)φ(r)d(u2n-1,Su2n-1)≤d(u2n-1,y).
Now either d(u2n-1,Su2n-1)≤d(y,Ty) or d(y,Ty)≤d(u2n-1,Su2n-1).
So in either case by (2.16),
(2.17)φ(r)min⁡⁡{d(u2n-1,Su2n-1),d(y,Ty)}≤d(u2n-1,y).
Hence by the assumption (2.1),
(2.18)d(u2n,Ty)≤H(Su2n-1,Ty)≤rM(Su2n-1,Ty)≤rmax⁡⁡{d(u2n-1,y),d(u2n-1,Su2n-1),d(y,Ty),d(u2n-1,Ty)+d(y,Su2n-1)2}.
Making n→∞,
(2.19)d(u,Ty)≤rmax⁡⁡{d(u,y),d(u,u),d(y,Ty),d(u,Ty)+d(y,u)2}≤rmax⁡⁡{d(u,y),d(y,Ty),d(u,Ty)}.
This yields (2.12).  Similarly, we can show (2.13).
Now, we show thatu∈Su∩Tu.
For0≤r<1/2, the following  cases  arise.Case 1.
Supposeu∉Su and u∉Tu.  Then  as  in  [8,  page  6], let  a∈Tu  be such that
(2.20)2rd(a,u)<d(u,Tu),
and  a∈Su be such that  2rd(a,u)<d(u,Su).
Sincea∈Tu implies a≠u, we have from (2.12) and (2.13),
(2.21)d(u,Ta)≤rmax⁡⁡{d(u,a),d(a,Ta)},(2.22)d(u,Sa)≤rmax⁡⁡{d(u,a),d(a,Sa)}.
On the other hand, since φ(r)d(u,Tu)≤d(u,Tu)≤d(a,u),
(2.23)φ(r)min⁡⁡{d(a,Sa),d(u,Tu)}≤d(a,u).
Therefore by the assumption (2.1),
(2.24)d(Sa,a)≤H(Sa,Tu)≤rmax⁡⁡{d(a,u),d(u,Tu),d(a,Sa),d(u,Sa)+d(a,Tu)2}=rmax⁡⁡{d(a,u),d(a,Sa),12d(u,Sa)}.
This gives  d(a,Sa)≤H(Sa,Tu)≤rd(a,u)<d(a,u).
So by (2.22), d(Sa,u)≤rd(a,u). Thus(2.25)d(u,Tu)≤d(u,Sa)+H(Sa,Tu)≤rd(a,u)+rd(a,u)=2rd(a,u)<d(u,Tu)(bytheassumptionofCase1).
This contradicts u∉Tu.  Consequently  u∈Tu.  Similarly  u∈Su.Case 2.
Letu∈Suandu∉Tu. Then  as  in  the previous case, let  a∈Tu  be such that
(2.26)2rd(a,u)<d(u,Tu).
Since a≠u, we have from (2.13),
(2.27)d(u,Sa)≤rmax⁡⁡{d(u,a),d(a,Sa)}.
On the other hand, Since φ(r)d(u,Tu)≤d(u,Tu)≤d(a,u),
(2.28)φ(r)min⁡⁡{d(a,Sa),d(u,Tu)}≤d(a,u).
Therefore by the assumption (2.1),
(2.29)d(Sa,a)≤H(Sa,Tu)≤rmax⁡⁡{d(a,u),d(u,Tu),d(a,Sa),d(u,Sa)+d(a,Tu)2}=rmax⁡{d(a,u),d(a,Sa),12d(u,Sa)}.
This gives  d(a,Sa)≤H(Sa,Tu)≤rd(a,u)<d(a,u).
So by (2.22), d(Sa,u)≤rd(a,u). Thus
(2.30)d(u,Tu)≤d(u,Sa)+H(Sa,Tu)≤rd(a,u)+rd(a,u)=2rd(a,u)<d(u,Tu)(bytheassumptionofCase2).
This contradicts u∉Tu.  Consequently  u∈Tu.Case 3.
u∈Tu and u∉Su. As  in  the  previous  case,  it  follows  that  u∈Su.
Now we consider the case1/2≤r<1.
First we show that(2.31)H(Sx,Tu)≤rmax⁡⁡{d(x,u),d(x,Sx),d(u,Tu),d(x,Tu)+d(u,Sx)2}.
Assume that x≠u. Then for every n∈N, there exists zn∈Sx such that
(2.32)d(u,zn)≤d(u,Sx)+1nd(x,u).
Therefore
(2.33)d(x,Sx)≤d(x,zn)≤d(x,u)+d(u,zn)≤d(x,u)+d(u,Sx)+1nd(x,u).
Using (2.13) with y=x, (2.33) implies
(2.34)d(x,Sx)≤d(x,u)+rmax⁡⁡{d(x,u),d(x,Sx)}+1nd(u,x).
If d(x,u)≥d(x,Sx),then (2.34) gives
(2.35)d(x,Sx)≤d(x,u)+rd(x,u)+1nd(u,x)=(1+r+1n)d(x,u).
Making n→∞,
(2.36)d(x,Sx)≤(1+r)d(x,u).
Thus φ(r)d(x,Sx)=(1-r)d(x,Sx)≤(1/(1+r))d(x,Sx)≤d(x,u).
Thenφ(r)min{d(x,Sx),d(u,Tu)}≤d(x,u),and by the assumption (2.1),
(2.37)H(Sx,Tu)≤rmax⁡⁡{d(x,u),d(x,Sx),d(u,Tu),d(x,Tu)+d(u,Sx)2}.
If d(x,u)<d(x,Sx),then (2.34) gives
(2.38)d(x,Sx)≤d(x,u)+rd(x,Sx)+1nd(u,x),
that is, (1-r)d(x,Sx)≤(1+(1/n))d(x,u).Makingn→∞,(2.39)φ(r)d(x,Sx)≤d(x,u).
Then φ(r)min{d(x,Sx),d(u,Tu)}≤d(x,u), and by the assumption, we get (2.37).Takingx=u2n+1 in (2.37) and passing to the limit, we obtain (2.40)d(u,Tu)≤rd(u,Tu).
This gives u∈Tu. Analogously, u∈Su.The following result generalizes Theorem1.2.Corollary 2.3.
LetX be a complete metric space and S,T maps from X into X. Suppose there exists r∈[0,1) such that for every x,y∈X,
(2.41)φ(r)min⁡⁡{d(x,Sx),d(y,Ty)}≤d(x,y)impliesd(Sx,Ty)≤rM(Sx,Ty).
Then S and T have a unique common fixed point.Proof.
For single-valued mapsS and T, it comes from Theorem 2.2 that they have a common fixed point. The uniqueness of the common fixed point follows easily.Remark 2.4.
Theorem1.1 is obtained as a particular case of Theorem 2.2 when S=T.Now we derive the following result due toĐorić and Lazović [9, Corollary 2.3].Corollary 2.5.
LetX be a complete metric space and T a map from X into X. Suppose there exists r∈[0,1) such that for every x,y∈X,
(2.42)φ(r)d(x,Tx)≤d(x,y)impliesd(Tx,Ty)≤rM(Tx,Ty).
Then T has a unique fixed point.Proof.
It comes from Corollary2.3 when S=T.The following example shows the generality of our results.Example 2.6.
LetX={(0,0),(0,4),(4,0),(0,5),(5,0),(4,5),(5,4)} be endowed with the metric d defined by
(2.43)d[(x1,x2),(y1,y2)]=|x1-y1|+|x2-y2|.
Let S and T be such that
(2.44)S(x1,x2)={(x1,0)ifx1≤x2(0,0)ifx1>x2,T(x1,x2)={(x2,0)ifx1≤x2(0,x2)ifx1>x2.
Then S and T do not satisfy the condition (1.6) of Theorem 1.2 at  x=(4,5),y=(5,4). However, this is readily verified that all the hypotheses of Corollary 2.3 are satisfied for the maps S and T.Theorem 2.7.
LetX be a complete metric space and P,Q:X→BN(X). Assume there exists r∈[0,1) such that for every x,y∈X,
(2.45)φ(r)min⁡⁡{ρ(x,Px),ρ(y,Qy)}≤d(x,y)
implies
(2.46)ρ(Px,Qy)≤rmax⁡⁡{d(x,y),ρ(x,Px),ρ(y,Qy),d(x,Qy)+d(y,Px)2}.
Then there exsits a unique point z∈X such that z∈Pz∩Qz.Proof.
Chooseλ∈(0,1). Define single-valued maps S,T:X→X as follows. For each x∈X, let Sx be a point of Px which satisfies
(2.47)d(x,Sx)≥rλρ(x,Px).
Similarly, for each y∈X, let Ty be a point of Qy such that
(2.48)d(y,Ty)≥rλρ(y,Qy).
Since Sx∈Px and Ty∈Qy,
(2.49)d(x,Sx)≤ρ(x,Px),d(y,Ty)≤ρ(y,Qy).
So, (2.45) gives
(2.50)φ(r)min⁡⁡{d(x,Sx),d(y,Ty)}≤φ(r)min⁡⁡{ρ(x,Px),ρ(y,Qy)}≤d(x,y),
and this implies (2.46). Therefore
(2.51)d(Sx,Ty)≤ρ(Px,Qy)≤r⋅r-λmax⁡⁡{rλd(x,y),rλρ(x,Px),rλρ(y,Qy),rλd(x,Qy)+rλd(y,Px)2}≤r1-λmax⁡⁡{d(x,y),d(x,Sx),d(y,Ty),d(x,Ty)+d(y,Sx)2}.
So (2.50), namely, φ(r′)min{d(x,Sx),d(y,Ty)}≤d(x,y) implies
(2.52)d(Sx,Ty)≤r′max⁡⁡{d(x,y),d(x,Sx),d(y,Ty),d(x,Ty)+d(y,Sx)2},
where r′=r1-λ<1.
Hence by Theorem2.2, S and Thave a unique point z∈X such that Sz=Tz=z. This implies z∈Pz∩Qz.Corollary 2.8.
LetX be a complete metric space and P:X→BN(X). Assume there exists r∈[0,1) such that for every x,y∈X,
(2.53)ρ(x,Px)≤(1+r)d(x,y)impliesρ(Px,Py)≤rmax⁡⁡{d(x,y),ρ(x,Px),ρ(y,Py),d(x,Py)+d(y,Px)2}.
Then there exists a unique point z∈X such that z∈Pz.Proof.
It comes from Theorem2.7 when Q=P.


## 3. Applications

Throughout this  section, we assume thatY and Z are Banach spaces, W⊆Y and D⊆Z. Let R denotes the field of reals, g1,g2:W×D→Rand G1,G2:W×D×R→R. Taking W and D  as the state and decision spaces, respectively, the problem of dynamic programming reduces to the problem of solving functional equations:(3.1)pi=sup⁡y∈D{gi(x,y)+Hi(x,y,pi(x,y))},x∈W,i=1,2.In the multistage process, some functional equations arise in a natural way (cf. [22, 23]; see also [21, 24, 28, 29]). In this section, we study the existence of common solution of the functional equations (3.1) arising in dynamic programming.LetB(W) denotes the set of all bounded real-valued functions on W. For an arbitrary h∈B(W), define ∥h∥=supx∈W|h(x)|. Then (B(W),∥·∥) is a Banach space. Suppose that the following conditions hold: (DP-1)
H1,H2,g1,and g2 are bounded.(DP-2)
There existsr∈[0,1) such that for every (x,y)∈W×D,h,k∈B(W)and t∈W,(3.2)φ(r)min⁡{|h(t)-A1h(t)|,|k(t)-A2k(t)|}≤|h(t)-k(t)|
implies(3.3)|H1(x,y,h(t))-H2(x,y,k(t))|≤rmax⁡⁡{|h(t)-k(t)|,|h(t)-A1h(t)|,|k(t)-A2k(t)|,|h(t)-A2k(t)|+|k(t)-A1h(t)|2},
where A1,A2 are defined as follows:(3.4)Aih(x)=sup⁡y∈DHi(x,y,h(x,y)),x∈W,h∈B(W),i=1,2.Theorem 3.1.
Assume the conditions (DP-1) and (DP-2). Then the functional equations (3.1), i=1,2, have a unique common solution in B(W).Proof.
For anyh,k∈B(W), let d(h,k)=sup{|h(x)-k(x)|:x∈W}. Then (B(W),d) is a complete metric space.
Letλ be any arbitrary positive number and h1,h2∈B(W). Pick x∈W and choose y1,y2∈D such that
(3.5)Aihi<Hi(x,yi,hi(xi))+λ,
where xi=(x,yi),i=1,2.
Further,(3.6)A1h1≥H1(x,y2,h1(x2)),(3.7)A2h2≥H2(x,y1,h2(x1)).
Therefore, the first inequality in (DP-2) becomes
(3.8)φ(r)min⁡⁡{|h1(x)-A1h1(x)|,|h2(x)-A2h2(x)|}≤|h1(x)-h2(x)|,
and this together with (3.5) and (3.7) implies
(3.9)A1h1-A2h2<H1(x,y1,h1(x1))-H2(x,y,h2(x1))+λ≤|H1(x,y1,h1(x1))-H2(x,y1,h2(x1))|+λ≤rM(H1h1,H2h2)+λ.
Similarly, (3.5), (3.6), and (3.8) imply
(3.10)A2h2(x)-A1h1(x)≤rM(A1h1,A2h2)+λ.
So, from (3.10) and (3.11), we obtain
(3.11)|A1h1(x)-A2h2(x)|≤rM(A1h1,A2h2)+λ.
Since this inequality is true for any x∈W, and λ>0 is arbitrary, on taking supremum, we find from (3.8) and (3.11) that
(3.12)φ(r)min⁡⁡{d(h1,A1h1),d(h2,A2h2)}≤d(h1,h2)
implies
(3.13)d(A1h1,A2h2)≤rM(A1h1,A2h2).
Therefore, Corollary 2.3 applies, wherein A1 and A2 correspond, respectively, to the maps S and T. So A1 and A2 have a unique common fixed point h*, that is, h*(x) is the unique bounded common solution of the functional equations (3.1), i=1,2.The following result generalizes a recent result of Singh and Mishra [12, Corollary  4.2] which in turn extends certain results from [21, 23, 24].Corollary 3.2.
Suppose that the following conditions hold.(i)
G and g are bounded.(ii)
There existsr∈[0,1) such that for every x,y∈W×D,h,k∈B(W) and t∈W,
(3.14)φ(r)|h(t)-Kh(t)|≤|h(t)-k(t)|implies|G(x,y,h(t))-G(x,y,k(t))|≤rmax⁡⁡M(K,h(t),k(t)),
where K is defined as(3.15)Kh(t)=sup⁡y∈D{g(t,y)+G(t,y,h(t,y))},t∈W,h∈B(W).
Then the functional equation (3.1) with  H1=H2=G and g1=g2=g possesses a unique bounded solution in W.Proof.
It comes from Theorem3.1 when  g1=g2=g and H1=H2=G.



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