Common fixed point results for new Ciric-type rational ...

J. Fixed Point Theory Appl. 54:02)81( https://doi.org/10.1007/s11784-018-0525-6 c Springer International Publishing AG, part of Springer Nature 2018

Journal of Fixed Point Theory and Applications

Common fixed point results for new Ciric-type rational multivalued F -contraction with an application Tahair Rasham, Abdullah Shoaib, Nawab Hussain, Muhammad Arshad and Sami Ullah Khan Abstract. In this article, common fixed point theorems for a pair of multivalued mappings satisfying a new Ciric-type rational F -contraction condition in complete dislocated metric spaces are established. An example is constructed to illustrate our results. An application to the system of integral equations is presented to support the usability of proved results. Our results combine, extend and infer several comparable results in the existing literature. Mathematics Subject Classification. 46S40, 47H10, 54H25. Keywords. Fixed point, complete dislocated metric space, proximinal sets, multivalued mappings, new Ciric-type rational F -contraction.

1. Introduction and preliminaries Let J : S −→ S be a mapping. A point x in S is a fixed point of J, if x = Jx. Banach [8] established the fundamental fixed point theorem, which has played an important role in various fields of applied mathematical analysis. Due to its importance, several authors have obtained many interesting extensions of his result (see [1–33]). Many authors have established fixed point theorems in complete dislocated metric space. The idea of dislocated topology has been applied in the field of logic programming semantics (see [13]). Dislocated metric space (metric-like space) (see [13]) is a generalization of partial metric space (see [22]). Wardowski [33] introduced a new type of contraction called F -contraction and proved a fixed point theorem concerning F -contraction. He generalized many fixed point results in a different aspects. Afterward, Secelean [29] proved fixed point theorems regarding F -contraction by iterated function systems. Hussain and Salimi [17] and Piri et al. [25] proved a fixed point result for F -Suzuki contractions for some weaker conditions on the self-map in

54

Page 2 of 16

T. Rasham et al.

a complete metric space. Acar et al. [3] introduced the concept of generalized multivalued F -contraction mappings. Furthermore, Acar et al. [2] extended the multivalued F -contraction with δ-distance and established fixed point results in complete metric space. Sgroi et al. [30] established fixed point theorems for multivalued F -contraction condition and obtained the solution of certain functional and integral equations, which was a proper generalization of some multivalued fixed point theorems including Nadler’s theorem [23]. Many other useful results on F -contractions can be seen in [5,6,16,21]. In this paper, we recall the concept of F -contraction to obtain some common fixed point results for multivalued mappings on proximinal sets satisfying a new Ciric-type rational F -contraction condition in the context of complete dislocated metric spaces. We give the following definitions and results which will be needed in the sequel. Definition 1. [13] Let X be a nonempty set. A mapping dl : X × X → [0, ∞) is called a dislocated metric (or simply dl -metric) if the following conditions hold, for any x, y, z ∈ X : (i) If dl (x, y) = 0 , then x = y; (ii) dl (x, y) = dl (y, x); (iii) dl (x, y) ≤ dl (x, z) + dl (z, y). Then, dl is called a dislocated metric on X, and the pair (X, dl ) is called dislocated metric space or dl metric space. It is clear that if dl (x, y) = 0, then from (i), x = y. But if x = y, dl (x, y) may not be 0. Example 2. [13] If X = R+ ∪ {0}, then dl (x, y) = x + y defines a dislocated metric dl on X. Definition 3. [13] Let (X, dl ) be a dislocated metric space. (i) A sequence {xn } in (X, dl ) is called a Cauchy sequence, if given ε > 0, there corresponds n0 ∈ N such that for all n, m ≥ n0 we have dl (xm , xn ) < ε or lim dl (xn , xm ) = 0. n,m→∞

(ii) A sequence {xn } dislocated-converges (for short dl -converges) to x if lim dl (xn , x) = 0. In this case, x is called a dl -limit of {xn }. n→∞

(iii) (X, dl ) is called complete if every Cauchy sequence in X converges to a point x ∈ X such that dl (x, x) = 0. Definition 4. [31] Let K be a nonempty subset of dislocated metric space X and let x ∈ X. An element y0 ∈ K is called a best approximation in K if dl (x, K) = dl (x, y0 ), where dl (x, K) = inf dl (x, y). y∈K

If each x ∈ X has at least one best approximation in K, then K is called a proximinal set. We denote P (X) be the set of all closed proximinal subsets of X. Definition 5. [31] The function Hdl : P (X) × P (X) → R+ , defined by Hdl (A, B) = max{sup dl (a, B), sup dl (A, b)} a∈A

is called dislocated Hausdorff metric on P (X).

b∈B

Common fixed point results for new Ciric-type rational

Page 3 of 16

45

Definition 6. [33] Let (X, d) be a metric space. A mapping T : X → X is said to be a F -contraction if there exists τ > 0 such that ∀x, y ∈ X, d(T x, T y) > 0 ⇒ τ + F (d(T x, T y)) ≤ F (d(x, y))

(1.1)

where F : R+ → R is a mapping satisfying the following conditions: (F1) F is strictly increasing, i.e., for all x, y ∈ R+ such that x < y, F (x) < F (y); (F2) for each sequence {αn }∞ n=1 of positive numbers, limn→∞ αn = 0 if and only if limn→∞ F (αn ) = −∞; (F3) there exists k ∈ (0, 1) such that limα→0+ αk F (α)=0 . We denote by F , the set of all functions satisfying the conditions (F1)–(F3). Example 7. [33] The Family F is not empty. 1. F (x) = ln(x); x > 0. 2. F (x) = x + ln(x); x > 0. −1 3. F (x) = √ ; x > 0. x Remark 8. [33] From (F1) and (1.1), it is easy to conclude that every F -contraction is necessarily continuous. Theorem 9. [33] Let (X, d) be a complete metric space and let T : X → X be an F -contraction. Then T has a unique fixed point x∗ ∈ X and for every x ∈ X the sequence {T n x}n∈N converges to x∗ . Lemma 10. [31] Let (X, dl ) be a dislocated metric space. Let (P (X), Hdl ) be a dislocated Hausdorff metric space on P (X). Then for all A, B ∈ P (X) and for each a ∈ A, there exists ba ∈ B that satisfies dl (a, B) = dl (a, ba ) and then Hdl (A, B) ≥ dl (a, ba ).

2. Main result Let (X, dl ) be a dislocated metric space, and x0 ∈ X and S, T : X → P (X) be the multifunctions on X. Let x1 ∈ Sx0 be an element such that dl (x0 , Sx0 ) = dl (x0 , x1 ). Let x2 ∈ T x1 be such that dl (x1 , T x1 ) = dl (x1 , x2 ). Let x3 ∈ Sx2 be such that dl (x2 , Sx2 ) = dl (x2 , x3 ). Continuing this process, we construct a sequence xn of points in X such that x2n+1 ∈ Sx2n and x2n+2 ∈ T x2n+1 , where n = 0, 1, 2, .... Also, dl (x2n , Sx2n ) = dl (x2n , x2n+1 ), dl (x2n+1 , T x2n+1 ) = dl (x2n+1 , x2n+2 ). We denote this iterative sequence by {T S(xn )}. We say that {T S(xn )} is a sequence in X generated by x0 . We begin with the following definition. Definition 11. Let (X, dl ) be a complete dislocated metric space and S, T : X → P (X) be two multivalued mappings. The pair (S, T ) is called a pair of new Ciric-type rational F -contractions, if for all x, y ∈ {T S(xn )}, we have τ + F (Hdl (Sx, T y)) ≤ F (Dl (x, y)),

(2.1)

54

Page 4 of 16

T. Rasham et al.

where F ∈ F and τ > 0, and dl (x, Sx) .dl (y, T y) , dl (x, Sx), dl (y, T y) . Dl (x, y) = max dl (x, y), 1 + dl (x, y) (2.2) The following theorem is one of our main results. Theorem 12. Let (X, dl ) be a complete dislocated metric space and (S, T ) be a pair of new Ciric-type rational multivalued F -contractions. Then, {T S(xn )} → u ∈ X. Moreover, if (2.1) also holds for u, then S and T have a common fixed point u in X and dl (u, u) = 0. Proof. If Dl (x, y) = 0, then clearly x = y is a common fixed point of S and T. Then there is nothing to prove and our proof is complete. Let Dl (y, x)) > 0 for all x, y ∈ {T S(xn )} with x = y. Then from contractive condition (2.1), and Lemma 10, we get F (dl (x2i+1 , x2i+2 )) ≤ F (Hdl (Sx2i , T x2i+1 )) ≤ F (Dl (x2i , x2i+1 )) − τ for all i ∈ N ∪ {0}, where Dl (x2i , x2i+1 ) dl (x2i , Sx2i ) .dl (x2i+1 , T x2i+1 ) = max dl (x2i , x2i+1 ), , 1 + dl (x2i , x2i+1 ) dl (x2i , Sx2i ), dl (x2i+1 , T x2i+1 ) dl (x2i , x2i+1 ) .dl (x2i+1 , x2i+2 ) = max dl (x2i , x2i+1 ), , 1 + dl (x2i , x2i+1 ) dl (x2i , x2i+1 ), dl (x2i+1 , x2i+2 ) = max{dl (x2i , x2i+1 ), dl (x2i+1 , x2i+2 )}. If, Dl (x2i , x2i+1 ) = dl (x2i+1 , x2i+2 ), then F (dl (x2i+1 , x2i+2 )) ≤ F (dl (x2i+1 , x2i+2 )) − τ, which is a contradiction due to (F1). Therefore, F (dl (x2i+1 , x2i+2 )) ≤ F (dl (x2i , x2i+1 )) − τ,

(2.3)

for all i ∈ N ∪ {0}. Similarly, we have F (dl (x2i , x2i+1 )) ≤ F (dl (x2i−1 , x2i )) − τ,

(2.4)

for all i ∈ N ∪ {0}. By using (2.4) in (2.3), we have F (dl (x2i+1 , x2i+2 )) ≤ F (dl (x2i−1 , x2i )) − 2τ. Repeating these steps, we get F (dl (x2i+1 , x2i+2 )) ≤ F (dl (x0 , x1 )) − (2i + 1)τ.

(2.5)

Similarly, we have F (dl (x2i , x2i+1 )) ≤ F (dl (x0 , x1 )) − 2iτ.

(2.6)


Page 5 of 16

45

Inequalities (2.5) and (2.6) can jointly be written as F (dl (xn , xn+1 )) ≤ F (dl (x0 , x1 )) − nτ.

(2.7)

On taking limit n → ∞, on both sides of (2.7), we have lim F (dl (xn , xn+1 )) = −∞.

(2.8)

lim dl (xn , xn+1 ) = 0.

(2.9)

n→∞

Since F ∈ F , n→∞

By (2.7), for all n ∈ N, we obtain (dl (xn , xn+1 ))k ((F (dl (xn , xn+1 )) − F (dl (x0 , x1 ))) ≤ −(dl (xn , xn+1 ))k nτ ≤ 0. (2.10) Considering (2.8), (2.9) and letting n → ∞ in (2.10), we have lim (n(dl (xn , xn+1 ))k ) = 0.

(2.11)

n→∞

Since (2.11) holds, there exist n1 ∈ N, such that n(dl (xn , xn+1 ))k ≤ 1 for all n ≥ n1 or, 1 for all n ≥ n1 . (2.12) dl (xn , xn+1 ) ≤ 1 nk Using (2.12), we get from m > n > n1 , dl (xn , xm ) ≤ dl (xn , xn+1 ) + dl (xn+1 , xn+2 ) + · · · + dl (xm−1 , xm ) =

m−1

dl (xi , xi+1 ) ≤

i=n

∞

dl (xi , xi+1 ) ≤

i=n

The convergence of the series

∞

i=n

∞ 1 1

i=n

1

1

ik

ik

.

entails limn,m→∞ dl (xn , xm ) = 0.

Hence, {T S(xn )} is a Cauchy sequence in (X, dl ). Since (X, dl ) is a complete dislocated metric space, there exists u ∈ X such that {T S(xn )} → u, that is, lim dl (xn , u) = 0.

n→∞

(2.13)

Now, by Lemma 10, we have τ + F (dl (x2n+1 , T u)) ≤ τ + F (Hdl (Sx2n , T u))).

(2.14)

As inequality (2.1) also holds for u, we have τ + F (dl (x2n+1 , T u)) ≤ F (Dl (x2n , u)),

(2.15)

where Dl (x2n , u) dl (x2n , Sx2n ) .dl (u, T u) = max dl (x2n , u), , dl (x2n , Sx2n ), dl (u, T u) 1 + dl (x2n , u) dl (x2n , x2n+1 ) .dl (u, T u) = max dl (x2n , u), , dl (x2n , x2n+1 ), dl (u, T u) . 1 + dl (x2n , u) Taking limit n → ∞, and using (2.13), we get lim Dl (x2n , u) = dl (u, T u).

n→∞

(2.16)

54

Page 6 of 16

T. Rasham et al.

Since F is strictly increasing, (2.15) implies dl (x2n+1 , T u) < Dl (x2n , u). Taking limit n → ∞, and using (2.16), we get dl (u, T u) < dl (u, T u), which is a contradiction; hence, dl (u, T u) = 0 or u ∈ T u. Similarly, using (2.13) and Lemma 10 and the inequality τ + F (dl (x2n+2 , Su)) ≤ τ + F (Hdl (T x2n+1 , Su)), we can show that dl (u, Su) = 0 or u ∈ Su. Hence S and T have a common fixed point u in X. Now, dl (u, u) ≤ dl (u, T u) + dl (T u, u) ≤ 0.

This implies that dl (u, u) = 0.

Example 13. Let X = {0} ∪ Q+ and dl (x, y) = x + y. Then, (X, dl ) is a complete dislocated metric space. Define the mappings S, T : X → P (X) as follows: 1 2 1 2 x, x and T (x) = x, x for all x ∈ X. S(x) = 3 3 5 5 Define the function F : R+ → R by F (x) = ln(x) for all x ∈ R+ and τ > 0. As x, y ∈ X, τ = ln(1.2), by taking x0 = 7, we define the sequence 7 7 , 45 , · · · } in X generated by x0 = 7. We have {T S(xn )} = {7, 73 , 15

Hdl (Sx, T y) = max

sup dl (a, T y), sup dl (Sx, b)

a∈Sx

b∈T y

y 2y x 2x , = max sup dl a, , , sup dl ,b 2 5 3 3 a∈Sx b∈T y

2x y x 2y , , , dl = max dl 3 5 3 5 2x y x 2y + , + = max , 3 5 3 5 where

).dl (y, y5 , 2y 5 ) , Dl (x, y) = max dl (x, y), 1 + dl (x, y) x 2x y 2y , dl x, , dl y, , 3 3 5 5 x y dl (x, x3 ).dl (y, y5 ) , dl x, , dl y, = max dl (x, y), 1 + dl (x, y) 3 5 4x 6y 8xy , , = max x + y, = x + y. 15(1 + x + y) 3 5 dl (x,

x

2x 3, 3


Case (i). If, max

2x 3

+ y5 , x3 +

2y 5

=

x 3

+

2y 5 ,

Page 7 of 16

45

and τ = ln(1.2), then we have

10x + 12y ≤ 25x + 25y 6 x 2y ( + ) ≤ x+y 5 3 5 x 2y ln(1.2) + ln( + ) ≤ ln(x + y), 3 5 which implies that τ + F (Hdl (Sx, T y) ≤ F (Dl (x, y)). 2x y y x 2y = 3 + 5 , and τ = ln(1.2), then Case (ii). Similarly, if max 2x 3 + 5, 3 + 5 we have 20x + 6y ≤ 25x + 25y

4x 2y + ≤ x+y 3 5

2x y + ≤ ln(x + y). ln(1.2) + ln 3 5 6 5

Hence, τ + F (Hdl (Sx, T y) ≤ F (Dl (x, y)). Hence, all the hypotheses of Theorem 12 are satisfied and so (S, T ) have a common fixed point. Corollary 14. Let (X, dl ) be a complete dislocated metric space and S : X → P (X) be a multivalued mapping such that τ + F (Hdl (Sx, Sy)) ≤ F (Dl (x, y))

(2.17)

for all x, y ∈ {SS(xn )}, where F ∈ F and τ > 0, and dl (x, Sx) .dl (y, Sy) , dl (x, Sx), dl (y, Sy) . Dl (x, y) = max dl (x, y), 1 + dl (x, y) Then, {SS(xn )} → u ∈ X. Moreover, if (2.17) also holds for u, then S has a fixed point u in X and dl (u, u) = 0. Remark 15. By setting the following different values of Dl (x, y) in equation (2.2), we can obtain different results on multivalued F -contractions as corollaries of Theorem 12: (1) Dl (x, y) = dl (x, y), dl (x, Sx) .dl (y, T y) , (2) Dl (x, y) = 1 + dl (x, y) (3) Dl (x, y) = dl (x, Sx), (4) Dl (x, y) = dl (y, T y), dl (x, Sx) .dl (y, T y) (5) Dl (x, y) = max dl (x, y), , 1 + dl (x, y) (6) Dl (x, y) = max {dl (x, y), dl (x, Sx)} , (7) Dl (x, y) = max {dl (x, y), dl (y, T y)} ,

54

Page 8 of 16

T. Rasham et al.

(8) Dl (x, y) (9) Dl (x, y) (10) Dl (x, y) (11) Dl (x, y) (12) Dl (x, y) (13) Dl (x, y)

dl (x, Sx) .dl (y, T y) , dl (x, Sx) , = max 1 + dl (x, y) dl (x, Sx) .dl (y, T y) , dl (y, T y) , = max 1 + dl (x, y) = max {dl (x, Sx), dl (y, T y)} , dl (x, Sx) .dl (y, T y) , dl (x, Sx) , = max dl (x, y), 1 + dl (x, y) dl (x, Sx) .dl (y, T y) , dl (y, T y) , = max dl (x, y), 1 + dl (x, y) = max {dl (x, y), dl (x, Sx), dl (y, T y)} .

Theorem 16. Let (X, dl ) be a complete dislocated metric space and S, T : X → P (X) be the multivalued mappings. Assume that if F ∈ F and τ ∈ R+ such that τ + F (Hdl (Sx, T y))

d2 (x, Sx).dl (y, T y) ≤ F a1 dl (x, y) + a2 dl (x, Sx) + a3 dl (y, T y) + a4 l 1 + d2l (x, y) (2.18) for all x, y ∈ {T S(xn )}, with x = y where a1 , a2 , a3 , a4 > 0 , a1 +a2 +a3 +a4 = 1 and a3 + a4 = 1. Then, {T S(xn )} → u ∈ X. Moreover, if (2.18) also holds for u, then S and T have a common fixed point u in X and dl (u, u) = 0. Proof. As x1 ∈ Sx0 and x2 ∈ T x1 , by using Lemma 10, τ + F (dl (x1 , x2 )) = τ + F (dl (x1 , T x1 )) ≤ τ + F (Hdl (Sx0 , T x1 )) ≤ F a1 dl (x0 , x1 ) + a2 dl (x0 , x1 ) + a3 dl (x1 , T x1 )

d2l (x0 , Sx0 ).dl (x1 , T x1 ) +a4 1 + d2l (x0 , x1 ) ≤ F a1 dl (x0 , x1 ) + a2 dl (x0 , x1 ) + a3 dl (x1 , x2 )

d2l (x0 , x1 ) +a4 dl (x1 , T x1 ) 1 + d2l (x0 , x1 ) ≤ F ((a1 + a2 )dl (x0 , x1 ) + (a3 + a4 )dl (x1 , x2 )). Since F is strictly increasing, we have dl (x1 , x2 ) < (a1 + a2 )dl (x0 , x1 ) + (a3 + a4 )dl (x1 , x2 )

a1 + a2 < dl (x0 , x1 ). 1 − a3 − a4 From a1 + a2 + a3 + a4 = 1 and a3 + a4 = 1, we deduce 1 − a3 − a4 > 0 and so dl (x1 , x2 ) < dl (x0 , x1 ). Consequently, F (dl (x1 , x2 )) ≤ F (dl (x0 , x1 )) − τ.


Page 9 of 16

45

As we have x2i+1 ∈ Sx2i and x2i+2 ∈ T x2i+1 , from contractive condition (2.18) and Lemma 10, we get τ + F (dl (x2i+1 , x2i+2 )) = τ + F (dl (x2i+1 , T x2i+1 )) ≤ τ + F (Hdl (Sx2i , T x2i+1 )) ≤ F a1 dl (x2i , x2i+1 ) + a2 dl (x2i , Sx2i )

d2l (x2i , Sx2i ).dl (x2i+1 , T x2i+1 ) + a3 dl (x2i+1 , T x2i+1 ) + a4 1 + d2l (x2i , x2i+1 ) ≤ F a1 dl (x2i , x2i+1 ) + a2 dl (x2i , x2i+1 ) + a3 dl (x2i+1 , x2i+2 )

d2l (x2i , x2i+1 ) + a4 dl (x2i+1 , x2i+2 ) 1 + d2l (x2i , x2i+1 ) ≤ F (a1 dl (x2i , x2i+1 ) + a2 dl (x2i , x2i+1 ) + a3 dl (x2i+1 , x2i+2 ) + a4 dl (x2i+1 , x2i+2 )). Since F is strictly increasing, and a1 + a2 + a3 + a4 = 1 where a3 + a4 = 1, we deduce 1 − a3 − a4 > 0 and obtain dl (x2i+1 , x2i+2 ) < a1 dl (x2i , x2i+1 ) + a2 dl (x2i , x2i+1 ) + a3 dl (x2i+1 , x2i+2 ) + a4 dl (x2i+1 , x2i+2 )) < (a1 + a2 )dl (x2i , x2i+1 ) + (a3 + a4 )dl (x2i+1 , x2i+2 ),

a1 + a2 dl (x2i+1 , x2i+2 ) < dl (x2i , x2i+1 ) = dl (x2i , x2i+1 ). 1 − a3 − a4 This implies that F (dl (x2i+1 , x2i+2 )) ≤ F (dl (x2i , x2i+1 )) − τ. Following similar arguments as given in Theorem 12, we have {T S(xn )} → u, that is, lim dl (xn , u) = 0. (2.19) n→∞

Now, by Lemma 10, we have τ + F (dl (x2n+1 , T u)) ≤ τ + F (Hdl (Sx2n , T u)). Using inequality (2.18), we have τ + F (dl (x2n+1 , T u)) ≤ F a1 dl (x2n , u) + a2 dl (x2n , Sx2n ) + a3 dl (u, T u)

d2l (x2n , Sx2n ).dl (u, T u) + a4 1 + d2l (x2n , u) ≤ F a1 dl (x2n , u) + a2 dl (x2n , x2n+1 ) + a3 dl (u, T u)

d2 (x2n , x2n+1 ).dl (u, T u) + a4 l . 1 + d2l (x2n , u)

54

Page 10 of 16

T. Rasham et al.

Since F is strictly increasing, we have dl (x2n+1 , T u) < a1 dl (x2n , u) + a2 dl (x2n , x2n+1 ) + a3 dl (u, T u) d2 (x2n , x2n+1 ).dl (u, T u) . +a4 l 1 + d2l (x2n , u) Taking limit n → ∞, and by using (2.19), we get dl (u, T u) < a3 dl (u, T u), which is a contradiction; hence, dl (u, T u) = 0 or u ∈ T u. Similarly, using (2.18), (2.19), Lemma 10 and the inequality τ + F (dl (x2n+2 , Su)) ≤ τ + F (Hdl (T x2n+1 , Su)) we can show that dl (u, Su) = 0 or u ∈ Su. Hence, S and T have a common fixed point u in (X, dl ). Now, dl (u, u) ≤ dl (u, T u) + dl (T u, u) ≤ 0. This implies that dl (u, u) = 0.

If we take S = T in Theorem 16, then we have the following result. Corollary 17. Let (X, dl ) be a complete dislocated metric space and S : X → P (X) be the multivalued mappings. Assume that if F ∈ F and τ ∈ R+ such that τ + F (Hdl (Sx, Sy))

d2 (x, Sx).dl (y, Sy) ≤ F a1 dl (x, y) + a2 dl (x, Sx) + a3 dl (y, Sy) + a4 l 1 + d2l (x, y) (2.20) for all x, y ∈ {SS(xn )}, with x = y where a1 , a2 , a3 , a4 > 0 , a1 +a2 +a3 +a4 = 1 and a3 + a4 = 1. Then {T S(xn )} → u ∈ X. Moreover, if (2.20) also holds for u, then S has a fixed point u in X and dl (u, u) = 0. If we take a2 = 0 in Theorem 16, then we have the following result. Corollary 18. Let (X, dl ) be a complete dislocated metric space and S, T : X → P (X) be the multivalued mappings. Assume that if F ∈ F and τ ∈ R+ such that

d2l (x, Sx).dl (y, T y) τ + F (Hdl (Sx, T y)) ≤ F a1 dl (x, y) + a3 dl (y, T y) + a4 1 + d2l (x, y) (2.21) for all x, y ∈ {T S(xn )}, with x = y where a1 , a3 , a4 > 0 , a1 + a3 + a4 = 1 and a3 + a4 = 1. Then, {T S(xn )} → u ∈ X. Moreover, if (2.21) also holds for u, then S and T have a common fixed point u in X and dl (u, u) = 0. If we take a3 = 0 in Theorem 16, then we have the following result.


Page 11 of 16

45

Corollary 19. Let (X, dl ) be a complete dislocated metric space and S, T : X → P (X) be the multivalued mappings. Assume that if F ∈ F and τ ∈ R+ such that

d2l (x, Sx).dl (y, T y) τ + F (Hdl (Sx, T y)) ≤ F a1 dl (x, y) + a2 dl (x, Sx) + a4 1 + d2l (x, y) (2.22) for all x, y ∈ {T S(xn )}, with x = y where a1 , a2 , a4 > 0 , a1 + a2 + a4 = 1 and a4 = 1. Then, {T S(xn )} → u ∈ X. Moreover, if (2.22) also holds for u, then S and T have a common fixed point u in X and dl (u, u) = 0. If we take a4 = 0 in Theorem 16, then we have the following result. Corollary 20. Let (X, dl ) be a complete dislocated metric space and S, T : X → P (X) be the multivalued mappings. Assume that if F ∈ F and τ ∈ R+ such that τ + F (Hdl (Sx, T y)) ≤ F (a1 dl (x, y) + a2 dl (x, Sx) + a3 dl (y, T y))

(2.23)

for all x, y ∈ {T S(xn )}, with x = y where a1 , a2 , a3 > 0 , a1 + a2 + a3 = 1 and a3 = 1. Then {T S(xn )} → u ∈ X. Moreover, if (2.23) also holds for u, then S and T have a common fixed point u in X and dl (u, u) = 0. If we take a1 = a2 = a3 = 0 in Theorem 16, then we have the following result. Corollary 21. Let (X, dl ) be a complete dislocated metric space and S, T : X → P (X) be the multivalued mappings. Assume that if F ∈ F and τ ∈ R+ such that 2

dl (x, Sx).dl (y, T y) τ + F (Hdl (Sx, T y)) ≤ F (2.24) 1 + d2l (x, y) for all x, y ∈ {T S(xn )}, with x = y. Then, {T S(xn )} → u ∈ X. Moreover, if (2.24) also holds for u, then S and T have a common fixed point u in X and dl (u, u) = 0. Remark 22. We can obtain the partial metric and metric version of all theorems proved above as the corollaries, which are still new results in the literature.

3. Application to systems of integral equations Fixed point theorems for operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [4,18,24,28] and references therein). In this section, we apply our result to the existence of a solution of system of integral equations. We begin this section with the following definition. Definition 23. Let (X, dl ) be a complete dislocated metric space. The mappings S, T : X → X are called as a pair of new Ciric-type rational F contraction, if for all x, y ∈ X, we have τ + F (dl (Sx, T y)) ≤ F (Dl (x, y)),

(3.1)

54

Page 12 of 16

T. Rasham et al.

where F ∈ F and τ > 0, and dl (x, Sx) .dl (y, T y) , dl (x, Sx), dl (y, T y) . Dl (x, y) = max dl (x, y), 1 + dl (x, y)

(3.2)

Theorem 24. Let (X, dl ) be a complete dislocated metric space and (S, T ) be a pair of new Ciric-type rational F -contractions. Then, S and T have a common fixed point u in X and dl (u, u) = 0.

Proof. The proof is similar to that of Theorem 12 and so omitted. We consider the following form of Volterra-type integral equations. t u(t) =

K1 (t, s, u(s))ds,

(3.3)

K2 (t, s, v(s))ds

(3.4)

0

t v(t) = 0

for all t ∈ [0, 1]. We find the solution of (3.3) and (3.4). Let X = C([0, 1], R+ ) be the set of all continuous functions on [0, 1], endowed with the complete dislocated quasi metric. For u ∈ C([0, 1], R+ ), define supremum norm as: u τ = sup {|u(t)| e−τ t }, where τ > 0 is taken arbitrary. Then, define t∈[0,1]

dτ (u, v) = sup {|u(t) + v(t)| e−τ t } = u + v τ t∈[0,1]

for all u, v ∈ C([0, 1], R+ ); with these settings, (C([0, 1], R+ ), dτ ) becomes a complete dislocated metric space. Now, we prove the following theorem to ensure the existence of common solution of integral equations. Theorem 25. Assume the following conditions are satisfied: (i) K1 , K2 : [0, 1] × [0, 1] × C([0, 1], R+ ) → R; (ii) define t K1 (t, s, u(s))ds,

Su(t) = 0

t T v(t) =

K2 (t, s, v(s))ds. 0

Suppose there exist τ > 0, such that |K1 (t, s, u) + K2 (t, s, v)| ≤

(τ

τ M (u, v) M (u, v) τ + 1)2

for all t, s ∈ [0, 1] and u, v ∈ C([0, 1], R), where |u(t) + Su(t)| |v(t) + T v(t)| , M (u, v) = max |u(t) + v(t)| , 1 + |u(t) + v(t)|


Page 13 of 16

45

× |u(t) + Su(t)| , |v(t) + T v(t)|

.

Then, integral Eqs. (3.3) and (3.4) have a unique common solution. Proof. By assumption (ii), t |Su(t) + T v(t)| =

|K1 (t, s, u(s) + K2 (t, s, v(s)))| ds, 0

t ≤ 0

(τ

τ M (u, v) τ + 1)2

t ≤ 0

≤ ≤

(τ (τ

(τ

τ M (u, v) τ + 1)2

([M (u, v)]e−τ s )eτ s ds,

M (u, v) τ eτ s ds,

t

τ M (u, v) τ M (u, v) τ + 1)2 M (u, v) τ M (u, v) τ + 1)2

eτ s ds, 0

eτ t .

This implies that |Su(t) + T v(t)| e−τ t ≤ Su(t) + T v(t) τ ≤

(τ (τ

M (u, v) τ M (u, v) τ + 1)2 M (u, v) τ M (u, v) τ + 1)2

, ,

M (u, v) τ + 1 1 ≤ , M (u, v) τ Su(t) + T v(t) τ 1 1 ≤ , τ+ M (u, v) τ Su(t) + T v(t) τ

τ

which further implies that τ−

1 Su(t) + T v(t) τ

≤

−1 M (u, v) τ

.

−1 So, all the conditions of Theorem 24 are satisfied for F (x) = √ ; x > 0 and x dτ (u, v) = u + v τ . Hence, integral equations given in (3.3) and (3.4) have a unique common solution.

Compliance with ethical standards Conflicts of interest The authors declare that they have no competing interests.

54

Page 14 of 16

T. Rasham et al.

References [1] Abbas, M., Ali, B., Romaguera, S.: Fixed and periodic points of generalized contractions in metric spaces. Fixed Point Theory Appl. 2013, 243 (2013) ¨ Altun, I.: A Fixed Point Theorem for Multivalued Mappings with [2] Acar, O., delta-Distance. Abstr. Appl. Anal. 2014, Article ID 497092 (2014). https://doi. org/10.1155/2014/497092 ¨ Durmaz, G., Minak, G.: Generalized multivalued F −contractions on [3] Acar, O., complete metric spaces. Bull. Iran. Math. Soc. 40, 1469–1478 (2014) [4] Agarwal, R.P., Hussain, N., Taoudi, M.-A.: Fixed Point Theorems in Ordered Banach Spaces and Applications to Nonlinear Integral Equations. Abstr. Appl. Anal. 2012, Article ID 245872 (2012). https://doi.org/10.1155/2012/245872 [5] Ahmad, J., Al-Rawashdeh, A., Azam, A.: Some new fixed point theorems for generalized contractions in complete metric spaces. Fixed Point Theory Appl. 2015, 80 (2015) [6] Arshad, M., Khan, S.U., Ahmad, J.: Fixed point results for F -contractions involving some new rational expressions. JP J. Fixed Point Theory Appl. 11(1), 79–97 (2016) [7] Azam, A., Arshad, M.: Fixed points of a sequence of locally contractive multivalued maps. Comp. Math. Appl. 57, 96–100 (2009) [8] Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux equations itegrales. Fund. Math. 3, 133–181 (1922) ´ c, L.B.: A generalization of Banach’s contraction principle. Proc. Am. [9] Ciri´ Math. Soc. 45, 267–273 (1974) [10] Dugundji, J., Granas, A.: Fixed Point Theory, vol. I. Monografite Mat, Warszawa (1982) [11] Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory, vol. 28. Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics (1990) [12] Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83. Marcel Dekker, New York (1984) [13] Hitzler, P., Seda, A.K.: Dislocated Topologies. J. Elect. Eng. 51(12/s), 3–7 (2000) [14] Hussain, N., Abdul, L., Iram, I.: Fixed point results for generalized Fcontractions in modular metric and fuzzy metric spaces. Fixed Point Theory Appl. 2015, 158 (2015) [15] Hussain, N., Ahmad, J., Ciric, L., Azam, A.: Coincidence point theorems for generalized contractions with application to integral equations, Fixed Point Theory Appl. 2015:78 (2015). https://doi.org/10.1186/s13663-015-0331-4 [16] Hussain, N., Ahmad, J., Azam, A.: On Suzuki-Wardowski type fixed point theorems. J. Nonlinear Sci. Appl. 8, 1095–1111 (2015) [17] Hussain, N., Salimi, P.: Suzuki-Wardowski type fixed point theorems for αGF -contractions. Taiwan. J. Math. 18(6), 1879–1895 (2014) [18] Hussain, N., Khan, A.R., Agarwal, Ravi P.: Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces. J. Nonlinear Convex Anal. 11(3), 475–489 (2010)


Page 15 of 16

45

[19] Hussain, A., Arshad, M., Khan, S.U.: τ −Generalization of fixed point results for F −contraction, Bangmod Int. J. Math. Comp. Sci. 1(1), 127–137 (2015) [20] Hussain, A., Arshad, M., Nazam, M., Khan, S.U.: New type of results involving closed ball with graphic contraction. J. Inequal. Spec. Funct. 7(4), 36–48 (2016) [21] Khan, S.U., Arshad, M., Hussain, A., Nazam, M.: Two new types of fixed point theorems for F -contraction. J. Adv. Stud. In. Topology 7(4), 251–260 (2016) [22] Matthews, S.G.: Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci. 728, 183–197 (1994) [23] Nadler, S.B.: Multivalued contraction mappings. Pac. J. Math. 30, 475–488 (1969) [24] Nieto, J.J., Rodr´ıguez-L´ opez, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223– 239 (2005) [25] Piri, H., Kumam, P.: Some fixed point theorems concerning F -contraction in complete metric spaces. Fixed Point Theory Appl. 2014, 210 (2014) [26] Reich, S.: Some remarks concerninig contraction mappings. Cand. Math Bull. 14, 121–124 (1971) [27] Reich, S.: Fixed points of contractive functions. Boll. Un. Math. Ital. 5, 26–42 (1972) [28] Salimi, P., Latif, A., Hussain, N.: Modified α-ψ-contractive mappings with applications. Fixed Point Theory Appl. 2013, 151 (2013) [29] Secelean, N.A.: Iterated function systems consisting of F -contractions. Fixed Point Theory Appl. (2013). https://doi.org/10.1186/1687-1812-2013-277 (Article ID 277) [30] Sgroi, M., Vetro, C.: Multi-valued F −contractions and the solution of certain functional and integral equations. Filomat 27(7), 1259–1268 (2013) [31] Shoaib, A., Hussain, A., Arshad, M., Azam, A.: Fixed point results for α∗ ψ-Ciric type multivalued mappings on an intersection of a closed ball and a sequence with graph. J. Math. Anal. 7(3), 41–50 (2016) [32] Shoaib, A.: Fixed point results for α∗ -ψ -multivalued mappings. Bull. Math. Anal. Appl. 8(4), 43–55 (2016) [33] Wardowski, D.: Fixed point theory of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012). https:// doi.org/10.1186/1687-1812-2012-94

Tahair Rasham and Muhammad Arshad Department of Mathematics International Islamic University H-10 Islamabad 44000 Pakistan e-mail: tahir [email protected]; [email protected]

54

Page 16 of 16

Abdullah Shoaib Department of Mathematics and Statistics Riphah International University Islamabad 44000 Pakistan e-mail: [email protected] Nawab Hussain Department of Mathematics King Abdulaziz University P.O. Box 80203Jeddah 21589 Saudi Arabia e-mail: [email protected] Sami Ullah Khan Department of Mathematics Gomal University Dera Ismail Khan Pakistan e-mail: [email protected]

T. Rasham et al.