Common fixed point theorems for six self-mappings under integral type inequality satisfying (E.A) and (CLR) properties in the context of complex valued metric space (not necessarily complete) are established. The derived results are new even for ordinary metric spaces. We prove existence result for optimal unique solution of the system of functional equations used in dynamical programming with complex domain.

1. Introduction and Preliminaries

Metric fixed point theory is the most impressive and active branch of modern mathematics that has vast applications in applied functional and numerical analysis. Banach contraction principle [1] is one of the best known results in this theory. This principle can be considered as the launch of metric fixed point theory that guarantees the existence and uniqueness of fixed points of mappings. In the following years, various efforts have been done to further generalize Banach contraction principle in different direction for a single map.

The exploration of common fixed point theory is an active field of research activity since 1976. The work of Jungck [2] is considered as major achievement in the field of common fixed point theory. Jungck presented the notion of commuting maps to introduce the common fixed point results for two self-maps on complete metric space. To improve common fixed point theorems, researchers began to utilize weaker conditions than commuting mappings such as weakly commuting maps, compatible mappings, compatible mappings of type (A), compatible mappings of type (B), compatible mappings of type (P), and compatible mappings of type (C). In the study of common fixed point results of weakly compatible mappings we often require the assumption of the continuity of mappings or the completeness of underlying space. As a consequence a natural question arises as to whether there exist common fixed point theorems, which do not enforce such conditions. Regarding this Aamri and El Moutawakil [3] relaxed these conditions by introducing the notion of (E.A) property and it was marked that (E.A) property does not require the condition of continuity of mappings and completeness of the underlying space. However, (E.A) property tolerates the condition of closeness of the range subspaces of the involved mappings. In 2011, the new notion of Common Limit in the range property (shortly (CLR) property) was given by Sintunavarat and Kumam [4] that does not enforce the above-mentioned conditions. Moreover, the significance of (CLR) property reveals that closeness of range subspaces is not essential. Using these two important notions many fixed point theorems were established [3–6].

One of the most pleasant generalizations of Banach principle is the Branciari [7] fixed point theorem for a single mapping satisfying an integral type inequality. After that, serval researchers ([8–11], etc.) generalize the result of Branciari in ordinary metric spaces.

On the other hand Azam et al. [12] studied complex valued metric space and proved common fixed point theorems for two self-mappings satisfying a rational type inequality. Manro et al. [13] generalized the theorem of Branciari [7] for two self-maps under contractive condition of integral type satisfying property (E.A) and (CLR) property in the setting of complex valued metric spaces. Bahadur Zada et al. [6] generalized the results of [13] for four self-maps in the context of complex valued metric spaces.

The aim of this paper is to prove common fixed point theorems for six self-maps, satisfying integral type contractive condition using property (E.A) and (CLR) property in complex valued metric spaces, which extends and generalizes many results of the existing literature.

Throughout the paper C+={z∈C:z≿(0,0)}, opt stand for sup or inf. Z and Y are Banach spaces, Ω⊆Z is the state space, D⊆Y is the decision space, Φ={ϕ:ϕ:[0,∞[→[0,∞[ is a Lebesgue integrable mapping which is summable on each compact subset of [0,∞[, nonnegative and nondecreasing such that, for each ε>0, ∫0εϕtdt>0}, and Φ∗={φ:Rn→C is a complex valued Lebesgue integrable mapping, which is summable and nonvanishing on each measurable subset of Rn, such that, for each ε≻0, ∫0εφtdt≻0}.

Definition 1 (see [<xref ref-type="bibr" rid="B5">12</xref>]).

Let C be the set of complex numbers and z,w∈C. Define a partial order ≾ on C as follows:(1)z≾wiif Rez≤Rew,Imz≤Imw,z≺wiif Rez<Rew,Imz<Imw. Note that

k1,k2∈R and k1≤k2⇒k1z≾k2z for all z∈C;

0≾z≾w⇒z<w for all z,w∈C;

z≾w and w≺w∗⇒z≺w∗ for all z,w,w∗∈C.

Definition 2 (see [<xref ref-type="bibr" rid="B20">14</xref>]).

The “max” function for the partial order relation “≾” is defined by the following:

max{w1,w2}=w2⇔w1≾w2.

if w1≾max{w2,w3}, then w1≾w2 or w1≾w3.

max{w1,w2}=w2⇔w1≾w2 or w1≤w2.

Definition 3 (see [<xref ref-type="bibr" rid="B5">12</xref>]).

Let X be a nonempty set and d:X×X→C be the mapping satisfying the following axioms:

0≾d(z1,z2), for all z1,z2∈X and d(z1,z2)=0 if and only if z1=z2.

d(z1,z2)=d(z2,z1), for all z1,z2∈X.

d(z1,z2)≾d(z1,z3)+d(z3,z2), for all z1,z2,z3∈X.

Then pair (X,d) is called a complex valued metric space.

Example 4.

Let z1,z2∈C and define the mapping d:C×C→C by (2)dz1,z2=0if z1=z2,ι˙if z1≠z2.

Then (C,d) is a complex valued metric space.

Definition 5 (see [<xref ref-type="bibr" rid="B5">12</xref>]).

Let {zn} be a sequence in complex valued metric (X,d) and z∈X. Then z is called the limit of {zn} if for every w∈C, with 0≺w, there is n0∈N such that d(zn,z)≺w for all n>n0 and one writes limn→∞zn=z.

Lemma 6 (see [<xref ref-type="bibr" rid="B5">12</xref>]).

Any sequence {zn} in complex valued metric space (X,d) converges to z if and only if |d(zn,z)|→0 as n→∞.

Definition 7 (see [<xref ref-type="bibr" rid="B17">4</xref>]).

Let X be a nonempty set and K,L:X→X be two self-maps. Then

z∈X is called a fixed point of L if Lz=z;

z∈X is called a coincidence point of K and L if Kz=Lz;

z∈X is called a common fixed point of K and L if Kz=Lz=z.

Jungck [2] initiated the concept of commuting maps in the following way.

Definition 8.

Two self-maps K and L of nonempty set X are commuting if LKz=KLz, for all z∈X.

Jungck [15] initiated the concept of weakly compatible maps in ordinary metric spaces while Bhatt et al. [16] refined this notion in the complex valued metric space in the following way.

Definition 9.

Two self-maps K and L on complex valued metric space X are weakly compatible if there exists point z∈X such that KLz=LKz whenever Kz=Lz.

Aamri and El Moutawakil [3] initiated the concept of (E.A) property in ordinary metric spaces while Verma and Pathak [14] defined this concept in complex valued metric space as follows.

Definition 10.

Two self-maps K and L on a complex valued metric space X satisfy property (E.A) if there exists sequence {zn} in X such that (3)limr→∞Lzn=limr→∞Kzn=zfor somez∈X.

Sintunavarat and Kumam [4] introduced the notion of (CLR) property in ordinary metric spaces, in a similar mode. Verma and Pathak [14] defined this notion in a complex valued metric space in the following way.

Definition 11.

Two self-maps K and L on a complex valued metric space X satisfy (CLRK) if there exists sequence {zn} in X such that (4)limn→∞Lzn=limr→∞Kzn=Kzfor somez∈X.

Remark 12 (see [<xref ref-type="bibr" rid="B21">6</xref>]).

Let φ∈Φ∗, such that Reφ, Imφ∈Φ and zn is a sequence in C+ converges to z, and then limn→∞∫0znφsds=∫0zφsds.

Lemma 13 (see [<xref ref-type="bibr" rid="B21">6</xref>]).

Let φ∈Φ∗, such that Reφ, Imφ∈Φ and zn is a sequence in C+, and then limn→∞∫0znφsds=0 if and only if zn→0,0, as n→∞.

2. Main Results

Let Ψ be the class of all functions ψ:C+→C+ that satisfy the following properties:

ψ is nondecreasing on C+.

ψ is upper semicontinuous on C+.

ψ(0)=0 and ψ(z)≺z for every z≻0.

Now, we present our first result.

Theorem 14.

Let (X,d) be a complex valued metric space and K,L,M,N,R,S:X→X be six self-mappings satisfying the following conditions:

One of pairs (K,NR) and (L,MS) satisfies property (E.A) such that K(X)⊆MS(X) and L(X)⊆NR(X).

where ψ∈Ψ,φ∈Φ∗ and(6)Δ1z1,z2=dMSz2,Lz21+dNRz1,Kz11+dNRz1,MSz2;Δ2z1,z2=dNRz1,Kz11+dMSz2,Lz21+dNRz1,MSz2;Δ3z1,z2=maxdNRz1,MSz2,dNRz1,Kz1,dMSz2,Lz2,12dKz1,MSz2+dLz2,NRz1.

If one of MS(X) and NR(X) is closed subspace of X such that pairs (K,NR) and (L,MS) are weakly compatible, then each pair of pairs (K,NR) and (L,MS) has a coincidence point in X. Moreover, if (K,S), (L,R), (MS,R), and (NR,S) are commuting pairs, then K,L,M,N,R, and S have a unique common fixed point in X.

Proof.

Let pair (K,NR) satisfy (E.A) property, so there exists sequence {zn} in X such that(7)limn→∞Kzn=limn→∞NRzn=zfor somez∈X.Since K(X)⊆MS(X), there exists {wn} in X such that Kzn=MSwn and thus, from (7), we get(8)limn→∞Kzn=limn→∞NRzn=limn→∞MSwn=z.

We assert that limn→∞Lwn=z. If limn→∞Lwn=w≠z, then, upon putting z1=zn and z2=wn in condition (2) of Theorem 14, we have(9)∫0dKzn,Lwnφtdt≾ψmax∫0Δjzn,wnφtdt:1≤j≤3,where(10)Δ1zn,wn=dMSwn,Lwn1+dNRzn,Kzn1+dNRzn,MSwn;Δ2zn,wn=dNRzn,Kzn1+dMSwn,Lwn1+dNRzn,MSwn;Δ3zn,wn=maxdNRzn,MSwn,dNRzn,Kzn,dMSwn,Lwn,12dKzn,MSwn+dLwn,NRzn.Taking upper limit as n→∞ in (9), we have(11)Δ1zn,wn⟶dz,w,Δ2zn,wn⟶0,Δ3zn,wn⟶dz,w,∫0dz,wφtdt=limsupn→∞∫0dKzn,Lwnφtdt≾limsupn→∞ψmax∫0Δjzn,wnφtdt:1≤j≤3≾ψlimsupn→∞max∫0Δjzn,wnφtdt:1≤j≤3=ψmax∫0dz,wφtdt,0,∫0dz,wφtdt=ψ∫0dz,wφtdt≺∫0dz,wφtdt⟹∫0dz,wφtdt<∫0dz,wφtdt,which contradict with our assumption; thus z=w and limn→∞Lwn=z. Therefore (8) becomes(12)limn→∞Kzn=limn→∞NRzn=limn→∞Lwn=limn→∞MSwn=z.Also, since MS(X) is closed subspace of X, there exists u∈X such that MSu=z and, using (12), we get(13)limn→∞Kzn=limn→∞NRzn=limn→∞Lwn=limn→∞MSwn=z=MSu.Now, we claim that Lu=MSu. To support the claim, let Lu≠MSu. Then, using condition (2) of Theorem 14 with z1=zn and z2=u, one can get(14)∫0dKzn,Luφtdt≾ψmax∫0Δjzn,uφtdt:1≤j≤3,where(15)Δ1zn,u=dMSu,Lu1+dNRzn,Kzn1+dNRzn,MSu;Δ2zn,u=dNRzn,Kzn1+dMSu,Lu1+dNRzn,MSu;Δ3zn,u=maxdNRzn,MSu,dNRzn,Kzn,dMSu,Lu,12dKzn,MSu+dLu,NRzn.Taking upper limit as n→∞ in (14), we have(16)Δ1zn,u⟶dz,Lu,Δ2zn,u⟶0,Δ3zn,u⟶dz,Lu,∫0dz,Luφtdt=limsupn→∞∫0dKzn,Luφtdt≾limsupn→∞ψmax∫0Δjzn,uφtdt:1≤j≤3≾ψlimsupn→∞max∫0Δjzn,uφtdt:1≤j≤3=ψmax∫0dz,Luφtdt,0,∫0dz,Luφtdt=ψ∫0dz,Luφtdt≺∫0dz,Luφtdt⟹∫0dz,Luφtdt<∫0dz,Luφtdt, which is a contradiction. Thus, Lu=z and hence(17)Lu=MSu=z.Since L(X)⊆NR(X), there exists v∈X such that Lu=NRv and it follows from (17) that(18)Lu=MSu=NRv=z.We show that Kv=NRv. Let on contrary Kv≠NRv; then, using condition (2) of Theorem 14 with z1=v and z2=u, we have(19)∫0dKv,Luφtdt≾ψmax∫0Δjv,uφtdt:1≤j≤3,where(20)Δ1v,u=dMSu,Lu1+dNRv,Kv1+dNRv,MSu=0;Δ2v,u=dNRv,Kv1+dMSu,Lu1+dNRv,MSu=dz,Kv;Δ3v,u=maxdNRv,MSu,dNRv,Kv,dMSu,Lu,12dKv,MSu+dLu,NRv=dz,Kv.Therefore,(21)∫0dKv,zφtdt≾ψmax0,∫0dz,Kvφtdt,∫0dz,Kvφtdt≾ψ∫0dz,Kvφtdt≺∫0dz,Kvφtdt, which is a contradiction to our assumption that Kv≠NRv. Thus Kv=NRv and hence, from (18), we get(22)Kv=Lu=MSu=NRv=z.Now, using the weak compatibility of pairs (K,NR), (L,MS), and (22), we have(23)Kv=NRv⟹NRKv=KNRv⟹Kz=NRz,(24)Lu=MSu⟹MSLu=LMSu⟹Lz=MSz.Hence z is the coincident point of each pair (K,NR) and (L,MS).

Next, we have to show that z is the common fixed point of K,L,M,N,R, and S. For this, we claim that Kz=z. If Kz≠z, then upon putting z1=z,z2=u in condition (2) of Theorem 14 and using (22) and (23) we have (25)∫0dKz,Luφtdt≾ψmax∫0Δjz,uφtdt:1≤j≤3, where(26)Δ1z,u=dMSu,Lu1+dNRz,Kz1+dNRz,MSu=0;Δ2z,u=dNRz,Kz1+dMSu,Lu1+dNRz,MSu=0;Δ3z,u=maxdNRz,MSu,dNRz,Kz,dMSu,Lu,12dKz,MSu+dLu,NRz=dKz,z.Therefore,(27)∫0dKz,zφtdt≾ψmax0,0,∫0dKz,zφtdt≾ψ∫0dKz,zφtdt≺∫0dKz,zφtdt, which is impossible. Thus Kz=z and hence, in view of (23), we get(28)Kz=NRz=z.Similarly, we can show that(29)Lz=MSz=z.Hence, from (28) and (29), we get(30)Kz=Lz=MSz=NRz=z.

Now, by commuting conditions of pairs (K,S) and (NR,S) and using (28) and (30), we have K(Sz)=S(Kz)=Sz and NRSz=SNRz=Sz; from here it follows that(31)KSz=NRSz=Sz.

Also, by commuting conditions of pairs (L,R) and (MS,R) and taking (29) and (30), we have L(Rz)=R(Lz)=Rz and MSRz=RMSz=Rz; from here it follows that(32)LRz=MSRz=Rz.

Further, assume the Sz≠z. Then upon putting z1=Sz,z2=z in condition (2) of Theorem 14 and using (29) and (31), we have (33)∫0dKSz,Lzφtdt≾ψmax∫0ΔjSz,zφtdt:1≤j≤3, where(34)Δ1Sz,z=dMSz,Lz1+dNRSz,KSz1+dNRSz,MSz=0;Δ2Sz,z=dNRSz,KSz1+dMSz,Lz1+dNRSz,MSz=0;Δ3Sz,z=maxdNRSz,MSz,dNRSz,KSz,dMSz,Lz,12dKSz,MSz+dLz,NRSz=maxdSz,z,dSz,Sz,dz,z,12dSz,z+dz,Sz=dSz,z. Therefore, (35)∫0dSz,zφtdt≾ψmax0,0,∫0Sz,zφtdt≺∫0Sz,zφtdt, which is a contradiction; thus Sz=z. Also Mz=z as MSz=z, so from (30) it follows that(36)Kz=Lz=Mz=Sz=NRz=z.

Similarly, using condition (2) of Theorem 14 with z1=z and z2=Rz and taking (28) and (32), one can easily obtain that Rz=z. Also Nz=z as NRz=z. Hence, from (36), we get(37)Kz=Lz=Mz=Nz=Rz=Sz=z.That is z is a common fixed point of K,L,M,N,R, and S in X.

Similarly, if (L,MS) satisfies property (E.A) and NR(X) is closed subspace of X, then we can prove that z is a common fixed point of K,L,M,N,R, and S in X in the same arguments as above.

Uniqueness. For the uniqueness of common fixed point, let z∗≠z be another fixed point of K,L,M,N,R, and S. Then, using condition (2) of Theorem 14, we have (38)∫0dz,z∗φtdt=∫0dKz,Lz∗φtdt≾ψmax∫0Δjz,z∗φtdt:1≤j≤3, where(39)Δ1z,z∗=dMSz∗,Lz∗1+dNRz,Kz1+dNRz,MSz∗=0;Δ2z,z∗=dNRz,Kz1+dMSz∗,Lz∗1+dNRz,MSz∗=0;Δ3z,z∗=maxdNRz,MSz∗,dNRz,Kz,dMSz∗,Lz∗,dKz,MSz∗+dLz∗,NRz2=dz,z∗.Thus,(40)∫0dz,z∗φtdt≾ψmax0,0,∫0dz,z∗φtdt≺∫0dz,z∗φtdt, which is a contradiction; hence z is a unique common fixed point of K,L,M,N,R, and S in X.

Now we present some corollaries; their proofs are easily followed from Theorem 14, so we omit the proofs.

Corollary 15.

Let (X,d) be a complex valued metric space and K,M,N,R,S:X→X be five self-mappings satisfying the following conditions:

One of pairs (K,NR) and (K,MS) satisfies property (E.A) such that K(X)⊆MS(X) and K(X)⊆NR(X).

where ψ∈Ψ,φ∈Φ∗ and(42)Δ1z1,z2=dMSz2,Kz21+dNRz1,Kz11+dNRz1,MSz2;Δ2z1,z2=dNRz1,Kz11+dMSz2,Kz21+dNRz1,MSz2;Δ3z1,z2=maxdNRz1,MSz2,dNRz1,Kz1,dMSz2,Kz2,12dKz1,MSz2+dKz2,NRz1.

If one of MS(X) and NR(X) is closed subspace of X such that pairs (K,NR) and (K,MS) are weakly compatible, then each pair of pairs (K,NR) and (K,MS) has a coincidence point in X. Moreover, if (K,S), (K,R), (MS,R), and (NR,S) are commuting pairs, then K,M,N,R, and S have a unique common fixed point in X.

Corollary 16.

Let (X,d) be a complex valued metric space and K,L,R,S:X→X be four self-mappings satisfying the following conditions:

One of the pairs (K,S) and (L,R) satisfies property (E.A) such that K(X)⊆R(X) and L(X)⊆S(X).

where ψ∈Ψ,φ∈Φ∗ and(44)Δ1z1,z2=dRz2,Lz21+dSz1,Kz11+dSz1,Rz2;Δ2z1,z2=dSz1,Kz11+dRz2,Lz21+dSz1,Rz2;Δ3z1,z2=maxdSz1,Rz2,dSz1,Kz1,dRz2,Lz2,12dKz1,Rz2+dLz2,Sz1.

If one of R(X) and S(X) is closed subspace of X, then pairs (K,S) and (L,R) have a coincidence point in X. Moreover, if (K,S) and (L,R) are weakly compatible, then K,L,R, and S have a unique common fixed point in X.

Corollary 17.

Let (X,d) be a complex valued metric space and K,L,R:X→X be three self-mappings satisfying the following conditions:

One of the pairs (K,R) and (L,R) satisfies property (E.A) such that K(X)⊆R(X) and L(X)⊆R(X).

where ψ∈Ψ,φ∈Φ∗ and(46)Δ1z1,z2=dRz2,Lz21+dRz1,Kz11+dRz1,Rz2;Δ2z1,z2=dRz1,Kz11+dRz2,Lz21+dRz1,Rz2;Δ3z1,z2=maxdRz1,Rz2,dRz1,Kz1,dRz2,Lz2,12dKz1,Rz2+dLz2,Rz1.

If R(X) is closed subspace of X, then pairs (K,R) and (L,R) have a coincidence point in X. Moreover, if (K,R) and (L,R) are weakly compatible, then K,L, and R have a unique common fixed point in X.

Corollary 18.

Let (X,d) be a complex valued metric space and K,L:X→X be two self-mappings satisfying the following conditions:

where ψ∈Ψ,φ∈Φ∗ and(48)Δ1z1,z2=dKz2,Lz21+dLz1,Kz11+dLz1,Kz2;Δ2z1,z2=dLz1,Kz11+dKz2,Lz21+dLz1,Kz2;Δ3z1,z2=maxdLz1,Kz2,dLz1,Kz1,dKz2,Lz2,12dKz1,Kz2+dLz2,Lz1.

If K(X) is closed subspace of X, then pair (K,L) has a coincidence point in X. Moreover, if (K,L) is weakly compatible, then mappings K and L have a unique common fixed point in X.

Similar to the arguments of Theorem 14, we conclude the following result and omit their proof.

Theorem 19.

Let (X,d) be a complex valued metric space and K,L,M,N,R,S:X→X be six self-mappings satisfying the following conditions:

One of pairs (K,NR) and (L,MS) satisfies property (E.A) such that K(X)⊆MS(X) and L(X)⊆NR(X).

∀z1,z2∈X. (49)∫0dKz1,Lz2φtdt≾ψ∫0Δ3z1,z2φtdt,

where ψ∈Ψ,φ∈Φ∗ and (50)Δ3z1,z2=maxdNRz1,MSz2,dNRz1,Kz1,dMSz2,Lz2,12dKz1,MSz2+dLz2,NRz1.

If one of MS(X) and NR(X) is closed subspace of X such that pairs (K,NR) and (L,MS) are weakly compatible, then each pair of pairs (K,NR) and (L,MS) has a coincidence point in X. Moreover, if (K,S), (L,R), (MS,R), and (NR,S) are commuting pairs, then K,L,M,N,R, and S have a unique common fixed point in X.

Theorem 20.

Let (X,d) be a complex valued metric space and K,L,M,N,R,S:X→X be six self-mappings satisfying condition (2) of Theorem 14 and either pair (K,NR) satisfies (CLRK) property or pair (L,MS) satisfies (CLRL) property such that K(X)⊆MS(X) and L(X)⊆NR(X). If pairs (K,NR) and (L,MS) are weakly compatible, then each pair of pairs (K,NR) and (L,MS) has a coincidence point in X. Moreover, if (K,S), (L,R), (MS,R), and (NR,S) are commuting pairs, then K,L,M,N,R, and S have a unique common fixed point in X.

Proof.

Suppose that pair (K,NR) satisfies (CLRK) property, then there exists sequence {zn} in X such that(51)limn→∞Kzn=limn→∞NRzn=Ktfor somet∈X.Since K(X)⊆MS(X), there exists u∈X such that Kt=MSu.

We claim that Lu=MSu. To support the claim, let Lu≠MSu. Then on using condition (2) of Theorem 14, with setting z1=zn and z2=u, we have(52)∫0dKzn,Luφtdt≾ψmax∫0Δjzn,uφtdt:1≤j≤3,where(53)Δ1zn,u=dMSu,Lu1+dNRzn,Kzn1+dNRzn,MSu;Δ2zn,u=dNRzn,Kzn1+dMSu,Lu1+dNRzn,MSu;Δ3zn,u=maxdNRzn,MSu,dNRzn,Kzn,dMSu,Lu,12dKzn,MSu+dLu,NRzn.Taking upper limit as n→∞ in (52) and using (51), we get(54)Δ1zn,u⟶dKt,Lu,Δ2zn,u⟶0,Δ3zn,u⟶dLu,Kt,∫0dKt,Luφtdt=limsupn→∞∫0dKzn,Luφtdt≾limsupn→∞ψmax∫0Δjzn,uφtdt:1≤j≤3≾ψlimsupn→∞max∫0Δjzn,uφtdt:1≤j≤3=ψmax∫0dKt,Luφtdt,0,∫0dLu,Ktφtdt=ψ∫0dLu,Ktφtdt≺∫0dLu,Ktφtdt⟹∫0dKt,Luφtdt<∫0dKt,Luφtdt,which is a contradiction. Thus Lu=Kt and hence(55)Lu=MSu=Kt.Also, since L(X)⊆NR(X), there exists v∈X such that Lu=NRv. Thus (55) becomes(56)Lu=MSu=NRv=Kt.Now, we assert that Kv=NRv. Let on contrary Kv≠NRv; then setting z1=v and z2=u, in condition (2) of Theorem 14, we get (57)∫0dKv,Luφtdt≾ψmax∫0Δjv,uφtdt:1≤j≤3, where(58)Δ1v,u=dMSu,Lu1+dNRv,Kv1+dNRv,MSu;Δ2v,u=dNRv,Kv1+dMSu,Lu1+dNRv,MSu;Δ3v,u=maxdNRv,MSu,dNRv,Kv,dMSu,Lu,12dKv,MSu+dLu,NRv.Using (56), we have(59)∫0dKv,Ktφtdt≾ψmax0,∫0dKt,Kvφtdt,∫0dKt,Kvφtdt≾ψ∫0dKt,Kvφtdt≺∫0dKt,Kvφtdt⟹∫0dKv,Ktφtdt<∫0dKt,Kvφtdt,which is impossible. Thus Kv=Kt and hence(60)Kv=NRv=Kt.Therefore, from (56) and (60), we get(61)Kv=Lu=MSu=NRv=Kt=zsay.

Finally, following the lines in the proof of Theorem 14 we can show that z is the coincident point of pairs (K,NR) and (L,MS) and is a unique common fixed point of the mappings K,L,M,N,R, and S.

Similar to the arguments of Theorem 20, we conclude the following results and omit their proofs.

Theorem 21.

Let (X,d) be a complex valued metric space and K,L,M,N,R,S:X→X be six self-mappings satisfying the following conditions:

Either pair (K,NR) satisfies (CLRK) property or pair (L,MS) satisfies (CLRL) property such that K(X)⊆MS(X) and L(X)⊆NR(X).

∀z1,z2∈X. (62)∫0dKz1,Lz2φtdt≾ψ∫0Δ3z1,z2φtdt,

where ψ∈Ψ,φ∈Φ∗ and (63)Δ3z1,z2=maxdNRz1,MSz2,dNRz1,Kz1,dMSz2,Lz2,12dKz1,MSz2+dLz2,NRz1.If pairs (K,NR) and (L,MS) are weakly compatible, then each pair of pairs (K,NR) and (L,MS) has a coincidence point in X. Moreover, if (K,S), (L,R), (MS,R), and (NR,S) are commuting pairs, then K,L,M,N,R, and S have a unique common fixed point in X. Corollary 22.

Let (X,d) be a metric space and K,L,M,N,R,S:X→X be six self-mappings satisfying the following conditions:

Either pair (K,NR) satisfies (CLRK) property or pair (L,MS) satisfies (CLRL) property such that K(X)⊆MS(X) and L(X)⊆NR(X).

∀z1,z2∈X. (64)∫0dKz1,Lz2φtdt≤α∫0Δ3z1,z2φtdt,

where 0≤α<1,φ∈Φ and(65)Δ3z1,z2=maxdNRz1,MSz2,dNRz1,Kz1,dMSz2,Lz2,12dKz1,MSz2+dLz2,NRz1.

If pairs (K,NR) and (L,MS) are weakly compatible, then each pair (K,NR) and (L,MS) has a coincidence point in X. Moreover, if (K,S), (L,R), (MS,R), and (NR,S) are commuting pairs, then K,L,M,N,R, and S have a unique common fixed point in X.

Similarly to Theorem 14 one can derive variant of corollaries from Theorems 19, 20, and 21.

Remark 23.

The conclusions of Theorems 14, 19, 20, and 21 are still valid if we replace Δ3 with Δ3∗, where (66)Δ3∗z1,z2=maxdNRx,MSy,dNRx,Kt,dMSy,Ly,dKt,MSy,dLy,NRx.

Remark 24.

Theorems 14 and 20 and Corollary 15 extends Theorem 2.1 of [11] in complex valued metric space. Corollary 16 generalizes the results of [8–11] in complex valued metric space. Moreover, the real valued metric space version of our main results generalizes the results of [8–11].

To support Theorem 21, we present the following example.

Example 25.

Let X={z=x+ι˙y:x,y∈0,1} be a complex valued metric space with metric d:X×X→C defined by (67)dz1,z2=z1-z2eiθfor a givenθ∈0,π2. Define self-maps K,L,M,N,R, and S on X by Kz=0, Lz=0, Mz=z/2, Nz=z/4, Rz=z/3, and Sz=z/6.

Then, (68)MSz=Mz6=z12,NRz=Nz3=z12. Also, we define φ:R2→C by φ(x,y)=2+0ι˙ and ψ:C+→C+ by ψ(z)=z/2.

Clearly K(X)=0⊆MS(X)=z=x+ι˙y:x,y∈0,1/12 and L(X)⊆NR(X).

Now, we construct sequence zn=xn+ι˙yn=1/(n+1)+ι˙/(n+1) in X such that(69)limn→∞Kzn=limn→∞K1n+1+ι˙n+1=0,limn→∞NRzn=limn→∞NR1n+1+ι˙n+1=limn→∞1121n+1+ι˙n+1=0.that is, there exists sequence {zn} in X such that (70)limn→∞Kzn=limn→∞NRzn=0=Kzforz=0+0ι˙∈X. Hence (K,NR) satisfies (CLRK) property.

Next, check the following condition(71)∫0dKz1,Lz2φtdt≾ψ∫0Δz1,z2φtdt=ψ2tΔz1,z2=Δz1,z2,where(72)Δz1,z2=maxdNRz1,MSz2,dNRz1,Kz1,dMSz2,Lz2,12dKz1,MSz2+dLz2,NRz1=maxz112-z212eiθ,z112eiθ,z212eiθ,12z136eiθ+z112eiθ.Since(73)0≾maxz112-z212eiθ,z112eiθ,z212eiθ,12z136eiθ+z112eiθ,therefore(74)∫0dKz1,Lz2φtdt≾maxdNRz1,MSz2,dNRz1,Kz1,dMSz2,Lz2,12dKz1,MSz2+dLz2,NRz1.Thus, from (71), (73), and (74) and by using the value of ψ, we have (75)∫0dKz1,Lz2φtdt≾ψ∫0Δz1,z2φtdt, where(76)Δz1,z2=maxdNRz1,MSz2,dNRz1,Kz1,dMSz2,Lz2,12dKz1,MSz2+dLz2,NRz1. Also pairs (K,NR) and (L,MS) are weakly compatible and (K,S), (L,R), (MS,R), and (NR,S) are commuting pairs. Hence from Theorem 21, 0 is a unique common fixed point of K,L,M,N,R, and S.

3. Applications

Many researchers study the applications of common fixed point theorems in complex valued metric spaces; see for instance [17, 18] and the references therein. On the other hand, Liu et al. [19] and Sarwar et al. [20] study the existence and uniqueness of common solution for the system of functional equations arising in dynamic programming with real domain. We apply Corollary 22 for the existence and uniqueness of a common solution for the following system of functional equations arising in dynamic programming with complex domain (see [21]).(77)p1z=optw∈Duz,w+Θ1z,w,p1τ1z,w∀z∈Ω,p2z=optw∈Duz,w+Θ2z,w,p2τ2z,w∀z∈Ω,p3z=optw∈Dvz,w+Θ3z,w,p3τ3z,w∀z∈Ω,p4z=optw∈Dvz,w+Θ4z,w,p4τ4z,w∀z∈Ω,p5z=optw∈Dvz,w+Θ5z,w,p5τ5z,w∀z∈Ω,p6z=optw∈Dvz,w+Θ6z,w,p6τ6z,w∀z∈Ω,where z and w signify the state and decision vectors, respectively, pi(z) denotes the optimal return functions with initial state z, τi:Ω×D→Ω,Θi:Ω×D×C→R∀i∈{1,2,3,4,5,6}, and u,v:Ω×D→C.

Let CΩ be the space of all continuous real valued functions on possibly complex domain Ω with metric (78)dh,k=supz∈Ωhz-kz∀h,k∈CΩ.

We prove the following result.

Theorem 26.

Let u,v and Θi:Ω×D×C→R, i=1,2,…,6, be bounded functions and let K,L,M,N,R,S:CΩ→CΩ be six operators defined as(79)Kh1z=optw∈Duz,w+Θ1z,w,h1τ1z,w∀z∈Ω,Lh2z=optw∈Duz,w+Θ2z,w,h2τ2z,w∀z∈Ω,Mh3z=optw∈Dvz,w+Θ3z,w,h3τ3z,w∀z∈Ω,Nh4z=optw∈Dvz,w+Θ4z,w,h4τ4z,w∀z∈Ω,Rh5z=optw∈Dvz,w+Θ5z,w,h5τ5z,w∀z∈Ω,Sh6z=optw∈Dvz,w+Θ6z,w,h6τ6z,w∀z∈Ω,for all hi∈CΩ and z∈Ω. Assume that the following conditions hold:

There exist hn∈CΩ such that limn→∞Khn=limn→∞NRhn=Kh∗, for some h∗∈CΩ.

KCΩ⊆MSCΩ such that pairs (K,NR) and (L,MS) are weakly compatible.

Pairs (K,S), (L,R), (MS,R), and (NR,S) are commuting.

For h1,h2∈CΩ. (80)∫0Θ1z,w,h1τz,w-Θ2z,w,h2τz,wφtdt≤α∫0Δ3h1,h2φtdt,

where (81)Δ3h1,h2=maxNRh1-MSh2,NRh1-Kh1,MSh2-Lh2,12Kh1-MSh2+Lh2-NRh1, where h1∈CΩ,0≤α<1, and ϕ:R+→R+ is a nonnegative summable Lebesgue integrable function such that (82)∫0εϕsds>0 for each ε>0. Then the system of functional equations (77) has a unique bounded solution. Proof.

Notice that the system of functional equations (77) has a unique bounded solution if and only if the system of operators (79) have a unique common fixed point. Now since u,v, and Θi are bounded, there exists positive number λ such that (83)supuz,w,vz,w,Θiz,w,w∗:z,w,w∗∈Ω×D×C,i=1,2,…,6≤λ. Now, by using properties of the theory of integration and definition of ϕ, we conclude that, for each positive number λ, there exists positive δλ, such that(84)∫Γϕsds≤λfor all Γ⊆[0,2λ] with mΓ≤δλ, where mΓ is the Lebesgue measure of Γ.

Now, we consider two possible cases.

Case 1. Suppose that optw∈D=supw∈D. Let z∈Ω and h1,h2∈CΩ; then for δλ>0 there exist w1,w2∈D such that(85)Kh1z<uz,w1+Θ1z,w1,h1τ1z,w1+δλ,(86)Lh2z<uz,w2+Θ2z,w2,h2τ2z,w2+δλ,(87)Kh1z≥uz,w2+Θ1z,w2,h1τ1z,w2,(88)Lh2z≥uz,w1+Θ2z,w1,h2τ2z,w1. From inequalities (85) and (88) it follows that (89)Kh1z-Lh2z<Θ1z,w1,h1τ1z,w1-Θ2z,w1,h2τ2z,w1+δλ≤Θ1z,w1,h1τ1z,w1-Θ2z,w1,h2τ2z,w1+δλ which gives(90)Kh1z-Lh2z<maxΘ1z,w1,h1τ1z,w1-Θ2z,w1,h2τ2z,w1+δλ,Θ1z,w2,h1τ1z,w2-Θ2z,w2,h2τ2z,w2+δλ. Similarly, using inequalities (86) and (87) we obtain(91)Lh2z-Kh1z<maxΘ1z,w1,h1τ1z,w1-Θ2z,w1,h2τ2z,w1+δλ,Θ1z,w2,h1τ1z,w2-Θ2z,w2,h2τ2z,w2+δλ. Therefore from (90) and (91) we get(92)Kh1z-Lh2z<maxΘ1z,w1,h1τ1z,w1-Θ2z,w1,h2τ2z,w1+δλ,Θ1z,w2,h1τ1z,w2-Θ2z,w2,h2τ2z,w2+δλ<maxA+δλ,B+δλ, where A=Θ1z,w1,h1(τ1(z,w1))-Θ2(z,w1,h2(τ2(z,w1))) and B=Θ1z,w2,h1(τ1(z,w2))-Θ2(z,w2,h2(τ2(z,w2))).

Case 2. Suppose that optw∈D=infw∈D. By following the procedure in Case 1, one can check that (92) holds.

Now, from 3.10, we have (93)∫0Kh1z-Lh2zϕtdt<∫0maxA+δλ,B+δλϕtdt=max∫0A+δλϕtdt,∫0B+δλϕtdt=max∫0Aφtdt+∫AA+δλφtdt,∫0Bφtdt+∫BB+δλφtdt=max∫0Aφtdt,∫0Bφtdt+max∫AA+δλφtdt,∫BB+δλφtdt. And, by condition (iv) of Theorem 26, we get (94)∫0Kh1z-Lh2zϕtdt<α∫0maxNRh1-MSh2,NRh1-Kh1,MSh2-Lh2,1/2Kh1-MSh2+Lh2-NRh1ϕtdt+max∫AA+δλφtdt,∫BB+δλφtdt, and using (84) we get (95)∫0Kh1z-Lh2zϕtdt<α∫0maxNRh1-MSh2,NRh1-Kh1,MSh2-Lh2,1/2Kh1-MSh2+Lh2-NRh1ϕtdt+λ. Since above inequality is true for each z∈Ω and λ>0 is taken arbitrarily, we deduce that (96)∫0dKh1,Lh2ϕtdt≤α∫0Δ3h1,h2ϕtdt, where (97)Δ3h1,h2=maxdNRh1,MSh2,dNRh1,Kh1,dMSh2,Lh2,12dKh1,MSh2+dLh2,NRh1. Also, from condition i of Theorem 26 pair K,NR satisfies (CLR) property. Thus all hypothesis of Corollary 22 are satisfied. Consequently operators (79) have a unique common fixed point, that is, system (77) of functional equations has a unique bounded solution.

Competing Interests

The authors declare that they have no competing interests regarding this manuscript.

Authors’ Contributions

All authors read and approved the final version.

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