Fractal and Fractional

Research

20 pages, 671 KiB

Open AccessArticle

On Novel Mathematical Modeling for Studying a Class of Nonlinear Caputo-Type Fractional-Order Boundary Value Problems Emerging in CGT

by Ali Turab, Wutiphol Sintunavarat and Jong-Suk Ro

Fractal Fract. 2023, 7(2), 99; https://doi.org/10.3390/fractalfract7020099 - 17 Jan 2023

Viewed by 1790

Abstract

Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. [...] Read more.

Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. Star graphs were used since their method requires a central point linked to other vertices but not to itself. Our objective is to expand the mechanism by introducing the idea of an isobutane graph that has the chemical formula

C_{4} H_{10}

and CAS number 75-28-5. By using the appropriate fixed point theory findings, this paper investigates the existence of solutions to fractional boundary value problems of Caputo type on such graphs. Additionally, two examples are provided to strengthen our important conclusions. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

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28 pages, 385 KiB

Open AccessArticle

Fixed Point for Almost Contractions in v-Generalized b-Metric Spaces

by Anshuka Kadyan, Savita Rathee, Anil Kumar, Asha Rani and Kenan Tas

Fractal Fract. 2023, 7(1), 60; https://doi.org/10.3390/fractalfract7010060 - 4 Jan 2023

Cited by 1 | Viewed by 2159

Abstract

We present a significant example to show that the class of v-generalized b-metric spaces properly contains the class of v-generalized metric spaces as well as b-metric spaces. This is accomplished because the example provided by Došenović et al. (2020) [...] Read more.

We present a significant example to show that the class of v-generalized b-metric spaces properly contains the class of v-generalized metric spaces as well as b-metric spaces. This is accomplished because the example provided by Došenović et al. (2020) is insufficient to expose the generality of v-generalized b-metric spaces over the existing related spaces. Therefore, we establish fixed point theorems by defining generalized almost contractions of rational type and Reich type in v-generalized b-metric spaces. Moreover, we compare the proven results with the already existing fixed point theorems in this space by presenting suitable examples. As a consequence of these fixed point theorems, we further develop some common fixed-point results that ensure the existence and uniqueness of coincidence points and common fixed points for a pair of self maps. Finally, we use the outcome to check that the given Fredholm integral equation has a solution and that it is also unique. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

15 pages, 383 KiB

Open AccessArticle

On the Solvability of Mixed-Type Fractional-Order Non-Linear Functional Integral Equations in the Banach Space C(I)

by Vijai Kumar Pathak, Lakshmi Narayan Mishra, Vishnu Narayan Mishra and Dumitru Baleanu

Fractal Fract. 2022, 6(12), 744; https://doi.org/10.3390/fractalfract6120744 - 16 Dec 2022

Cited by 17 | Viewed by 1578

Abstract

This paper is concerned with the existence of the solution to mixed-type non-linear fractional functional integral equations involving generalized proportional (

κ, ϕ

)-Riemann–Liouville along with Erdélyi–Kober fractional operators on a Banach space

C ([1, T])

arising [...] Read more.

This paper is concerned with the existence of the solution to mixed-type non-linear fractional functional integral equations involving generalized proportional (

κ, ϕ

)-Riemann–Liouville along with Erdélyi–Kober fractional operators on a Banach space

C ([1, T])

arising in biological population dynamics. The key findings of the article are based on theoretical concepts pertaining to the fractional calculus and the Hausdorff measure of non-compactness (MNC). To obtain this goal, we employ Darbo’s fixed-point theorem (DFPT) in the Banach space. In addition, we provide two numerical examples to demonstrate the applicability of our findings to the theory of fractional integral equations. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

13 pages, 292 KiB

Open AccessArticle

New Results for Weakly Compatible (WC) and R-Weakly Commuting (RWC) Mappings with an Applicationin Dynamic Programming

by Penumarthy Parvateesam Murthy, Sanjay Kumar, Rajesh Kumar, Pusplata Sahu, Zoran D. Mitrović and Reny George

Fractal Fract. 2022, 6(12), 733; https://doi.org/10.3390/fractalfract6120733 - 10 Dec 2022

Viewed by 1367

Abstract

The aim of this paper is to obtain some new results about common fixed points. Our results use weaker conditions than those previously used. We have relaxed the conditions for commutating pair mappings and compatible mappings of the type

(A)

, [...] Read more.

The aim of this paper is to obtain some new results about common fixed points. Our results use weaker conditions than those previously used. We have relaxed the conditions for commutating pair mappings and compatible mappings of the type

(A)

, which were introduced in 1976. The theorems are enriched by using the concept of

W C

and various types of weakly commuting pairs of maps in metric spaces. To discuss the existence and uniqueness of the common solutions, we have obtained an application to the functional equations in dynamic programming. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

12 pages, 315 KiB

Open AccessArticle

Common Fixed Point for Meir–Keeler Type Contraction in Bipolar Metric Space

by Penumarthy Parvateesam Murthy, Chandra Prakash Dhuri, Santosh Kumar, Rajagopalan Ramaswamy, Muhannad Abdullah Saud Alaskar and Stojan Radenovi’c

Fractal Fract. 2022, 6(11), 649; https://doi.org/10.3390/fractalfract6110649 - 4 Nov 2022

Cited by 7 | Viewed by 1396

Abstract

In mathematical analysis, the Hausdorff derivatives or the fractal derivatives play an important role. Fixed-point theorems and metric fixed-point theory have varied applications in establishing a unique common solution to differential equations and integral equations. In the present work, some fixed-point theorems using [...] Read more.

In mathematical analysis, the Hausdorff derivatives or the fractal derivatives play an important role. Fixed-point theorems and metric fixed-point theory have varied applications in establishing a unique common solution to differential equations and integral equations. In the present work, some fixed-point theorems using the extension of Meir–Keeler contraction in the setting of bipolar metric spaces have been proved. The derived results have been supplemented with non-trivial examples. Our results extend and generalise the results established in the past. We have provided an application to find an analytical solution to an Integral Equation to supplement the derived result. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

48 pages, 421 KiB

Open AccessArticle

Analysis of a Hybrid Coupled System of ψ-Caputo Fractional Derivatives with Generalized Slit-Strips-Type Integral Boundary Conditions and Impulses

by Zhiwei Lv, Ishfaq Ahmad, Jiafa Xu and Akbar Zada

Fractal Fract. 2022, 6(10), 618; https://doi.org/10.3390/fractalfract6100618 - 21 Oct 2022

Cited by 7 | Viewed by 1767

Abstract

In the current paper, we analyzed the existence and uniqueness of a solution for a coupled system of impulsive hybrid fractional differential equations involving

ψ

-Caputo fractional derivatives with generalized slit-strips-type integral boundary conditions. We also study the Ulam–Hyers stability for the considered [...] Read more.

In the current paper, we analyzed the existence and uniqueness of a solution for a coupled system of impulsive hybrid fractional differential equations involving

ψ

-Caputo fractional derivatives with generalized slit-strips-type integral boundary conditions. We also study the Ulam–Hyers stability for the considered system. For the existence and uniqueness of the solution, we use the Banach contraction principle. With the help of Schaefer’s fixed-point theorem and some assumptions, we also obtain at least one solution of the mentioned system. Finally, the main results are verified with an appropriate example. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

22 pages, 385 KiB

Open AccessArticle

Discussion on the Approximate Controllability of Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators

by Chandrabose Sindhu Varun Bose, Ramalingam Udhayakumar, Ahmed M. Elshenhab, Marappan Sathish Kumar and Jong-Suk Ro

Fractal Fract. 2022, 6(10), 607; https://doi.org/10.3390/fractalfract6100607 - 18 Oct 2022

Cited by 25 | Viewed by 1745

Abstract

This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost sectorial operators, fractional differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove the result. We start by emphasizing the existence of [...] Read more.

This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost sectorial operators, fractional differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove the result. We start by emphasizing the existence of a mild solution and demonstrate the approximate controllability of the fractional system. In addition, an example is presented to demonstrate the principle. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

8 pages, 299 KiB

Open AccessArticle

Positive Solutions for Perturbed Fractional p-Laplacian Problems

by Mengfei Tao and Binlin Zhang

Fractal Fract. 2022, 6(10), 571; https://doi.org/10.3390/fractalfract6100571 - 8 Oct 2022

Cited by 2 | Viewed by 1511

Abstract

In this article, we consider a class of quasilinear elliptic equations involving the fractional p-Laplacian, in which the nonlinear term satisfies subcritical or critical growth. Based on a fixed point result due to Carl and Heikkilä, we can well overcome the lack [...] Read more.

In this article, we consider a class of quasilinear elliptic equations involving the fractional p-Laplacian, in which the nonlinear term satisfies subcritical or critical growth. Based on a fixed point result due to Carl and Heikkilä, we can well overcome the lack of compactness which has been a key difficulty for elliptic equations with critical growth. Moreover, we establish the existence and boundedness of the weak solutions for the above equations. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

19 pages, 345 KiB

Open AccessArticle

Best Proximity Point Theorems for the Generalized Fuzzy Interpolative Proximal Contractions

by Khalil Javed, Maha M. A. Lashin, Muhammad Nazam, Hamed H. Al Sulami, Aftab Hussain and Muhammad Arshad

Fractal Fract. 2022, 6(8), 455; https://doi.org/10.3390/fractalfract6080455 - 21 Aug 2022

Viewed by 1694

Abstract

The idea of best proximity points of the fuzzy mappings in fuzzy metric space was intorduced by Vetro and Salimi. We introduce a new type of proximal contractive condition that ensures the existence of best proximity points of fuzzy mappings in the fuzzy [...] Read more.

The idea of best proximity points of the fuzzy mappings in fuzzy metric space was intorduced by Vetro and Salimi. We introduce a new type of proximal contractive condition that ensures the existence of best proximity points of fuzzy mappings in the fuzzy complete metric spaces. We establish certain best proximity point theorems for such proximal contractions. We improve and generalize the fuzzy proximal contractions by introducing

(Ψ, Φ)

-fuzzy proximal contractions and

(Ψ, Φ)

-fuzzy proximal interpolative contractions. The obtained results improve and generalize many best proximity point theorems published earlier. Moreover, we provide many nontrivial examples to validate our best proximity point theorem. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

13 pages, 1284 KiB

Open AccessArticle

Stability Analysis and Computational Interpretation of an Effective Semi Analytical Scheme for Fractional Order Non-Linear Partial Differential Equations

by Javed Iqbal, Khurram Shabbir and Liliana Guran

Fractal Fract. 2022, 6(7), 393; https://doi.org/10.3390/fractalfract6070393 - 16 Jul 2022

Cited by 7 | Viewed by 2116

Abstract

In this study we will check the stability of the semi analytical technique, the Laplace variational iteration (LVI) scheme, which is the combination of a variational iteration technique and the Laplace transform method. Then, we will apply it to solve some non-linear fractional [...] Read more.

In this study we will check the stability of the semi analytical technique, the Laplace variational iteration (LVI) scheme, which is the combination of a variational iteration technique and the Laplace transform method. Then, we will apply it to solve some non-linear fractional order partial differential equations. Since the Laplace transform cannot be applied to non-linear problems, the combination of the variational iteration technique with it will give a better and rapidly convergent sequence. Exact solutions may also exist, but we will show that the coupled technique is much better to approximate the exact solutions. The Caputo–Fabrizio fractional derivative will be used throughout the study. In addition, some possible implications of the results given here are connected with fixed point theory. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

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13 pages, 839 KiB

Open AccessArticle

On the Global Well-Posedness of Rotating Magnetohydrodynamics Equations with Fractional Dissipation

by Muhammad Zainul Abidin, Muhammad Marwan, Humaira Kalsoom and Omer Abdalrhman Omer

Fractal Fract. 2022, 6(6), 340; https://doi.org/10.3390/fractalfract6060340 - 17 Jun 2022

Cited by 1 | Viewed by 1888

Abstract

This work considers the three-dimensional incompressible rotating magnetohydrodynamics equation spaces with fractional dissipation

{(- Δ)}^{℘}

for

\frac{1}{2} < ℘ \leq 1

. Furthermore, we use the Littlewood–Paley decomposition and frequency localization techniques to establish the global well-posedness of fractional [...] Read more.

This work considers the three-dimensional incompressible rotating magnetohydrodynamics equation spaces with fractional dissipation

{(- Δ)}^{℘}

for

\frac{1}{2} < ℘ \leq 1

. Furthermore, we use the Littlewood–Paley decomposition and frequency localization techniques to establish the global well-posedness of fractional rotating magnetohydrodynamics equations in a more generalized Besov spaces characterized by the time evolution semigroup related to the generalized linear Stokes–Coriolis operator. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

22 pages, 415 KiB

Open AccessArticle

Neutrosophic Double Controlled Metric Spaces and Related Results with Application

by Fahim Uddin, Umar Ishtiaq, Aftab Hussain, Khalil Javed, Hamed Al Sulami and Khalil Ahmed

Fractal Fract. 2022, 6(6), 318; https://doi.org/10.3390/fractalfract6060318 - 6 Jun 2022

Cited by 10 | Viewed by 2097

Abstract

In this paper, the authors introduce the notion of neutrosophic double controlled metric spaces as a generalization of neutrosophic metric spaces. For this purpose, two non-comparable functions, ξ and Γ, are used in triangle inequalities. The authors prove several interesting results for contraction [...] Read more.

In this paper, the authors introduce the notion of neutrosophic double controlled metric spaces as a generalization of neutrosophic metric spaces. For this purpose, two non-comparable functions, ξ and Γ, are used in triangle inequalities. The authors prove several interesting results for contraction mappings with non-trivial examples. At the end of the paper, the authors prove the existence, and the uniqueness, of the integral equation to support the main result. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

13 pages, 342 KiB

Open AccessArticle

Fixed Point Results for Generalized F-Contractions in b-Metric-like Spaces

by Huaping Huang, Kastriot Zoto, Zoran D. Mitrović and Stojan Radenović

Fractal Fract. 2022, 6(5), 272; https://doi.org/10.3390/fractalfract6050272 - 17 May 2022

Cited by 11 | Viewed by 2813

Abstract

The purpose of this paper is to introduce several generalized F-contractions in b-metric-like spaces and establish some fixed point theorems for such contractions. Moreover, some nontrivial examples are given to illustrate the superiority of our results. In addition, as an application, [...] Read more.

The purpose of this paper is to introduce several generalized F-contractions in b-metric-like spaces and establish some fixed point theorems for such contractions. Moreover, some nontrivial examples are given to illustrate the superiority of our results. In addition, as an application, we find the existence and uniqueness of a solution to a class of integral equations in the context of b-metric-like spaces. Full article

(This article belongs to the Special Issue New Trends on Fixed Point Theory)

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