Nonunique Fixed Point Results via Kannan F-Contraction on Quasi-Partial b-Metric Space

This paper is aimed at acquainting with a new Kannan F-expanding type mapping by the approach of Wardowski in the complete metric space. We establish some fixed point results for Kannan F-expanding type mapping and F-contractive type mappings which satisfy F-contraction conditions. Additionally, some new results are given which generalize several results present in the literature. Moreover, some applications and examples are provided to show the practicality of our results.


Introduction and Preliminaries
In 1922, Banach [1] commenced one of the most essential and notable results called the Banach contraction principle, i.e., let P be a self-mapping on a nonempty set X and d be a complete metric, if there exists a constant k ∈ ½0, 1Þ such that for all u, v ∈ X. Then, it has a unique fixed point in X. Due to its significance, in 1968, Kannan [2] introduced a different intuition of the Banach contraction principle which removes the condition of continuity, i.e., for all u, v ∈ ½0, 1/2, there exists a constant ρ ∈ ½0, 1Þ such that On the other hand, the notion of metric space has been generalized in several directions, and the abovementioned contraction principle has been enhanced in the new settings by considering the concept of convergence of functions. In 1989, Bakhtin [3] introduced the notion of b-metric space which was revaluated by Czerwik [4] in 1993.

Definition 1.
A b-metric space on a nonempty set X is a function d : X × X ⟶ ½0,∞Þ such that for all u, v, w ∈ X and for some real number s ≥ 1, it satisfies the following: (M1) If dðu, vÞ = 0, then u = v (M2) dðu, vÞ = dðv, uÞ (M3) dðu, wÞ ≤ s½dðu, vÞ + dðv, wÞ Then, the pair ðX, d, sÞ is called the b-metric space. Motivated by this, many researchers [5][6][7][8] generalized the concept of metric spaces and established on the existence of fixed points in the setting of b-metric space keeping in mind that, unlike standard metric, b-metric is not necessarily continuous due to the modified triangle inequality. In general, a b-metric does not induce a topology on X.
Partial metric space is one of the attempts to generalize the notion of the metric space. In 1994, Matthews [9] introduced the notion of a partial metric space in which dðu, uÞ are no longer necessarily zero.

Definition 2.
A partial metric on a nonempty set X is a function p : X × X ⟶ ½0,∞Þ such that for all u, v, w ∈ X, it satisfies the following: Then, the pair ðX, pÞ is called the partial metric space.
Definition 3. Let ðX, pÞ be a partial metric space. Then, several topological concepts for partial metric space can be easily defined as follows: (1) A sequence fu n g in the partial metric space ðX, pÞ converges to the limit u if pðu, uÞ = lim n⟶∞ pðu, u n Þ (2) It is said to be a Cauchy sequence if lim n⟶∞ pðu n , u m Þ exists and is finite (3) A partial metric space ðX, pÞ is called complete if every Cauchy sequence fu n g in X converges with respect to τ p , to a point u ∈ X such that pðu, uÞ = lim n⟶∞ pðu n , u m Þ For more details, see, for example, [10][11][12], and the related references therein. The following definition gives room for the lack of symmetry in the spaces under study. In 2013, Karapinar et al. [13] introduced quasi-partial metric space that satisfies the same axioms as metric spaces.
Definition 5. A quasi-partial b-metric on a nonempty set X is a function qp b : X × X ⟶ ½0,∞Þ such that for some real number ρ ≥ 1, it satisfies the following: for all u, v, w ∈ X: The infimum over all reals ρ ≥ 1 satisfying ðQPb 4 Þ is called the coefficient of ðX, qp b Þ and represented by RðX, qp b Þ. Lemma 6. Let ðX, qp b Þ be a quasi-partial b-metric space. Then, the following hold: (iii) the quasi-partial b-metric space ðX, qp b Þ is said to be complete if every Cauchy sequence fu n g ⊂ X converges with respect to τqp b to a point u ∈ X such that (iv) a mapping f : X ⟶ X is said to be continuous at u 0 ∈ X, if for every ε > 0, there exists δ > 0 such that f ðBðu 0 , δÞÞ ⊂ Bðf ðu 0 Þ, εÞ The extensive application of the Banach contraction principle has motivated many researchers to study the possibility of its generalization. A great number of generalizations of this famous result have appeared in the literature. In 2012, Wardowski [16] established a new notion of F-contraction and proved the fixed point theorem which generalized the Banach contraction principle.
Definition 8 (see [16]). Let ðX, dÞ be a metric space, and there exists a mapping F : ð0,∞Þ ⟶ ℝ which satisfies the following condition: (F 1 ) F is strictly increasing (F 2 ) For any sequence fx n g n∈N , lim n⟶∞ x n = 0 if and for all u, v ∈ X Theorem 9 (see [16]). Let ðX, dÞ be a complete metric space and T : X ⟶ X an F-contraction. Then, T has a unique fixed point x * ∈ X, and for every x ∈ X, the sequence fT n xg n∈N converges to x * .

Journal of Function Spaces
In 2012, Samet et al. [17] established the class of α -admissible mappings as follows: Definition 10 (see [17]). Let α : X × X ⟶ ½0,∞Þ be given mapping where X is a nonempty set. A self-mapping T is called α-admissible if for all x, y ∈ X, we have Motivated by this, Aydi et al. [18] extended the notion of F-contraction and prove the following result.
Theorem 11 (see [18]). Let ðX, dÞ be a metric space. A selfmapping T : X ⟶ X is said to be a modified F-contraction via α-admissible mappings. Suppose that Then, T has a fixed point. In 2015, Kumam et al. [19] generalized the contraction condition by adding four new values dðT 2 x, xÞ, dðT 2 x, TxÞ, dðT 2 x, yÞ, dðT 2 x, TyÞ and introduced F-Suzuki contraction mappings in complete metric space. The Suzuki-type generalization can be said to have many applications, as in computer science, game theory, biosciences, and in other areas of mathematical sciences such as in dynamic programming, integral equations, and data dependence. Recently, Wardowski [20] proposed the replacement of the positive constant δ in equation (6) by a function ϕ and relaxed the conditions on F.
Consider a function F B : ℝ + ⟶ ℝ by F B ðuÞ = ln u. Note that with F = F B , the F-contraction reduces to a Banach contraction. Therefore, the Banach contractions are a particular case of F-contractions. Meanwhile, there exist F-contractions which are not Banach contractions.
ðF 4 Þ For some δ > 0 and any sequence fx n g, we have for all n ∈ N, s ∈ ℝ In 2017, Gornicki [36] established F-expanding type mappings.
The concept of F-expanding type mappings was redefined as Kannan F-expanding type mappings by Goswami et al. [37].
Definition 14 (see [37]). A mapping P : and dðu, PuÞdðv, PvÞ = 0 implies for all u, v ∈ X. Following this direction, we have established a new type of mapping, i.e., Kannan F-expanding type mapping, and proved some fixed point results for F-contractive type mappings as well as Kannan F-expanding type mappings in the setting of quasi-partial b-metric space without using the continuity of mapping. Also, we attain the nonunique fixed point in quasi-partial b-metric space which lacks symmetry property.
The main motive behind this study is that today, this field of research has vast literature. The significance of the Kannan type mapping is that it characterizes completeness which the Banach contraction does not; also, it does not require continuous mapping. In this paper, some examples and applications for the solution of a certain integral equation and the existence of a bounded solution of the functional equation are also given to represent the practicality of the results obtained. The application shows the role of fixed point theorems in dynamic programming, which is used in computer programming and optimization.
The future aspect of this study is to prove the existence of a unique fixed point in Kannan F-expanding type mapping. Another field of research can be the existence of a common fixed point for the same. The notion of interpolative F 3 Journal of Function Spaces -contraction as well as interpolation for Kannan F-expanding type mapping can also be future studies concerning the present manuscript.

Fixed Point for F-Contractive Type Mappings
In this section, the existence of a fixed point for F-contractive type mappings in a quasi-partial b-metric space is obtained.
Definition 15. For a quasi-partial b-metric space ðX, qp b Þ, a mapping P : X ⟶ X is said to be an F-contractive type mapping if there exists δ > 0 such that, if qp b ðu, PuÞqp b ðv, PvÞ ≠ 0, then and if qp b ðu, PuÞqp b ðv, PvÞ = 0, then for all u, v, w ∈ X and ρ ≥ 1.
A self-mapping P on X is called an F-contraction if there exist τ ∈ ℝ + such that for all u, v ∈ X with qp b ðPu, PvÞ > 0.
Here, F satisfies (F1)-(F3) for any k ∈ ð0, 1Þ. Each mapping P : X ⟶ X satisfying Definition 16 is an F-contraction such that for all u, v ∈ X, Pu ≠ Pv.
It is clear that for u, v ∈ X such that Pu = Pv, the previous inequality also holds, and hence, P is a contraction as shown in Figure 1.
for all u, v ∈ X, Pu ≠ Pv. Hence, P is a contraction as shown in Figure 2.
Theorem 19. Let ðX, qp b Þ be a quasi-partial b-metric space and P : X ⟶ X be an F-contractive type mapping. Then, P has a unique fixed point u * ∈ X, and for every u 0 ∈ X, a sequence fP n x 0 g n∈ℕ is convergent to u * .
Proof. Let u 0 be an arbitrary and fixed point in X, and we assume a sequence fu n g n∈N ⊂ X such that u n+1 = Pu n , n = 0 , 1, ⋯. To prove P has a fixed point, we need to show that if u n 0 +1 = u n 0 , then Pu n 0 = u n 0 for all n 0 ∈ ℕ. Suppose that u n+1 ≠ u n for every n ∈ ℕ, then qp b ðu n+1 , u n Þ > 0, and using equation (6), we have which implies Using ðF 2 Þ, we get Also, using ðF 3 Þ, there exists k ∈ ð0, 1Þ such that Let us denote qp b ðu n+1 , u n Þ by α n . From inequality (18), the following holds Also, if there exists n 1 < n ∈ ℕ such that nα k n ≤ 1, we have To prove fu n g n∈ℕ is a Cauchy sequence, let us consider m, n ∈ ℕ such that m > n ≥ n 1 . From the definition of quasi-partial b-metric space and equation (24), we have Using the convergence of series, we get that fu n g n∈ℕ is a Cauchy sequence. Since X is complete, there exists u * ∈ X such that lim n⟶∞ u n = u * , and the continuity of P implies Hence, P has a unique fixed point.☐ Theorem 20. For a quasi-partial b-metric space ðX, qp b Þ, we say X is complete if for every closed subset Y of X, P : Y ⟶ Y is an F-contractive type mapping having a fixed point.
Proof. Suppose that there does not exist any Cauchy sequence in X which has a convergent subsequence and we have a sequence θ u n ð Þ = inf qp b u n , u m ð Þ: m > n f g > 0 ð27Þ for all n ∈ ℕ where θðu n Þ ≤ θðu m Þ for m ≥ n. Also, we con-sider a subsequence fu n k g such that for any a with 0 < a < 1 and for all i, j ≥ n k . Then, Y = fu n k : k ∈ ℕg is a closed subset of X. Define P : X ⟶ X by for all k ∈ ℕ, which implies P has no fixed point. Now, By definition, which implies for some δ > 0. Hence, it proves that P is an F-contractive type mapping on a closed subset of X which has no fixed point. Thus, this is a contradiction and X is complete.☐ Theorem 21. Let ðX, qp b Þ be a quasi-partial b-metric space and P : X × CðXÞ be a closed F-contraction. Then, P has a fixed point.
Proof. Let u 0 ∈ X be an arbitrary point of X, and we have u 1 ∈ Pu 0 . If u 1 = u 0 , then u 1 is a fixed point of P, and hence, the proof is completed. Now, assume that u 1 ≠ u 0 . Since P is a F-contraction, there exists u 2 ∈ Pu 1 such that where and u 2 ≠ u 1 . Also, there exists u 3 ∈ Pu 2 such that and u 3 ≠ u 2 . With the recurrence of the same process, we get where u ∈ ð0, 10Þ, v ∈ ð0, 10Þ and w ∈ ℝ + . 5 Journal of Function Spaces for all n ∈ ℕ. It implies Assume that qp b n = qp b ðx n , x n+1 Þ > 0 for all n ∈ ℕ ∪ f0g. By equation (37), we have for all n ∈ ℕ. Letting n ⟶ ∞, property ðF 2 Þ implies Let k ∈ ð0, 1Þ such that By equation (38), the following holds for all n ∈ ℕ. Letting n ⟶ ∞, we get This implies lim n⟶∞ n 1/k qp b n = 0 and ∑ +∞ n=1 qp b n is convergent. Hence, fu n g is Cauchy sequence. Since X is complete, there exists x ∈ X such that u n = x as n ⟶ +∞. Since P is closed, ðu n , u n+1 Þ ⟶ ðx, xÞ, we get x ∈ Px, and hence, x is the fixed point of P.☐ Corollary 22. Let ðX, qp b Þ be a quasi-partial b-metric space and P : X ⟶ CðXÞ be an upper semicontinuous F-contraction. Then, P has a fixed point. Example 23. Consider the quasi-partial b-metric space ðX, qp b Þ where X = f0, 2, 4,⋯g and qp b : X × X ⟶ ½0,∞Þ is given by which is also shown in Figure 3, and P : X ⟶ CðXÞ is defined by Now, we show that P satisfies Definition 16, where ρ = 2, τ = 2 and FðuÞ = log u + u for each u ∈ ℝ + . Let for all u, v ∈ X with v ∈ Pu, we have w = 0 ∈ Pv. Here, qp b ðv, wÞ > 0 iff x ≥ 4 and w > 0. If it is true, then This implies Hence, for all u, v ∈ X and qp b ðv, wÞ > 0. Then, by Theorem 21, P has a fixed point.

Fixed Point for Kannan F-Expanding Type Mapping
In this section, we prove the fixed point results for Kannan F -expanding type mappings in a quasi-partial b-metric space.
Definition 24. Let us consider a mapping P : X ⟶ X; it is said to be Kannan F-expanding type mapping if there exists Δ > 0 such that qp b ðu, PuÞqp b ðv, PvÞ ≠ 0 implies for all u, v, w ∈ X.
Lemma 25. Let ðX, qp b Þ be a quasi-partial b-metric space and P : X ⟶ X be surjective. Then, there exists a mapping P * : X ⟶ X such that P ∘ P * is the identity map on X.
Proof. For any point u ∈ X, let v u ∈ X be any point such that Pv u = u. Let P * u = v u for all u ∈ X. Then, ðP ∘ P * ÞðuÞ = Pð P * uÞ = Pv u = u for all u ∈ X.☐ Theorem 26. Let ðX, qp b Þ be a quasi-partial b-metric space and P : X ⟶ X be surjective and a Kannan F-expanding type mapping. Then, P has a unique fixed point γ ∈ X.
Proof. Assume that there exists a mapping P * : X ⟶ X such that P ∘ P * is the identity map on X. Let u, v be arbitrary points of X such that u ≠ v and x = P * u, y = P * v which also implies that x ≠ y. Applying equation (48) for qp b ðx, PxÞqp b ðy, PyÞ = 0. Since Px = PðP * ðuÞÞ = u and P y = PðP * ðvÞÞ = v, we get for qp b ðu, PuÞqp b ðv, PvÞ ≠ 0 and for qp b ðu, PuÞqp b ðv, PvÞ = 0, which implies P * is Kannan F -contractive type mapping. Also, we know that P * has a unique fixed point γ ∈ X, and for every u 0 ∈ X, the sequence fP * n u 0 g converges to γ. In particular, γ is also a fixed point of P since P * γ = γ implies that Finally, if γ 0 = Pγ 0 is another fixed point, then from equation (49), which is not possible, and hence, P has a unique fixed point.☐

Applications of F-Contraction
In this section, we discuss the applications of the results obtained to prove the existence of the solution of an integral equation and a functional equation.