Journal of Mathematical Analysis ISSN: 2217-3412, URL: www.ilirias.com/jma Volume 9 Issue 2 (2018), Pages 158-166.
FIXED POINT THEOREMS FOR MULTI-VALUED GENERALIZED CONTRACTION IN b-METRIC SPACES SADDAM HUSSAIN, MUHAMMAD SARWAR AND CEMIL TUNC∗
Abstract. The purpose of this article is to study the existence and uniqueness of fixed points and common fixed points for multi valued mappings via rational type contraction in the setting of b-complete metric spaces. For the usability of the established result an appropriate example is also given.
1. Introduction and Preliminaries The Bananch contraction theorem has very important role due to its fruitful applications within as well as outside mathematics. Many authors have extended this theorem employing relatively more general contractive conditions ensuring the existence and uniqueness of the fixed points. The concept of the well known b-metric space was initially discovered by Bakhtin [4] and Czerwik [6], whose another name is metric type space [11]. This b-metric space specially solved the problem of convergence of measurable functions regarding a measure. Applying this concept, Czerwik in [6] and [7] generalized the Banach contraction theorem in b-metric spaces. Many researchers have done a lot of works in b-metric spaces, see [13, 16, 17]. Edelstein [10] has generalized the Banach contraction theorem to mappings satisfying a less restrictive Lipschitz inequality like local contraction. Nadler [14] generalized this result to a multi valued version. Zoguchi and Takahashi [18] have improved Reich’s result [15] and derived the existence of fixed points for multivalued mappings when its values are closed bounded instead of compact. The study of multi-valued mappings got very much importance in the last decades, due to its applications in mathematical optimization, control theory and differential inclusions [9, 12]. In other words one can state that this theory lies at the junction of topology, theory of functions and non-linear functional analysis. In particular, recalling that Nadler [14] combined the concepts of contraction and multi-valued mapping and proved some fixed point theorems. Abdou [1] introduced the notion of occasionally weakly compatible mappings (the (owc)-property) 2000 Mathematics Subject Classification. 47H10; 54H25. Key words and phrases. b-Complete metric space, common fixed point, multi-valued mapping, rational type contractive mappings. Cemil TUNC*-Corresponding author. c
2018 Ilirias Research Institute, Prishtin¨ e, Kosov¨ e. Submitted February 20, 2018. Published April 26, 2018. Communicated by S.A. Mohiuddine. 158
FIXED POINT THEOREMS FOR MULTI−VALUED GENERALIZED CONTRACTION IN b-METRIC SPACES 159
and the common limit in the range (the (CLR)- property) for four single and multivalued maps in metric spaces and proved coincidence point as well as common fixed point results for the hybrid contraction with (owc)-property and (CLR)-property. Chifu and Petrusel [8] established fixed point theorems for multi-valued operators in b-metric spaces equipped with graph and also investigated some existence theorems of multi-valued fractals in b-metric spaces, using Ciric type contractive conditions regarding a functional H. The purpose of this article is to derive the existence and uniqueness of common fixed point theorems for multi valued maps using rational contractive condition in the light of complete b-metric spaces. In the whole paper, R denotes real numbers and R+ positive real numbers. Definition 1.1. [14] Let X1 and X2 be nonempty sets, a function T from X1 to the power set of X2 is called a multi-valued mapping, denoted by: T : X1 → 2X2 . Definition 1.2. [14] A point xκ0 ∈ X1 is known as a fixed point of a multi-valued mapping T : X1 → 2X2 , if xκ0 ∈ T xκ0 . Definition 1.3. [3] Let (X1 , dτ ) be a metric space. A mapping T : X1 → X2 is called contraction if dτ (T xκ , T y κ ) ≤ rdτ (xκ , y κ ) for each xκ , y κ ∈ X1 , where 0 ≤ r < 1. Definition 1.4. [14] Suppose (X1 , dτ ) is a metric space. Hausdorff metric on CB(X1 ) produced by dτ is defined as: H(E, F ) = max{sup dτ (xκ , F ), sup dτ (y κ , E): xκ ∈ E, y κ ∈ F } for all E, F ∈ CB(X1 ), where CB(X1 ) is the class of closed as well as bounded nonempty subsets of X1 and dτ (xκ , F ) = inf {dτ (xκ , b) : b ∈ F }, for each xκ ∈ X1 . Definition 1.5. [14] Let (X1 , dτ ) be a metric space. A map T : X1 → CB(X1 ) is called a multi-valued contraction if H(T xκ , T y κ ) ≤ rdτ (xκ , y κ ) for all xκ , y κ ∈ X1 , where 0 ≤ r < 1. Lemma 1.6. [14] If E, F ∈ CB(X1 ) and a ∈ E, then for every > 0, there exists b ∈ F such that dτ (a, b) ≤ H(E, F ) + . If E, F are in C(X1 ) is the class of non-empty and compact subsets of X1 , then one can choose b ∈ F such that dτ (a, b) ≤ H(E, F ). Definition 1.7. [4, 5] Suppose X1 is a non-empty set. A given function dτ : X1 × X1 → R+ is known as b-metric, if for each xκ , y κ , z κ ∈ X1 , the stated conditions are hold: (1) dτ (xκ , y κ ) = 0 if and only if xκ = y κ ; (2) dτ (xκ , y κ ) = dτ (y κ , xκ ); (3) dτ (xκ , z κ ) ≤ µ[dτ (xκ , y κ ) + dτ (y κ , z κ )]. Then (X1 , dτ ) with the parameter µ ≥ 1, µ ∈ R, is called b-metric space. P κp Example 1.8. [5] Consider the lp space (0 < p < 1), lp = {(ynκ ) ∈ R : |yn | < ∞}, and the function dτ : lp × lp → R defined by P 1 dτ (xκ , y κ ) = ( |xκn − ynκ |p ) p , xκ = (xκn ), y κ = (ynκ ) ∈ lp . Then (X, dτ ) is 1 1 known as b-metric space via parameter µ = 2 2 , also dτ (xκ , z κ ) ≤ 2 2 [dτ (xκ , y κ ) + κ κ dτ (y , z )].
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Example 1.9. [2] Suppose X1 = {0, 1, 2} and let dτ : X1 ×X1 → R+ be a function, defined by: dτ (2, 2) = dτ (1, 1) = dτ (0, 0) = 0, dτ (0, 1) = dτ (1, 2) = dτ (0, 1) = dτ (2, 1) = 1 and dτ (0, 2) = dτ (2, 0) = m ≥ 2. Then, the inequality dτ (xκ , y κ ) ≤ m κ κ κ κ κ κ κ 2 [dτ (x , z ) + dτ (z , y )] holds for each x , y , z ∈ X1 . This inequality does not satisfy for m > 2. Definition 1.10. [5] Suppose (X1 , dτ ) be a given b-metric space. A sequence {ynκ } is known to be converge to y κ ∈ X1 if there exists l(δ) ∈ N for all δ > 0, such that dτ (ynκ , y κ ) < δ for each n ≥ l(δ). Definition 1.11. [5] Suppose (X1 , dτ ) is b-metric space. The sequence {ynκ } is known as Cauchy sequence if one can find l(δ) ∈ N for all δ > 0, such that dτ (ynκ , xκm ) < δ for each n, m ≥ l(δ). 2. Main Results In this section, we will use the following contraction to prove the main result. Definition 2.1. Assume X1 is a non empty set and (X1 , dτ ) is a b-metric space with constant µ ≥ 1. The mappings T, S : X1 → CB(X1 ) are called a multi-valued generalized contraction if H(T xκ , Sy κ ) ≤ α1 dτ (xκ , T xκ ) + α2 dτ (y κ , Sy κ ) + α3 dτ (xκ , Sy κ ) + α4 dτ (y κ , T xκ ) + α5 dτ (xκ , y κ ) dτ (xκ , T xκ ) + dτ (y κ , Sy κ ) dτ (xκ , T xκ )(1 + dτ (xκ , T xκ )) + α +α6 7 1 + dτ (xκ , y κ ) + dτ (xκ , T xκ ) 1 + dτ (T xκ , Sy κ )dτ (xκ , y κ ) κ κ κ κ dτ (x , Sy )dτ (T x , Sy ) dτ (T xκ , Sy κ ) + dτ (xκ , y κ ) +α8 + α9 κ κ κ κ 1 + dτ (x , y ) + dτ (y , Sy ) 1 + dτ (xκ , T xκ ) + dτ (y κ , Sy κ ) κ κ for all x , y ∈ X1 and αi ≥ 0, i = 1, 2, 3, · · · , 9, with α1 + α2 + α5 + α6 + 2(α7 + α9 ) + 2µ(α3 + α8 ) < 1 and α1 + α3 + α4 + α5 + α8 + 2α9 < 1. Now first new result is the following theorem. Theorem 2.2. Suppose (X1 , dτ ) is a b-complete metric space with parameter µ ≥ 1 and assume T, S : X1 → CB(X1 ) be a multi-valued generalized contraction. Then, the mappings T and S have a unique common fixed point. Proof. Take xκ ∈ X1 , define xκ0 = xκ , assume xκ1 ∈ T xκ0 , xκ2 ∈ Sxκ1 such that xκ2n+1 ∈ T xκ2n , xκ2n+2 ∈ Sxκ2n+1 . By using Lemma 1.6, we can choose xκ2 ∈ Sxκ1 such that dτ (xκ1 , xκ2 ) ≤ H(T xκ0 , Sxκ1 ) + (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ). Then, by using Definition 2.1, we have dτ (xκ1 , xκ2 ) ≤ α1 dτ (xκ0 , T xκ0 )+α2 dτ (xκ1 , Sxκ1 )+α3 dτ (xκ0 , Sxκ1 )+α4 dτ (xκ1 , T xκ0 )+α5 dτ (xκ0 , xκ1 ) dτ (xκ0 , T xκ0 )(1 + dτ (xκ0 , T xκ0 )) dτ (xκ0 , T xκ0 ) + dτ (xκ1 , Sxκ1 ) + α7 κ κ κ κ 1 + dτ (x0 , x1 ) + dτ (x0 , T x0 ) 1 + dτ (T xκ0 , Sxκ1 )dτ (xκ0 , xκ1 ) dτ (xκ0 , Sxκ1 )dτ (T xκ0 , Sxκ1 ) dτ (T xκ0 , Sxκ1 ) + dτ (xκ0 , xκ1 ) +α8 + α9 κ κ κ κ 1 + dτ (x0 , x1 ) + dτ (x1 , Sx1 ) 1 + dτ (xκ0 , T xκ0 ) + dτ (xκ1 , Sxκ1 ) +(α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ) κ κ = α1 dτ (x0 , x1 ) + α2 dτ (xκ1 , xκ2 ) + α3 dτ (xκ0 , xκ2 ) + α4 dτ (xκ1 , xκ1 ) + α5 dτ (xκ0 , xκ1 ) +α6
FIXED POINT THEOREMS FOR MULTI−VALUED GENERALIZED CONTRACTION IN b-METRIC SPACES 161
dτ (xκ0 , xκ1 ) + dτ (xκ1 , xκ2 ) dτ (xκ0 , xκ1 )(1 + dτ (xκ0 , xκ1 )) + α 7 1 + dτ (xκ0 , xκ1 ) + dτ (xκ0 , xκ1 ) 1 + dτ (xκ1 , xκ2 )dτ (xκ0 , xκ1 ) κ κ κ κ dτ (x0 , x2 )dτ (x1 , x2 ) dτ (xκ1 , xκ2 ) + dτ (xκ0 , xκ1 ) +α8 + α 9 1 + dτ (xκ0 , xκ1 ) + dτ (xκ1 , xκ2 ) 1 + dτ (xκ0 , xκ1 ) + dτ (xκ1 , xκ2 ) +(α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ) κ κ ≤ α1 dτ (x0 , x1 ) + α2 dτ (xκ1 , xκ2 ) + α3 dτ (xκ0 , xκ2 ) + α5 dτ (xκ0 , xκ1 ) + α6 dτ (xκ0 , xκ1 ) +α7 dτ (xκ0 , xκ1 ) + α7 dτ (xκ1 , xκ2 ) + α8 dτ (xκ0 , xκ2 ) + α9 dτ (xκ1 , xκ2 ) +α9 dτ (xκ0 , xκ1 ) + (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ) +α6
or dτ (xκ1 , xκ2 ) ≤ α1 dτ (xκ0 , xκ1 )+α2 dτ (xκ1 , xκ2 )+µα3 [dτ (xκ0 , xκ1 )+dτ (xκ1 , xκ2 )]+α5 dτ (xκ0 , xκ1 ) +α6 dτ (xκ0 , xκ1 ) + α7 dτ (xκ0 , xκ1 ) + α7 dτ (xκ1 , xκ2 ) + µα8 [dτ (xκ0 , xκ1 ) + dτ (xκ1 , xκ2 )] +α9 dτ (xκ0 , xκ1 ) + α9 dτ (xκ1 , xκ2 ) + (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ) (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )dτ (xκ0 , xκ1 ) +(α2 + µα3 + α7 + µα8 + α9 )dτ (xκ1 , xκ2 ) +(α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ) so that dτ (xκ1 , xκ2 ) ≤ (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )dτ (xκ0 , xκ1 ) +(α2 + µα3 + α7 + µα8 + α9 )dτ (xκ1 , xκ2 ) +(α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ). Hence, we have dτ (xκ1 , xκ2 ) ≤ (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ) dτ (xκ0 , xκ1 ) 1 − (α2 + µα3 + α7 + µα8 + α9 ) (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ) . + 1 − (α2 + µα3 + α7 + µα8 + α9 ) Similarly, again by using Lemma 1.6, we can choose xκ3 ∈ T xκ2 such that
(2.1)
(α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )2 . 1 − (α2 + µα3 + α7 + µα8 + α9 ) Then, by using Definition 2.1, we have dτ (xκ2 , xκ3 ) ≤ H(Sxκ1 , T xκ2 ) +
dτ (xκ2 , xκ3 ) ≤ α1 dτ (xκ1 , Sxκ1 )+α2 dτ (xκ2 , T xκ2 )+α3 dτ (xκ1 , T xκ2 )+α4 dτ (xκ2 , Sxκ1 )+α5 dτ (xκ1 , xκ2 ) dτ (xκ1 , Sxκ1 ) + dτ (xκ2 , T xκ2 ) dτ (xκ1 , Sxκ1 )(1 + dτ (xκ1 , Sxκ1 )) + α 7 1 + dτ (xκ1 , xκ2 ) + dτ (xκ1 , Sxκ1 ) 1 + dτ (Sxκ1 , T xκ2 )dτ (xκ1 , xκ2 ) κ κ κ κ dτ (x1 , T x2 )dτ (Sx1 , T x2 ) dτ (Sxκ1 , T xκ2 ) + dτ (xκ1 , xκ2 ) +α8 + α9 κ κ κ κ 1 + dτ (x1 , x2 ) + dτ (x2 , T x2 ) 1 + dτ (xκ1 , Sxκ1 ) + dτ (xκ2 , T xκ2 ) (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )2 + 1 − (α2 + µα3 + α7 + µα8 + α9 ) κ κ = α1 dτ (x1 , x2 ) + α2 dτ (xκ2 , xκ3 ) + α3 dτ (xκ1 , xκ3 ) + α4 dτ (xκ2 , xκ2 ) + α5 dτ (xκ1 , xκ2 ) dτ (xκ1 , xκ2 )(1 + dτ (xκ1 , xκ2 )) dτ (xκ1 , xκ2 ) + dτ (xκ2 , xκ3 ) + α +α6 7 1 + dτ (xκ1 , xκ2 ) + dτ (xκ1 , xκ2 ) 1 + dτ (xκ2 , xκ3 )dτ (xκ1 , xκ2 ) κ κ κ κ dτ (x1 , x3 )dτ (x2 , x3 ) dτ (xκ2 , xκ3 ) + dτ (xκ1 , xκ2 ) +α8 + α9 κ κ κ κ 1 + dτ (x1 , x2 ) + dτ (x2 , x3 ) 1 + dτ (xκ1 , xκ2 ) + dτ (xκ2 , xκ3 ) +α6
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(α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )2 1 − (α2 + µα3 + α7 + µα8 + α9 ) ≤ α1 dτ (xκ1 , xκ2 ) + α2 dτ (xκ2 , xκ3 ) + α3 dτ (xκ1 , xκ3 ) + α5 dτ (xκ1 , xκ2 ) + α6 dτ (xκ1 , xκ2 ) +α7 dτ (xκ1 , xκ2 ) + α7 dτ (xκ2 , xκ3 ) + α8 dτ (xκ1 , xκ3 ) + α9 dτ (xκ2 , xκ3 ) (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )2 , +α9 dτ (xκ1 , xκ2 ) + 1 − (α2 + µα3 + α7 + µα8 + α9 ) dτ (xκ2 , xκ3 ) ≤ α1 dτ (xκ1 , xκ2 )+α2 dτ (xκ2 , xκ3 )+µα3 [dτ (xκ1 , xκ2 )+dτ (xκ2 , xκ3 )]+α5 dτ (xκ1 , xκ2 ) +α6 dτ (xκ1 , xκ2 ) + α7 dτ (xκ2 , xκ3 ) + α7 dτ (xκ1 , xκ2 ) + µα8 [dτ (xκ1 , xκ2 ) + dτ (xκ2 , xκ3 )] (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )2 +α9 dτ (xκ1 , xκ2 ) + α9 dτ (xκ2 , xκ3 ) + 1 − (α2 + µα3 + α7 + µα8 + α9 ) = (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )dτ (xκ1 , xκ2 ) +(α2 + µα3 + α7 + µα8 + α9 )dτ (xκ2 , xκ3 ) (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )2 + 1 − (α2 + µα3 + α7 + µα8 + α9 ) so that +
dτ (xκ2 , xκ3 ) ≤ (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )dτ (xκ1 , xκ2 ) +(α2 + µα3 + α7 + µα8 + α9 )dτ (xκ2 , xκ3 ) (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 )2 . + 1 − (α2 + µα3 + α7 + µα8 + α9 ) Hence, we have dτ (xκ2 , xκ3 ) ≤ (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ) dτ (xκ1 , xκ2 ) 1 − (α2 + µα3 + α7 + µα8 + α9 ) 2 α1 + µα3 + α5 + α6 + α7 + µα8 + α9 + . 1 − (α2 + µα3 + α7 + µα8 + α9 ) Using inequality (2.1) in (2.2), we get dτ (xκ2 , xκ3 ) ≤ 2 α1 + µα3 + α5 + α6 + α7 + µα8 + α9 (xκ0 , xκ1 ) 1 − (α2 + µα3 + α7 + µα8 + α9 ) 2 α1 + µα3 + α5 + α6 + α7 + µα8 + α9 . +2 1 − (α2 + µα3 + α7 + µα8 + α9 ) Proceeding this way, with induction process we have a sequence {xκn }, with xκ2n+1 ∈ T xκ2n and xκ2n+2 ∈ Sxκ2n+1 , such that (α1 + µα3 + α5 + α6 + α7 + µα8 + α9 ) dτ (xκn−1 , xκn ) 1 − (α2 + µα3 + α7 + µα8 + α9 ) n α1 + µα3 + α5 + α6 + α7 + µα8 + α9 + . 1 − (α2 + µα3 + α7 + µα8 + α9 )
dτ (xκn , xκn+1 ) ≤
For all n N , let λ=
α1 + µα3 + α5 + α6 + α7 + µα8 + α9 1 − (α2 + µα3 + α7 + µα8 + α9 )
dτ (xκn , xκn+1 ) ≤ λdτ (xκn−1 , xκn ) + λn ≤ λ[λdτ (xκn−2 , xκn−1 ) + λn−1 ] + λn
(2.2)
FIXED POINT THEOREMS FOR MULTI−VALUED GENERALIZED CONTRACTION IN b-METRIC SPACES 163
= λ2 dτ (xκn−2 , xκn−1 ) + λλn−1 + λn = λ2 dτ (xκn−2 , xκn−1 ) + 2λn ≤ λ3 dτ (xκn−3 , xκn−2 ) + 3λn .. . κ κ n dτ (xn , xn+1 ) ≤ λ dτ (xκ0 , xκ1 ) + nλn or
n α1 + µα3 + α5 + α6 + α7 + µα8 + α9 dτ (xκn , xκn+1 ) ≤ dτ (xκ0 , xκ1 ) 1 − (α2 + µα3 + α7 + µα8 + α9 ) n α1 + µα3 + α5 + α6 + α7 + µα8 + α9 +n . 1 − (α2 + µα3 + α7 + µα8 + α9 ) As 0 < λ < 1, radius of convergence of Σλn and Σnλn is same. So, {xκn } is b-Cauchy sequence. And (X1 , dτ ) is b-complete metric space, so {xκn }∞ n=0 is b-convergent, and lim xκn = uκ .
n→∞
Now, dτ (uκ , Suκ ) ≤ µ[dτ (uκ , xκ2n+1 ) + dτ (xκ2n+1 , Suκ )] = µdτ (uκ , xκ2n+1 ) + µdτ (T xκ2n , Suκ ) ≤ µdτ (uκ , xκ2n+1 ) + µH(T xκ2n , Suκ ). Using Definition 2.1, we obtain dτ (uκ , Suκ ) ≤ µdτ (uκ , xκ2n+1 ) + µα1 dτ (xκ2n , T xκ2n ) + µα2 dτ (uκ , Suκ ) +µα3 dτ (xκ2n , Suκ ) + µα4 dτ (uκ , T xκ2n ) + µα5 dτ (xκ2n , uκ ) dτ (xκ2n , T xκ2n ) + dτ (uκ , Suκ ) dτ (xκ2n , T xκ2n )(1 + dτ (xκ2n , T xκ2n )) + µα +µα6 7 1 + dτ (xκ2n , uκ ) + dτ (xκ2n , T xκ2n ) 1 + dτ (T xκ2n , Suκ )dτ (xκ2n , uκ ) κ κ κ κ dτ (T xκ2n , Suκ ) + dτ (xκ2n , uκ ) dτ (x2n , Su )dτ (T x2n , Su ) +µα8 + µα 9 1 + dτ (xκ2n , uκ ) + dτ (uκ , Suκ ) 1 + dτ (xκ2n , T xκ2n ) + dτ (uκ , Suκ ) κ κ κ κ = µdτ (u , x2n+1 ) + µα1 dτ (x2n , x2n+1 ) + µα2 dτ (uκ , Suκ ) +µα3 dτ (xκ2n , Suκ ) + µα4 dτ (uκ , xκ2n+1 ) + µα5 dτ (xκ2n , uκ ) dτ (xκ2n , xκ2n+1 ) + dτ (uκ , Suκ ) dτ (xκ2n , xκ2n+1 )(1 + dτ (xκ2n , xκ2n+1 )) + µα +µα6 7 1 + dτ (xκ2n , uκ ) + dτ (xκ2n , xκ2n+1 ) 1 + dτ (xκ2n+1 , Suκ )dτ (xκ2n , uκ ) κ κ κ κ dτ (xκ2n+1 , Suκ ) + dτ (xκ2n , uκ ) dτ (x2n , Su )dτ (x2n+1 , Su ) + µα . +µα8 9 1 + dτ (xκ2n , uκ ) + dτ (uκ , Suκ ) 1 + dτ (xκ2n , xκ2n+1 ) + dτ (uκ , Suκ ) Taking limit n → ∞, one can have dτ (uκ , Suκ ) ≤ µα2 dτ (uκ , Suκ ) + µα3 dτ (uκ , Suκ ) + µα7 dτ (uκ , Suκ ) dτ (uκ , Suκ )dτ (uκ , Suκ ) dτ (uκ , Suκ ) + µα 9 1 + dτ (uκ , Suκ ) 1 + dτ (uκ , Suκ ) κ κ κ κ ≤ µα2 dτ (u , Su ) + µα3 dτ (u , Su ) + µα7 dτ (uκ , Suκ ) +µα8 dτ (uκ , Suκ ) + µα9 dτ (uκ , Suκ ) dτ (uκ , Suκ ) ≤ [µα2 + µα3 + µα7 + µα8 + µα9 ]dτ (uκ , Suκ ) [1 − (µα2 + µα3 + µα7 + µα8 + µα9 )]dτ (uκ , Suκ ) ≤ 0, which indicates that dτ (uκ , Suκ ) = 0. Thus, uκ = Suκ . Which implies that uκ ∈ Suκ . Similarly one can show that, uκ ∈ T uκ . Hence S has a fixed point. +µα8
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Next to prove the uniqueness of fixed point, assume v κ is another fixed point of T and S. Consider dτ (uκ , v κ ) = dτ (T uκ , Sv κ ) ≤ H(T uκ , Sv κ ) or dτ (uκ , v κ ) ≤ α1 dτ (uκ , T uκ )+α2 dτ (v κ , Sv κ )+α3 dτ (uκ , Sv κ )+α4 dτ (v κ , T uκ )+α5 dτ (uκ , v κ ) +α6 +α8
dτ (uκ , T uκ )(1 + dτ (uκ , T uκ )) dτ (uκ , T uκ ) + dτ (v κ , Sv κ ) + α7 κ κ κ κ 1 + dτ (u , v ) + dτ (u , T u ) 1 + dτ (T uκ , Sv κ )dτ (uκ , v κ )
dτ (T uκ , Sv κ ) + dτ (uκ , v κ ) dτ (uκ , Sv κ )dτ (T uκ , Sv κ ) + α 9 1 + dτ (uκ , v κ ) + dτ (v κ , Sv κ ) 1 + dτ (uκ , T uκ ) + dτ (v κ , Sv κ )
= α1 dτ (uκ , v κ ) + α2 dτ (v κ , v κ ) + α3 dτ (uκ , v κ ) + α4 dτ (uκ , v κ ) + α5 dτ (uκ , v κ ) +α6 +α8
dτ (uκ , uκ ) + dτ (v κ , v κ ) dτ (uκ , uκ )(1 + dτ (uκ , uκ )) + α 7 1 + dτ (uκ , v κ ) + dτ (uκ , uκ ) 1 + dτ (uκ , v κ )dτ (uκ , v κ )
dτ (uκ , v κ )dτ (uκ , v κ ) dτ (uκ , v κ ) + dτ (uκ , v κ ) + α 9 1 + dτ (uκ , v κ ) + dτ (v κ , v κ ) 1 + dτ (uκ , uκ ) + dτ (v κ , v κ )
≤ α1 dτ (uκ , v κ ) + α3 dτ (uκ , v κ ) + α4 dτ (uκ , v κ ) + α5 dτ (uκ , v κ ) + α8 dτ (uκ , v κ ) +2α9 dτ (uκ , v κ ), dτ (uκ , v κ ) ≤ (α1 + α3 + α4 + α5 + α8 + 2α9 )dτ (uκ , v κ ) [1 − (α1 + α3 + α4 + α5 + α8 + 2α9 )]dτ (uκ , v κ ) ≤ 0. Since α1 + α3 + α4 + α5 + α8 + 2α9 < 1, then 1 − (α1 + α3 + α4 + α5 + α8 + 2α9 ) > 0. Which shows that dτ (uκ , v κ ) ≤ 0. So dτ (uκ , v κ ) = 0, thus uκ = v κ . Hence T and S have unique common fixed point.
With the same procedure of Theorem 2.2, one can derive the next theorem. Theorem 2.3. Let (X1 , dτ ) be a b-complete metric space with the parameter µ ≥ 1. Suppose T : X1 → CB(X1 ) be a multi valued mapping satisfies the condition: H(T xκ , T y κ ) ≤ α1 dτ (xκ , T xκ ) + α2 dτ (y κ , T y κ ) + α3 dτ (xκ , T y κ ) + α4 dτ (y κ , T xκ ) + α5 dτ (xκ , y κ ) +α6
dτ (xκ , T xκ )(1 + dτ (xκ , T xκ )) dτ (xκ , T xκ ) + dτ (y κ , T y κ ) + α7 κ κ κ κ 1 + dτ (x , y ) + dτ (x , T x ) 1 + dτ (T xκ , T y κ )dτ (xκ , y κ )
dτ (xκ , T y κ )dτ (T xκ , T y κ ) dτ (T xκ , T y κ ) + dτ (xκ , y κ ) + α 9 1 + dτ (xκ , y κ ) + dτ (y κ , T y κ ) 1 + dτ (xκ , T xκ ) + dτ (y κ , T y κ ) κ κ for each x , y ∈ X1 and αi ≥ 0, i = 1, 2, 3, · · · , 9, with +α8
α1 + α2 + α5 + α6 + 2(α7 + α9 ) + 2µ(α3 + α8 ) < 1 and α1 + α3 + α4 + α5 + α8 + 2α9 < 1. Then T has a unique fixed point. The following example verifies Theorem 2.2
FIXED POINT THEOREMS FOR MULTI−VALUED GENERALIZED CONTRACTION IN b-METRIC SPACES 165
Example 2.4. Suppose X1 =R. A metric dτ : X1 × X1 → X1 is defined by dτ (xκ , y κ ) = |xκ − y κ |, for all xκ , y κ ∈ X1 , with the parameter µ = 1. The pair (X1 , dτ ) is b-complete metric space. We define h κi κ T , S: X1 → CB(X1 ) as T xκ = 0, x2 and Sy κ = 0, y2 for each xκ , xκ ∈ X1 . Then H(T xκ , Sy κ ) ≤ α1 dτ (xκ , T xκ ) + α2 dτ (y κ , Sy κ ) + α3 dτ (xκ , Sy κ ) + α4 dτ (y κ , T xκ ) + α5 dτ (xκ , y κ ) dτ (xκ , T xκ )(1 + dτ (xκ , T xκ )) dτ (xκ , T xκ ) + dτ (y κ , Sy κ ) + α +α6 7 1 + dτ (xκ , y κ ) + dτ (xκ , T xκ ) 1 + dτ (T xκ , Sy κ )dτ (xκ , y κ ) κ κ κ κ dτ (x , Sy )dτ (T x , Sy ) dτ (T xκ , Sy κ ) + dτ (xκ , y κ ) +α8 + α9 , κ κ κ κ 1 + dτ (x , y ) + dτ (y , Sy ) 1 + dτ (xκ , T xκ ) + dτ (y κ , Sy κ ) 1 1 where, (α1 = α2 = α3 = α4= α6 = α 8 = α9 = 20 ), α5 = 2 . h 7 =καi κ Also H(T xκ , Sy κ ) = H 0, x2 , 0, y2 = 21 |xκ − y κ | = 12 dτ (xκ , y κ ). Thus, a point 0 ∈ X1 , is the unique common fixed point of T and S. Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this manuscript. References [1] A. A. N. Abdou, Fixed points of multivalued contraction mappings in modular metric spaces, J. Nonl. Sci. Appl. 9 (2016), 787-798. [2] H. Aydi, M. F. Bota, E. Karapinar, S. Mitrovi´ c, A fixed point theorem for set-valued quasicontractions in b-metric spaces, Fixed Point Theory Appl., 2012 (2012), 8 pp. [3] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fundam. Math., 3 :133-181 (1922). [4] I. A. Bakhtin, The Contraction mapping Principle in quasimetric spaces, Funtional Analysis, 30 ,26–37 (1989). [5] M. Boriceanu, Fixd point theory for multivalued generalized contraction on a set with two b-metrices, Studia Univ. Babes-Bolyani Math., LIV(3) (2009),1-14. [6] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis,1 (1993),5-11. [7] S. Czerwik,Non-linear Set-Valud Contraction mappings in b-metric spaces,Atti. sem. math. fis. Univ. modena,46, (1998),263-276. [8] C. Chifu, G. Petrusel, fixed points for multivalued contractions in b-metric spaces with applications to fractals, Taiwanese Journal of Math., 18, No. 5, pp. 1365-1375, Oct (2014). [9] H. W. Corley, Optimality conditions for maximization of set-valued functions, J. Optim. Theory Appl., 58, 1-10 (1988). [10] M. Edelstein: An extension of Banachs contraction principle, Proc. Am. Math. Soc., 12, 7-10 (1961). [11] N. Hussain, D. Dori´ c, Z. Kadelburg, S. Radenovi´ c,, Suzuki-type fixed point result in metic type spaces,Fixed point Theory. Appl., 2012 (2012) , 12 pp. [12] J. M. Joseph, D. D . Roselin and M. Marudai, Fixed point theorems on multi valued mappings in b-metric spaces,SpringerPlus,(2016) 5:217. [13] M. Javahernia, A. Razani, and F. Khojasteh, Fixed point of multi-valued contractions via manageable functions and Lius generalization, Cogent Mathematics, Vol. 3, 1, 2016 [14] S. B. Nadler Multi-valued contraction mappings, Pac. J. Math., 30 :475-488 (1969). [15] S. Reich, Some fixed point problems, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 57, 194-198 (1974). [16] S. Shukla, S. Radenovic, C. Vetro, Set-valued Hardy-Rogers type ontraction in 0-complete partial metric spaces, Int. J. Math. Mat. Sci., Volume 2014, Article ID 652925, 9 pages. [17] J. Tiammee, P. Charoensawan, and S. Suantai, Fixed Point Theorems for Multivalued Nonself G-Almost Contractions in Banach Spaces Endowed with Graphs, Journal of Function Spaces, Volume 2017 (2017), Article ID 7053849, 5 pages
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[18] M. Zoguchi and N. Takahashi, W: Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl., 141(1), 177-188 (1989). Saddam Hussain 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, Van Yuzuncu Yil University,65080, Turkey E-mail address:
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