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# On Some Novel Fixed Point Results for Generalized -Contractions in -Metric-Like Spaces with Application

1 Department of Mathematics and Computer Sciences, Faculty of Natural Sciences, University of Gjirokastra, Gjirokastra, 6001, Albania

2 Department of Mathematics, Faculty of Natural Sciences, University of Tirana, Tirana, 1001, Albania

3 Faculty of Electrical Engineering, University of Banja Luka, Banja Luka, 78000, Bosnia and Herzegovina

4 Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Beograd, 11120, Serbia

* Corresponding Author: Zoran D. Mitrović. Email:

(This article belongs to the Special Issue: Computational Aspects of Nonlinear Operator and Fixed Point Theory with Applications)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(1), 673-686. https://doi.org/10.32604/cmes.2022.022878

**Received** 30 March 2022; **Accepted** 17 May 2022; **Issue published** 29 September 2022

## Abstract

The focus of our work is on the most recent results in fixed point theory related to contractive mappings. We describe variants of -contractions that expand, supplement and unify an important work widely discussed in the literature, based on existing classes of interpolative and -contractions. In particular, a large class of contractions in terms of and*F*for both linear and nonlinear contractions are defined in the framework of -metric-like spaces. The main result in our paper is that --weak contractions have a fixed point in -metric-like spaces if function

*F*or the specified contraction is continuous. As an application of our results, we have shown the existence and uniqueness of solutions of some classes of nonlinear integral equations.

## Keywords

Fixed point theory has been studied for a long time. Its application relies on the existence of solutions to mathematical problems that are based on the contraction principle. An interesting generalization of the Banach contraction principle was given by Wardowski [1,2] using a different type of contraction called

In this section, we list some well-known definitions and lemmas in terms of

Definition 2.1 [9]. Let V be a nonempty set and

The pair

In a

Definition 2.2 [10]. Let

(a) The sequence

(b) The sequence

(c) The pair

Definition 2.3 [10]. Let

Note that in a

Lemma 2.1 [11,12]. Let

Lemma 2.2. [9]. Let

for all

Lemma 2.3. [9]. Let

(a) If

(b) If

(c) If

Lemma 2.4. [13]. Let

We begin the main section with a definition that is an expanding outlook of Wardowski type

Definition 3.1. Let

(a) F is strictly increasing;

(b)

(c) For all

Remark 3.1. In the above definitions property F and conditions (1) yield

The continuity of the mapping f follows from the inequality

Remark 3.2.

• Our definition generalizes the previous definitions given in [14–16]. It contains a reduced number of conditions compared with the previous definitions.

• The definition of

• If

• For

The following is the first fixed point theorem for

Theorem 3.1. Let

Proof. Let be

Further from inequality (2), we get

which implies

In view of Lemma 2.1, the corresponding Picard sequence

According to the (1), it follows:

that from property of F we get

From triangular property and (4), we have

Since f is continuous and using (3), (4) we obtain

Since

which implies

Previous inequality is a contradiction, so

Corollary 3.1. Let

for all

Proof. Inequality (1) implies (6) if we set

Example 3.1. Let

Taking the logarithms in the above inequality and fixing

With the aim of expanding the initiated Definition 2.1 and starting a result that includes Theorem 3.1 and its respective corollaries, we will use a class of implicit relations, which makes simultaneously effective enormous literature on this topic.

Let

(a) g is non-decreasing with respect to each variable:

(b)

Definition 3.2. Let

(a)

(b)

for all

Remark 3.3.

• The above definition reduces to a generalized

• Fixing the parameter

• Fixing

Theorem 3.2. Let

Proof. Let

If we assume that

for all

which is a contradiction. Therefore

for all

which is a contradiction. Therefore, we conclude that

Next, we show that

From condition (7), we get

Taking the upper limit in (10) as

Hence, the acquired inequality

is a contradiction since

Let

which implies

Taking the upper limit in (12), and using Lemma 2.1 and result (9), it follows that

Since

Let u and v be two fixed points of f, where

Since this is a contradiction, it implies

Theorem 3.3. Let

(a) F is strictly increasing;

(b)

Then f has a unique fixed point in V.

Proof. The proof follows from Theorem 3.2. by setting

Corollary 3.2. Let

(a) F is strictly increasing;

(b)

(c)

for all

Proof. The proof follows from Theorem 3.2 by taking

Corollary 3.3. Let

(a) F is strictly increasing;

(b)

(c)

for all

Proof. The proof follows from Theorem 3.2. by taking

Recently, many authors have studied new types of contractions known as interpolative contractions and hybrid contractions. The reader can refer to [3,11,17–21]. The rest of the paper deals with this type of contractions extended in the setting of

Theorem 3.4. Let

(a) F is strictly increasing;

(b)

(c)

for all

Proof. The proof follows from Theorem 3.2 by taking

where

Theorem 3.5. Let

(a) F is strictly increasing;

(b)

(c)

Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking

Theorem 3.6. Let

(a) F is strictly increasing;

(b)

(c)

Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking

where

Theorem 3.7. Let

(a) F is strictly increasing;

(b)

(c)

Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking

Corollary 3.4. Let

(a)

(b)

Then f has unique fixed point in V.

Proof. The proof follows from Theorem 3.2 by taking

Remark 3.4.

• Varieties of further results can be obtained by extending the set

• Many significant fixed point theorems that were established for types of interpolative and hybrid contractive conditions essentially belong to the class of generalized

The study of the existence, nonexistence and uniqueness of the solution of differential and integral equations, plays a fundamental role in the research on nonlinear analysis and engineering mathematics. One of the main tools developed in this area consists of the application of a fixed point method.

Let us study the existence of solution for the nonlinear integral equation

where

Let

It is obvious that

Consider the mapping

for all

Theorem 4.1. Consider the integral Eq. (1) via the following assertions:

i. The mapping

ii.

iii. The constants

for

Proof. For all

Hence, by taking logarithms in inequality (25) we get

Further, fixing

Therefore, f is a

The Definitions 2.1 and 3.2 not only a large class of contractions in terms of

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The author declares that they have no conflicts of interest to report regarding the present study.

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**APA Style**

*Computer Modeling in Engineering & Sciences*,

*135*

*(1)*, 673-686. https://doi.org/10.32604/cmes.2022.022878

**Vancouver Style**

**IEEE Style**

*Comput. Model. Eng. Sci.*, vol. 135, no. 1, pp. 673-686. 2023. https://doi.org/10.32604/cmes.2022.022878