# A New Type of Coincidence and Common Fixed Point Theorem with Applications

Authors: Shyam Lal Singh; Raj Kamal; Manuel De la Sen; Renu Chugh
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Coincidence and common fixed point theorems for a class of Ćirić-Suzuki hybrid contractions involving a multivalued and two single-valued maps in a metric space are obtained. Some applications including the existence of a common solution for certain class of functional equations arising in a dynamic programming are also discussed.


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## Body

## 1. Introduction

Consistent with [1] (see also [2, 3]), Y denotes an arbitrary nonempty set, (X,d) a metric space, and CL(X) (resp., CB(X)), the collection of all nonempty closed (resp., closed bounded) subsets of X. The hyperspace (CL(X),H) (resp., (CB(X),H)) is called the generalized Hausdorff (resp., the 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)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}.For the sake of brevity, we follow the following notations, whereinS, f, and g are maps to be defined specifically in a particular context, while x and y are elements of some specific domain:
(2)M(S;fx,gy)=max⁡{d(Sx,gy)+d(Sy,fx)2d(fx,gy),d(fx,Sx),d(gy,Sy),d(Sx,gy)+d(Sy,fx)2};M(S;fx,fy)=max⁡{d(Sx,fy)+d(Sy,fx)2d(fx,fy),d(fx,Sx),d(fy,Sy),d(Sx,fy)+d(Sy,fx)2};M(Sx,Sy)=max⁡{d(y,Sx)+d(x,Sy)2d(x,y),d(x,Sx),d(y,Sy),d(y,Sx)+d(x,Sy)2};M1(fx,gy)=max⁡{d(y,fx)+d(x,gy)2d(x,y),d(x,fx),d(y,gy),d(y,fx)+d(x,gy)2}.The Banach contraction principle (Bcp) plays an important role in nonlinear analysis and has numerous generalizations and several applications (see, e.g., [1–21] and others). Nadler Jr. [1] (see also [22]) initiated the study of multivalued Banach contractions in metric spaces. In view of its numerous applications, the Nadler multivalued contraction theorem received enormous attention (see, e.g., [2, 3, 7, 8, 11–15, 17–21, 23–36] and references thereof).The following result [13, p. 250] extends and generalizes many results due to Fisher [37], Goebel [38], Kubiak [29], and others.Theorem 1.
LetS:Y→CL(X) and f,g:Y→X be such that S(Y)⊆f(Y)∩g(Y), and one of S(Y), f(Y) or g(Y) is a complete subspace of X. Assume there exists r∈[0,1) such that for every x,y∈Y,
(3)H(Sx,Sy)≤rM(S;fx,gy).
Then(i)
Sand f have a coincidence point v in Y,(ii)
S and g have a coincidence point w in Y.Further, ifY=X, then(iii)
S and f have a common fixed point v provided that fv is a fixed point of f, and f and S commute at v;(iv)
S and g have a common fixed point w provided that gw is a fixed point of g, and g and S commute at w;(v)
S, f, and g have a common fixed point provided that (iii) and (iv) both are true.We remark that certain contractive conditions studied forS:Y→CL(X) and f,g:Y→X by Ćirić [5], Covitz and Nadler Jr. [16], Czerwik [6], Fisher [37], Goebel [38], Jungck [17], Kubiak [29], Naimpally et al. [8], Pathak [15], Pathak et al. [9], Petrusel and Rus [10], Reich [11], and Rus [3] are included in the following condition:
(4)H(Sx,Sy)≤rM(S;fx,gy),
for every x,y∈Y, where 0≤r<1.In particular, (4) with Y=X and f=g= the identity map on X was studied by Ćirić [5].Recently, Suzuki [39, Th. 2] obtained a remarkable generalization of the Bcp. The same has been extended to multivalued maps by Kikkawa and Suzuki [30] in the following manner.Theorem 2.
Define a strictly decreasing functionη:[0,1)→((1/2),1] by
(5)η(r)=11+r.
Let (X,d) be a complete metric space and S:Y→CB(X). Assume there exists r∈[0,1) such that for every x,y∈Y,
(6)η(r)d(x,Sx)≤d(x,y)impliesH(Sx,Sy)≤rd(x,y).
Then there exists z∈X such that z∈Sz.Subsequently, some interesting extensions and generalizations of Theorem2 were obtained among others by Abbas et al. [23], Dhompongsa and Yingtaweesittikul [24], Dorić and Lazović [25], Kamal et al. [18], Moţ and Petruşel [26], Singh and Mishra [27, 31, 36], and Singh et al. [28, 32, 33].The importance of Suzuki contraction theorem [39, Th. 2] and subsequently obtained coincidence and fixed point theorems (cf. [23–28, 30–33, 36] and others) for maps in metric spaces satisfying Suzuki-type contractive conditions is that the contractive conditions are required to be satisfied not for all points of the domain.In all that follows we take a nonincreasing functionφ from [0,1) onto (0,1] defined by
(7)φ(r)={1if0≤r<121-rif12≤r<1.Recently, Singh et al. [33] obtained the following coincidence and common fixed point theorem which is a generalization of a result of Dorić and Lazović [25].Theorem 3.
LetS:Y→CL(X) and f:Y→X be such that S(Y)⊆f(Y). Assume there exists r∈[0,1) such that for every x,y∈Y,
(8)φ(r)d(fx,Sx)≤d(fx,fy)impliesH(Sx,Sy)≤rM(S;fx,fy).
If one of S(Y) or f(Y) is a complete subspace of X, then there exists a point z∈Y such that fz∈Sz.
Further, ifY=X and fz is a fixed point of f, then fz is a fixed point of S provided that f is IT-commuting with S at z.In this paper, we obtain a coincidence and common fixed point theorem (cf. Theorem6) extending and generalizing Theorems 1, 2, 3, and several others. We also deduce the existence of common solution for a certain class of functional equations arising in dynamic programming. Examples are given to justify theorems and applications.


## 2. Main Results

The following definition is due to Itoh and Takahashi [19] (see also [27]).Definition 4.
LetS:X→CL(X) and f:X→X. Then the hybrid pair (S,f) is IT-commuting at z∈X if fSz⊆Sfz.We remark that IT-commuting maps are more general than commuting maps [34, p. 2]. However, a pair of maps f,g:X→X are IT-commuting (also called weakly compatible by Jungck and Rhoades [20]) at x∈X if fgx=gfx when fx=gx.We will need the following lemma essentially due to Nadler Jr. [1] (see also [5], [2, p. 61], [35, p. 4], [3, p. 76]).Lemma 5.
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.LetC(S,f) denote the collection of all coincidence points of S and f; that is, C(S,f)={z∈Y:fz∈Sz} when S:Y→CL(X) and f:Y→X; and C(S,f)={z∈Y:fz=Sz} when S,f:Y→X.The following is the main result of this section.Theorem 6.
LetS:Y→CL(X) and f,g:Y→X be such that S(Y)⊆f(Y)∩g(Y). Assume there exists r∈[0,1) such that for every x,y∈Y,(9a)φ(r)min⁡{d(fx,Sx),d(gy,Sy)}≤d(fx,gy)
implies
(9b)H(Sx,Sy)≤rM(S;fx,gy).If one of S(Y), f(Y), or g(Y) is a complete subspace of X, then(I)
C(S,f) is nonempty; that is, there exists a point z∈Y such that fz∈Sz.(II)
C(S,g) is nonempty; that is, there exists a point z1∈Y such that gz1∈Sz1.Further if,Y=X, then(III)
S and f have a common fixed point provided that the maps S and f are IT-commuting just at coincidence point z and fz is fixed point of f;(IV)
S and g have a common fixed point provided that the maps S and g are IT-commuting just at coincidence point z1 and gz1 is fixed point of g;(V)
S, f, and g have a common fixed point provided that both (III) and (IV) are true.Proof.
Without loss of generality, we may taker>0 and f, g nonconstant maps.
Letε>0 be such that β=r+ε<1. We construct two sequences {xn} in Y and {yn} in X as follows.
Letx0∈Y and y0=gx1∈Sx0. By Lemma 5, there exists y1=fx2∈Sx1 such that
(10)d(fx2,gx1)≤H(Sx0,Sx1)+εM(S;fx0,gx1).
Similarly, there exists y2=gx3∈Sx2 such that
(11)d(fx2,gx3)≤H(Sx2,Sx1)+εM(S;fx2,gx1).
Continuing in this manner, we find a sequence {yn} in X such that
(12)y2n=gx2n+1∈Sx2n,y2n+1=fx2n+2∈Sx2n+1,d(fx2n,gx2n+1)≤H(Sx2n,Sx2n-1)+εM(S;fx2n,gx2n-1),d(fx2n+2,gx2n+1)≤H(Sx2n,Sx2n+1)+εM(S;fx2n,gx2n+1).
Now, we show that for any n∈N,
(13)d(y2n,y2n-1)≤βd(y2n-1,y2n-2).
Suppose if d(gx2n-1,Sx2n-1)≥d(fx2n,Sx2n), then
(14)φ(r)min⁡{d(fx2n,Sx2n),d(gx2n-1,Sx2n-1)}≤d(fx2n,gx2n-1).
Therefore, by the assumption,
(15)d(fx2n,gx2n+1)≤H(Sx2n,Sx2n-1)+εM(S;fx2n,gx2n-1)≤rM(S;fx2n,gx2n-1)+εM(S;fx2n,gx2n-1)=βM(S;fx2n,gx2n-1)=βmax⁡{d(gx2n-1,Sx2n)+d(fx2n,Sx2n-1)2d(fx2n,gx2n-1),d(fx2n,Sx2n),d(gx2n-1,Sx2n-1),d(gx2n-1,Sx2n)+d(fx2n,Sx2n-1)2}.
This yields (13).
Suppose ifd(fx2n,Sx2n)≥d(gx2n-1,Sx2n-1), then
(16)φ(r)min⁡⁡{d(fx2n,Sx2n),d(gx2n-1,Sx2n-1)}≤d(fx2n,gx2n-1).
Therefore, by the assumption,
(17)d(fx2n,gx2n+1)≤H(Sx2n,Sx2n-1)+εM(S;fx2n,gx2n-1)≤rM(S;fx2n,gx2n-1)+εM(S;fx2n,gx2n-1)=βM(S;fx2n,gx2n-1)=βmax⁡{d(gx2n-1,Sx2n)+d(fx2n,Sx2n-1)2d(fx2n,gx2n-1),d(fx2n,Sx2n),d(gx2n-1,Sx2n-1),d(gx2n-1,Sx2n)+d(fx2n,Sx2n-1)2}≤βmax⁡{d(fx2n,gx2n-1),d(fx2n,gx2n+1)},
yielding (13). So, in both cases, we obtain (13). In an analogous manner, we show that
(18)d(y2n+1,y2n)≤βd(y2n,y2n-1).
We conclude from (13) and (18) that for any n∈N,
(19)d(yn+1,yn)≤βd(yn,yn-1).
Therefore the sequence {yn} is Cauchy. Assume that the space g(Y) is complete. Notice that the sequence {y2n} is contained in g(Y) and has a limit in g(Y). Call it u. Let z∈f-1u. Then z∈Y and fz=u. The subsequence {y2n+1} also converges to u. Let z1∈g-1u. Then
(20)gz1=u.
Now we show that for any gy∈X-{fz},
(21)d(u,Sy)≤rmax⁡{d(u,gy),d(gy,Sy)},
and for any fy∈X-{gz},
(22)d(u,Sy)≤rmax⁡{d(u,fy),d(fy,Sy)}.
Since fx2n→fz, there exists n0∈N (naturals) such that
(23)d(fx2n,fz)≤13d(fz,gy)forgy≠fzandalln≥n0.
Also, since gx2n+1→fz, there exists n1∈N such that
(24)d(gx2n+1,fz)≤13d(fz,gy)forgy≠fzandalln≥n1.
Then, as in [39, p. 1862] (see also [25]),
(25)φ(r)d(fx2n,Sx2n)≤d(fx2n,Sx2n)≤d(fx2n,gx2n+1)≤23d(fz,gy)=d(fz,gy)-13d(fz,gy)≤d(fz,gy)-d(fx2n,fz)≤d(fx2n,gy).
Therefore,
(26)φ(r)d(fx2n,Sx2n)≤d(fx2n,gy).
Now, either d(fx2n,Sx2n)≤d(gy,Sy) or d(gy,Sy)≤d(fx2n,Sx2n).
In each case, by (26) and the assumption,
(27)d(fx2n+1,Sy)≤H(Sx2n,Sy)≤rM(S;fx2n,gy).≤rmax⁡{d(fx2n,Sy)+d(gy,Sx2n)2d(fx2n,gy),d(fx2n,Sx2n),d(gy,Sy),d(fx2n,Sy)+d(gy,Sx2n)2}.
Making n→∞,
(28)d(u,Sy)≤rmax⁡{d(u,Sy)+d(u,gy)2d(u,gy),d(u,u),d(gy,Sy),d(u,Sy)+d(u,gy)2}≤rmax⁡{d(u,gy),d(gy,Sy),d(u,Sy)+d(u,gy)2}=rmax⁡{d(u,gy),d(gy,Sy)}.
This yields (21); that is,
(29)d(fz,Sy)≤rmax⁡{d(fz,gy),d(gy,Sy)}.
Analogously, we can prove (22); that is,
(30)d(gz1,Sy)≤rmax⁡{d(gz1,fy),d(fy,Sy)}.
Now, we show that C(S,f) is nonempty.
We first consider the case0≤r<1/2.
Supposefz∉Sz. Then as in [24, p. 6], let ga∈Sz be such that 2rd(ga,fz)<d(Sz,fz).
Sincega∈Sz implies ga≠fz, we have from (21) and (22),
(31)d(fz,Sa)≤rmax⁡{d(fz,ga),d(ga,Sa)}.
On the other hand, since φ(r)d(fz,Sz)≤d(fz,Sz)≤d(fz,ga),
(32)φ(r)min⁡{d(fz,Sz),d(ga,Sa)}≤d(fz,ga).
Therefore, by the assumption (13),
(33)d(ga,Sa)≤H(Sz,Sa)≤rmax⁡{d(fz,Sa)+d(ga,Sz)2d(fz,ga),d(fz,Sz),d(ga,Sa),d(fz,Sa)+d(ga,Sz)2}=rmax⁡{d(fz,ga),d(ga,Sa)}.
This gives d(ga,Sa)≤H(Sz,Sa)≤rd(fz,ga)<d(fz,ga).
So by (31), d(fz,Sa)≤rd(fz,ga). Thus, by the assumption,
(34)d(fz,Sz)≤d(fz,Sa)+H(Sz,Sa)≤rd(fz,ga)+rd(fz,ga)=2rd(fz,ga)<d(fz,Sz).
This contradicts fz∉Sz. Consequently, fz∈Sz, and C(S,f) is nonempty.
In an analogous manner, we can prove in the case0≤r<1/2 that C(S,g) is nonempty.
We now consider the case1/2≤r<1. We first show that
(35)H(Sz,Sy)≤rmax⁡{d(gy,Sz)+d(fz,Sy)2d(fz,gy),d(fz,Sz),d(gy,Sy),d(gy,Sz)+d(fz,Sy)2}.
Assume that fz≠gy. Then for every n∈N, there exists zn∈Sy such that
(36)d(fz,zn)≤d(fz,Sy)+1nd(fz,gy).
Therefore,
(37)d(gy,Sy)≤d(gy,zn)≤d(gy,fz)+d(fz,zn)≤d(gy,fz)+d(fz,Sy)+1nd(fz,gy).
So using (31), the inequality (37) implies
(38)d(gy,Sy)≤d(fz,gy)+rmax⁡{d(fz,gy),d(gy,Sy)}+1nd(fz,gy).
If d(fz,gy)≥d(gy,Sy), then (38) gives
(39)d(gy,Sy)≤d(fz,gy)+rd(fz,gy)+1nd(fz,gy)=(1+r+1n)d(fz,gy).
Making n→∞,
(40)d(gy,Sy)≤(1+r)d(fz,gy).
Thus,
(41)φ(r)d(gy,Sy)=(1-r)d(gy,Sy)≤(11+r)d(gy,Sy)≤d(fz,gy).
Then
(42)φ(r)min⁡{d(fz,Sz),d(gy,Sy)}≤d(fz,gy),
and by the assumption,
(43)H(Sz,Sy)≤rmax⁡{d(gy,Sz)+d(fz,Sy)2d(fz,gy),d(fz,Sz),d(gy,Sy),d(gy,Sz)+d(fz,Sy)2}.
If d(fz,gy)<d(gy,Sy), then (38) gives
(44)d(gy,Sy)≤d(fz,gy)+rd(gy,Sy)+1nd(fz,gy);
that is, (1-r)d(gy,Sy)≤(1+1/n)d(fz,gy).
Makingn→∞, φ(r)d(gy,Sy)≤d(fz,gy).
Thenφ(r)min⁡{d(fz,Sz),d(gy,Sy)}≤d(fz,gy), and by the assumption, we get (43).
Sinced(Sz,fx2n+2)≤H(Sz,Sx2n+1), taking y=x2n+1 in (43) and passing to the limit, we obtain
(45)d(Sz,fz)≤rd(fz,Sz).
This gives fz∈Sz; that is, z is a coincidence point of f and S. Analogously, gz∈Sz. Thus, (I) and (II) are completely proved.
Further, ifY=X, fz is a fixed point of f, and S and f are IT-commuting at z, then fSz⊆Sfz. Therefore, fz∈Sz implies ffz∈fSz⊆Sfz, so fz∈Sfz. This proves that u=fz is a common fixed point of f and S. This proves (III). Analogously, S and g have a common fixed point gz1. Therefore (20) implies that u is a common fixed point of S and g. This proves (IV). Now (V) is immediate.Remark 7.
In Theorem6, the hypothesis “fz is a fixed point of f” is essential for the existence of a common fixed point of S and f (see also [8]). Similarly, the hypothesis “gz1 is a fixed point of g” is essential for the existence of a common fixed point of S and g. Further, the contractive condition for three maps S:Y→CL(X) and f,g:Y→X studied by Abbas et al. [23] are included in the assumptions of Theorem 6.Corollary 8.
Theorem2.Proof.
It comes from Theorem6 when g=f.The following result due to Dorić and Lazović [25] generalizing many fixed point theorems is obtained as a special case from Theorem 6 when Y=X and f and g are the identity map on X.Corollary 9.
Let(X,d) be a complete metric space and S:X→CL(X). Assume there exists r∈[0,1) such that for every x,y∈X,
(46)φ(r)d(x,Sx)≤d(x,y)impliesH(Sx,Sy)≤rM(Sx,Sy).
Then there exists an element z∈X such that z∈Sz.The following result extends and generalizes coincidence and fixed point theorems of Fisher [37], Goebel [38], Jungck [17], and others.Corollary 10.
Letf,g,P:Y→X be such that P(Y)⊆f(Y)∩g(Y). Let P(Y) or f(Y) or g(Y) be a complete subspace of X. Assume there exists r∈[0,1) such that for every x,y∈Y,
(47)φ(r)min⁡{d(fx,Px),d(gy,Py)}≤d(fx,gy),
implies
(48)d(Px,Py)≤rM(P;fx,gy).
Then C(P,f) and C(P,g) are nonempty. Further, if  Y=X and if P commutes with f and g at a common coincidence point, then f, g, and P have a unique common fixed point; that is, there exists a unique point z∈X such that fz=gz=Pz=z.Proof.
SetSx={Px} for every x∈Y. Then it easily comes from Theorem 6 that C(P,f) and C(P,g) are nonempty. Further, if Y=X and P commutes with f and g at z, then ffz=fPz=Pfz and ggz=gPz=Pgz.
Alsoφ(r)min⁡{d(fz,Pz),d(ffz,Pfz)}=0≤d(fz,ffz), and this implies
(49)d(Pz,Pfz)≤rmax⁡{d(fz,Pfz)+d(ffz,Pz)2d(fz,ffz),d(fz,Pz),d(ffz,Pfz),d(fz,Pfz)+d(ffz,Pz)2}=rd(Pz,Pfz).
This says that fz is fixed point of f and P. Analogously gz is fixed point of g and P. The uniqueness of the common fixed point follows easily.Corollary 11.
Let(X,d) be a complete metric space and let f,g:X→X be an onto maps. Assume there exists r∈[0,1) such that for every x,y∈X,
(50)φ(r)min⁡⁡{d(x,fx),d(y,gy)}≤d(fx,gy)impliesd(x,y)≤rM1(fx,gy).
Then f and g have a unique common fixed point.Proof.
It comes from Corollary10 when Y=X and P is the identity map on X.Corollary 12.
Let(X,d) be a complete metric space and let f:X→X be onto maps. Assume there exists r∈[0,1) such that for every x,y∈X,
(51)φ(r)d(x,fx)≤d(fx,fy)impliesd(x,y)≤rM(fx,fy).
Then f has a unique fixed point.Proof.
It comes from Corollary11 when f=g.The following example shows that Theorem6 is indeed more general than Theorem 1.Example 13.
Consider a metric spaceX={(0,0),(0,1),(1,0),(1,2),(2,1)}, where d is defined by
(52)d[(x1,x2),(y1,y2)]=|x1-y1|+|x2-y2|.
Let S, f and g:X→X be such that
(53)S(x1,x2)={(0,0)if(x1,x2)≠(1,2),(2,1)(0,1)if(x1,x2)=(1,2)(1,0)if(x1,x2)=(2,1),f(x1,x2)=(x2,x1)∀(x1,x2)∈X,g(x1,x2)={(x1,x2)if(x1,x2)≠(1,0)(0,1)if(x1,x2)=(1,0).
It is readily verified that
(54)d(Sx,Sy)≤12max⁡{d(Sx,gy)+d(Sy,fx)2d(fx,gy),d(fx,Sx),d(gy,Sy),d(Sx,gy)+d(Sy,fx)2},
for all (x,y)∈X except for x,y∈{(1,2),(2,1)} with r=1/2.
Forx,y∈{(1,2),(2,1)}, condition (3) yields 2≤2r, which contradicts 0≤r<1. Therefore, the condition (3) of Theorem 1 is not satisfied. So, in order to see that the maps S, f, and g satisfy the assumption of Theorem 6, we notice that the condition (9a) of Theorem 6 does not hold for x,y∈{(1,2),(2,1)}. Indeed, for (x,y)=((1,2),(2,1)),
(55)φ(r)min⁡{d(fx,Sx),d(gy,Ty)}=φ(r)min⁡{d(f(1,2),S(1,2)),d(g(2,1),T(2,1))}=φ(r)min⁡{2,2}=2φ(r).
That is, φ(r)min⁡{d(fx,Sx),d(gy,Ty)}=1>0=d(fx,gy).
This violates (9a) when φ(r)=1/2 (as r=1/2). Similarly (9a) is also not true for (x,y)=((2,1),(1,2)). It is easily seen that all other hypotheses of Theorem 6 are also true.Now we give an application of Corollary10.Theorem 14.
LetS:Y→BN(X) and f,g:Y→X be such that S(Y)⊆f(Y)∩g(Y), and let one of S(Y), f(Y), or g(Y) be a complete subspace of X. Assume there exists r∈[0,1) such that for every x,y∈Y,
(56)φ(r)min⁡{ρ(fx,Sx),ρ(gy,Sy)}≤d(fx,gy),
implies
(57)ρ(Sx,Sy)≤rmax⁡{d(fx,Sy)+d(gy,Sx)2d(fx,gy),ρ(fx,Sx),ρ(gy,Sy),d(fx,Sy)+d(gy,Sx)2}.
Then C(S,f) and C(S,g) are nonempty.Proof.
Chooseλ∈(0,1). Define single-valued maps h1,h2:X→X as follows. For each x∈X, let h1x be a point of Sx which satisfies
(58)d(fx,h1x)≥rλρ(fx,Sx).
Similarly, for each y∈X, let h2y be a point of Sy such that
(59)d(gy,h2y)≥rλρ(gy,Sy).
Since h1x∈Sx and h2y∈Sy,
(60)d(fx,h1x)≤ρ(fx,Sx),d(gy,h2y)≤ρ(gy,Sy).
So (56) gives
(61)φ(r)min⁡{d(fx,h1x),d(gy,h2y)}≤φ(r)min⁡{ρ(fx,Sx),ρ(gy,Sy)}≤d(fx,gy),
and this implies (57). Therefore,
(62)d(h1x,h2y)≤ρ(Sx,Sy)≤r·r-λmax⁡{rλd(fx,Sy)+rλd(gy,Sx)2rλd(fx,gy),rλρ(fx,Sx),rλρ(gy,Sy),rλd(fx,Sy)+rλd(gy,Sx)2}≤r1-λmax⁡{d(fx,h2y)+d(gy,h1x)2d(fx,gy),d(fx,h1x),d(gy,h2y),d(fx,h2y)+d(gy,h1x)2}.
So (61), namely, φ(r′)min⁡{d(fx,h1x),d(gy,h2y)}≤d(fx,gy), implies
(63)d(h1x,h2y)≤r′max⁡{d(fx,h2y)+d(gy,h1x)2d(fx,gy),d(fx,h1x),d(gy,h2y),d(fx,h2y)+d(gy,h1x)2},
where r′=r1-λ<1.
Hence, by Corollary10, there exist z1,z2∈Y such that h1z1=fz1 and h2z2=gz2. This implies that z1 is a coincidence point of f and S, and z2 is a coincidence point of g and S.Corollary 15.
LetS:Y→BN(X) and f:Y→X be such that S(Y)⊆f(Y), and let S(Y) or f(Y) be a complete subspace of X. Assume there exists r∈[0,1) such that for every x,y∈Y,
(64)φ(r)ρ(fx,Sx)≤d(fx,fy),
implies
(65)ρ(Sx,Sy)≤rmax⁡{d(fx,Sy)+d(fy,Sx)2d(fx,fy),ρ(fx,Sx),ρ(fy,Sy),d(fx,Sy)+d(fy,Sx)2}.
Then there exists z∈Y such that fz∈Sz.Proof.
It comes from Theorem14 when g=f.Corollary 16.
LetX be a complete metric space and let S:X→BN(X). Assume there exists r∈[0,1) such that for every x,y∈X,
(66)φ(r)ρ(x,Sx)≤d(x,y),
implies
(67)ρ(Sx,Sy)≤rmax⁡{d(x,Sy)+d(y,Sx)2d(x,y),ρ(x,Sx),ρ(y,Sy),d(x,Sy)+d(y,Sx)2}.
Then there exists a unique point z∈X such that z∈Sz.Proof.
It comes from Theorem14 that S has a fixed point when f=g is the identity map on X. The uniqueness of the fixed point follows easily.


## 3. Applications

Throughout this section, we assume thatU and V are Banach spaces, W⊆U, and D⊆V. Let R denote the field of reals, τ:W×D→W, g,g′:W×D→R, and G,F1,F2:W×D×R→R. Considering W and D as the state and decision spaces, respectively, the problem of dynamic programming reduces to the problem of solving the functional equations:(68a)p=sup⁡y∈D{g(x,y)+G(x,y,p(τ(x,y)))},x∈W,(68b)qi=sup⁡y∈D{g′(x,y)+Fi(x,y,q(τ(x,y)))},x∈W,i=1,2.Indeed, in the multistage process, some functional equations arise in a natural way (cf. Bellman [40] and Bellman and Lee [41]; see also [6, 9, 15, 28, 33, 42–45]). In this section, we study the existence of a common solution of the functional equations (68a) and (68b) arising in the dynamic programming.LetB(W) denote the set of all bounded real-valued functions on W. For an arbitrary h∈B(W), define ∥h∥=sup⁡x∈W|h(x)|. Then (B(W),∥·∥) is a Banach space. Suppose that the following conditions hold:(DP-1)G, F1, F2, g, and g′ are bounded.(DP-2) Letφ(r) be considered as in the previous sections. Assume that there exists r∈[0,1) such that for every (x,y)∈W×D, h,k∈B(W) and t∈W,
(69)φ(r)min⁡⁡{|J1h(t)-Ah(t)|,|J2k(t)-Ak(t)|}≤|J1h(t)-J2k(t)|,
implies
(70)|G(x,y,h(t))-G(x,y,k(t))|≤rM(A;J1h,J2k),
where
(71)M(A;J1h,J2k)=max⁡{|J1h(t)-Ak(t)|+|J2k(t)-Ah(t)|2|J1h(t)-J2k(t)|,|J1h(t)-Ah(t)|,|J2k(t)-Ak(t)|,|J1h(t)-Ak(t)|+|J2k(t)-Ah(t)|2},
and A, J1, and J2 are defined as follows:
(72)Ah(x)=sup⁡y∈D{g(x,y)+G(x,y,h(τ(x,y)))},x∈W,h∈B(W),Jih(x)=qi=sup⁡y∈D{g′(x,y)+Fi(x,y,h(τ(x,y)))},x∈W,h∈B(W),i=1,2.(DP-3) For anyh,k∈B(W), there exists u,v∈B(W) such that
(73)Ah(x)=J1u(x),Ak(x)=J2v(x),x∈W.(DP-4) There existsh,k∈B(W) such that
(74)J1h(x)=Ah(x)impliesJ1Ah(x)=AJ1h(x),J2k(x)=Ak(x)impliesJ2Ak(x)=AJ2k(x).Theorem 17.
Assume the conditions (DP-1)–(DP-4). LetJ(B(W)) be a closed convex subspace of B(W). Then the functional equations (68a) and (68b), i=1,2, have a unique bounded 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. By virtue of (DP-3) and (DP-4), A(B(W))⊆J1(B(W))∩J2(B(W)) and the map A is IT-commuting with J1 and J2 at coincidence points.
Letλ be an arbitrary positive number and h1,h2∈B(W). Pick x∈W, and choose y1,y2∈D such that
(75)Ahj<g(x,yj)+G(x,yj,hj(xj))+λ,j=1,2,
where xj=τ(x,yj). Further,
(76)Ah1≥g(x,y2)+G(x,y2,h1(x2)),(77)Ah2≥g(x,y1)+G(x,y1,h2(x1)).
Therefore, the first inequality in (DP-2) becomes
(78)φ(r)min⁡{|J1h1(x)-Ah1(x)|,|J2h2(x)-Ah2(x)|}≤|J1h1(x)-J2h2(x)|,
and this together with (75), (77), and (78) implies
(79)Ah1-Ah2<G(x,y1,h1(x1))-G(x,y1,h2(x1))+λ≤|G(x,y1,h1(x1))-G(x,y1,h2(x1))|+λ≤rM(A;J1h1,J2h2)+λ.
Similarly, (75), (76), and (78) imply
(80)Ah2(x)-Ah1(x)≤rM(A;J1h1,J2h2)+λ.
So, from (79) and (80), we obtain
(81)|Ah1(x)-Ah2(x)|≤rM(A;J1h1,J2h2)+λ.
As λ>0 is arbitrary and (81) is true for any x∈W, taking supremum, we find from (78) and (81) that
(82)φ(r)min⁡{d(J1h1,Ah1),d(J2h2,Ah2)}≤d(J1h1,J2h2),
implies
(83)d(Ah1,Ah2)≤rM(A;J1h1,J2h2).
Therefore, Corollary 10 applies, wherein A, J1 and J2 correspond, respectively, to the maps P, f, and g. So (A,J1) and (A,J2) have a unique common fixed point h*; that is, h*(x) is the unique bounded common solution of the functional equations (68a) and (68b), i=1,2.Now we furnish an example in support of Theorem17.Example 18.
LetX=Y=R be a Banach space endowed with the standard norm ∥·∥ defined by ∥x∥=|x|, for all x∈X.
SupposeW=[0,1]⊆X be the state space and D=[0,∞)⊆Y the decision space. Define τ:W×D→W by
(84)τ(x,y)=xy2+1,x∈W,y∈D.
For any h,k∈B(W) and i=1,2, define p,qi:W→R by
(85)p(x)=qi(x)=x2+12.
Define G,F1,F2,g,g′:W×D×R→R by
(86)G(x,y,t)=14{x(x+1)(y+1)sinyy+1+2},F1(x,y,t)=12x+y+1+12sint,F2(x,y,t)=12x+3y+1+12sint,g(x,y)=x2y2x+y2,g′(x,y)=x2y5x+y5.
Notice that G, F1, F2, g, and g′ are bounded. Also
(87)J1h(x)=sup⁡y∈D{g′(x,y)+F1(x,y,h(τ(x,y)))},=x2+12=q1(x);x∈W,h∈B(W),J2k(x)=sup⁡y∈D{g′(x,y)+F2(x,y,k(τ(x,y)))},=x2+12=q2(x);x∈W,h∈B(W),Ah(x)=sup⁡y∈D{g(x,y)+G(x,y,h(τ(x,y)))},=x2+12=p(x);x∈W,h∈B(W),Ak(x)=sup⁡y∈D{g(x,y)+G(x,y,k(τ(x,y)))},=x2+12=p(x);x∈W,h∈B(W).
Now
(88)φ(r)min⁡{|J1h(t)-Ah(t)|,|J2k(t)-Ak(t)|}=φ(r)min⁡{|q1(x)-p(x)|,|q2(x)-p(x)|}=0=|J1h(t)-J2k(t)|.
Thus,
(89)φ(r)min⁡{|Jh(t)-Ah(t)|,|Jk(t)-Ak(t)|}=|Jh(t)-Jk(t)|,
and this implies
(90)|G(x,y,h(t))-G(x,y,k(t))|=0≤rM(A;Jh(t),Jk(t)).
Finally, for any h,k∈B(W) with Ah=Jh, we have
(91)AJh=p(x)=q(x)=JJh=JAh;
that is, JAh=AJh, and with Ak=Jk, we have AJk=p(x)=q(x)=JJk=JAk; that is, JAk=AJk.
Thus, all the hypotheses of Theorem17 are satisfied. So the system of (68a) and (68b) has a unique solution in B(W).Corollary 19.
Suppose that the following conditions hold.
(i)G, F, g, and g′ are bounded.
(ii) Assume there existsr∈[0,1) such that for every (x,y)∈W×D, h,k∈B(W) and t∈W,
(92)φ(r)|Jh(t)-Ah(t)|≤|Jh(t)-Jk(t)|,
implies
(93)|G(x,y,h(t))-G(x,y,k(t))|≤rmax⁡{|Jh(t)-Ak(t)|+|Jk(t)-Ah(t)|2|Jh(t)-Jk(t)|,|Jh(t)-Ah(t)|,|Jk(t)-Ak(t)|,|Jh(t)-Ak(t)|+|Jk(t)-Ah(t)|2},
where A and J are defined as follows:
(94)Ah(x)=sup⁡y∈D{g(x,y)+G(x,y,h(τ(x,y)))},x∈W,h∈B(W),Jh(x)=q=sup⁡y∈D{g′(x,y)+F(x,y,h(τ(x,y)))},x∈W,h∈B(W).
(iii) For anyh,k∈B(W), there exists u,v∈B(W) such that
(95)Ah(x)=Ju(x),Ak(x)=Jv(x),x∈W.
(iv) There existsh,k∈B(W) such that
(96)Jh(x)=Ah(x)impliesJAh(x)=AJh(x),Jk(x)=Ak(x)impliesJAk(x)=AJk(x).
Then the functional equations (68a) and (68b) with F1=F2=F possess a unique bounded common solution in W.Proof.
It comes from Theorem17 when F1=F2=F.Now we derive the the following result due to Dorić and Lazović [25], which in turn extends certain results from [41, 42].Corollary 20.
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,
(97)φ(r)|h(t)-Kh(t)|≤|h(t)-k(t)|
implies
(98)|G(x,y,h(t))-G(x,y,k(t))|≤rmax⁡M(K,K;h(t),k(t)),
where K is defined as
(99)Ah(x)=sup⁡y∈D{g(x,y)+G(x,y,h(τ(x,y)))},x∈W,h∈B(W).
Then the functional equation (68a) with G1=G2=G possesses a unique bounded solution in W.Proof.
It comes from Corollary19 when g′=0, τ(x,y)=x and F(x;y;t)=t as the assumption (DP-3) becomes redundant in this context.



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*Source: 642378-2014-04-30.xml*