USING IMPLICIT RELATION TO PROVE COMMON COUPLED FIXED POINT THEOREMS FOR TWO HYBRID PAIRS OF MAPPINGS

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USING IMPLICIT RELATION TO PROVE COMMON COUPLED FIXED POINT THEOREMS FOR TWO HYBRID PAIRS OF

MAPPINGS

BHAVANA DESHPANDE1_{, AMRISH HANDA}2_{, DEEPMALA}3_,_§

Abstract. Using implicit relation we establish two common coupled fixed point

theo-rems under the conditions of weakly commutativity andw−compatibility on a complete metric space, which is not partially ordered. We do not use the condition of continuity of any mapping for proving the existence of coupled coincidence and common coupled fixed point.

Keywords:Coupled fixed point, coupled coincidence point, implicit relation,w−compatibility, weakly commuting mappings.

AMS Subject Classification: 47H10, 54H25.

1. Introduction and Preliminaries

Let (X, d) be a metric space and CB(X) be the set of all nonempty closed bounded subsets of X. Let D(x, A) denote the distance from x to A ⊂ X and H denote the Hausdorff metric induced byd,that is,

D(x, A) = inf

a∈Ad(x, a),

H(A, B) = max

sup

a∈A

D(a, B), sup

b∈B

D(b, A)

, for all A, B ∈CB(X).

The study of fixed points for multivalued contractions and non-expansive mappings using the Hausdorff metric was studied by many authors under different conditions. This theory has found application in control theory, convex optimization, differential inclusions and economics. There exists considerable literature about fixed point properties for two hybrid pairs of mappings, including [2, 10, 11, 12, 20, 25, 26, 32].

Bhaskar and Lakshmikantham [7] introduced the concept of coupled fixed point for single-valued mappings and established some coupled fixed point results and found its application in the existence and uniqueness of solution for periodic boundary value prob-lems. Lakshmikantham and Ciric [18] proved coupled coincidence and common coupled fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces and extended the results established in [7]. Many authors focused on cou-pled fixed point theory for single-valued mappings and proved remarkable results including [5, 9, 13, 15, 16, 21, 33].

1

Department of Mathematics, Govt. P. G. Arts & Science College, Ratlam (M. P.) India. e-mail: [email protected];

2 _{Department of Mathematics, Govt. P. G. Arts & Science College, Ratlam (M. P.) India.}

e-mail: [email protected];

3 _{SQC & OR Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata, 700 108, India.}

e-mail: [email protected];

§ Manuscript received: June 26, 2014; Accepted: August 26, 2015.

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Recently Samet et al. [28] claimed that most of the coupled fixed point theorems for single-valued mappings on ordered metric spaces are consequences of well-known fixed point theorems.

The concepts related to coupled fixed point for single valued mappings have been ex-tended by Abbas et al. [1] for multivalued mappings to obtained coupled coincidence point and common coupled fixed point theorems involving hybrid pair of mappings in complete metric spaces. At present, very few authors have been studied coupled fixed point theorems for hybrid pair of mappings including [1, 19].

In [1], Abbas, Ciric, Damjanovic and Khan introduced the following concept:

Definition 1.1. Let X be a nonempty set,F :X×X →2X (a collection of all nonempty subsets ofX) and g be a self-mapping on X. An element(x, y)∈X×X is called

(1) a coupled fixed point of F if x∈F(x, y) and y∈F(y, x).

(2) a coupled coincidence point of hybrid pair {F, g} ifgx∈F(x, y) and gy ∈F(y, x). (3) a common coupled fixed point of hybrid pair {F, g} if x = gx ∈ F(x, y) and

y=gy ∈F(y, x).

We denote the set of coupled coincidence points of mappings F and g by CF, g}. Note that if (x, y)∈C(F, g), then(y, x) is also in C(F, g).

Definition 1.2. Let F :X×X→2X be a multivalued mapping and g be a self-mapping on X. The mapping g is called F−weakly commuting at some point (x, y) ∈ X×X if

g2x∈F(gx, gy) and g2y∈F(gy, gx).

Definition 1.3. Let F :X×X→2X be a multivalued mapping and g be a self-mapping on X. The hybrid pair {F, g} is called w−compatible if gF(x, y) ⊆F(gx, gy) whenever

(x, y)∈C(F, g).

Lemma 1.1. [14] Let (X, d) be a metric space. Then, for each a∈X and B ∈CB(X),

there is b0 ∈B such that D(a, B) =d(a, b0),where D(a, B) = infb∈Bd(a, b).

On the other hand, several authors have been studied fixed point theorems satisfying an implicit relation for single-valued and multivalued mappings under different conditions including [3, 4, 6, 8, 17, 22, 23, 24, 27, 29, 30, 31, 34].

In this paper, we establish two common coupled fixed point theorems for two hybrid pairs of mappings satisfying an implicit relation under the conditions of weakly commuta-tivity andw−compatibility on a complete metric space, which is not partially ordered. To prove our theorems we do not use condition of continuity of any mapping. We improve, extend and generalize the result of Sedghi et al. [29].

2. Implicit relation

LetR+ be the set of all non-negative real numbers and let Ψ be the set of all continuous functionsψ: (R+)11→R satisfying the following conditions:

ψ1 :ψ(t1, t2, ..., t11) is non-decreasing in t1 and non-increasing in t2, t3, ..., t11. ψ2 : There exists 0< k <1 such that for everyu, v, p, q ∈R+ such that

ψ(u, v, v, u, u+v, 0, q, q, p, p+q, 0)≤0, or

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then max{u, p} ≤kmax{v, q}. ψ3 : For allu, v >0,

ψ(u, u, 0, 0, u, u, v, 0, 0, v, v)>0.

Example 2.1. Letψ(t1, t2, ..., t11) =t1−hmax{t2, t3, t4, t5+t6

2 , t7, t8, t9,

t10+t11

2 }where 0< h <1.

(ψ1) Obvious. (ψ2) Let max{u, p} > 0 and ψ(u, v, v, u, u+v, 0, q, q, p, p+q, 0) = u −hmax{u, v, p, q} ≤ 0. Thus u ≤ hmax{max{u, p}, max{v, q}}. Similarly

p ≤ hmax{max{u, p}, max{v, q}}. Thus max{u, p} ≤ hmax{max{u, p}, max{v, q}}.

Now, if max{u, p} ≥max{v, q},then max{u, p} ≤hmax{u, p}<max{u, p},which is a contradiction. Thusmax{u, p}<max{v, q} andmax{u, p} ≤hmax{v, q}.Similarly, let

max{u, p} >0 and ψ(u, v, u, v, 0, u+v, q, p, q, 0, p+q) =u−hmax{u, v, p, q} ≤0,

then we havemax{u, p} ≤hmax{v, q}.Thus(ψ2) is satisfying withk=h <1.If max{u, p}= 0,thenmax{u, p} ≤kmax{v, q}.(ψ3) ψ(u, u,0,0, u, u, v,0,0, v, v) =u−hmax{u, v}= max{u−hu, u−hv}= max{u(1−h), u−hv}>0.Therefore ψ∈Ψ.

Example 2.2. Let ψ(t1, t2, ..., t11) = t1−αmax{t2, t3, t4, t7, t8, t9} −βmax{t5+t6, t10+t11} where α, β≥0 and α+ 2β <1.

(ψ1) Obvious. (ψ2) Let max{u, p} > 0 and ψ(u, v, v, u, u+v, 0, q, q, p, p+q, 0) =u−αmax{v, u, q, p} −βmax{u+v, p+q} ≤0,thenu≤αmax{max{u, p},max{v, q}}+β[max{u, p}+ max{v, q}],it follows thatu≤max{(α+β) max{u, p}+βmax{v, q}, (α+β) max{v, q}+βmax{u, p}}. Similarly p≤max{(α+β) max{u, p}+βmax{v, q}, (α+β) max{v, q}+βmax{u, p}}.Thusmax{u, p} ≤max{(α+β) max{u, p}+βmax{v, q}, (α+β) max{v, q}+βmax{u, p}}. Now, if max{u, p} ≥ max{v, q}, then max{u, p} ≤(α+2β) max{u, p}<max{u, p},which is a contradiction. Thusmax{u, p}<max{v, q} and max{u, p} ≤ (α+ 2β) max{v, q}. Similarly, let max{u, p}>0 and ψ(u, v, u, v, 0, u+v, q, p, q, 0, p+q) = u−αmax{v, u, q, p} −βmax{u+v, p+q} ≤ 0, then we have max{u, p} ≤ (α+ 2β) max{v, q}. Thus (ψ2) is satisfying with k = α+ 2β < 1. If max{u, p} = 0, then max{u, p} ≤kmax{v, q}. (ψ3) ψ(u, u, 0, 0, u, u, v, 0, 0, v, v) = u−αmax{u, v}−βmax{2u,2v}= max{u−αu−2βu, u−αv−2βv}= max{u(1−α−2β), u−(α+ 2β)v}>0. Therefore ψ∈Ψ.

Example 2.3. Let ψ(t1, t2, ..., t11) = t1 −amax{t2, t7} −bmax{t3 +t4, t8 +t9} −

cmax{t5+t6, t10+t11}, where a, b, c∈[0, 1)and a+ 2b+ 2c <1.

(ψ1) Obvious. (ψ2) Let max{u, p} > 0 and ψ(u, v, v, u, u+v, 0, q, q, p, p+q, 0) =u−amax{v, q} −bmax{u+v, p+q} −cmax{u+v, p+q} ≤0,thenu≤amax{v, q}+b[max{u, p}+ max{v, q}] +c[max{u, p}+ max{v, q}]. Similarly p≤amax{v, q}+ b[max{u, p} + max{v, q}] +c[max{u, p} + max{v, q}]. Thus max{u, p} ≤ amax{v, q}+b[max{u, p}+ max{v, q}] +c[max{u, p}+ max{v, q}].Now, if max{u, p} ≥max{v, q},thenmax{u, p} ≤(a+ 2b+ 2c) max{u, p}<max{u, p},which is a contradiction. Thus

max{u, p} <max{v, q} and max{u, p} ≤(a+ 2b+ 2c) max{v, q}.Similarly, let max{u, p} > 0 and ψ(u, v, u, v, 0, u+v, q, p, q, 0, p+q) = u−amax{v, q} −bmax{u+v, p+q} − cmax{u +v, p +q} ≤ 0, then we have max{u, p} ≤ (a+ 2b+ 2c) max{v, q}. Thus (ψ2) is satisfying with k = a+ 2b+ 2c < 1. If max{u, p} = 0, then max{u, p} ≤kmax{v, q}. (ψ3) ψ(u, u, 0, 0, u, u, v, 0, 0, v, v) =u−amax{u, v} −cmax{2u, 2v}= max{u−au−2cu, u−av−2cu}= max{u(1−a−2c), u−(a+ 2c)v}>0.Therefore

ψ∈Ψ.

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(ψ1) Obvious. (ψ2) Let max{u, p} > 0 and ψ(u, v, v, u, u+v, 0, q, q, p, p+q, 0) =u−hmax{v, q} ≤0,thenu≤hmax{v, q}.Similarlyp≤hmax{v, q}.Thusmax{u, p} ≤ hmax{v, q}. Similarly, let max{u, p} > 0 and ψ(u, v, u, v, 0, u+v, q, p, q, 0, p+q) = u−hmax{v, q} ≤ 0, then we have max{u, p} ≤ hmax{v, q}. Thus (ψ2) is

satisfying with k = h < 1. If max{u, p} = 0, then max{u, p} ≤kmax{v, q}. (ψ3) ψ(u, u, 0, 0, u, u, v, 0, 0, v, v) =u−hmax{u, v}= max{u−hu, u−hv} = max{u(1−h), u−hv}>0.Therefore ψ∈Ψ.

Example 2.5. ψ(t1, t2, ..., t11) =t1−hmax{t2, t3, t4, t5+t6

2 , t7, t8, t9,

t10+t11

2 }−Lmin{t3, t4, t5, t6, t8, t9, t10, t11},where h∈[0,1)and L≥0.

(ψ1) Obvious. (ψ2) Let max{u, p} > 0 and ψ(u, v, v, u, u+v, 0, q, q, p, p+q, 0) = u −hmax{u, v, p, q} ≤ 0, then u ≤ hmax{max{u, p}, max{v, q}}. Similarly

p ≤ hmax{max{u, p}, max{v, q}}. Thus max{u, p} ≤ hmax{max{u, p}, max{v, q}.

Now, if max{u, p} ≥max{v, q},then max{u, p} ≤hmax{u, p}<max{u, p},which is a contradiction. Thusmax{u, p}<max{v, q} andmax{u, p} ≤hmax{v, q}.Similarly, let

max{u, p} >0 and ψ(u, v, u, v, 0, u+v, q, p, q, 0, p+q) =u−hmax{u, v, p, q} ≤0,

then we havemax{u, p} ≤hmax{v, q}.Thus(ψ2) is satisfying withk=h <1.If max{u, p}= 0,thenmax{u, p} ≤kmax{v, q}.(ψ3) ψ(u, u,0,0, u, u, v,0,0, v, v) =u−hmax{u, v}= max{u−hu, u−hv}= max{u(1−h), u−hv}>0.Therefore ψ∈Ψ.

Example 2.6. Let ψ(t1, t2, ..., t11) =t1−hmax{t2, t7},where h∈[0, 1).

(ψ1) Obvious. (ψ2) Let max{u, p} > 0 and ψ(u, v, v, u, u+v, 0, q, q, p, p+q, 0} = u −hmax{v, q} ≤ 0. Thus u ≤ hmax{v, q}. Similarly p ≤ hmax{v, q}. Thus

max{u, p} ≤hmax{v, q}. Similarly, let max{u, p}> 0 and ψ(u, v, u, v, 0, u+v, q, p, q, 0, p+q) = u−hmax{v, q} ≤0, then we have max{u, p} ≤ hmax{v, q}. Thus (ψ2)

is satisfying with k=h <1.If max{u, p}= 0, thenmax{u, p} ≤kmax{v, q}.(ψ3) ψ(u, u, 0, 0, u, u, v, 0, 0, v, v) =u−hmax{u, v}= max{u−hu, u−hv} = max{u(1−h), u−hv}>0.Therefore ψ∈Ψ.

Example 2.7. Let ψ(t1, t2, ..., t11) =t1−bmax{t3+t4 2 ,

t8+t9

2 } where b∈[0, 1 2).

(ψ1) Obvious. (ψ2) Let max{u, p} > 0 and ψ(u, v, v, u, u+v, 0, q, q, p, p+q, 0} = u−bmax{u+v

2 ,

p+q

2 } ≤ 0. Thus u ≤

b

2[max{u, p}+ max{v, q}]. Similarly p ≤

b

2[max{u, p}+ max{v, q}].Thusmax{u, p} ≤

b

2[max{u, p}+ max{v, q}].Now, if max{u, p} ≥max{v, q}, then max{u, p} ≤ bmax{u, p} < max{u, p}, which is a contradiction. Thusmax{u, p}<max{v, q}and max{u, p} ≤bmax{v, q}.Similarly, letmax{u, p}>0

and ψ(u, v, u, v, 0, u+v, q, p, q, 0, p+q) = u−bmax{u+v

2 ,

p+q

2 } ≤ 0, then we have max{u, p} ≤bmax{v, q}. Thus(ψ2) is satisfying with k=b <1. If max{u, p}= 0, then

max{u, p} ≤kmax{v, q}.(ψ3) ψ(u, u,0,0, u, u, v, 0,0, v, v) =u >0.Therefore ψ∈Ψ. Example 2.8. Let ψ(t1, t2, ..., t11) =t1−cmax{t5+t6, t10+t11} where c∈[0, 1₂).

(ψ1) Obvious. (ψ2) Let max{u, p} > 0 and ψ(u, v, v, u, u+v, 0, q, q, p, p+q, 0} = u −cmax{u +v, p+ q} ≤ 0. Thus u ≤ c[max{u, p}+ max{v, q}]. Similarly

p ≤ c[max{u, p}+ max{v, q}]. Thus max{u, p} ≤ c[max{u, p} + max{v, q}]. Now, if max{u, p} ≥ max{v, q}, then max{u, p} ≤ 2cmax{u, p} < max{u, p}, which is a contradiction. Thus max{u, p} <max{v, q} and max{u, p} ≤ 2cmax{v, q}. Similarly, letmax{u, p}>0 andψ(u, v, u, v,0, u+v, q, p, q,0, p+q) =u−cmax{u+v, p+q} ≤0,

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Example 2.9. Let ψ(t1, t2, ..., t11) =t1−hmax{t2, t3+₂t4, t5+₂t6, t7, t8+₂t9, t10+₂t11} where h∈[0,1).

(ψ1) Obvious. (ψ2) Let max{u, p} > 0 and ψ(u, v, v, u, u+v, 0, q, q, p, p+q, 0} = u −hmax{v, q} ≤ 0. Thus u ≤ hmax{v, q}. Similarly p ≤ hmax{v, q}. Thus

max{u, p} ≤hmax{v, q}. Similarly, let max{u, p}> 0 and ψ(u, v, u, v, 0, u+v, q, p, q, 0, p+q) = u−hmax{v, q} ≤0, then we have max{u, p} ≤ hmax{v, q}. Thus (ψ2)

is satisfying with k=h <1.If max{u, p}= 0, thenmax{u, p} ≤kmax{v, q}.(ψ3) ψ(u, u, 0, 0, u, u, v, 0, 0, v, v) =u−hmax{u, v}= max{u−hu, u−hv} = max{u(1−h), u−hv}>0.Therefore ψ∈Ψ.

3. Main Results

Theorem 3.1. Let (X, d) be a complete metric space. Assume F, G:X×X →CB(X)

andf, g :X→X be mappings satisfying

(i) F(X×X)⊆g(X), G(X×X)⊆f(X), (ii) for all x, y, u, v∈X, where ψ∈Ψ,

ψ 

    

H(F(x, y), G(u, v)),

d(f x, gu), D(f x, F(x, y)), D(gu, G(u, v)), D(f x, G(u, v)), D(gu, F(x, y)), d(f y, gv), D(f y, F(y, x)), D(gv, G(v, u)),

D(f y, G(v, u)), D(gv, F(y, x))



    

≤0,

(iii) f(X) and g(X) are closed subsets of X, then

(a) F and f have a coupled coincidence point,

(b) G and g have a coupled coincidence point,

(c) F andf have a common coupled fixed point, if f isF−weakly commuting at(x, y)

andf2x=f x and f2y=f y for (x, y)∈C{F, f},

(d) Gand g have a common coupled fixed point, if g is G−weakly commuting at (_ex,y)_e

andg2ex=gxe andg 2

e

y=gey for (x,e y)e ∈C{G, g},

(e) F, G, f and g have common coupled fixed point provided that both (c) and (d) are true.

Proof. Let x0, y0 ∈ X be arbitrary. Chooseu1 =gx1 ∈ F(x0, y0) and v1 =gy1 ∈ F(y0, x0),asF(X×X)⊆g(X).SinceF, G:X×X →CB(X),therefore by Lemma 1.1, there existu2∈G(x1, y1) andv2 ∈G(y1, x1) such that

d(u1, u2) ≤ H(F(x0, y0), G(x1, y1)), d(v1, v2) ≤ H(F(y0, x0), G(y1, x1)).

Since G(X×X) ⊆ f(X), there exist x2, y2 ∈ X such that u2 = f x2 ∈ G(x1, y1) and v2 =f y2 ∈G(y1, x1).Then we choose u3∈F(x2, y2) and v3 ∈F(y2, x2) such that

d(u2, u3) ≤ H(G(x1, y1), F(x2, y2)), d(v2, v3) ≤ H(G(y1, x1), F(y2, x2)).

Continuing this process, we obtain sequences{un}, {vn},{xn} and {yn} in X such that

for alln≥0,we have

u2n = f x2n∈G(x2n−1, y2n−1), u2n+1=gx2n+1 ∈F(x2n, y2n),

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and

d(u2n−1, u2n) ≤ H(F(x2n−2, y2n−2), G(x2n−1, y2n−1)),

d(u2n, u2n+1) ≤ H(G(x2n−1, y2n−1), F(x2n, y2n)),

d(v2n−1, v2n) ≤ H(F(y2n−2, x2n−2), G(y2n−1, x2n−1)), d(v2n, v2n+1) ≤ H(G(y2n−1, x2n−1), F(y2n, x2n)).

Then by condition (ii),we get

ψ 

        

H(F(x2n, y2n), G(x2n−1, y2n−1)), d(f x2n, gx2n−1),

D(f x2n, F(x2n, y2n)), D(gx2n−1, G(x2n−1, y2n−1)),

D(f x2n, G(x2n−1, y2n−1)), D(gx2n−1, F(x2n, y2n)),

d(f y2n, gy2n−1),

D(f y2n, F(y2n, x2n)), D(gy2n−1, G(y2n−1, x2n−1)), D(f y2n, G(y2n−1, x2n−1)), D(gy2n−1, F(y2n, x2n))



        

≤0.

Using (ψ1),we get

ψ 

    

d(u2n+1, u2n),

d(u2n, u2n−1), d(u2n, u2n+1), d(u2n−1, u2n),

0, d(u2n−1, u2n+1),

d(v2n, v2n−1), d(v2n, v2n+1), d(v2n−1, v2n),

0, d(v2n−1, v2n+1)



    

≤0,

which implies that

ψ 

    

d(u2n+1, u2n),

d(u2n, u2n−1), d(u2n, u2n+1), d(u2n−1, u2n),

0, d(u2n−1, u2n) +d(u2n, u2n+1), d(v2n, v2n−1), d(v2n, v2n+1), d(v2n−1, v2n),

0, d(v2n−1, v2n) +d(v2n, v2n+1)



    

≤0.

By (ψ2),we get

max{d(u2n+1, u2n), d(v2n+1, v2n)} ≤ kmax{d(u2n, u2n−1), d(v2n, v2n−1)}.

Similarly, we can obtain

max{d(u2n, u2n−1), d(v2n, v2n−1)}

≤ kmax{d(u2n−1, u2n−2), d(v2n−1, v2n−2)}.

Thus, we have for alln∈N,

max{d(un, un+1), d(vn, vn+1)} ≤ kmax{d(un−1, un), d(vn−1, vn)} ≤ knmax{d(u0, u1), d(v0, v1)}

≤ knδ. Thus

max{d(un, un+1), d(vn, vn+1)} ≤knδ, (1) where

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Thus, form, n∈_Nwithm > n,by triangle inequality and (1), we get

max{d(un, um+n), d(vn, vm+n)} ≤ max{d(un, un+1), d(vn, vn+1)}

+ max{d(un+1, un+2), d(vn+1, vn+2)}

+...+ max{d(um+n−1, um+n), d(vm+n−1, vm+n)} ≤ knδ+kn+1δ+...+kn+m−1δ

≤ kn(1 +k+k2+...+km−1)δ

≤ k

n₍₁₋_km₎

1−k δ →0 as n, m→ ∞,

which shows that{un} and {vn} are Cauchy sequences in X. Since X is complete, there

existu, v ∈X such that

lim

n→∞un = nlim→∞f x2n= limn→∞gx2n+1 =u, (2)

lim

n→∞vn = nlim→∞f y2n= limn→∞gy2n+1=v.

Sincef(X) and g(X) are closed subsets of X, then there exist x, y,x,e ye∈X, we have

u=f x=g_ex, v=f y=g_ey. (3)

Now, since f x2n ∈ G(x2n−1, y2n−1) and f y2n ∈ G(y2n−1, x2n−1), therefore by using condition (ii),we get

ψ 

    

H(F(x, y), G(x2n−1, y2n−1)),

d(f x, gx2n−1), D(f x, F(x, y)), D(gx2n−1, G(x2n−1, y2n−1)),

D(f x, G(x2n−1, y2n−1)), D(gx2n−1, F(x, y)),

d(f y, gy2n−1), D(f y, F(y, x)), D(gy2n−1, G(y2n−1, x2n−1)),

D(f y, G(y2n−1, x2n−1)), D(gy2n−1, F(y, x))



    

≤0,

which implies, by (ψ1),that

ψ 

    

D(F(x, y), f x2n),

d(f x, gx2n−1), D(f x, F(x, y)), d(gx2n−1, f x2n),

d(f x, f x2n), D(gx2n−1, F(x, y)),

d(f y, gy2n−1), D(f y, F(y, x)), d(gy2n−1, f y2n),

d(f y, f y2n), D(gy2n−1, F(y, x))



    

≤0.

Lettingn→ ∞ in the above inequality, using the continuity of ψ,(2) and (3), we obtain

ψ 



D(F(x, y), f x),

0, D(f x, F(x, y)), 0, 0, D(f x, F(x, y)), 0, D(f y, F(y, x)), 0, 0, D(f y, F(y, x))



≤0.

Thus, by (ψ2),we obtain

D(f x, F(x, y)) = 0 andD(f y, F(y, x)) = 0,

which implies that

(8)

that is, (x, y) is a coupled coincidence point of F and f. This proves (a). Again, since gx2n+1 ∈F(x2n, y2n) and gy2n+1∈F(y2n, x2n),therefore by using condition (ii),we get

ψ 

    

H(F(x2n, y2n), G(x,e ey)),

d(f x2n, gx), D(f xe 2n, F(x2n, y2n)), D(gex, G(x,e y)),e D(f x2n, G(ex, ey)), D(gex, F(x2n, y2n)), d(f y2n, gy), D(f ye 2n, F(y2n, x2n)), D(gy, G(e y,e x)),e

D(f y2n, G(ey, x)), D(ge ey, F(y2n, x2n))



    

≤0,

which implies, by (ψ1),that

ψ 

    

D(gx2n+1, G(ex, ey)),

d(f x2n, gx), d(f xe 2n, gx2n+1), D(gex, G(x,e y)),e D(f x2n, G(ex, ey)), d(gx, gx2e n+1), d(f y2n, gey), d(f y2n, gy2n+1), D(gey, G(y,e ex)),

D(f y2n, G(y,e ex)), d(gey, gy2n+1)



    

≤0.

Lettingn→ ∞ in the above inequality, using the continuity of ψ,(2) and (3), we obtain

ψ 



D(gx, G(e x,e y)),e

0, 0, D(gx, G(_e _ex, _ey)), D(gx, G(_e _ex, y)),_e 0, 0, 0, D(gy, G(e y,e x)), D(ge ey, G(y,e ex)), 0



≤0.

Thus, by (ψ2),we obtain

D(gx, G(_e _ex, _ey)) = 0 andD(gy, G(_e _ey, _ex)) = 0, which implies that

gx_e∈G(_ex, y) and_e g_ey∈G(y,_e _ex),

that is, (x,e ey) is a coupled coincidence point of Gand g. This proves (b).

Furthermore, from condition (c),we have f is F−weakly commuting at (x, y),that is, f2x∈F(f x, f y), f2y∈F(f y, f x) and f2x=f x, f2y =f y.Thusf x=f2x∈F(f x, f y) and f y=f2y ∈F(f y, f x),that is, u=f u∈F(u, v) and v =f v∈F(v, u).This proves (c).A similar argument proves (d).Then (e) holds immediately.

Put f =g in the Theorem 3.1, we get the following result:

Corollary 3.1. Let (X, d) be a complete metric space. Assume F, G:X×X →CB(X)

andg:X →X be mappings satisfying

(i) F(X×X)⊆g(X), G(X×X)⊆g(X), (ii) for all x, y, u, v∈X andψ∈Ψ,

ψ 

    

H(F(x, y), G(u, v)),

d(gx, gu), D(gx, F(x, y)), D(gu, G(u, v)), D(gx, G(u, v)), D(gu, F(x, y)), d(gy, gv), D(gy, F(y, x)), D(gv, G(v, u)),

D(gy, G(v, u)), D(gv, F(y, x))



    

≤0,

(iii) g(X) is a closed subset of X, then

(a) F and g have a coupled coincidence point,

(c) F and g have a common coupled fixed point, if g is F−weakly commuting at (x, y)

andg2x=gx andg2y=gy for (x, y)∈C(F, g),

(d) Gand g have a common coupled fixed point, if g is G−weakly commuting at (_ex,y)_e

andg2ex=gxe andg 2

e

y=gey for (x,e y)e ∈C{G, g},

(9)

Put F =Gand f =g in the Theorem 3.1, we get the following result:

Corollary 3.2. Let (X, d) be a complete metric space. Assume F :X×X → CB(X)

andg:X →X be mappings satisfying

(i) F(X×X)⊆g(X),

(ii) for all x, y, u, v∈X andψ∈Ψ,

ψ 

    

H(F(x, y), F(u, v)),

d(gx, gu), D(gx, F(x, y)), D(gu, F(u, v)), D(gx, F(u, v)), D(gu, F(x, y)), d(gy, gv), D(gy, F(y, x)), D(gv, F(v, u)),

D(gy, F(v, u)), D(gv, F(y, x))



    

≤0.

If (iii) of Corollary 3.1 holds, then

(b) F and g have a common coupled fixed point, if g is F−weakly commuting at (x, y)

andg2x=gx andg2y=gy for (x, y)∈C(F, g).

Examples 2.1-2.9 and Theorem 3.1 imply the following:

andf, g :X→X be mappings satisfying(i) of Theorem 3.1 and

(i) for allx, y, u, v ∈X, where 0< h <1, H(F(x, y), G(u, v))

≤ hmax  



d(f x, gu), D(f x, F(x, y)), D(gu, G(u, v)), d(f y, gv), D(f y, F(y, x)), D(gv, G(v, u)),

D(f x, G(u, v))+D(gu, F(x, y))

2 ,

D(f y, G(v, u))+D(gv, F(y, x)) 2

 

 ,

or for allx, y, u, v ∈X, where α, β≥0 and α+ 2β <1, H(F(x, y), G(u, v))

≤ αmax

d(f x, gu), D(f x, F(x, y)), D(gu, G(u, v)), d(f y, gv), D(f y, F(y, x)), D(gv, G(v, u))

+βmax

D(f x, G(u, v)) +D(gu, F(x, y)), D(f y, G(v, u)) +D(gv, F(y, x))

,

or for allx, y, u, v ∈X, where a, b, c∈[0, 1)and a+ 2b+ 2c <1, H(F(x, y), G(u, v))

≤ amax{d(f x, gu), d(f y, gv)}

+bmax

D(f x, F(x, y)) +D(gu, G(u, v)), D(f y, F(y, x)) +D(gv, G(v, u))

+cmax

,

or for allx, y, u, v ∈X, where h∈[0,1) and L≥0, H(F(x, y), G(u, v))

≤ hmax{d(f x, gu), d(f y, gv)}

+Lmax    

  

D(f x, F(x, y)), D(gu, G(u, v)), D(f x, G(u, v)), D(gu, F(x, y)), D(f y, F(y, x)), D(gv, G(v, u)), D(f y, G(v, u)), D(gv, F(y, x))

   

(10)

or for allx, y, u, v ∈X, where h∈[0,1) and L≥0,

H(F(x, y), G(u, v))

≤ hmax  



d(f x, gu), D(f x, F(x, y)), D(gu, G(u, v)), d(f y, gv), D(f y, F(y, x)), D(gv, G(v, u)),

2 ,

D(f y, G(v, u))+D(gv, F(y, x)) 2

 



+Lmax    

  

D(f x, F(x, y)), D(gu, G(u, v)), D(f x, G(u, v)), D(gu, F(x, y)), D(f y, F(y, x)), D(gv, G(v, u)), D(f y, G(v, u)), D(gv, F(y, x))

   

   ,

or for allx, y, u, v ∈X, where h∈[0,1),

H(F(x, y), G(u, v))≤hmax{d(f x, gu), d(f y, gv)},

or for allx, y, u, v ∈X, where b∈[0, 1₂),

H(F(x, y), G(u, v))≤bmax (

D(f x, F(x, y))+D(gu, G(u, v))

2 ,

D(f y, F(y, x))+D(gv, G(v, u)) 2

) ,

or for allx, y, u, v ∈X, where c∈[0, 1₂), H(F(x, y), G(u, v))≤cmax

,

or for allx, y, u, v ∈X, where 0< h <1,

H(F(x, y), G(u, v))

≤ hmax  



d(f x, gu), d(f y, gv),

D(f x, F(x, y))+D(gu, G(u, v))

2 ,

D(f y, F(y, x))+D(gv, G(v, u))

2 ,

D(f y, G(v, u))+D(gv, F(y, x)) 2

 

 .

If (i) of Theorem 3.1 holds, then

(a) F and f have a coupled coincidence point,

(c) F andf have a common coupled fixed point, if f isF−weakly commuting at(x, y)

andf2x=f x and f2y=f y for (x, y)∈C{F, f},

(d) Gand g have a common coupled fixed point, if g is G−weakly commuting at (ex,y)e

andg2_ex=gx_e andg2y_e=g_ey for (_ex,y)_e ∈C{G, g},

(e) F, G, f and g have common coupled fixed point provided that both (c) and (d) are true.

Examples 2.1-2.9 and Corollary 3.1 imply the following:

andg:X →X be mappings satisfying (i) of Corollary 3.1 and

(i) for allx, y, u, v ∈X, where 0< h <1,

H(F(x, y), G(u, v))

≤ hmax  



d(gx, gu), D(gx, F(x, y)), D(gu, G(u, v)), d(gy, gv), D(gy, F(y, x)), D(gv, G(v, u)),

D(gx, G(u, v))+D(gu, F(x, y))

2 ,

D(gy, G(v, u))+D(gv, F(y, x)) 2

 

(11)

or for allx, y, u, v ∈X, where α, β≥0 and α+ 2β <1,

H(F(x, y), G(u, v))

≤ αmax

d(gx, gu), D(gx, F(x, y)), D(gu, G(u, v)), d(gy, gv), D(gy, F(y, x)), D(gv, G(v, u))

+βmax

D(gx, G(u, v)) +D(gu, F(x, y)), D(gy, G(v, u)) +D(gv, F(y, x))

,

or for allx, y, u, v ∈X, where a, b, c∈[0, 1)and a+ 2b+ 2c <1,

H(F(x, y), G(u, v))

≤ amax{d(gx, gu), d(gy, gv)}

+bmax

D(gx, F(x, y)) +D(gu, G(u, v)), D(gy, F(y, x)) +D(gv, G(v, u))

+cmax

,

H(F(x, y), G(u, v))

≤ hmax{d(gx, gu), d(gy, gv)}

+Lmax    

  

D(gx, F(x, y)), D(gu, G(u, v)), D(gx, G(u, v)), D(gu, F(x, y)), D(gy, F(y, x)), D(gv, G(v, u)), D(gy, G(v, u)), D(gv, F(y, x))

   

   ,

H(F(x, y), G(u, v))

≤ hmax  



d(gx, gu), D(gx, F(x, y)), D(gu, G(u, v)), d(gy, gv), D(gy, F(y, x)), D(gv, G(v, u)),

2 ,

D(gy, G(v, u))+D(gv, F(y, x)) 2

 



+Lmax    

  

D(gx, F(x, y)), D(gu, G(u, v)), D(gx, G(u, v)), D(gu, F(x, y)), D(gy, F(y, x)), D(gv, G(v, u)), D(gy, G(v, u)), D(gv, F(y, x))

   

   ,

H(F(x, y), G(u, v))≤hmax{d(gx, gu), d(gy, gv)},

or for allx, y, u, v ∈X, where b∈[0, 1₂),

H(F(x, y), G(u, v))≤bmax

( _D₍_{gx, F}₍_{x, y}₎₎₊_D₍_{gu, G}₍_{u, v}₎₎

2 ,

D(gy, F(y, x))+D(gv, G(v, u)) 2

) ,

or for allx, y, u, v ∈X, where c∈[0, 1₂), H(F(x, y), G(u, v))≤cmax

(12)

or for allx, y, u, v ∈X, where 0< h <1, H(F(x, y), G(u, v))

≤ hmax  



d(gx, gu), d(gy, gv),

D(gx, F(x, y))+D(gu, G(u, v))

2 ,

D(gy, F(y, x))+D(gv, G(v, u))

2 ,

D(gy, G(v, u))+D(gv, F(y, x)) 2

 

 .

(c) F and g have a common coupled fixed point, if g is F−weakly commuting at (x, y)

andg2x=gx andg2y=gy for (x, y)∈C(F, g),

(d) Gand g have a common coupled fixed point, if g is G−weakly commuting at (ex,y)e

andg2_ex=gx_e andg2y_e=g_ey for (_ex,y)_e ∈C{G, g},

(e) F, Gandg have common coupled fixed point provided that both(c) and(d) are true.

andg:X →X be mappings satisfying (i) of Corollary 3.2 and

(i) for allx, y, u, v ∈X, where 0< h <1, H(F(x, y), F(u, v))

≤ hmax  



d(gx, gu), D(gx, F(x, y)), D(gu, F(u, v)), d(gy, gv), D(gy, F(y, x)), D(gv, F(v, u)),

D(gx, F(u, v))+D(gu, F(x, y))

2 ,

D(gy, F(v, u))+D(gv, F(y, x)) 2

 

 ,

or for allx, y, u, v ∈X, where α, β≥0 and α+ 2β <1, H(F(x, y), F(u, v))

≤ αmax

d(gx, gu), D(gx, F(x, y)), D(gu, F(u, v)), d(gy, gv), D(gy, F(y, x)), D(gv, F(v, u))

+βmax

D(gx, F(u, v)) +D(gu, F(x, y)), D(gy, F(v, u)) +D(gv, F(y, x))

,

or for allx, y, u, v ∈X, where a, b, c∈[0, 1)and a+ 2b+ 2c <1, H(F(x, y), F(u, v))

≤ amax{d(gx, gu), d(gy, gv)}

+bmax

D(gx, F(x, y)) +D(gu, F(u, v)), D(gy, F(y, x)) +D(gv, F(v, u))

+cmax

,

or for allx, y, u, v ∈X, where h∈[0,1) and L≥0, H(F(x, y), F(u, v))

≤ hmax{d(gx, gu), d(gy, gv)}

+Lmax    

  

D(gx, F(x, y)), D(gu, F(u, v)), D(gx, F(u, v)), D(gu, F(x, y)), D(gy, F(y, x)), D(gv, F(v, u)), D(gy, F(v, u)), D(gv, F(y, x))

   

(13)

or for allx, y, u, v ∈X, where h∈[0,1) and L≥0, H(F(x, y), F(u, v))

≤ hmax  



d(gx, gu), D(gx, F(x, y)), D(gu, F(u, v)), d(gy, gv), D(gy, F(y, x)), D(gv, F(v, u)),

2 ,

D(gy, F(v, u))+D(gv, F(y, x)) 2

 



+Lmax    

  

D(gx, F(x, y)), D(gu, F(u, v)), D(gx, F(u, v)), D(gu, F(x, y)), D(gy, F(y, x)), D(gv, F(v, u)), D(gy, F(v, u)), D(gv, F(y, x))

   

   ,

H(F(x, y), F(u, v))≤hmax{d(gx, gu), d(gy, gv)},

or for allx, y, u, v ∈X, where b∈[0, 1₂), H(F(x, y), F(u, v))≤bmax

(

D(gx, F(x, y))+D(gu, F(u, v))

2 ,

D(gy, F(y, x))+D(gv, F(v, u)) 2

) ,

or for allx, y, u, v ∈X, where c∈[0, 1₂), H(F(x, y), F(u, v))≤cmax

,

or for allx, y, u, v ∈X, where 0< h <1, H(F(x, y), F(u, v))

≤ hmax  



d(gx, gu), d(gy, gv),

D(gx, F(x, y))+D(gu, F(u, v))

2 ,

D(gy, F(y, x))+D(gv, F(v, u))

2 ,

D(gy, F(v, u))+D(gv, F(y, x)) 2

 

 .

(b) F and g have a common coupled fixed point, if g is F−weakly commuting at (x, y)

andg2x=gx andg2y=gy for (x, y)∈C(F, g).

Theorem 3.2. Let (X, d) be a complete metric space. Assume F, G:X×X →CB(X)

andf, g :X→X be mappings satisfying(i),(ii) of Theorem 3.1 and

(i) {F, f} and {G, g} are w−compatible,

(ii) f(X) or g(X) is a closed subset of X,

then F, G, f and g have a common coupled fixed point.

Proof. We can prove like Theorem 3.1 that {un} and {vn} are Cauchy sequences in X.

Since X is complete, there exist u, v ∈ X satisfying (2). Suppose that f(X) is a closed subset ofX, then there exist x, y ∈X, we have

u=f x and v=f y. (4) As in Theorem 3.1, we can prove that

(14)

f2y ∈ F(f y, f x), that is, f u ∈ F(u, v) and f v ∈ F(v, u). Now, by condition (ii) of Theorem 3.1, we get

ψ 

    

H(F(u, v), G(x2n−1, y2n−1)),

d(f u, gx2n−1), D(f u, F(u, v)), D(gx2n−1, G(x2n−1, y2n−1)), D(f u, G(x2n−1, y2n−1)), D(gx2n−1, F(u, v)),

d(f v, gy2n−1), D(f v, F(v, u)), D(gy2n−1, G(y2n−1, x2n−1)), D(f v, G(y2n−1, x2n−1)), D(gy2n−1, F(v, u))



    

≤0.

From (ψ1) and triangle inequality, we have

ψ 



d(f u, u2n),

d(f u, u2n−1), 0, d(u2n−1, u2n), d(f u, u2n), d(u2n−1, f u), d(f v, v2n−1), 0, d(v2n−1, v2n), d(f v, v2n), d(v2n−1, f v)



≤0.

Lettingn→ ∞ in the above inequality, we get

ψ 



d(f u, u),

d(f u, u), 0, 0, d(f u, u), d(f u, u), d(f v, v), 0, 0, d(f v, v), d(f v, v)



≤0.

Hence, by (ψ3),we have d(f u, u) =d(f v, v) = 0.Thus

u=f u∈F(u, v) and v=f v∈F(v, u).

SinceF(X×X)⊆g(X),then there existx,e ye∈X such thatgex=u=f u∈F(u, v) and gy_e=v=f v∈F(v, u).Now, by condition (ii) of Theorem 3.1, we get

ψ 

    

H(F(u, v), G(_ex, y)),_e

d(f u, g_ex), D(f u, F(u, v)), D(gx, G(_e _ex, _ey)), D(f u, G(_ex, y)), D(g_e _ex, F(u, v)), d(f v, gey), D(f v, F(v, u)), D(gey, G(y,e x)),e

D(f v, G(y,_e x)), D(g_e y, F_e (v, u))



    

≤0,

and so we have

ψ 



D(u, G(x,_e y)),_e

0, 0, D(u, G(ex, y)), D(u, G(e ex, ey)), 0 0, 0, D(v, G(y,_e x)), D(v, G(_e _ey, x)),_e 0



≤0.

Hence, by (ψ2),we have D(u, G(_ex,y)) =_e D(v, G(y,_e x)) = 0._e Thus u=g_ex∈G(x,_e y) and_e v=gy_e∈G(y,_e _ex),

that is, (_ex, _ey) is a coupled coincidence point of G and g. Hence (x,_e y)_e ∈C{G, g}. From w−compatibility of {G, g}, we have gG(x,e y)e ⊆ G(gx, ge ey), hence g

2 e

x ∈ G(gex, gey) and g2y_e ∈ G(g_ey, gx),_e that is, gu ∈ G(u, v) and gv ∈ G(v, u). Again, by condition (ii) of Theorem 3.1, we get

ψ 

    

H(F(u, v), G(u, v)),

d(f u, gu), D(f u, F(u, v)), D(gu, G(u, v)), D(f u, G(u, v)), D(gu, F(u, v)), d(f v, gv), D(f v, F(v, u)), D(gv, G(v, u)),

D(f v, G(v, u)), D(gv, F(v, u))



    

≤0,

and so by triangle inequality, we have

ψ 



d(u, gu),

d(u, gu), 0, 0, d(u, gu), d(u, gu), d(v, gv), 0, 0, d(v, gv), d(v, gv)



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Hence, by (ψ3),we have d(u, gu) =d(v, gv) = 0.Thus

u=gu∈G(u, v) and v=gv∈G(v, u).

Therefore (u, v) is a common coupled fixed point of F, G, f and g. The proof is similar wheng(X) is assumed to be a closed subset of X.

Put f =g in the Theorem 3.2, we get the following result:

andg:X →X be mappings satisfying (i), (ii) of Corollary 3.1 and

(i) {F, g} and{G, g} are w−compatible.

If (iii) of Corollary 3.1 holds, then F, G and g have a common coupled fixed point.

Put F =Gand f =g in the Theorem 3.2, we get the following result:

andg:X →X be mappings satisfying (i), (ii) of Corollary 3.2 and

(i) {F, g} isw−compatible.

If (iii) of Corollary 3.1 holds, then F andg have a common coupled fixed point.

Examples 2.1-2.9 and Theorem 3.2 imply the following:

and f, g : X → X be mappings satisfying (i) of Theorem 3.1 , (i) of Corollary 3.3, (i)

and(ii) of Theorem 3.2, then F, G, f andg have a common coupled fixed point.

and f, g :X → X be mappings satisfying (i), (iii) of Corollary 3.1, (i) of Corollary 3.4, and(i) of Corollary 3.6, then F, G andg have a common coupled fixed point.

Corollary 3.10. Let(X, d)be a complete metric space. Assume F, G:X×X →CB(X)

and g :X → X be mappings satisfying (iii) of Corollary 3.1, (i) of Corollary 3.2, (i) of Corollary 3.5 and (i) of Corollary 3.7, then F andg have a common coupled fixed point.

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Bhavana Deshpandefor the photography and short autobiography, see TWMS J. Appl. Eng. Math. V.5, No.1, 2015.

Amrish Handa received his M.Sc. in Mathematics degree from Vikram University of Ujjain, Madhya Pradesh, India in 2005. He secured first position in the whole university. Thenafter he cleared National Eligibility Test in Mathematical Sciences organized by Council of Scientific and Industrial Research, India in 2008. He worked as a Junior Research Fellow on Research Project entitled ”A Study on Fixed Point Theory and its Applications.” sanctioned by Madhya Pradesh Council of Science and Technology with Principal investigator Dr. Bhavana Deshpande from 2012 to 2014. His research area is Fixed point theory and its applications. His 34 research papers are published and 14 research papers are accepted for publication in reputed international journals