A new contraction based on ℋ-simulation functions in the frame of extended <em>b</em>-metric spaces and application

Abstract

&lt;p&gt;The conceptions of &lt;em&gt;b&lt;/em&gt;-metric spaces and metric spaces play a remarkable role in proving many theorems of uniqueness and existence solution of such equations as integral or differential equations. The conception of extended &lt;em&gt;b&lt;/em&gt;-metric spaces is considered as a generalization concept of &lt;em&gt;b&lt;/em&gt;-metric spaces and metric spaces and this concept was employed to unify some new fixed point results in the literature. On the other hand, a new concept of simulation functions founded in 2020 in the name of ℋ-simulation functions and employed this class of functions to unify some fixed point results in the literature. The main objective of this manuscript, is to establish a new contraction namely, (γ, ϕ, θ) ℋ-contraction, this contraction based on the concepts of extended &lt;em&gt;b&lt;/em&gt;-metric spaces and the class of functions (ℋ-simulation functions) and the class of functions Θ-functions and the class of functions Φ-functions, We utilize this new contraction to unify the uniqueness and existence of fixed point results in the literature. In addition, some illustrative examples and an interesting application were established to show the novelty of our work.&lt;/p&gt;

On sentinel method of one-phase Stefan problem

“…which represents the solution of (26). Consequently, due to B * is injective, then (30) yields that σ * = 0. In light of the previous discussion and to show the existence and uniqueness of ( 8) and ( 9), we can have ∂J ∂σ , δσ…”

“…In this section, we transform the considered problem into a fixed point problem, and then we prove the local convergence. However, to get an overview about the fixed point problem, the reader may refer to the references [28][29][30]. Now, in order to carry such a transformation, we differentiae S (α) = S ( α, α) to obtain…”