Generally, the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models. ^{*}^{*}^{*}^{*}^{*}^{*}^{*}^{*}

^{*}-algebra-valued quasi controlled K-metric spaces

One of the most relevant theories marking the passage from classical to modern analysis is the fixed point theory which was implemented by Banach [

The

In [

In this work, we introduce the notion of

Throughout this paper,

Note that

To prove our main results, it will be useful to introduce the following lemma.

if

if

for all

The triplet

In this section, by omitting the symmetry condition, we introduce the notion of

Define a

Given the

Then,

Let the

We take

Thus,

Next, we introduce some topological concepts on

Then, it is evident that

The open ball

If every Cauchy sequence

Let

Then,

Let us define the

The condition (

Therefore,

This prove that

We deduce

We conclude that the sequence

We will fix the notion of a continuous metric in the context presented in this paper since in the literature during the proof of the results in fixed point certain problems arise due to the possible discontinuity of the

As

Our main result runs as follows.

Now we prove that

Since

Thus, the above inequality implies

Letting

Remains to see that

Therefore,

Then, we get a contradiction, as a result

Dynamic programming is a powerful technique for solving some complex problems in computer sciences. We illustrate Theorem 3.2 by studying the existence and uniqueness of the solutions of the functional equation presented in the following example.

It is easy to get

Therefore, the

Let the

We define a mapping

It is easy to get

We find

By applying the previous results and involving the

Let

We consider

We now claim

Assume now the function

Define the operator

Moreover, under the conditions

Hence,

We prove the existence of solutions to problem 4 utilising our deduced fixed point theorems. Now, let

Similar to the Example 6, one can easily verify the completeness of

Since

Note that for all

Assume that

Now let

We see that

The results obtained are supported by non-trivial examples and complement and extend some of the most recent results from the literature. We have made a contribution by establishing some basic fixed-point problems considering a

Future study is to investigate the sufficient conditions to guarantee the existence of a unique positive definite solution of the nonlinear matrix equations in the setting of

The authors Thabet Abdeljawad and Aziz Khan would like to thank Prince Sultan University for the support through the TAS research lab.

^{*}-algebra valued metric spaces and related fixed point theorems

^{*}-algebra valued

^{*}-algebra valued extended

^{*}-algebra valued asymmetric spaces

^{*}-algebra valued b-asymmetric metric spaces

^{*}-algebra valued contractive type mappings

^{*}-algebra valued partial