Hybrid Multivalued Type Contraction Mappings in α K-Complete Partial b-Metric Spaces and Applications

In this paper, we initiate the notion of generalized multivalued (αK, Υ, Λ)-contractions and provide some new common fixed point results in the class of αK-complete partial b-metric spaces. The obtained results are an improvement of several comparable results in the existing literature. We set up an example to elucidate our main result. Moreover, we present applications dealing with the existence of a solution for systems either of functional equations or of nonlinear matrix equations.


Introduction and Preliminaries
Fixed point theory plays an essential role in functional and nonlinear analysis.Banach [1] proved a significant result for contraction mappings.Since then, many works dealing with fixed point results have been provided by various authors (see, for example, ).
On the one hand, Bakhtin [43] and Czerwik [34,35] gave generalizations of the known Banach fixed point theorem in the class of b-metric spaces.In 1994, Matthews [23,24] introduced the notion of a partial metric space, which is a generalization of metric spaces.Very recently, Shukla [41] introduced the notion of partial b-metric spaces by combining partial metric spaces and b-metric spaces.
On the other hand, Popescu [22] introduced triangular α-orbital admissible maps.Karapinar [42] gave some fixed point results for a generalized α-ψ-Geraghty contraction type mappings using triangular α-admissibility.Recently, Ameer et al. [32] initiated the concept of generalized α * -ψ-Geraghty type multivalued contraction mappings and developed new common fixed point results in the class of α-complete b-metric spaces.
In this paper, we initiate the notion of generalized multivalued (α * K , Υ, Λ)-contraction pair of mappings.Some new common fixed point results are established for these mappings in the setting of α K -complete partial b-metric spaces.Examples are also given to support the obtained results.Finally, we apply the obtained results to ensure the existence of a solution of either a pair of functional equations or nonlinear matrix equations.
(1) Every Cauchy sequence in (ω, d P b ) is also Cauchy in (ω, P b , K) and vice versa.
(2) (ω, P b , K) is complete if and only if (ω, d P b ) is a complete metric space.
Piri and Kumam [17] modified the set of functions F ∈ .Definition 7. [17] Let (ω, d) be a MS.T : ω → ω is said to be a F-contraction self-mapping if there exist F ∈ F and τ > 0 such that where F is the set of functions F : (0, ∞) → (−∞, ∞) satisfying the following conditions: (F1) F is strictly increasing, i.e., for all (F3) F is continuous.
On the other hand, recently Jleli and Samet [9,10] initiated the concept of θ-contractions.
For examples of functions in Φ, see [13].For a MS (ω, d), CB(ω) stands for the collection of all closed and bounded subsets in ω.Theorem 4. Let S : ω −→ CB(ω) be a multivalued mapping on the complete MS (ω, d).The two statements are equivalent: (i) S is a multivalued θ-contraction mapping with θ ∈ Ξ.
(ii) S is a multivalued F-contraction mapping with F ∈ F .
Proof.The proof of this theorem follows immediately from the proof of Theorem 3.

Main Results
We start with the following definitions.
Now, we initiate the concept of generalized (α * K , Υ, Λ)-contraction multivalued pair of mappings as follows: Definition 18.Let (ω, P b , K) be a partial b-metric space and α K : ω × ω −→ [0, ∞) be a function.Given S, T : ω −→ CB P b (ω).The pair (S, T) is called a generalized (α * K , Υ, Λ)-contraction multivalued pair of mappings if there exist a comparison function Υ and a function where Our first main result is the following.
Theorem 5. Let (ω, P b , K) be a partial b-metric space.Given If Υ is continuous, then there exists a common fixed point of S and T, e.g.
Since Λ is nondecreasing, we have Hence, from Equation (3), where )), then from (5), we have . By Equation ( 5), we get that Similarly, for ζ 2 ∈ T (ζ 1 ) and ζ 3 ∈ S (ζ 2 ).We have This implies that By continuing in this manner, we build a sequence {ζ n } in ω in order that (S, T) is triangular α * K -orbital admissible.By Lemma 7, we have where ) , then from (7) we have By Equation ( 7), we get that This implies that , for all n ∈ N ∪ {0} , which implies Letting n −→ ∞ in the above inequality, we get From (Φ2) and Lemma 2, we get We claim that that {ζ n } is Cauchy.We argue by contradiction.Suppose that there exist ε > 0 and a sequence ĥn Taking n → ∞ in Equation ( 9), we get From triangular inequality, we have and Applying the upper limit when n → ∞ in (2.11) and applying Equation ( 8) together with Equation (10), Again, the upper limit in Equation ( 12) yields that Similarly, By triangular inequality, we have On letting n → ∞ in Equation ( 15) and using the inequalities in Equations ( 8) and ( 13), we get Similarly, lim From Equations ( 16) and ( 17), we get From Equations ( 8) and (10), we can choose a positive integer n 0 ≥ 1 such that for all n ≥ n 0 , from Equation (1), we get, where Taking the limit as n → ∞ and using Equations ( 8), ( 10), ( 13) and ( 14), we get From Equation ( 16), (Φ2), and by Lemma 7 since α K (ζ ĥ This is a contradiction.Therefore, {ζ n } is Cauchy.The α K -completeness of the partial b-metric space (ω, P b , K) implies the α K -completeness of the b-metric space (ω, d P b ).Thus, there exists ζ * ∈ ω so that lim By Lemma 1, lim Since Thus, from Equation ( 8) and axiom (P b 2) with Equation ( 19), we have Combining Equations ( 20) and ( 22)), we get Hence, lim where From Equation ( 23), Letting k −→ ∞ in the above inequality and by continuity of Λ and Υ, we obtain that Corollary 1.Let (ω, P b , K) be a partial b-metric space.Given α K : ω × ω −→ [0, ∞) and S : ω −→ CB P b (ω).Suppose that: (i) (ω, P b , K) is an α K -complete partial b-metric space.
(ii) S is a generalized (α * K , Υ, Λ)-contraction multivalued mapping, that is, if there exist a comparison function Υ and and a function (iii) S is triangular α * K -orbital admissible.(iv) There exists Proof.Set S = T in Theorem 5.
In addition, we define the function α If the sequence {ζ n } is Cauchy with α K (ζ n , ζ n+1 ) ≥ K 2 for each integer n, then {ζ n } ⊆ 0, 1 2 .Since 0, 1  2 , P b , K is a complete partial b-metric space, {ζ n } converges in 0, 1 2 ⊆ ω.Thus (ω, P b , K) is an Therefore, (S, T) is triangular α * K -orbital admissible.Let {ζ n } be a Cauchy sequence so that lim Suppose, without any loss of generality, that all ζ, η are nonzero and ζ < η.Then, Hence, all the hypotheses of Theorem 5 hold, and so S and T have a common fixed point.

Some Consequences
In this section, we obtain some fixed point results for singlevalued mappings when applying the corresponding results of Section 2. Definition 20.Let (ω, P b , K) be a partial b-metric space.Given α K : ω × ω −→ [0, ∞) and S, T : ω −→ ω are two self-mappings.(S, T) is called a generalized (α K , Υ, Λ)-contraction pair of mappings if there exist a comparison function Υ and a function If Υ is continuous, then S and T have a common fixed point ζ * ∈ ω.
The proof follows from the proof of Theorem 5.
Corollary 5. Let (ω, P b , K) be a complete partial b-metric space.Let S, T : ω −→ ω be two self-mappings such that: (i) (S, T) is a generalized (Υ, Λ)-contraction pair of mappings, i.e., there exist a comparison function Υ and a function Λ ∈ Φ such that for ζ, η ∈ ω, (ii) S and T are P b -continuous.
If Υ is continuous, there exists a common fixed point, e.g.ζ * ∈ ω.
Proof.It follows as the same lines in proof of Theorem 5.

Application to Nonlinear Matrix Equations
Denote by J(n) the set of all n × n Hermitian matrices, Q(n) by the set of all n × n Hermitian positive definite matrices and S(n) by the set of all n × n positive semi-definite matrices.A > 0 (respectively, A ≥ 0) means A ∈ Q(n) (respectively, A ∈ S(n)).The spectral norm is denoted by ., i.e., where µ + (E * E) is the greatest eigenvalue of the matrix E * E. The Ky Fan norm is given as where {S 1 (E), S 2 (E), • • • , S n (E)} is the set of the singular values of E.Moreover, , The set (J(n), P b ) is a complete partial b-metric space, where Take the system of nonlinear matrix equations: where π is a positive definite matrix, E 1 ,..., E m are n × n matrices and γ, δ are mappings from J(n) to where Then, the matrix in Equation ( 24) has a solution in Q(n).
Proof.Define Γ, Φ : Then, a common fixed point of Γ and Φ is a solution of Equation (24).Let X, Y ∈ J(n) with X = Y.Then, for P b (X, Y) > 0, we have and so Therefore, all conditions of Corollary 5 immediately hold.Thus, Γ and Φ have a common fixed point and hence the system in Equation (24) of matrix equations has a solution in Q(n).

Conclusions
In this paper, we have provided common fixed theorems for generalized (α * K , Υ, Λ)-contraction multivalued pair of mappings in α K -complete partial b-metric spaces.Our results are extensions of recent fixed point theorems of Wardowski [21], Piri and Kumam [17], Jleli et al. [9,10] and Liu et al. [13] and some other results.Moreover, we applied our main results to solve systems of functional equations and nonlinear matrix equations.It would be interesting to apply our given concepts and results for generalized metric spaces.
Author Contributions: All authors read and approved the manuscript.
For A ∈ CB P b (ω) and ζ ∈ ω, then D P b (ζ, A) = P b (ζ, ζ) if and only if ζ ∈ A, where A is the closure of A.