Suantai et al. Fixed Point Theory and Applications (2015) 2015:224 DOI 10.1186/s13663-015-0473-4

R E S E A R C H Open Access

Common coupled fixed point theoremsfor θ -ψ-contraction mappings endowed witha directed graphSuthep Suantai1, Phakdi Charoensawan1* and Tatjana Aleksic Lampert2

*Correspondence:phakdi@hotmail.com1Department of Mathematics,Faculty of Science, Chiang MaiUniversity, Chiang Mai, 50200,ThailandFull list of author information isavailable at the end of the article

AbstractIn this paper, we present some existence and uniqueness results for coupledcoincidence point and common fixed point of θ -ψ -contraction mappings incomplete metric spaces endowed with a directed graph. Our results generalize theresults obtained by Kadelburg et al. (Fixed Point Theory Appl. 2015:27, 2015,doi:10.1007/s11590-013-0708-4). We also have an application to some integral systemto support the results.

MSC: coupled fixed point; coupled coincidence point; common fixed point;Geraghty-type condition; edge preserving; metric spaces; connected graph;monotone; partially ordered set

1 Introduction and preliminariesFor F : X × X → X and g : X → X, a concept of coupled coincidence point (x, y) ∈ X × Xsuch that gx = F(x, y) and gy = F(y, x) was first introduced by Lakshimikantham and Ćirić[]. Their results extended the result in [, ]. Also, the existence and uniqueness of acoupled coincidence point for such a mapping that satisfies the mixed monotone propertyin a partially ordered metric space were studied. Consequently, a number of coupled fixedpoint and coupled coincidence point results have been shown recently. For example, see[–].

Choudhury and Kundu [] give a notion of compatibility.

Definition . ([]) Let (X, d) be a metric space, and let g : X → X and F : X × X → X.The mappings g and F are said to be compatible if

limn→∞ d

(gF(xn, yn), F(gxn, gyn)

)= and lim

n→∞ d(gF(yn, xn), F(gyn, gxn)

)=

whenever {xn} and {yn} are sequences in X such that limn→∞ F(xn, yn) = limn→∞ gxn andlimn→∞ F(yn, xn) = limn→∞ gyn.

Let � denote the class of all functions θ : [,∞) × [,∞) → [, ) that satisfy the follow-ing conditions:

(θ) θ (s, t) = θ (t, s) for all s, t ∈ [,∞);

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Suantai et al. Fixed Point Theory and Applications (2015) 2015:224 Page 2 of 11

(θ) for any two sequences {sn} and {tn} of nonnegative real numbers,

θ (sn, tn) → ⇒ sn, tn → .

In , Kadelburg et al. [] used the monotone and g-monotone properties to obtainedcommon coupled fixed point theorems for Geraghty-type contraction with compatibilityof F and g .

Let (X, d) be a metric space, � be a diagonal of X ×X, and G be a directed graph with noparallel edges such that the set V (G) of its vertices coincides with X and � ⊆ E(G), whereE(G) is the set of the edges of the graph. That is, G is determined by (V (G), E(G)). We willuse this notation of G throughout this work.

The fixed point theorem using the context of metric spaces endowed with a graph wasfirst studied by Jachymski []. The result generalized the Banach contraction principle tomappings on metric spaces with a graph. Since then, many authors studied the problem ofexistence of fixed points for single-valued mappings and multivalued mappings in severalspaces with graphs; see [–].

Recently, Chifu and Petrusel [] give the concept of G-continuity for a mapping F :X → X and the property A as follows.

Definition . Let (X, d) be a complete metric space, G be a directed graph, and F : X →X be a mapping. Then

(i) F is called G-continuous if for all (x∗, y∗) ∈ X and for any sequence (ni)i ∈N ofpositive integers such that F(xni , yni ) → x∗, F(yni , xni ) → y∗ as i → ∞ and(F(xni , yni ), F(xni+, yni+)), (F(yni , xni ), F(yni+, xni+)) ∈ E(G), we have that

F(F(xni , yni ), F(yni , xni )

) → F(x∗, y∗) and

F(F(yni , xni ), F(xni , yni )

) → F(y∗, x∗) as i → ∞;

(ii) (X, d, G) has property A if for any sequence (xn)n∈N ⊂ X with xn → x as n → ∞ and(xn, xn+) ∈ E(G) for n ∈N, then (xn, x) ∈ E(G).

Their results generalized the result in [] by using the context of metric spaces endowedwith a directed graph.

The aim of this work is to prove some existence and uniqueness results for a coupledcoincidence point and a common fixed point of θ -ψ contraction mappings in completemetric spaces endowed with a directed graph. The results generalize the results obtainedby Kadelburg et al. []. An application to some integral system is provided to support theresults.

2 Common coupled fixed pointWe define the set CcFix(F) of all coupled coincidence points of mappings F : X → X andg : X → X and the set (X)F

g as follows:

CcFix(F) ={

(x, y) ∈ X : F(x, y) = gx and F(y, x) = gy}

and

(X)F

g ={

(x, y) ∈ X :(gx, F(x, y)

),(gy, F(y, x)

) ∈ E(G)}

.

Suantai et al. Fixed Point Theory and Applications (2015) 2015:224 Page 3 of 11

Now, we give some definitions that are useful for our main results.

Definition . We say that F : X → X and g : X → X are G-edge preserving if

[(gx, gu), (gy, gv) ∈ E(G)

] ⇒ [(F(x, y), F(u, v)

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

) ∈ E(G)].

Definition . Let (X, d) be a complete metric space, and E(G) be the set of the edges ofthe graph. We say that E(G) satisfies the transitivity property if and only if, for all x, y, a ∈ X,

(x, a), (a, y) ∈ E(G) → (x, y) ∈ E(G).

Let � denote the class of all functions ψ : [,∞) → [,∞) that satisfy the followingconditions:

(ψ) ψ is nondecreasing;(ψ) ψ(s + t) ≤ ψ(s) + ψ(t);(ψ) ψ is continuous;(ψ) ψ(t) = ⇔ t = .

Definition . Let (X, d) be a complete metric space endowed with a directed graph G.The mappings F : X → X and g : X → X are called a θ -ψ-contraction if:

() F and g is G-edge preserving;() there exist θ ∈ � and ψ ∈ � such that for all x, y, u, v ∈ X satisfying

(gx, gu), (gy, gv) ∈ E(G),

ψ(d(F(x, y), F(u, v)

)) ≤ θ(d(gx, gu), d(gy, gv)

)ψ

(M(gx, gu, gy, gv)

), ()

where M(gx, gu, gy, gv) = max{d(gx, gu), d(gy, gv)}.

Lemma . Let (X, d) be a complete metric space endowed with a directed graph G, and letF : X → X and g : X → X be a θ -ψ-contraction. Assume that there exist x, y, a, b ∈ Xand F(X × X) ⊂ g(X). Then:

(i) There exists sequences {xn}, {yn}, {an}, {bn} in X for which

gxn = F(xn–, yn–) and gyn = F(yn–, xn–),

gan = F(an–, bn–) and gbn = F(bn–, an–) for n = , , . . . .()

(ii) If (gxn, gan) and (gyn, gbn) ∈ E(G) for all n ∈ N, then

limn→∞ dn = lim

n→∞ M(gxn, gan, gyn, gbn) = .

Proof (i) Let x, y, a, b ∈ X. By the assumption that F(X × X) ⊂ g(X) and F(x, y),F(y, x), F(a, b), F(b, a) ∈ X, it easy to construct sequences {xn}, {yn}, {an}, and {bn}in X for which

gxn = F(xn–, yn–) and gyn = F(yn–, xn–),

gan = F(an–, bn–) and gbn = F(bn–, an–) for n = , , . . . .

Suantai et al. Fixed Point Theory and Applications (2015) 2015:224 Page 4 of 11

(ii) Let (gxn, gan) and (gyn, gbn) ∈ E(G) for all n ∈ N. Using the θ -ψ-contraction () and(), we obtain that

ψ(d(gxn+, gan+)

)= ψ

(d(F(xn, yn), F(an, bn)

))

≤ θ(d(gxn, gan), d(gyn, gbn)

)ψ

(M(gxn, gan, gyn, gbn)

)()

and

ψ(d(gyn+, gbn+)

)= ψ

(d(F(yn, xn), F(bn, an)

))

≤ θ(d(gyn, gbn), d(gxn, gan)

)ψ

(M(gyn, gbn, gxn, gan)

)

= θ(d(gxn, gan), d(gyn, gbn)

)ψ

(M(gxn, gan, gyn, gbn)

)()

for all n ∈N. From () and () we get

ψ(M(gxn+, gan+, gyn+, gbn+)

)

= ψ(max

{d(gxn+, gan+), d(gyn+, gbn+)

})

≤ θ(d(gxn, gan), d(gyn, gbn)

)ψ

(M(gxn, gan, gyn, gbn)

)

< ψ(M(gxn, gan, gyn, gbn)

)()

for all n ∈N, that is,

ψ(M(gxn+, gan+, gyn+, gbn+)

)< ψ

(M(gxn, gan, gyn, gbn)

).

Regarding the properties of ψ , we conclude that

M(gxn+, gan+, gyn+, gbn+) < M(gxn, gan, gyn, gbn).

It follows that dn := M(gxn, gan, gyn, gbn) is decreasing. Then dn → d as n → ∞ for somed ≥ . We claim that d = . Suppose not. Using (), we have

ψ(M(gxn+, gan+, gyn+, gbn+))ψ(M(gxn, gan, gyn, gbn))

≤ θ(d(gxn, gan), d(gyn, gbn)

)< .

Taking the limit as n → ∞, we have

θ(d(gxn, gan), d(gyn, gbn)

) → .

Since θ ∈ �,

d(gxn, gan) → and d(gyn, gbn) →

as n → ∞. Therefore,

limn→∞ dn = lim

n→∞ M(gxn, gan, gyn, gbn) = ,

Suantai et al. Fixed Point Theory and Applications (2015) 2015:224 Page 5 of 11

which is a contradiction. Hence,

limn→∞ dn = lim

n→∞ M(gxn, gan, gyn, gbn) = �

Next, we will prove the main result.

Theorem . Let (X, d) be a complete metric space endowed with a directed graph G, andlet F : X → X and g : X → X be a θ -ψ-contraction. Suppose that:

(i) g is continuous, and g(X) is closed;(ii) F(X × X) ⊂ g(X), and g and F are compatible;

(iii) F is G-continuous, or the tripled (X, d, G) has property A;(iv) E(G) satisfies the transitivity property.

Under these conditions, CcFix(F) �= ∅ if and only if (X)Fg �= ∅.

Proof Let CcFix(F) �= ∅. Then there exists (u, v) ∈ CcFix(F) such that (gu, F(u, v)) = (gu, gu)and (gv, F(v, u)) = (gv, gv) ∈ � ⊂ E(G). Thus, (gu, F(u, v)) and (gv, F(v, u)) ∈ E(G). It followsthat (u, v) ∈ (X)F

g , so that (X)Fg �= ∅.

Now, suppose that (X)Fg �= ∅. Let x, y ∈ X be such that (x, y) ∈ (X)F

g . Then(gx, F(x, y)) and (gy, F(y, x)) ∈ E(G). From Lemma .(i) we have sequences {xn} and{yn} in X for which

gxn = F(xn–, yn–) and gyn = F(yn–, xn–) for n = , , . . . .

Since (gx, F(x, y)) = (gx, gx) and (gy, F(y, x)) = (gy, gy) ∈ E(G) and F and gare G-edge preserving, we have (F(x, y), F(x, y)) = (gx, gx) and (F(y, x), F(y, x)) =(gy, gy) ∈ E(G). By induction we shall obtain (gxn–, gxn) and (gyn–, gyn) ∈ E(G) for eachn ∈N. By Lemma .(ii) we have

dn := M(gxn–, gxn, gyn–, gyn) → as n → ∞. ()

Now, we shall show that {gxn} and {gyn} are Cauchy sequences. Applying a similar argu-ment as in the proof of Theorem . in [] and using (), condition (iv), and property of ψ ,it follows that {gxn} and {gyn} are Cauchy sequences. By condition (i) there exist u, v ∈ g(X)such that

limn→∞ gxn = lim

n→∞ F(xn, yn) = u and limn→∞ gyn = lim

n→∞ F(yn, xn) = v.

By the compatibility of g and F we have that

limn→∞ d

(gF(xn, yn), F(gxn, gyn)

)= and lim

n→∞ d(gF(yn, xn), F(gyn, gxn)

)= . ()

Now, suppose that (a) F is G-continuous. It is easy to see that

d(gu, F(gxn, gyn)

) ≤ d(gu, gF(xn, yn)

)+ d

(gF(xn, yn), F(gxn, gyn)

).

Taking the limit as n → ∞ and using (), the continuity of g , and G-continuity of F , wehave that d(gu, F(u, v)) = , that is, gu = F(u, v). Using a similar idea, we also have thatgv = F(v, u). Therefore, CcFix(F) �= ∅.

Suantai et al. Fixed Point Theory and Applications (2015) 2015:224 Page 6 of 11

Suppose now that (b) the tripled (X, d, G) with property A. Let gx = u and gy = v for somex, y ∈ X. Then we have (gxn, gx) and (gyn, gy) ∈ E(G) for each n ∈N. By () we have

ψ(d(gx, F(x, y)

)+ d

(gy, F(y, x)

))

≤ ψ(d(gx, gxn+) + d

(gxn+, F(x, y)

)+ d(gy, gyn+) + d

(gyn+, F(y, x)

))

≤ ψ(d(F(xn, yn), F(x, y)

))+ ψ

(d(F(yn, xn), F(y, x)

))

+ ψ(d(gx, gxn+)

)+ ψ

(d(gy, gyn+)

)

≤ θ(d(gxn, gx), d(gyn, gy)

)ψ

(M(gxn, gx, gyn, gy)

)

+ ψ(d(gx, gxn+)

)+ ψ

(d(gy, gyn+)

).

Letting n → ∞, we have ψ(d(gx, F(x, y)) + d(gy, F(y, x))) = . By properties of ψ , we cansee that d(gx, F(x, y)) + d(gy, F(y, x)) = . Finally, gx = F(x, y) and gy = F(y, x). �

We denote by CmFix(F) the set of all common fixed points of mappings F : X → X andg : X → X, that is,

CmFix(F) ={

(x, y) ∈ X : F(x, y) = gx = x and F(y, x) = gy = y}

.

Theorem . In addition to hypotheses of Theorem ., assume that(vi) for any two elements (x, y), (u, v) ∈ X × X , there exists (a, b) ∈ X × X such that

(gx, ga), (gu, ga), (gy, gb), (gv, gb) ∈ E(G).Then, CmFix(F) �= ∅ if and only if (X)F

g �= ∅.

Proof Theorem . implies that there exists (x, y) ∈ X × X such that gx = F(x, y) and gy =F(y, x). Suppose that there exists another (u, v) ∈ X × X such that gu = F(u, v) and gv =F(v, u). We will show that gx = gu and gy = gv.

By condition (vi) there exists (a, b) ∈ X × X such that (gx, ga), (gu, ga), (gy, gb), (gv, gb) ∈E(G). Set a = a, b = b, x = x, y = y, u = u, and v = v. By Lemma .(i) we have se-quences {an}, {bn} {xn}, {yn}, {un}, and {vn} in X for which

gan = F(an–, bn–) and gbn = F(bn–, an–),

gxn = F(xn–, yn–) and gyn = F(yn–, xn–),

gun = F(un–, vn–) and gvn = F(vn–, un–)

for n ∈N. By the properties of coincidence points, xn = x, yn = y and un = u, vn = v, that is,

gxn = F(x, y), gyn = F(y, x) and gun = F(u, v), gvn = F(v, u) for all n ∈N.

Since (gx, ga), (gy, gb) ∈ E(G), we have (gx, ga) and (gy, gb) ∈ E(G). Because F and gare G-edge preserving, we have (F(x, y), F(a, b)) = (gx, ga) and (F(y, x), F(b, a)) =(gy, gb) ∈ E(G). Similarly, (gx, gan) and (gy, gbn) ∈ E(G). By Lemma .(ii) we obtain

limn→∞ dn = lim

n→∞ M(gx, gan, gy, gbn) = ,

Suantai et al. Fixed Point Theory and Applications (2015) 2015:224 Page 7 of 11

and then

limn→∞ d(gx, gan) = and lim

n→∞ d(gy, gbn) = .

Similarly, from (gu, ga), (gv, gb) ∈ E(G) we have

limn→∞ d(gu, gan) = and lim

n→∞ d(gv, gbn) = .

By the triangle inequality we have

d(gx, gu) ≤ d(gx, gan) + d(gan, gu) and d(gy, gv) ≤ d(gy, gbn) + d(gbn, gv)

for all n ∈ N. Letting n → ∞ in these two inequalities, we get that d(gx, gu) = andd(gy, gv) = . Therefore, we have gx = gu and gy = gv.

The proof of the existence and uniqueness of a common fixed point can be derived usinga similar argument as in Theorem . in []. �

Remark . In the case where (X, d,�) is a partially ordered complete metric space, let-ting E(G) = {(x, y) ∈ X × X : x � y} and ψ(t) = t, we obtain Theorem . and Theorem .in [].

3 ApplicationsIn this section, we apply our theorem to the existence theorem for a solution of the fol-lowing integral system:

x(t) =∫ T

f(t, s, x(s), y(s)

)ds + h(t),

y(t) =∫ T

f(t, s, y(s), x(s)

)ds + h(t),

()

where t ∈ [, T] with T > .Let X := C([, T],Rn) with ‖x‖ = maxt∈[,T] |x(t)|, for x ∈ C([, T],Rn).We define the graph G with partial order relation by

x, y ∈ X, x ≤ y ⇔ x(t) ≤ y(t) for any t ∈ [, T].

Thus, (X,‖ · ‖) is a complete metric space endowed with a directed graph G.Let E(G) = {(x, y) ∈ X × X : x ≤ y}. Then E(G) satisfies the transitivity property, and

(X,‖ · ‖, G) has property A.

Theorem . Consider system (). Suppose that(i) f : [, T] × [, T] ×R

n ×Rn → R

n and h : [, T] → Rn are continuous;

(ii) for all x, y, u, v ∈Rn with x ≤ u, y ≤ v, we have f (t, s, x, y) ≤ f (t, s, u, v) for all

t, s ∈ [, T];(iii) there exist ≤ k < and T > such that

∣∣f (t, s, x, y) – f (t, s, u, v)∣∣ ≤ k

T(|x – u| + |y – v|)

for all t, s ∈ [, T], x, y, u, v ∈Rn, x ≤ u, y ≤ v;

Suantai et al. Fixed Point Theory and Applications (2015) 2015:224 Page 8 of 11

(iv) there exists (x, y) ∈ X × X such that

x(t) ≤∫ T

f(t, s, x(s), y(s)

)ds + h(t) and

y(t) ≤∫ T

f(t, s, y(s), x(s)

)ds + h(t),

where t ∈ [, T].Then there exists at least one solution of the integral system ().

Proof Let F : X × X → X, (x, y) �→ F(x, y), where

F(x, y)(t) =∫ T

f(t, s, x(s), y(s)

)ds + h(t), t ∈ [, T],

and define g : X → X by gx(t) = x(t) .

System () can be written as

x = F(x, y) and y = F(y, x).

Let x, y, u, v ∈ X be such that gx ≤ gu and gy ≤ gv. We have x ≤ u, y ≤ v and

F(x, y)(t) =∫ T

f(t, s, x(s), y(s)

)ds + h(t)

≤∫ T

f(t, s, u(s), v(s)

)ds + h(t) = F(u, v)(t) for all t ∈ [, T]

and

F(y, x)(t) =∫ T

f(t, s, y(s), x(s)

)ds + h(t)

≤∫ T

f(t, s, v(s), u(s)

)ds + h(t) = F(v, u)(t) for all t ∈ [, T].

Thus, F and g are G-edge preserving.By condition (iv) it follows that (X)F

g = {(x, y) ∈ X ×X : gx ≤ F(x, y) and gy ≤ F(y, x)} �= ∅.On the other hand,

∣∣F(x, y)(t) – F(u, v)(t)∣∣

≤∫ T

∣∣f(t, s, x(s), y(s)

)– f

(t, s, u(s), v(s)

)∣∣ds

=∫ T

∣∣f

(t, s, x(s), y(s)

)– f

(t, s, u(s), v(s)

)∣∣ds

≤ kT

∫ T

(∣∣x(s) – u(s)∣∣ +

∣∣y(s) – v(s)∣∣)ds

≤ k(‖gx – gu‖ + ‖gy – gv‖

)

≤ kM(gx, gu, gy, gv) for all t ∈ [, T].

Suantai et al. Fixed Point Theory and Applications (2015) 2015:224 Page 9 of 11

Then, there exist ψ(t) = t and θ ∈ �, where θ (s, t) = k for s, t ∈ [,∞) with k ∈ [, ), suchthat

ψ(∥∥F(x, y) – F(u, v)

∥∥) ≤ θ(‖gx – gu‖,‖gy – gv‖)ψ(

M(gx, gu, gy, gv)),

where M(gx, gu, gy, gv) = max{‖gx – gu‖,‖gy – gv‖}. Hence, F and g are a θ -ψ-contraction.Thus, there exists a coupled common fixed point (x∗, y∗) ∈ X × X of the mapping F and

g , which is the solution of the integral system (). �

Theorem . Consider system (). Suppose that(i) f : [, T] × [, T] ×R

n ×Rn → R

n and h : [, T] → Rn are continuous;

(ii) for all x, y, u, v ∈Rn with x ≤ u, y ≤ v, we have f (t, s, x, y) ≤ f (t, s, u, v) for all

t, s ∈ [, T];(iii) for all t, s ∈ [, T], x, y, u, v ∈R

n, x ≤ u, y ≤ v,

∣∣f (t, s, x, y) – f (t, s, u, v)∣∣ ≤

Tln

( + max

{|x – u|, |y – v|});

(iv) there exists (x, y) ∈ X × X such that

x(t) ≤∫ T

f(t, s, x(s), y(s)

)ds + h(t),

y(t) ≤∫ T

f(t, s, y(s), x(s)

)ds + h(t),

where t ∈ [, T].Then there exists at least one solution of the integral system ().

Proof Let F : X × X → X, (x, y) �→ F(x, y), where

F(x, y)(t) =∫ T

f(t, s, x(s), y(s)

)ds + h(t), t ∈ [, T],

and define g : X → X by gx(t) = x(t). As in Theorem ., we have that F and g are G-edgepreserving.

By condition (iv) it follows that (X)Fg = {(x, y) ∈ X ×X : gx ≤ F(x, y) and gy ≤ F(y, x)} �= ∅.

On the other hand,

∣∣F(x, y)(t) – F(u, v)(t)∣∣

≤∫ T

∣∣f(t, s, x(s), y(s)

)– f

(t, s, u(s), v(s)

)∣∣ds

=∫ T

∣∣f

(t, s, x(s), y(s)

)– f

(t, s, u(s), v(s)

)∣∣ds

≤ T

∫ T

ln

( + max

{∣∣x(s) – u(s)∣∣,

∣∣y(s) – v(s)∣∣})ds

≤ ln(

+ max{

maxt∈[,T]

∣∣x(t) – u(t)∣∣, max

t∈[,T]

∣∣y(t) – v(t)∣∣})

Suantai et al. Fixed Point Theory and Applications (2015) 2015:224 Page 10 of 11

≤ ln( + max

{‖x – u‖,‖y – v‖})

= ln( + M(gx, gu, gy, gv)

)for all t ∈ [, T],

where M(gx, gu, gy, gv) = max{‖gx – gu‖,‖gy – gv‖}, which yields

ln(∣∣F(x, y)(t) – F(u, v)(t)

∣∣ +

)

≤ ln(ln

( + M(gx, gu, gy, gv)

)+

)

=ln(ln( + M(gx, gu, gy, gv)) + )

ln( + M(gx, gu, gy, gv))ln

( + M(gx, gu, gy, gv)

).

Hence, there exist ψ(x) = ln(x + ) and θ ∈ � defined by

θ (s, t) =

⎧⎨

⎩

ln(ln(+max{s,t}))ln(+max{s,t}) , s > or t > ,

r ∈ [, ), s = , t = ,

such that

ψ(d(F(x, y), F(u, v)

))= ψ

(∥∥F(x, y) – F(u, v)∥∥)

≤ θ(d(gx, gu), d(gy, gv)

)ψ

(M(gx, gu, gy, gv)

).

Hence, we see that F and g are a θ -ψ-contraction. Thus, there exists a coupled commonfixed point (x∗, y∗) ∈ X × X of the mapping F and g , which is a solution for the integralsystem (). �

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsThe author read and approved the final manuscript.

Author details1Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand. 2Department ofMathematics, Faculty of Sciences, Radoja Domanovica 12, Kragujevac, 34000, Serbia.

AcknowledgementsThis research was supported by Thailand Research Fund under the project RTA5780007 and Chiang Mai University.

Received: 30 March 2015 Accepted: 20 November 2015

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