Fixed Point of Orthogonal F-Suzuki Contraction Mapping on O-Complete b-Metric Spaces with Applications

Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan Department of Mathematics, Sri Sankara Arts and Science College (Autonomous), Affiliated to Madras University, Enathur, Kanchipuram, Tamil Nadu 631 561, India Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kattankulathur 603203, Kanchipuram, Chennai, Tamil Nadu, India


Introduction and Preliminaries
Banach contraction principle is one of the famous and useful results in mathematics. In last 100 years, it is extended in many different directions. The substitution of the metric space by other generalized metric spaces is one normal way to strengthen the Banach contraction principle. The concept of a b-metric space was introduced by Bakhtin [1] and Czerwik [2]. They also established the fixed point result in the setting of b-metric spaces which is a generalization of the Banach contraction principle. In 2015, Alsulami et al. [3] introduced the concepts of generalized F-Suzuki type contraction mappings and proved the fixed point theorems on complete b-metric spaces. On the other hand, Gordji et al. [4] introduced the new concept of an orthogonality in metric spaces and proved the fixed point result for contraction mappings in metric spaces endowed with this new type of orthogonality. Furthermore, they gave the application of this results for the existence and uniqueness of the solution of a first-order ordinary differential equation, while the Banach contraction mapping principle cannot be applied in this situation. This new concept of an orthogonal set has many applications, and there are also many types of the orthogonality. Afterward Eshaghi Gordji and Habibi [5] proved fixed point in generalized orthogonal metric spaces. Recently, Sawangsup et al. [6] introduced the new concept of an orthogonal F-contraction mappings and proved the fixed point theorems on orthogonal-complete metric spaces. Subsequently, many other researchers [7][8][9][10][11] studied the orthogonal contractive type mappings and obtained significant results. This paper is in continuation of these studies; first, we introduced the new concepts of generalized orthogonal F-Suzuki contraction mappings on an orthogonal bmetric space and then prove the fixed point theorems on orthogonal b-complete metric space with examples and applications to differential equations. Over results generalize/extend several results from the existing literature.
In this paper, we denote by ℕ, ℝ + , and W the set of positive integers, the set of positive real numbers, and the nonempty set, respectively. Next, we state the concept of a control function which was introduced by Wardowski [12].
Let I denote the family of all functions F : ℝ + ⟶ ℝ satisfying the following properties: (F 1 ) F is strictly increasing (F 2 ) for each sequence fα n g of positive numbers, we have Bakhtin [1] and Czerwik [2] gave the concept of a bmetric space as follows.
Definition 1 (see [2]). Let W be a nonempty set and s ≥ 1. Suppose that the mapping d : W × W ⟶ ℝ + satisfies the following conditions for all u, v, w ∈ W: Then, ðW, dÞ is called a b-metric space with the coefficient s.
Example 2 (see [2]). Define a mapping d : The idea of generalized F-Suzuki type contraction and F-Suzuki type contraction in complete b-metric spaces is due to Alsulami et al. [3]. Gordji et al. [4] introduced the notion of an orthogonal set (or O-set).
Definition 3 (see [4]). Let W ≠ ϕ and ⊥⊆ W × W be a binary relation. If ⊥ satisfies the following condition: then it is called an orthogonal set (briefly O-set). We denote this O-set by ðW, ⊥Þ.
Example 4 (see [4]). Let W be the set all people in the world. Define the binary relation ⊥ on W by v⊥u if v can give blood to u. According to the Table 1, if u 0 is a person such that his (her) blood type is O-, then we have u 0 ⊥u for all u ∈ W. This means that ðW, ⊥Þ is an O-set. In this O-set, u 0 (in Definition 3) is not unique. Note that, in this example, u 0 may be a person with blood type AB + . In this case, we have u⊥u 0 for all u ∈ W. Next, we give some basic definition for subsequent use.
Definition 7 (see [4]). Let ðW,⊥,dÞ be an orthogonal metric space. Then, a mapping G : W ⟶ W is said to be orthogonally continuous (or ⊥-continuous) in u ∈ W if for each O-sequence fu n g in W with u n ⟶ u as n ⟶ ∞, we have Gðu n Þ ⟶ GðuÞ as n ⟶ ∞. Also, G is said to be ⊥-con- Remark 8 (see [4]). Every continuous mapping is ⊥-continuous and the converse is not true.
Definition 9 (see [4]). Let ðW,⊥,dÞ be an orthogonal metric space. Then, W is said to be orthogonally complete (briefly, O-complete) if every Cauchy O-sequence is convergent.
Remark 10 (see [4]). Every complete metric space is Ocomplete and the converse is not true.
Definition 11 (see [4]). Let ðW, ⊥Þ be an O-set. A mapping G : W ⟶ W is said to be ⊥-preserving if Gu⊥Gv when ever u⊥v. An orthogonal sequence fu n g is said to satisfy the condition ðTÞ if ∀n, m ∈ ℕ, u n ⊥u m ð Þ or ∀n, m ∈ ℕ, u m ⊥u n ð Þ : Throughout the paper, we assume that an orthogonal sequence fu n g satisfies the condition ðTÞ:

Main Results
In this section, inspired by Alsulami et al. [3], the notion of a generalized F-Suzuki type contraction mapping, we introduce a new generalized F-Suzuki contraction mapping and obtain fixed point results for this contraction mapping on an orthogonal b-metric space.
Definition 12. Let ðW,⊥,dÞ be an orthogonal b -metric space with constant s ≥ 1 . A map G : W ⟶ W is said to be a generalized orthogonal F -Suzuki contraction mapping (briefly, generalized G ⊥ -Suzuki contraction) on ðW,⊥,dÞ if there are F ∈ I and τ > 0 such that the following condition holds: where c ∈ ½0, 1Þ and a, b ∈ ½0, 1 are real numbers with a + b + c = 1.
In the above Definition 12, if we take a = 1 and b = c = 0; we obtain the following Definition.
Definition 13. Let ðW,⊥,dÞ be an orthogonal b -metric space with constant s ≥ 1 . A map G : W ⟶ W is said to be an orthogonal F -Suzuki contraction mapping (briefly, G ⊥ -Suzuki contraction) on ðW,⊥,dÞ if there are F ∈ I and τ > 0 such that the following condition holds: Theorem 14. Let ðW,⊥,dÞ be an O -complete orthogonal bmetric space with constant s ≥ 1 and an orthogonal element u 0 and a mapping G : W ⟶ W . Suppose that there exist F ∈ I and τ > 0 such that the following conditions hold: Then, G has a fixed point t ∈ W, and for every u ∈ W, the sequence fG n ug converges to t.
It follows that u 0 ⊥Gu 0 or Gu 0 ⊥u 0 . Let for all n ∈ ℕ ∪ f0g. If u n = u n+1 for any n ∈ ℕ ∪ f0g, then it is clear that u n is a fixed point of G. Assume that u n ≠ u n+1 for all n ∈ ℕ ∪ f0g. Thus, we have dðGu n , Gu n+1 Þ > 0 for all for all n ∈ ℕ ∪ f0g. This implies that fu n g is an O-sequence.
Since G is generalized G ⊥ -Suzuki contraction mapping, we have Thus, by hypothesis of theorem, we have which is equivalent to Since a + b + c = 1, the inequality (12) turns into From ðF 1 Þ, we conclude that Therefore, fdðu n , Gu n Þg ∞ n=1 is a decreasing sequence of real numbers which is bounded below. Therefore, fdðu n , Gu n Þg ∞ n=1 converges and We will show that δ = 0. On the contrary, suppose that δ > 0. In other words, for every ε > 0 there exits k ∈ ℕ, such that From ðF 1 Þ, we find that On the other hand, we have due to (10). Since G is generalized G ⊥ -Suzuki contraction, we derive which yields that taking the fact that a + b + c = 1 into account, we obtain Analogously, again by (10), we have ð1/2sÞdðGu k , G 2 u k ÞÞ < dðGu k , G 2 u k Þ. Owing to the fact that G is generalized G ⊥ -Suzuki contraction, we conclude that

Journal of Function Spaces
It implies that since a + b + c = 1. Furthermore, by combining (21) and (23), we get Iteratively, we obtain that By letting n ⟶ ∞, we find that Consequently, from ðF 2 Þ, we derive that lim n→∞ dðG n u k , and from (8), we get This is a contradiction with the definition of δ. Hence, we have It what follows, we will prove that Suppose, on the contrary, that there exists ε > 0 and sequences fiðnÞg ∞ n=1 and fjðnÞg ∞ n=1 of natural numbers such that From the triangle inequality, we have Owing to (29), there exists and N 2 ∈ ℕ such that Taking (33) into account, (32) yields that So from ðF 2 Þ, we obtain On the other hand, we can easily get that Next, we claim that Arguing by contradiction, there exists k ≥ ℕ such that It follows from (31), (36), and (38) that This contradiction establishes the relation (37). Since G is ⊥-preserving, we have Regarding (29) and ðF 2 Þ, we obtain that From ðF 2 Þ, we get that This is a contradiction with relations in (31). Hence, lim k,n→∞ dðu n , u k Þ = 0; that is fu n g ∞ n=1 is a Cauchy sequence in W. On the account of completeness of ðW, dÞ, there exists t ∈ W such that We claim that, for every n ∈ ℕ, We will prove the claim above by the method of reductio ad absurdum. Suppose, on the contrary, that there exists k ∈ ℕ such that From (14) and ðF 1 Þ, we have It follows from (48) and (49) that This is a contradiction. Hence, (46) holds. Since G is generalized G ⊥ -Suzuki contraction, (46) yields that, for every n ∈ ℕ, either holds. On account of ðF 2 Þ, the limits in (29) and (45) imply that By letting n ⟶ ∞ in the inequality above together with the limits in (45) and (55), we conclude that dðt, GtÞ = 0. Thus, t is a fixed point of G; that is t = Gt. Let us analyze the second case (51). Regarding (12), we have As it was discussed above, from (29), (45), and ðF 2 Þ, we conclude that

Journal of Function Spaces
From ðF 2 Þ equivalently, we get Again by the triangle inequality together with (8), we find that By letting n ⟶ ∞ in the inequality above together with the limits in (45) and (59), we obtain that dðt, GtÞ = 0. Thus, t is a fixed point of G. Let h, g ∈ W be two fixed points of G and suppose that G n h = h ≠ g = G n g for all n ∈ ℕ. By choice of u 0 , we obtain u 0 ⊥h and u 0 ⊥g ð Þ or h⊥u 0 and g⊥u 0 ð Þ : Since G is ⊥-preserving, we have G n u 0 ⊥G n h and G n u 0 ⊥G n g ð Þ or G n h⊥G n u 0 and G n g⊥G for all n ∈ ℕ. Now, As n ⟶ ∞, we obtain dðh, gÞ ≤ 0. Thus, h = g. Hence, G has a unique fixed point in W.
Theorem 19. Let ðW,⊥,dÞ be an O -complete orthogonal bmetric space with constant s ≥ 1 and an orthogonal element 6 Journal of Function Spaces u 0 and a mapping G : W ⟶ W . Suppose that there exist F ∈ I and τ > 0 such that the following conditions hold: Then, G has a unique fixed point t ∈ W, and for every u ∈ W, the sequence fG n ug converges to t.
Proof. By taking a = 1 and b = c = 0 in Theorem 14, the proof is complete.
Corollary 24. Let G be a self mapping on an O -complete metric space ðW,⊥,dÞ . Assume that there exist F ∈ I and τ > 0 such that the following condition hold: where c ∈ ½0, 1Þ and a, b ∈ ½0, 1 are real numbers with a + b + c = 1.
Then, G has a unique fixed point t ∈ W.
Proof. Since any metric space is a b-metric space with constant s = 1, so from Theorem 14 the proof is complete.
Corollary 25. Let G be a self mapping on an O -complete metric space ðW,⊥,dÞ . Assume that there exist F ∈ I and τ > 0 such that the following condition hold: Then, G has a unique fixed point t ∈ W.
Proof. Since any metric space is a b-metric space with constant s = 1, so by taking a = 1 and b = c = 0 from Theorem 14, the proof is complete.
Proof. By taking a = 1 and b = c = 0 in Theorem 14, the proof is complete. Next, we give the fixed point theorem for orthogonal continuous mapping on an O-complete b-metric space ðW,⊥,dÞ.

(iii) G is ⊥-continuous
Then, G has a unique fixed point t ∈ W.
If u n = u n+1 for any n ∈ ℕ ∪ f0g, then it is clear that u n is a fixed point of G. Assume that u n ≠ u n+1 for all n ∈ ℕ ∪ f0g. Therefore, we have dðGu n , Gu n+1 Þ > 0 for all n ∈ ℕ ∪ f0g. As G is ⊥-preserving, we have for all n ∈ ℕ ∪ f0g. It implies that fu n g is an O-sequence. By condition (ii), we have Thus, As a + b + c = 1, we get From ðF 1 Þ, we conclude that Thus, fdðu n , Gu n Þg ∞ n=1 is a decreasing sequence of real numbers which is bounded below. Therefore, fdðu n , Gu n Þg ∞ n=1 converges and lim n→∞ d u n , Gu n ð Þ= δ = inf d u n , Gu n ð Þ: ∀n ∈ ℕ f g : Next, we show that δ = 0. Arguing by contradiction, we get δ > 0. For every ε > 0, there exits k ∈ ℕ, such that From ðF 1 Þ, we get On the other hand (78), we have So we obtain and thus Since a + b + c = 1, we obtain Also from (78), we have 0 < dðGu k , Gu k+1 ÞÞ < dðGu k , G 2 u k Þ, and thus, by assumption of theorem, we have and therefore, Since a + b + c = 1, we get Journal of Function Spaces Now by using (85) and continuing similar method as used in (89) and (92), we obtain Letting n ⟶ ∞, we find that