Novel results of α∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha _{\ast }$\end{document}-ψ-Λ-contraction multivalued mappings in F-metric spaces with an application

The objective of this paper is to introduce a new motif of α∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha _{\ast }$\end{document}-ψ-Λ-contraction multivalued mappings, some novel fixed-point and coincidence-point results for this contraction will be investigated in the scope of F-metric spaces, and some examples are given to illustrate our main results and we derive the existence and uniqueness of a solution of a functional equation to support our main result.

is called a metric on ϒ and the pair (ϒ, d) is said to be a MS.
Many writers used the motif of F-MS to investigate powerful fixed-point results; for instance, Alnaser et al. [4] defined relation theoretic contractions and proved some generalized fixed-point theorems in F-metric spaces. Hussain and Kanwal [20] considered the notion of α-ψ-contraction and presented some fixed-and coupled fixed-point results in the setting of F-MSs. Lateef and Ahmad [24] defined Dass and Gupta's contraction in the context of F-MSs and then proved some new fixed-point theorems to generalize and elaborate several known literature results. Mitrović et al. [26] proved certain common fixedpoint theorems and some consequences to obtain the results of Banach, Jungck, Reich, and Berinde in F-MSs with an application for dynamic programming. Hussain [19] introduced the idea of fractional convex-type contraction and established some new fixed-point results for Reich-type α-η-contraction and Kannan-type α-η-contraction mappings in F-MS. He derived some consequences for Suzuki-type contractions, orbitally T-complete, and orbitally continuous mappings.
BCP [12] appeared in 1922 as the basis of functional analysis and plays a main role in several branches of mathematics and applied sciences, which asserts that every contraction mapping defined in complete MS has a fixed point. In many directions, this principle has been extended and generalized either by relaxing the contractive stipulations or imposing some more stipulations on space. Jungck [22] studied coincidence and common fixed points of commuting mappings and improved the BCP. In [35], coincidence-point and common fixed-point theorems for a class of Ćirić-Suzuki hybrid contractions involving a multivalued and two single-valued maps in an MS are obtained. Coincidence-point theorems for Geraghty contraction mappings have been introduced in different spaces [27-29, 33, 34, 37-39]. Theorem 1.7 ([12]) Let (ϒ, Q) be a complete MS and : ϒ − → ϒ be a contraction mapping, that is ∀γ , δ ∈ ϒ, and k ∈ (0, 1), Then, has a unique fixed point.

Lemma 1.11
Mappings and are called weakly compatible if γ ∈ γ for some γ ∈ ϒ implies (γ ) ⊆ (γ ).  Asif et al. [10] obtain fixed points and common fixed-point results for Reich-type Fcontractions for both single and set-valued mappings in F-MSs. Alansari et al. [3] studied a few fuzzy fixed-point theorems and discussed the corresponding fixed-point theorems of multivalued and single-valued mappings on F-complete F-MSs.

Lemma 1.15 ([3]) Let A and B be nonempty closed and compact subsets of an F-metric
Let be the family of nondecreasing functions ψ : ( 3) is continuous.
We denote by the set of functions : We modify the Definition 1.17 by adding a general condition ( 4) that is given in the following way:  (1) (t) = at, a > 0; (2) (t) = |t|. Then ∈ . Now, we state and prove our main result.

Main results
In this section, we shall introduce a generalization of Geraghty contraction type mappings and establish some novel fixed-point theorems for α * --ψ-contraction multivalued mappings in the setting of F-MS.
Then, and have a unique point of coincidence. Indeed, if and are weakly compatible, then and have a unique common fixed point γ * ∈ ϒ.
Then, and g have a unique point of coincidence. Indeed, if and are weakly compatible, then and have a unique common fixed point γ * ∈ ϒ.
If γ = δ, then we have Otherwise, we have that (3.1) trivially holds. Therefore, all stipulations of Theorem 3.1 are satisfied. Since ϒ0 = 0 = 0, thus γ = 0 is a common fixed point of and .

Application for the existence of a solution to a functional equation
In this section, we use our main results to verify the existence and uniqueness of a solution to the functional equation: where : × ϒ → R and : × ϒ × R → R are bounded, μ : × ϒ → , and ϒ are BSs. Equations of the type (4.1) have applications in mathematical optimization, computer programming, and in dynamic programming, giving tools for solutions to boundary value problems arising in engineering and physical sciences. Bhakta and Mitra [13] introduced the existence theorems that proved the existence and uniqueness of the solution of a functional equation under certain conditions in Banach spaces. Deepmala [15] utilized the fixed-point theorems to establish the existence, uniqueness, and iterative approximation of the solution for a functional equation in Banach spaces and complete metric spaces. In [30,32], common solutions of certain functional equations arising in dynamic programming and common fixed-point theorems for a quadruple of self-mappings satisfying weak compatibility and JH-operator pairs on a complete metric space were discussed.
Note that and are bounded; this implies that stipulations (S 1 ) and (S 2 ) of Theorem 4.1 are satisfied. Now, Thus, all the assertions of Theorem 4.1 are satisfied and the functional Eq. (4.1) has a bounded solution in L =(S).

Conclusion
In this paper, we introduced a new notion of α * -ψ--contraction multivalued mappings and proved some novel fixed-point theorems for such contraction in F-MSs. Some consequences are studied to investigate coincidence-point results for this contraction in F-MSs. Also, we gave some examples to clarify our obtained results; we utilized the main results to discuss the existence and uniqueness of a solution to a functional equation. The new concepts lead to further investigations and applications.