Common Fixed-Point Theorems in the Partial b-Metric Spaces and an Application to the System of Boundary Value Problems

In this paper, we investigate the conditions for the existence of the common fixed points of generalized contractions in the partial b-metric spaces endowed with an arbitrary binary relation. We establish some unique common fixed-point theorems. The obtained results may generalize and improve earlier fixed-point results. We provide examples to illustrate our findings. As an application, we discuss the common solution to the system of boundary value problems.


Introduction, Preliminaries, and Motivations
The b-metric space was introduced by Czerwik [1]. It is obtained by modifying the triangle property of the metric space. Every metric is a b-metric, but the converse is not true. Almost all the fixed-point theorems in the metric spaces have been proved true in the b-metric spaces; for example, see [2][3][4][5][6][7][8][9][10] and references therein.
Matthews [11] introduced the notion of the partial metric space as a part of the study of denotational semantics of the dataflow network. In this space, the usual metric is replaced by a partial metric having a property that the selfdistance of any point of the space may not be zero. Every metric is a partial metric, but the converse is not true. Matthews [11] also initiated the fixed-point theory in the partial metric space. He proved the Banach contraction principle in this space to be applied in program verification. We can find so many fixed-point theorems in the metric spaces which have been proved in the partial metric spaces by many fixed-point theorists ( [12,26] and references therein).
Shukla [13] introduced the concept of partial b-metric by modifying the triangle property of the partial metric and investigated fixed points of Banach contraction and Kannan contraction in the partial b-metric spaces. Mustafa et al. [14] modified the triangle property of partial b-metric and established a convergence criterion and some working rules in partial b-metric spaces. Moreover, Mustafa et al. [14] investigated common fixed-point results for ðϕ, ψÞ-weakly contractive mappings. Dolicanin-Ðekic [15] obtained the fixed-point theorems for Ciric-type contractions in the partial b-metric spaces. Singh et al. [16] investigated some conditions to show the existence of the common fixed points of power graphic ðF, ψÞ-contractions defined on the partial b -metric space endowed with directed graphs. More results on F-contractions can be seen in [8,17,18].
Let X be a nonempty set, then the nonempty binary relation R is a subset of X 2 : The set X 2 itself is known as universal relation, and the empty set is known as an empty relation; both are trivial relations. If any two elements α, β ∈ X are related with respect to R, then we shall write ðα, βÞ ∈ R. We shall use the notation ½α, β ∈ R if either ðα, βÞ ∈ R or ðβ, αÞ ∈ R. R is reflexive if ðα, αÞ ∈ R, for all α ∈ X:R is symmetric if ðα, βÞ ∈ R implies ðβ, αÞ ∈ R, for all α, β ∈ X: R is antisymmetric if ðα, βÞ ∈ R and ðβ, αÞ ∈ R implies α = β, for all α, β ∈ X:R is transitive if ðα, βÞ ∈ R and ðβ, γÞ ∈ R implies ðα, γÞ ∈ R, for all α, β ∈ X. The inverse, transpose, or dual of binary relation R is denoted by R −1 and defined as follows: R −1 = fðα, βÞ ∈ X | ðβ, αÞ ∈ Rg. Let R s = R ∪ R −1 , then it is easy to prove that ðα, βÞ ∈ R s if and only if ½α, β ∈ R.
Definition 1 (see [19]). Let T be a self-mapping on a nonempty set X. A binary relation R on X is said to be T -closed if for all α, β ∈ X, Definition 2 (see [19]). Let R be a binary relation on X. A path in R from α to β is a sequence fα 0 , α 1 , α 2 , α 3 , ⋯, α n g ⊆ X such that The set of all paths from α to β in R is denoted by Γð α, β, RÞ. The path of length n involves n + 1 element of X: Definition 3 (see [19]). A metric space ðX, dÞ equipped with the binary relation R is called R-regular (or d-self-closed) if for each sequence fα n ginX, whenever ðα n , α n+1 Þ ∈ R and α n ⟶ d α, we have ðα n , αÞ ∈ R, for all n ∈ ℕ ∪ f0g.
Alam and Imdad [19] used nonempty binary relation on the nonempty set X to obtain the following relationtheoretic contraction principle.
Theorem 4 (see [19]). Let ðX, dÞ be a complete metric space and R be a binary relation on X. Let T be a self-mapping defined on ðX, dÞ satisfying the following conditions: (a) There exists α 0 ∈ X such that ðα 0 , Tðα 0 ÞÞ ∈ R and R is T-closed (b) Either T is continuous or ðX, dÞ is R − regular (c) There exists k ∈ ½0, 1Þ such that dðTðαÞ, TðβÞÞÞ ≤ kd ðα, βÞ for α, β ∈ X with ðα, βÞ ∈ R Then T admits a fixed point in X: Moreover, if Γðα, β, R s Þ is a nonempty set for all α, β ∈ X, then the fixed point is unique.
Theorem 5 (see [20]). Let ðX, dÞ be a complete metric space and R be a binary relation on X: Let T be a self-mapping defined on ðX, dÞ satisfying the following conditions: (a) There exists α 0 ∈ X such that ðα 0 , Tðα 0 ÞÞ ∈ R and R is T-closed There exists k ∈ ½0, 1Þ such that θðdðTðαÞ, TðβÞÞÞ ≤ ½θðdðα, βÞÞ k for α, β ∈ X with ðα, βÞ ∈ R Then T admits a fixed point in X: Moreover, if Γðα, β, R s Þ is a nonempty set for all α, β ∈ X, then the fixed point is unique.
Definition 6 (see [21]). Let T and S be two self-mappings on a nonempty set X: A binary relation R on X is said to be ð T, SÞ-closed if for all α, β ∈ X, Zada and Sarwar [21] generalized Theorem 4 by replacing Banach contraction with F-contraction as follows.
Theorem 7 (see [21]). Let ðX, dÞ be a complete metric space and R be a binary relation on X. If the self-mappings T and S defined on ðX, dÞ satisfy the following conditions: (a) There exists α 0 ∈ X such that ðα 0 , Tðα 0 ÞÞ ∈ R and R is ðT, SÞ-closed Then T and S have a unique common fixed point in X. Moreover, if Γðα, β, R s Þ is nonempty for all α, β ∈ X, then the common fixed point is unique.
Liu et al. [22] introduced the ðD, CÞ-contractions where the mapping D maps positive real numbers to positive real numbers and satisfies the conditions ðD 1 Þ − ðD 3 Þ: (D 1 ) D is nondecreasing (D 2 ) lim n⟶∞ Dðt n Þ = 0 ⟺ lim n⟶∞ t n = 0, for each positive sequence ft n g (D 3 ) D is continuous C : ð0,∞Þ ⟶ ð0,∞Þ is a comparison function; that is, it satisfies the following conditions: (i) C is monotone increasing, that is, (ii) lim n⟶∞ C n ðtÞ = 0 for all t > 0, where C n stands for the n th iterate of C Let D = fD : ð0,∞Þ ⟶ ð0,∞ÞjD satisfies ðD 1 Þ − ðD 3 Þg: If D is defined by DðtÞ = t ; DðtÞ = ln t, then D belongs to D: The mapping T is called ðD, CÞ-contraction if it satisfies the following condition: Theorem 10 (see [22]). Every generalized ðD, CÞ-contraction has a unique fixed point in a complete metric space ðX, dÞ.
In this paper, in Section 3, we investigate common fixedpoint results for generalized contractions in the partial b -metric spaces endowed with binary relation R. The obtained results generalize Theorems 4, 5, 7, 10. We support the results with a nontrivial example and counter the remarks given in [23].

Basic Notions in the Partial b-Metric Spaces
Let X be a nonempty set, and the mapping P : X × X ⟶ ⟶½0,∞Þ satisfies the following axioms: According to Matthews [11], if the mapping P satisfies axioms (1-4), we say that it is a partial metric on the set X and ðX, PÞ is called partial metric space. According to Shukla [13], if P satisfies axioms (1, 2, 3, and 5), then it is a partial b -metric on the set X and ðX, P b Þ is called partial b -metric space. For convenience, we denote partial b-metric by P b : The following relation can be observed.
Example 1 (see [13]). Let X = ½0,∞Þ, l > 1, be a constant and P b : X × X ⟶ ½0,∞Þ be defined by Then ðX, P b Þ is a partial b-metric space with coefficient s = 2 l > 1, but it is neither a b-metric space nor a partial metric space.
Definition 12 (see [13]). A sequence fx n g n∈N in the partial b-metric space ðX, P b , sÞ is called a convergent sequence if there exists x ∈ X such that The uniqueness of the limit of a convergent sequence may not be guaranteed in the partial b-metric spaces (see [23]).
Definition 13 (see [13]). A sequence fx n g n∈N in a partial b -metric space ðX, P b , sÞ is called the Cauchy sequence if The partial b-metric space ðX, P b , sÞ is said to be complete if every Cauchy sequence fx n g n∈N in X converges to a point x ∈ X.
(1) Every Cauchy sequence in the b -metric space is also Cauchy in the partial b-metric space and vice versa

Common Fixed-Point Theorems in the Partial b-Metric Spaces
This section is the main part of this paper. It contains some new common fixed-point theorems in the partial b-metric spaces. The existence theorems given in [12, 15, 19-22, 24, 27] can be seen as a special case of the results proved in this section. The results in this paper are based on the following contractive condition.
Definition 15. Let T and S be two self-mappings on the partial b-metric space ðX, P b , sÞ and R be a binary relation on X. Let The mappings T and S form a DC -contraction if there exists a continuous comparison function C and D ∈ D such that In [23], it was remarked that some contraction conditions on partial b-metric spaces imply contraction conditions on b -metric spaces (see Theorem 2.6 in [23]). In the following example, we show that the contraction condition (14) is independent of these remarks.
Example 4. Let X = ½0,∞Þ and R = X 2 . Let P b : X × X ⟶ ½ 0,∞Þ be defined by Then ðX, P b Þ is a partial b-metric space with coefficient s = 4: The associated b-metric is given by Define T ≡ S : ½0, 1 ⟶ ½0, 1 by TðxÞ = x/5 ðif x ∈ ½0, 1ÞÞ and Tð1Þ = 0: Consider This implies, a contradiction to the definition of mapping D ∈ D: On the other hand, for partial b-metric, we have Note that we have taken ð1, 5/6Þ ∈ R. Similarly, it can be shown that the above conclusion holds for all other cases.
Since, in general, b-metric is discontinuous mapping (see [5]), so by Example 2, the partial b-metric is not continuous in general. The following lemma is necessary for the upcoming results.
Lemma 16 (see [14]). Let ðX, P b , sÞ be a partial b-metric space. If there exists a fx n g in ðX, P b , sÞ and x * , y * such that lim n⟶∞ x n = x * . Then Partial metric space Partial b-metric space Figure 1 4

Journal of Function Spaces
Taking limit k ⟶ ∞, we have Also, we have the following information: Taking limit k ⟶ ∞, we have By axiom (5), we have Taking limit k ⟶ ∞ and using (35), we have By using (31), we have the following information from (33), (35), (37), and (39): Since ðα 2m k , α 2n k −1 Þ ∈ I, by (14), we have This is a contradiction to the definition of function D.
Theorem 18. Let ðX, P b Þ be a complete partial b-metric space and R be a transitive binary relation on X. Let T and S form a DC-contraction. Suppose that Γðα, β, RÞ ≠ ∅ and statement of Theorem 17 holds, then the mappings T and S admit a unique common fixed point in X.
Proof. We have proved the existence of a common fixed point in Theorem 17 : On the contrary, suppose that v and v * are two distinct common fixed points of T and S in X.
Case 2. If fα n g is not constant and arbitrary, we claim that P b ðα * , Sðα * ÞÞ = 0. Let P b ðα * , Sðα * ÞÞ > 0. It is proved in Theorem 17 that lim i⟶∞ α 2i+1 = α * , so there must be an integer n 0 > 0, such that It is assumed that ðX, P b Þ is R − regular, and by Theorem 17, we know that α 2i ⟶ α * as i ⟶ ∞;thus, ðα 2i , α * Þ ∈ R. By contractive condition (2.1), monotonicity of D, and Lemma 16, we have This is a contradiction to the definition of mapping D: Thus, P b ðα * , Sðα * ÞÞ = 0. Also, we have the following information: Thus, α * = Sðα * Þ: By interchanging roles of S and T, we have α * = Tðα * Þ: Hence, Tðα * Þ = Sðα * Þ = α * ; that is, α * is a common fixed point of T and S in X: The following is the most general theorem of this section. (1) The results in this section are independent of the observation made in [23], and hence, our results are a real generalization of the related results in literature (see [12,[19][20][21][22]) (2) Theorem 21 remains true if P b ðα, βÞ is replaced by Mðα, βÞ The following example explains the main results.

Example 5.
Let X = f a n = nðn + 1Þ/2 : n ∈ ℕg:Define the partial b-metric function P b : X × X ⟶ ½0,∞Þ by Then (X, P b , 2) is a complete partial b-metric space. Define D∶ð0,∞Þ ⟶ ð0,∞Þ by DðaÞ = ae a for each a > 0, then D ∈ D. Let the function C : ð0,∞Þ ⟶ ð0,∞Þ be 7 Journal of Function Spaces defined by CðrÞ = r/2 for all r ∈ ð0,∞Þ: Then C is continuous comparison. Define the binary relation R on X by R = a n , a m ð Þ : a n + a m ≥ 2 for each m ≥ n f g : ð56Þ Define the mappings T, S∶X ⟶ X by T a n ð Þ= a 1 , n = 1, We observe that there exists a 1 ∈ X such that ða 1 , T ð a 1 ÞÞ ∈ R ðsince ða 1 + T ða 1 Þ = 2Þ by definition of R, so assumption (a) is satisfied in Theorem 17. Let ða n , a m Þ ∈ R, then we have Tða n Þ + Sða m Þ ≥ 2 for each m ≥ n, so ð Tða n Þ, Sða m ÞÞ ∈ R. Thus, R is ðT, SÞ-closed (this verifies assumption (b) of Theorem 17. Also, T, S are continuous (assumption (c) is satisfied). Now, we show that T and S form DC-contraction. It is noted that the mappings T, S do not form Banach contraction in the partial b-metric sense. Indeed, lim n⟶∞ P b T a n ð Þ, S a 1 ð Þ ð Þ P b a n , a 1 Þ = lim n⟶∞ n 2 − n 2 We noticed that P b ðTða n Þ, Sða m ÞÞ > 0 for each m ≥ n: Thus, ða n , a m Þ ∈ I:Consider 4P b T a n ð Þ, S a m ð Þ ð Þ e 4P b T a n ð Þ,S a m ð Þ ð Þ ≤ 1 2 P b a n , a m ð Þ e P b a n ,a m ð Þ : This implies 8P b T a n ð Þ, S a m ð Þ ð Þ P b a n , a m ð Þ ≤ e P b a n ,a m ð Þ −4P b T a n ð Þ,S a m ð Þ ð Þ : ð60Þ For n = 1 and m = 2, the inequality (41) reduces to e 5 ≥ 8/9. Thus, (41) holds for this case. For n = 2 and m = 3, the inequality (41) gets the form e 32 ≥ 2/9. Similarly, for each m ≥ n, (41) holds true. Thus, we have Þ , for all α, β ∈ X: We note that a 1 = Tða 1 Þ = Sða 1 Þ.

Discussions.
In this part of the current section, we state some corollaries which are themselves prominent fixedpoint theorems in the literature.
The following corollary generalizes the results presented by Jleli and Samet [6] and al-Sulami et al. [20]. Corollary 23. Let ðX, P b Þ be a complete partial b-metric space and R be a transitive and antisymmetric binary relation on X. If the self-mappings T and S defined on ðX, P b Þ satisfy the following conditions: (a) Γðα, β, RÞ is nonempty for all α, β ∈ X (b) There exists α 0 ∈ X such that ðα 0 , Tðα 0 ÞÞ ∈ R and R is ðT, SÞ-closed (c) Either T, S are continuous or ðX, P b Þ is R − regular (d) There exists a function θ ∈ Θ and k ∈ ð0, 1Þ, such that for all α, β ∈ I, Then the mappings T and S admit a unique common fixed point.
Proof. Setting CðtÞ = ðln kÞt and DðtÞ = θðs 2 tÞ in Theorem 17 and following the proofs of Theorems 17, 18, and 20 respectively, we obtain the required result.☐ The following corollary generalizes and improves the results presented by Zada and Sarwar [21] and Wardowski [25].

Corollary 24.
Suppose that the self-mappings T and S defined on the complete partial b-metric space ðX, P b Þ satisfy the following conditions: (a) Γðα, β, RÞ is nonempty for all α, β ∈ X (b) There exists α 0 ∈ X such that ðα 0 , Tðα 0 ÞÞ ∈ R and R is ðT, SÞ-closed (c) Either T, S are continuous or ðX, P b Þ is R − regular (d) There exists F ∈ F and τ > 0,such that for all α, β ∈ I, If R is a transitive and antisymmetric binary relation on X, then the mappings T, S admit a unique common fixed point.
Proof. Setting CðtÞ = e −τ t and DðtÞ = e s 2 FðtÞ in Theorem 17 and following the proofs of Theorems 17, 18, and 20, respectively, we obtain the required result.☐ Corollary 25 (see [21]). Let ðX, P b Þ be a complete partial b -metric space and R be a transitive and antisymmetric binary relation on X. If the self-mappings T and S defined on ðX, P b Þ satisfy the following conditions: (a) Γðα, β, RÞ is nonempty for all α, β ∈ X (b) There exists α 0 ∈ X such that ðα 0 , Tðα 0 ÞÞ ∈ R and R is ðT, SÞ-closed