coupled fixed point theorem for multi-valued mapping

Journal of Mathematical Analysis ISSN: 2217-3412, URL: www.ilirias.com/jma Volume 8 Issue 5 (2017), Pages 27-39.

COUPLED FIXED POINT THEOREM FOR MULTI-VALUED MAPPING VIA GENERALIZED CONTRACTION IN PARTIALLY ORDERED METRIC SPACES WITH APPLICATIONS MUHAMMAD SHOAIB, MUHAMMAD SARWAR AND CEMIL TUNC

Abstract. In this manuscript, using generalized type contraction, multi-valued coupled fixed point results in the set up of partially ordered metric space are studied. For the validity of the derived result appropriate example and application are also provided.

1. Introduction and Preliminaries All through the article R+ , N and N0 will denote the sets of non-negative real numbers, positive integer and non-negative whole numbers respectively. Assume (Λ, %) be a metric space, also consider CB(Λ) be the class of all non-empty bounded and closed subsets of Λ. For ϑ ∈ Λ and Ω1 ⊂ Λ, denote d(ϑ, Ω1 ) = inf{%(ϑ, ζ) : ζ ∈ Ω1 }, and for Ω1 , Ω2 ⊂ Λ define δ(Ω1 , Ω2 ) = sup{%(ς, ζ) : ς ∈ Ω1 , ζ ∈ Ω2 }. Fixed point theory is an outstanding source which gives authentic techniques for the presence of fixed point, common fixed point, coincidence point and coupled fixed point for self maps under different conditions. Its main application comes forward by showing the uniqueness and existences of solution of a mathematical model like, functional equations, matrix equations, differential equations, integral equations, linear inequalities([3], [18], [21]). The pioneering and fruitful result regarding the said area were brought by Banach [5] entitled Banach contraction principle(BCP). Different researchers generalized BCP in numerous spaces via different contractive conditions([11], [14]) etc. Mainly, in 1969, Nadler [19] further modified the Banach contraction principle to multi-valued mapping with the Hausdorff metric. In partially ordered metric space, metric fixed point theory has been flourished in the near past. Ran and Reurings [21] elaborated the Banach contraction principle(BCP) in partially ordered sets and also discussed its applications to non-linear and linear matrix equations. Nieto and Lopez [20] modified the consequence of Ran 2000 Mathematics Subject Classification. 47H10; 54H25. Key words and phrases. Generalized Contraction, Coupled Fixed Point Theorem, Multi-valued mapping, Integral equations, Partially ordered metric space. c

2017 Ilirias Research Institute, Prishtin¨ e, Kosov¨ e. Submitted May 4, 2017. Published September 14, 2017. Communicated by S. A. Mohiuddine. 27

28

M. SHOAIB, M. SARWAR AND C. TUNC

and Reurings and used their well-formed results to find a solution for first order ODEs. Jaggi [14] introduce the concept of rational type contraction in complete metric space and established unique fixed point theorems. The established result of Jaggi [14] extended by Harjani et. al [11] for the study fixed points in complete partial ordered metric space. For further details (see[2],[4],[6],[11],[15],[16]). The notion of mixed monotone property of mappings established by Bhaskar and Lakshmikantham [7] for the existences of coupled fixed point results. Additionally, they applied their results on a first order ODEs (with along periodic boundary ´ c [18], generalized the idea of mixed conditions) [12]. Lakshmikanthem and Ciri´ monotone mapping and established a coupled fixed point theorem on behalf of non-linear contractions in partially ordered metric spaces. Abbas et. al [1] further developed coupled coincidence point and common fixed point theorem in complete metric space for multi-valued and single valued mappings. Huange and Fang [13] studied applications to integral inclusion with the help of fixed point results for multi-valued mixed increasing operator in ordered Banach spaces. In partially ordered metric spaces, Choudhery and Metya [9] established some result for multi-valued and single mapping. Yin and Gueo [22] defined the new notion of g-monotone mapping and established results for single and multi-valued mapping. After this the main results of [22] improved and extended by Choudhury and Metiya [8, 9]. Klanarong and Suantai [16] introduced a new type of mixed monotone multi-valued mapping in partially metric space. Furthermore, Kim and Sumit Chandok [17] proved coupled common fixed point theorems for generalized non-linear contraction in partially ordered metric spaces. In the concerned work, using the idea of generalized rational type contraction condition, coupled fixed point theorem for multi-valued hybrid maps in the structure of complete partially ordered metric spaces have been investigated. Definition 1.1. Assume Ψ signify the set of functions κ : R+ → (0, 1] with (i) R+ = {θ ∈ R/θ > 0}, (ii)κ(θn ) → 1 ⇒ θn → 0. Example 1.2. [16] Let ϕ : R+ → [0, 1) define by, ( 3 1 − ω2 if ω ≤ 1 ϕ(ω) = β < 1 if ω > 1. The following definition can be found in [16] . Definition 1.3. Let (Λ, ) be a partially ordered set. Let ∆1 and ∆2 be two nonempty subsets of Λ. Then denote (i) ∆1 ς(µs(ϑ) ) by equation (2.5) and equation (2.1) we have %(ς(µr(ϑ)+1 ), ς(µs(ϑ)+1 )) ≤ δ(ξ(µr(ϑ) , νr(ϑ) ), ξ(µs(ϑ) , νs(ϑ) ))  %(ς(µ   %(ς(µ r(ϑ) ), ς(µs(ϑ) )) + %(ς(νr(ϑ) ), ς(νs(ϑ) )) r(ϑ) ), ς(µs(ϑ) )) + %(ς(νr(ϑ) ), ς(νs(ϑ) )) . ≤ϕ 2 2 Which implies that γϑ
γϑ
. (2.20) %(ς(µr(ϑ)+1 ), ς(µs(ϑ)+1 )) ≤ ϕ 2 2 Similarly we have %(ς(νr(ϑ)+1 ), ς(νs(ϑ)+1 )) ≤ δ(ξ(νr(ϑ) , µr(ϑ) ), ξ(νs(ϑ) , µs(ϑ) ))  %(ς(ν   %(ς(ν r(ϑ) ), ς(νs(ϑ) )) + %(ς(µr(ϑ) ), ς(µs(ϑ) )) r(ϑ) ), ς(νs(ϑ) )) + %(ς(µr(ϑ) ), ς(µs(ϑ) )) ≤ϕ . 2 2

32

M. SHOAIB, M. SARWAR AND C. TUNC

%(ς(νr(ϑ)+1 ), ς(νs(ϑ)+1 )) ≤ ϕ

γϑ
γϑ
. 2 2

Adding (2.20) and (2.21) and using (2.19) we have γϑ
γϑ
γϑ ≤ αr(ϑ) + αs(ϑ) + 2ϕ . 2 2

(2.21)

(2.22)

By using (2.13) and (2.17) together with property of ϕ, we get contradiction. Thus {ς(µn )} and {ς(νn )} are Cauchy sequence. Since ς(Λ) is complete, there exist µ, ν ∈ Λ such that lim ς(µn ) = ς(µ), lim ς(νn ) = ς(ν).

n→∞

n→∞

(2.23)

Finally we claim ς(µ) ∈ ξ(µ, ν) and ς(ν) ∈ ξ(ν, µ). From equation (2.23) and (ii), we have ς(µn ) < ς(µ), ς(νn ) > ς(ν) ∀ n ≥ 0 by (2.1) we have %(ξ(µ, ν), ς(µn+1 )) ≤ δ(ξ(µ, ν), ξ(µn , νn ))  %(ς(µ ), ς(µ)) + %(ς(ν ), ς(ν))  %(ς(µ ), ς(µ)) + %(ς(ν ), ς(ν))  n n n n . ≤ϕ 2 2 This implies that %(ξ(µ, ν), ςµ) = 0. Hence ςµ ∈ ξ(µ, ν). Similarly, we can show that ς(ν) ∈ ξ(ν, µ). Thus (µ, ν) is a coupled coincidence point of ξ and ς.  Corollary 2.2. Let (Λ, ) is a partially ordered set. Further, suppose that there is a metric % on Λ such the (Λ, %) is complete metric space. Suppose ξ : Λ×Λ → CB(Λ) and ς : Λ → Λ are such that ξ is mixed ς- monotone of type (A) and  %(ςµ, ςu) + %(ςν, ςv)  %(ςµ, ςu) + %(ςν, ςv)  δ(ξ(µ, ν), ξ(u, v)) ≤ ϕ , (2.24) 2 2 ∀ µ, ν, u, v ∈ Λ ςµ > ςu and ςν < ςv and ϕ ∈ Ψ. Suppose that ξ(Λ × Λ) ⊂ ς(Λ) is a complete subset of Λ and suppose that (i) There exist ∀ µ0 , ν0 ∈ Λ such that {ςµ0 } < ξ(µ0 , ν0 ), {ςν0 } < ξ(ν0 , µ0 );

(2.25)

(ii) Λ hold the assumption given as; (a) If an increasing sequences µn → µ, νn → ν. Then µn < µ, νn < ν for all n. Then ∃ µ, ν ∈ Λ such that ς(µ) ∈ ξ(µ, ν), ς(ν) ∈ ξ(ν, µ).

(2.26)

. Example 2.3. Let Λ = [0, 1] and assume the natural ordered relation in Λ. Let ξ : Λ × Λ → CB(Λ), ς : Λ → Λ and ϕ : R+ → [0, 1) define by, ( 1 [ 2 , 1] if (µ, ν) = (1, 1) ξ(µ, ν) = µ 1 { 25 + 25(1+ν) } if (µ, ν) 6= (1, 1).

ϕ(t) =

ς(µ) = µ2 . ( 3 1 − t2 if t ≤ 1 β 1.

COUPLED FIXED POINT THEOREM FOR MULTI-VALUED MAPPING VIA GENERALIZED CONTRACTION 33

We discuss the below cases Case 1.1 (µ, ν) = (0, 0), (u, v) = (0, 1) or (µ, ν) = (1, 1), (u, v) = (0, 1) it is clear that (ςµ, ςν)  (ςu, ςv) or (ςµ, ςν)  (ςu, ςv) and
1 1 1 we have δ (ξ(0, 0), ξ(0, 1))) = δ { 25 }, { 50 }) = 50 and    %(ςµ, ςu) + %(ςν, ςv) %(ςµ, ςu) + %(ςν, ςv) ϕ . 2 2    %(0, 0) + %(0, 1) %(0, 0) + %(0, 1) 15 =ϕ = . 2 2 32 The second choice is clearly hold. Case 1.2 (µ, ν) = (1, 0), (u, v) = (0, 0) it is clear that (ςµ, ςν)  (ςu, ςv)or (ςµ, ςν)  (ςu, ςv) and
1 1 2 }, { 25 }) = 25 and we have δ (ξ(1, 0), ξ(0, 0))) = δ { 25    %(ςµ, ςu) + %(ςν, ςv) %(ςµ, ςu) + %(ςν, ςv) ϕ . 2 2    %(1, 0) + %(0, 0) %(ςµ, ςu) + %(ςν, ςv) 15 ϕ = . 2 2 32 Case 1.3 (µ, ν) = (1, 0), (u, v) = (0, 1) it is clear that (ςµ, ςν)  (ςu, ςv) or (ςµ, ςν)  (ςu, ςv) and
3 2 1 we have δ (ξ(1, 0), ξ(0, 1)) = δ { 25 }, { 50 }) = 50 and    %(ςµ, ςu) + %(ςν, ςv) %(ςµ, ςu) + %(ςν, ςv) ϕ . 2 2    %(1, 0) + %(0, 1) 1 %(1, 0) + %(0, 1) = . ϕ 2 2 2 Case 1.4 (µ, ν) = (1, 0), (u, v) = (1, 1) again it is clear that (ςµ, ςν)  (ςu, ςv) or (ςµ, ςν)  (ςu, ςv)and
2 3 1 we have δ (ξ(1, 0), ξ(1, 1)) = δ { 25 }, { 50 }) = 50 and    %(ςµ, ςu) + %(ςν, ςv) %(ςµ, ςu) + %(ςν, ςv) ϕ . 2 2    %(1, 0) + %(1, 1) %(1, 0) + %(1, 1) 15 ϕ = . 2 2 32 Clearly for (ςµ, ςν)  (ςu, ςv) or (ςµ, ςν)  (ςu, ςv) all the conditions of Theorem 2.1 holds. So (1, 1) is the coupled fixed point of ξ and ς. Theorem 2.4. Suppose that the hypothesis of Theorem 2.1 are useable. In addition assume there be (ςw, ςz) ∈ Λ × Λ for each (ςµ, ςν), (ςu, ςv) ∈ Λ × Λ which is comparable to (ςµ, ςν) and (ςu, ςv) also ς is w-commutative with ξ. Then the coupled fixed point of ξ and ς has unique, that is there exist a unique (µ, ν) ∈ Λ × Λ such that µ = ςµ ∈ ξ(µ, ν) and ν = ςν ∈ ξ(ν, µ). Proof. By Theorem 2.1 the set of coupled coincidence point is non-empty. Let (µ, ν) and (u, v) are two coupled coincidence points, that is, ς(µ) ∈ ξ(µ, ν), ς(ν) ∈ ξ(ν, µ), ς(u) ∈ ξ(u, v), ς(v) ∈ ξ(v, u).

34

M. SHOAIB, M. SARWAR AND C. TUNC

By given hypotheses, there exists a (ς(s), ς(t)) ∈ Λ × Λ such that (ς(s), ς(t)) is comparable to both (ς(µ), ς(ν)) and (ς(u), ς(v)). Take s0 = s and t0 = t and define sequences sn and tn as follows ς(sn+1 ) ∈ ξ(sn , tn ) and ς(tn+1 ) ∈ ξ(tn , sn ) ∀ n ≥ 0. Since (ς(s), ς(t)) is comparable to (ς(µ), ς(ν)), we may assume that (ς(s0 ), ς(t0 )) = (ς(s), ς(t))  (ς(µ), ς(ν)) (the other case is similar). Now, by using the mixed ςmonotone property of ξ and induction we obtain (ς(sn ), ς(tn ))  (ς(µ), ς(ν)) for all n. From equation (2.1) we have %(ς(µ), ς(sn+1 )) ≤ δ(ξ(µ, ν), ξ(sn , tn ))  %(ς(µ), ς(s )) + %(ς(ν), ς(t ))  %(ς(µ), ς(s )) + %(ς(ν), ς(t ))  n n n n , ≤ϕ 2 2 hence we have αn−1
αn−1
%(ς(µ), ς(sn+1 )) ≤ ϕ . (2.27) 2 2 Similarly we have %(ς(ν), ς(tn+1 )) ≤ δ(ξ(ν, µ), ξ(tn , µn ))  %(ς(ν), ς(t )) + %(ς(µ), ς(s ))  %(ς(ν), ς(t )) + %(ς(µ), ς(s ))  n n n n , ≤ϕ 2 2 which implies that αn−1
αn−1
%(ς(tn ), ς(tn+1 )) ≤ ϕ . (2.28) 2 2 Adding equations (2.27) and (2.28) we have αn−1
αn−1
αn ≤ 2ϕ , 2 2 which implies αn ≤ αn−1 . It follow that αn is a monotone decreasing sequence of non-negative real numbers. Therefor, there is some α ≥ 0 such that lim αn = α.

n→∞

(2.29)

Further, we show that α = 0. Suppose α > 0, then from αn αn−1
≤ϕ < 1, αn−1 2 by taking limit we have αn−1
lim ϕ = 1. n→∞ 2 By using ϕ function we have %(ς(µ), ς(sn )) → 0, %(ς(ν), ς(tn )) → 0. Similarly we can show that

(2.30)

%(ς(u), ς(sn )) → 0, %(ς(v), ς(tn )) → 0.

(2.31)

Together with equations (2.30) and (2.31) we get ς(µ) = ς(u), ς(ν) = ς(v).

(2.32)

Since ς(µ) ∈ ξ(µ, ν) and ς(ν) ∈ ξ(ν, µ), by w-commutativity of ξ and ς we have ς(ς(µ)) ∈ ς(ξ(µ, ν)) ⊆ ξ(ς(µ), ς(ν)), ς(ς(ν)) ∈ ς(ξ(ν, µ)) ⊆ ξ(ς(ν), ς(µ)).

COUPLED FIXED POINT THEOREM FOR MULTI-VALUED MAPPING VIA GENERALIZED CONTRACTION 35

By replacing ς(µ) and ς(ν) by γ1 and γ2 , respectively, we obtain ς(γ1 ) ∈ ξ(γ1 , γ2 ) and ς(γ2 ) ∈ ξ(γ2 , γ1 ). Hence (γ1 , γ2 ) is a coupled coincidence point. So u and v can be replaced by γ1 and γ2 in (2.32), respectively, which implies that ς(γ1 ) = ς(µ) and ς(γ2 ) = ς(ν), that is, γ1 = ς(γ1 ) ∈ ξ(γ1 , γ2 ) and γ2 = ς(γ2 ) ∈ ξ(γ2 , γ1 ). Thus (γ1 , γ2 ) is a coupled fixed point of ξ and ς. To prove the uniqueness, let (z, w) be another coupled fixed point. Then (2.32) implies that γ1 = ς(γ1 ) = ς(z) = z and γ2 = ς(γ2 ) = ς(w) = w.  3. Applications to Integral Equations In this section, we give an existence theorem for the solution of the integral equations given as:  Z ζ2  (ξ1 (r, ω) + ξ2 (r, ω))[f (r, α(ω), β(ω)) + g(r, α(ω), β(ω))]dω + h(r), r ∈ [ζ1 , ζ2 ]; α(r) =   ζ1 ζ2

Z β(r) =

ζ1

   (ξ1 (r, ω) + ξ2 (r, ω))[f (r, β(ω), α(ω)) + g(r, β(ω), α(ω))]dω + h(r), r ∈ [ζ1 , ζ2 ].

(3.1) The class of all continuous function denoted by Λ := C(I, R) defined on [ζ1 , ζ2 ]. Λ is the partially ordered set. The order relation in Λ is defined as: α, β ∈ C(I, R), α ≤ β ⇐⇒ α(r) ≤ β(r) ∀r ∈ I. Define % : Λ × Λ → R+ , by %(α, β) =

sup |α(r) − β(r)|∀ α, β ∈ Λ. t∈[ζ1 ,ζ2 ]

Then (Λ, %) is a complete metric space on Λ. Now defined the following partial order on Λ × Λ. For (α, β), (γ, v) ∈ Λ × Λ (α, β) ≤ (γ, v) ⇐⇒ α(r) ≥ γ(r) and β(r) ≤ v(r) ∀r ∈ I. For any (α, β) ∈ Λ × Λ, upper and lower bound of α, β are (max{α, β}), min{α, β}). Therefore for (α, β), (γ, v) ∈ Λ × Λ there exist (max{α, γ}), max{β, v}) which is comparable to (γ, v) and (α, β). Theorem 3.1. Suppose the following hypothesis hold: (1) ξi : [ζ1 , ζ2 ] × [ζ1 , ζ2 ] → R,i = 1, 2 and f, g : [ζ1 , ζ2 ] × R × R → R and h : [ζ1 , ζ2 ] → R are continuous; (2) ξ1 (r, ω) ≥ 0 , ξ2 (r, ω) ≥ 0; (3) There exist λ1 > 0, λ2 > 0 and ϕ ∈ Ψ such that for all α, γ, β, γ ∈ R with α ≥ γ and β ≤ γ, the following hypothesis holds:  %(α, γ) + %(β, γ)  %(α, γ) + %(β, γ)    0 ≤ f (ω, α, β) − f (ω, γ, γ) ≤ λ1 ϕ , 2 2  %(α, γ) + %(β, γ)  %(α, γ) + %(β, γ)   −λ2 ϕ ≤ g(ω, γ, γ) − g(ω, α, β) ≤ 0. 2 2 (3.2)

36

M. SHOAIB, M. SARWAR AND C. TUNC

R ζ sup ζ12 (ξ1 (r, ω) + ξ2 (r, ω))dω ≤ 21 ;

(4)

t∈[ζ1 ,ζ2 ]

(5) max{λ1 , λ2 } ≤ 1; (6) There exist (η1 , η2 ) ∈ Λ × Λ coupled upper and lower solution of the integral equation if η1 (r) ≤ η2 (r) and ζ2

Z η1 (r) ≤

h i (ξ1 (r, ω)+ξ2 (r, ω)) f (r, η1 (ω), η2 (ω))+g(r, η1 (ω), η2 (ω)) dω+h(r), r ∈ [ζ1 , ζ2 ];

ζ1

and ζ2

Z η2 (r) ≥

h i (ξ1 (r, ω)+ξ2 (r, ω)) f (r, η2 (ω), η1 (ω))+g(r, η2 (ω), η1 (ω)) dω+h(r), r ∈ [ζ1 , ζ2 ].

ζ1

Then system of integral equations 3.1 has a solution in C([ζ1 , ζ2 ], R). Proof. Define ξ : C([ζ1 , ζ2 ], R) × C([ζ1 , ζ2 ], R) → C([ζ1 , ζ2 ], R) by, Z

ζ2

ξ(α, β)(r) = ζ1

Z

ζ2

+ ζ1

h i   ξ1 (r, ω) f (ω, α(ω), β(ω)) + g(ω, α(ω), β(ω)) dω   

(3.3)

h i    ξ2 (r, ω) f (ω, α(ω)) + g(r, α(ω), β(ω)) dω + h(r), r ∈ [ζ1 , ζ2 ].

Now to show that ξ has mixed monotone property. For this if α1 ≤ α2 we have Z

ζ2

ξ(α1 , β) − ξ(α2 , β) =

h i ξ1 (r, ω) f (ω, α1 (ω), β(ω)) + g(ω, α1 (ω), β(ω)) dω

ζ1

Z

ζ2

+

h i ξ2 (r, ω) f (ω, α1 (ω), β(ω)) + g(ω, α1 (ω), β(ω)) dω + h(r)

ζ1

Z

ζ2

−

h i ξ1 (r, ω) f (ω, α2 (ω), β(ω)) + g(ω, α2 (ω), β(ω)) dω

ζ1

Z

ζ2

−

h i ξ2 (r, ω) f (ω, α2 (ω), β(ω)) + g(ω, α2 (ω), β(ω)) dω − h(r)

ζ1

=

hZ

ζ2

h

i (ξ1 (r, ω) f (ω, α1 (ω), β(ω))−f (ω, α2 (ω), β(ω)) + g(ω, α1 (ω), β(ω))−g(ω, α2 (ω), β(ω)) dω

ζ1

Z

ζ2

+

h

i i ξ2 (r, ω) f (ω, α1 (ω), β(ω))−f (ω, α2 (ω), β(ω)) + g(ω, α1 (ω), β(ω))−g(ω, α2 (ω), β(ω)) dω ≤ 0.

ζ1

By using given assumptions we get ξ(α1 , β(t)) ≤ ξ(α2 , β(t)), ∀s ∈ I.

COUPLED FIXED POINT THEOREM FOR MULTI-VALUED MAPPING VIA GENERALIZED CONTRACTION 37

Similarly, if β1 ≥ β2 Z

ζ2

ξ(α, β1 ) − ξ(α, β2 ) =

h i ξ1 (r, ω) f (ω, α(ω), β1 (ω)) + g(ω, α(ω), β1 (ω)) dω

ζ1 ζ2

Z

h i ξ2 (r, ω) f (ω, α(ω), β1 (ω)) + g(ω, α(ω), β1 (ω)) dω + h(r)

+ ζ1

Z

ζ2

−

h i ξ1 (r, ω) f (ω, α(ω), β2 (ω)) + g(ω, α(ω), β2 (ω)) dω

ζ1

Z

ζ2

h i ξ2 (r, ω) f (ω, α(ω), β2 (ω)) + g(ω, α(ω), β2 (ω)) dω − h(r).

− ζ1

Z

ζ2

= ζ1 ζ2

Z +

h

i (ξ1 (r, ω) f (ω, α(ω), β1 (ω))−f (ω, α(ω), β2 (ω)) + g(ω, α(ω), β1 (ω))−g(ω, α(ω), β2 (ω)) dω

h

i ξ2 (r, ω) f (ω, α(ω), β1 (ω))−f (ω, α(ω), β2 (ω)) + g(ω, α(ω), β1 (ω))−g(ω, α(ω), β2 (ω)) dω ≤ 0.

ζ1

By using given assumptions we obtained ξ(α, β1 (t)) ≤ ξ(α, β2 (t)), ∀ω ∈ I. Now using assumption of equation (3.2) we have Z ζ2 h i ξ1 (r, ω) f (ω, α(ω), β(ω))+g(ω, α(ω), β(ω)) dω %(ξ(α, β)−ξ(γ, γ)) = sup t∈[ζ1 ,ζ2 ]

ζ2

Z +

ζ1

h i ξ2 (r, ω) f (ω, α(ω), β(ω)) + g(ω, α(ω)), β(ω) dω + h(r)

ζ1

Z

ζ2

−

h i ξ1 (r, ω) f (ω, γ(ω), γ(ω)) + g(ω, α(ω)), β(ω) dω

ζ1

Z

ζ2

− ζ1

=

Z sup

t∈[ζ1 ,ζ2 ]

Z

ζ2

+ ζ1

ζ2

h i ξ2 (r, ω) f (ω, γ(ω), γ(ω)) + g(ω, γ(ω), γ(ω)) dω − h(r) .

h

i ξ1 (r, ω) f (ω, α(ω), β(ω))−f (ω, γ(ω), γ(ω)) + g(ω, α(ω), β(ω))−g(ω, γ(ω), γ(ω)) dω

ζ1


i ξ2 (r, ω) f (ω, α(ω), β(ω))−f (ω, γ(ω), γ(ω)) + g(ω, α(ω), β(ω))−g(ω, γ(ω), γ(ω)) dω . h

Z sup




ζ2

  %(α, γ) + %(β, γ)  %(α, γ) + %(β, γ)  ≤ ξ1 (r, ω) λ1 ϕ 2 2 t∈[ζ1 ,ζ2 ] ζ1  %(α, γ) + %(β, γ)  %(α, γ) + %(β, γ)  + λ2 ϕ dω 2 2  Z ζ2  %(α, γ) + %(β, γ)  %(α, γ) + %(β, γ)  + ξ2 (r, ω) λ1 ϕ 2 2 ζ1  %(α, γ) + %(β, γ)  %(α, γ) + %(β, γ)  + λ2 ϕ dω . 2 2

38

M. SHOAIB, M. SARWAR AND C. TUNC

Z ζ2  %(α, γ) + %(β, γ)  %(α, γ) + %(β, γ)  ((ξ1 (r, ω)+ξ2 (r, ω))ϕ dω ≤ 2 max{λ1 , λ2 } sup 2 2 t∈[ζ1 ,ζ2 ] ζ1  Z ζ2 %(α, γ) + %(β, γ)  %(α, γ) + %(β, γ)  (ξ1 (r, ω)+ξ2 (r, ω))dω ϕ = 2 max{λ1 , λ2 } sup . 2 2 t∈[ζ1 ,ζ2 ] ζ1 By using assumptions (4) and (5), which implies that  %(α, γ) + %(β, γ)  %(α, γ) + %(β, γ)  %(ξ(α, β) − ξ(γ, γ)) ≤ ϕ . 2 2 It is easy to verify condition [(ii)] of Theorem 2.1. Let (η1 , η2 ) be coupled upper and lower solution of the integral equation then, by assumption (6) we have η1 (r) ≤ ξ(η1 , η2 ) and ξ(η2 , η1 ) ≤ η2 (r). Since η1 ≤ η2 . By defining ς : Λ → Λ by ς(x) = x. By Theorem 2.4 and proposition 1.6 there exist solutions of the system 3.1 in C([ζ1 , ζ2 ], R).  Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this article. References [1] M. Abbas, L. Ciric, B. Damjanovic, and M. A. Khan, Coupled coincidence and common fixed point theorems for hybrid pair of mappings, Fixed Point Theory and Applications, vol. 2012, article 4, 2012. [2] R. P. Agarwal, Z. Kadelburg and S. Radenovi´ c, On coupled fixed point results in asymmetric G-metric spaces, Fixed Point Theory Appl. 2013, 2013:528. [3] A. Aghajani, M. Abbas and E. P. Kallehbasti, Coupled fixed point theorems in partially ordered metric spaces and application, Math. Commun. 17, 497-509, (2012). [4] H. Aydi, E. Karapinar, and W. Shatanawi, Coupled fixed point results for (ψ, φ)-weakly contractive condition in ordered partial metric spaces, Computers Mathematics with Applications, vol. 62, no. 12, pp. 4449-4460, 2011. [5] S. Banach, Sur les op´ erations dans les ensembles abstraits et leur application aux equations int´ egrales, Fund. Math., 3, 133-181, (1922). [6] V. Berinde, Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces, Nonlinear Analysis: Theory, Methods Applications, vol. 74, no. 18, pp. 7347-7355, (2011). [7] T. G. Bhaskar and V. Lakshmikanthan,Fixed point theorems in patially ordered metric spaces and application, Nonlinear Anal. 65, 1379-1393, (2006). [8] B. S. Choudhury and N. Metiya, Coincidence point theorems for a family of multivalued mappings in partially ordered metric spaces, Acta Universitatis Matthiae Belii, Series Mathematics, no. 2013, pp. 10-23, 2013. [9] B. S. Choudhury and N. Metiya, Multivalued and singlevalued fixed point results in partially ordered metric spaces, Arab Journal of Mathematical Sciences, vol. 17, no. 2, pp. 135-151, 2011. [10] M. A Geraghty, On contractive mappings. Proc. Am. Math. Soc. 40, 604-608 (1973). [11] J. Harjani, B. Lopez and K. Sadarangani, A fixed point theorem for a mapping satisfying a contractive condition of rational type on a partially ordered metric spaces, Abstr. Appl. Anal. 2010, 1-8, (2010). [12] J. Harjani and K. Sadarangani, Generalized contraction in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72, 1188-1197, (2010). [13] N. J. Huang and Y. P. Fang, Fixed points for multi-valued mixed increasing operators in ordered Banach spaces with applications to integral inclusions, Journal of Analysis and its Applications, vol. 22, no. 2, pp. 399-410, 2003. [14] D. S. Jaggi, Some unique fixed point theorems,Indian J. Pure appl. Anal. 8, 223-230, (1977). [15] Z. Kadelburg, H. K. Nashine, and S. Radenovi´ c, Common coupled fixed point results in partially ordered G-metric space, Bull. Math. Anal. Appl. (4) (2) (2012), 51-63.

COUPLED FIXED POINT THEOREM FOR MULTI-VALUED MAPPING VIA GENERALIZED CONTRACTION 39

[16] C. Klanarong and S. Suantai, Coupled coincidence point theorems for new types of mixed monotone multivalued mappings in partially ordered metric spaces, Abstract and Applied Analysis 2013, Article ID 604578, 7 pages. [17] J. K. Kim and S. Chandok, Coupled common fixed point theorems for generalized nonlinear contraction mappings with the mixed monotone property in partially ordered metric spaces, Fixed Point Theory and Applications 2013, 2013:307. ´ c, Coupled fixed point thorem for nonlinear contractions in [18] V. Lakshmikanthan and L. Ciri´ partially orderedmetric spaces, Nonlinear Anal., 70, 4341-4349, (2009). [19] S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math., 20(1969), 475-488, . [20] J. J. Nieto and R. R. Lopez, Contractive mapping theorems inpartially ordered sets and applications to ordinary differential equations, Order, 22, 223-239, (2005). [21] A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132, 1435-1443, (2004). [22] J. Yin and T. Guo, Some fixed point results for a class of g-monotone increasing multi-valued mappings, Arab Journal of Mathematical Sciences, vol. 19, no. 1, pp. 35-47, 2013. Muhammad Shoaib Department of Mathematics, University of Malakand, Chakdara Dir(L), Pakistan E-mail address: [email protected] Muhammad Sarwar Department of Mathematics, University of Malakand, Chakdara Dir(L), Pakistan E-mail address: [email protected] Cemil Tunc Department of Mathematics Faculty of Sciences Yuzuncu Yil University Turkey E-mail address: [email protected]