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Advances in Nonlinear Functional Analysis on Fractional Differential Equations

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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (21 March 2024) | Viewed by 16528

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Prof. Dr. Bipan Hazarika

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Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India
Interests: sequence spaces; application of fixed point theory; measure of non-compactness; fractional calculus, nonlinear analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear functional analysis is one of the techniques of nonlinear mapping between infinite dimensional vector space and a certain class of nonlinear spaces. The subject of nonlinear functional analysis is of interest in its own right, and it also serves to lay the foundations for different fields of pure and applied mathematics. Researchers across the world are actively involved in analyzing and developing different theories of nonlinear functional analysis and fractional differential equations which are applicable to real-world problems.
The fractional differential equations describe different types of nonlocal dynamic systems in scientific and engineering fields such as biology, physics, chemistry, control theory, economics, and signal processing, etc. Fractional derivatives play a significant role in the formulation of models of nonlinear systems in real-life phenomena. Fractional models are useful to report different chaotic behaviour. There exist many theoretical results for checking the existence and uniqueness of fractional differential equations, but there is still significant scope to discuss the nonlinear fractional differential equations.

In this Special Issue, we will share the recent progress and advances in the different fields of nonlinear functional analysis and fractional differential equations, with the goal of identifying fruitful research directions and inspiring collaborations in this field.

The aim of this Special Issue is to focus on recent developments and achievements in nonlinear functional analysis and fractional differential equations, and identify their various applications. We invite authors to submit original research and review articles describing new methods and applications which are directly or indirectly related to nonlinear functional analysis, fractional differential equations, Banach spaces, function spaces, and sequence spaces, etc. We also welcome research including fixed-point theory, nonlinear operator theory, and nonlinear fractional dynamic equations. Potential topics include:

Nonlinear functional analysis and applications
Control theory and applications
Nonlinear fractional dynamic equations on timescale
Fixed point theory and applications
Modelling in ecological systems
Nonlinear dynamical systems and fractional calculus
Solvability of infinite systems of nonlinear fractional differential, integral, and Integro-differential equations on sequence spaces, function spaces
Nonlinear functionals and variational methods for non-linear operators

Prof. Dr. Bipan Hazarika
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

nonlinear functional analysis
control theory and applications
nonlinear fractional dynamic equations on timescale
fixed point theory
ecological systems
nonlinear dynamical systems
fractional calculus
solvability of infinite systems
sequence spaces, function spaces
nonlinear functionals and variational methods
nonlinear operators

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Related Special Issue

Advances in Nonlinear Functional Analysis on Fractional Differential Equations, 2nd Edition in Fractal and Fractional

Published Papers (12 papers)

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Research

15 pages, 304 KiB

Open AccessArticle

New Multiplicity Results for a Boundary Value Problem Involving a ψ-Caputo Fractional Derivative of a Function with Respect to Another Function

by Yankai Li, Dongping Li, Fangqi Chen and Xiangjing Liu

Fractal Fract. 2024, 8(6), 305; https://doi.org/10.3390/fractalfract8060305 - 22 May 2024

Viewed by 772

Abstract

This paper considers a nonlinear impulsive fractional boundary value problem, which involves a

ψ

This paper considers a nonlinear impulsive fractional boundary value problem, which involves a

ψ

-Caputo-type fractional derivative and integral. Combining critical point theory and fractional calculus properties, such as the semigroup laws, and relationships between the fractional integration and differentiation, new multiplicity results of infinitely many solutions are established depending on some simple algebraic conditions. Finally, examples are also presented, which show that Caputo-type fractional models can be more accurate by selecting different kernels for the fractional integral and derivative. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

21 pages, 786 KiB

Open AccessArticle

A Novel Technique for Solving the Nonlinear Fractional-Order Smoking Model

by Abdelhamid Mohammed Djaouti, Zareen A. Khan, Muhammad Imran Liaqat and Ashraf Al-Quran

Fractal Fract. 2024, 8(5), 286; https://doi.org/10.3390/fractalfract8050286 - 10 May 2024

Cited by 4 | Viewed by 961

Abstract

In the study of biological systems, nonlinear models are commonly employed, although exact solutions are often unattainable. Therefore, it is imperative to develop techniques that offer approximate solutions. This study utilizes the Elzaki residual power series method (ERPSM) to analyze the fractional nonlinear smoking model concerning the Caputo derivative. The outcomes of the proposed technique exhibit good agreement with the Laplace decomposition method, demonstrating that our technique is an excellent alternative to various series solution methods. Our approach utilizes the simple limit principle at zero, making it the easiest way to extract series solutions, while variational iteration, Adomian decomposition, and homotopy perturbation methods require integration. Moreover, our technique is also superior to the residual method by eliminating the need for derivatives, as fractional integration and differentiation are particularly challenging in fractional contexts. Significantly, our technique is simpler than other series solution techniques by not relying on Adomian’s and He’s polynomials, thereby offering a more efficient way of solving nonlinear problems. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

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20 pages, 406 KiB

Open AccessArticle

New Study on the Controllability of Non-Instantaneous Impulsive Hilfer Fractional Neutral Stochastic Evolution Equations with Non-Dense Domain

by Gunasekaran Gokul, Barakah Almarri, Sivajiganesan Sivasankar, Subramanian Velmurugan and Ramalingam Udhayakumar

Fractal Fract. 2024, 8(5), 265; https://doi.org/10.3390/fractalfract8050265 - 27 Apr 2024

Cited by 1 | Viewed by 1173

Abstract

The purpose of this work is to investigate the controllability of non-instantaneous impulsive (NII) Hilfer fractional (HF) neutral stochastic evolution equations with a non-dense domain. We construct a new set of adequate assumptions for the existence of mild solutions using fractional calculus, semigroup theory, stochastic analysis, and the fixed point theorem. Then, the discussion is driven by some suitable assumptions, including the Hille–Yosida condition without the compactness of the semigroup of the linear part. Finally, we provide examples to illustrate our main result. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

15 pages, 301 KiB

Open AccessArticle

Fixed Point Theorems: Exploring Applications in Fractional Differential Equations for Economic Growth

by Afrah Ahmad Noman Abdou

Fractal Fract. 2024, 8(4), 243; https://doi.org/10.3390/fractalfract8040243 - 22 Apr 2024

Cited by 4 | Viewed by 1201

Abstract

The aim of this research is to introduce two new notions,

Θ

Ξ

,h)-contraction and rational (

α

η)

ψ

-interpolative contraction, in the setting of

F

-metric space and to establish corresponding fixed point theorems. To [...] Read more.

The aim of this research is to introduce two new notions,

Θ

Ξ

,h)-contraction and rational (

α

η)

ψ

-interpolative contraction, in the setting of

F

-metric space and to establish corresponding fixed point theorems. To reinforce understanding and highlight the novelty of our findings, we provide a non-trivial example that not only supports the obtained results but also illuminates the established theory. Finally, we apply our main result to discuss the existence and uniqueness of solutions for a fractional differential equation describing an economic growth model. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

13 pages, 344 KiB

Open AccessArticle

On Some Impulsive Fractional Integro-Differential Equation with Anti-Periodic Conditions

by Ymnah Alruwaily, Kuppusamy Venkatachalam and El-sayed El-hady

Fractal Fract. 2024, 8(4), 219; https://doi.org/10.3390/fractalfract8040219 - 10 Apr 2024

Cited by 2 | Viewed by 1069

Abstract

We investigate a class of boundary value problems (BVPs) involving an impulsive fractional integro-differential equation (IF-IDE) with the Caputo–Hadamard fractional derivative (C-HFD). We employ some fixed-point theorems (FPTs) to study the existence of this fractional BVP and its unique solution. The boundary conditions (BCs) established in this study are of a more general type and can be reduced to numerous specific examples by defining the parameters involved in the conditions. In this way, we extend some recent nice results. At the end, we use an example to verify our results. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

8 pages, 254 KiB

Open AccessArticle

A Study of an IBVP of Fractional Differential Equations in Banach Space via the Measure of Noncompactness

by Mouataz Billah Mesmouli, Amjad E. Hamza and Doaa Rizk

Fractal Fract. 2024, 8(1), 30; https://doi.org/10.3390/fractalfract8010030 - 29 Dec 2023

Cited by 2 | Viewed by 1432

Abstract

In this article, we are concerned with a very general integral boundary value problem of Riemann–Liouville derivatives. We will study the problem in Banach space. To be more specific, we are interested in proving the existence of a solution to our problem via the measure of noncompactness and Mönch fixed-point theorem. Our study in Banach space contains two nonlinear terms and two different orders of derivatives,

ς

and

τ

, such that

ς \in (1, 2]

and

τ \in (0, ς]

. Our paper ends with a conclusion. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

25 pages, 486 KiB

Open AccessArticle

Controlled Extended Branciari Quasi-b-Metric Spaces, Results, and Applications to Riesz-Caputo Fractional Differential Equations and Nonlinear Matrix Equations

by Reena Jain, Hemant Kumar Nashine and Reny George

Fractal Fract. 2024, 8(1), 20; https://doi.org/10.3390/fractalfract8010020 - 26 Dec 2023

Viewed by 1215

Abstract

We introduce the concept of controlled extended Branciari quasi-b-metric spaces, as well as a

G_{q}

-implicit type mapping. Under this new space setting, we derive some new fixed points, periodic points, right and left Ulam–Hyers stability, right and left weak [...] Read more.

We introduce the concept of controlled extended Branciari quasi-b-metric spaces, as well as a

G_{q}

-implicit type mapping. Under this new space setting, we derive some new fixed points, periodic points, right and left Ulam–Hyers stability, right and left weak well-posed properties, and right and left weak limit shadowing results. Additionally, we use these findings to solve the fractional differential equations of a Riesz–Caputo type with integral anti-periodic boundary values, as well of nonlinear matrix equations. All ideas, results, and applications are properly illustrated with examples. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

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26 pages, 740 KiB

Open AccessArticle

Solitary Waves Propagation Analysis in Nonlinear Dynamical System of Fractional Coupled Boussinesq-Whitham-Broer-Kaup Equation

by M. Mossa Al-Sawalha, Safyan Mukhtar, Rasool Shah, Abdul Hamid Ganie and Khaled Moaddy

Fractal Fract. 2023, 7(12), 889; https://doi.org/10.3390/fractalfract7120889 - 18 Dec 2023

Cited by 10 | Viewed by 1638

Abstract

The primary goal of this study is to create and characterise solitary wave solutions for the conformable Fractional Coupled Boussinesq-Whitham-Broer-Kaup Equations (FCBWBKEs), a model that governs shallow water waves. Through wave transformations and the chain rule, the authors used the modified Extended Direct Algebraic Method (mEDAM) for transforming FCBWBKEs into a more manageable Nonlinear Ordinary Differential Equation (NODE). This accomplishment is particularly noteworthy because it surpasses the drawbacks linked to both the Caputo and Riemann–Liouville definitions in complying to the chain rule. The study uses visual representations such as 3D, 2D, and contour graphs to demonstrate the dynamic nature of solitary wave solutions. Furthermore, the investigation of diverse wave phenomena such as kinks, shock waves, periodic waves, and bell-shaped kink waves highlights the range of knowledge obtained in the study of shallow water wave behavior. Overall, this study introduces novel methodologies that produce valuable and consistent results for the problem under consideration. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

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29 pages, 646 KiB

Open AccessArticle

Exploring Integral ϝ-Contractions with Applications to Integral Equations and Fractional BVPs

by Zubair Nisar, Nayyar Mehmood, Akbar Azam, Faryad Ali and Mohammed A. Al-Kadhi

Fractal Fract. 2023, 7(12), 833; https://doi.org/10.3390/fractalfract7120833 - 24 Nov 2023

Viewed by 1405

Abstract

In this article, two types of contractive conditions are introduced, namely extended integral

Ϝ

-contraction and

(ϰ, Ω

Ϝ)

-contraction. For the case of two mappings and their coincidence point theorems, a variant of

(ϰ, Ω

- [...] Read more.

In this article, two types of contractive conditions are introduced, namely extended integral

Ϝ

-contraction and

(ϰ, Ω

Ϝ)

-contraction. For the case of two mappings and their coincidence point theorems, a variant of

(ϰ, Ω

Ϝ)

-contraction has been introduced, which is called

(ϰ, Γ_{1, 2}, Ω

Ϝ)

-contraction. In the end, the applications of an extended integral

Ϝ

-contraction and

(ϰ, Ω

Ϝ)

-contraction are given by providing an existence result in the solution of a fractional order multi-point boundary value problem involving the Riemann–Liouville fractional derivative. An interesting existence result for the solution of the nonlinear Fredholm integral equation of the second kind using the

(ϰ, Γ_{1, 2}, Ω

Ϝ)

-contraction has been proven. Herein, an example is established that explains how the Picard–Jungck sequence converges to the solution of the nonlinear integral equation. Examples are given for almost all the main results and some graphs are plotted where required. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

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18 pages, 336 KiB

Open AccessArticle

Certain Interpolative Proximal Contractions, Best Proximity Point Theorems in Bipolar Metric Spaces with Applications

by Fahad Jahangeer, Salha Alshaikey, Umar Ishtiaq, Tania A. Lazăr, Vasile L. Lazăr and Liliana Guran

Fractal Fract. 2023, 7(10), 766; https://doi.org/10.3390/fractalfract7100766 - 19 Oct 2023

Viewed by 1329

Abstract

In this manuscript, we present several types of interpolative proximal contraction mappings including Reich–Rus–Ciric-type interpolative-type contractions and Kannan-type interpolative-type contractions in the setting of bipolar metric spaces. Further, taking into account the aforementioned mappings, we prove best proximity point results. These results are an extension and generalization of existing ones in the literature. Furthermore, we provide several nontrivial examples, an application to find the solution of an integral equation, and a nonlinear fractional differential equation to show the validity of the main results. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

24 pages, 438 KiB

Open AccessArticle

Solvability of a ϱ-Hilfer Fractional Snap Dynamic System on Unbounded Domains

by Sabri T. M. Thabet, Miguel Vivas-Cortez, Imed Kedim, Mohammad Esmael Samei and M. Iadh Ayari

Fractal Fract. 2023, 7(8), 607; https://doi.org/10.3390/fractalfract7080607 - 7 Aug 2023

Cited by 20 | Viewed by 1187

Abstract

This paper is devoted to studying the

ϱ

-Hilfer fractional snap dynamic system under the

ϱ

-Riemann–Liouville fractional integral conditions on unbounded domains

[a, \infty), a \geq 0

, for the first time. The results concerning the existence and [...] Read more.

This paper is devoted to studying the

ϱ

-Hilfer fractional snap dynamic system under the

ϱ

-Riemann–Liouville fractional integral conditions on unbounded domains

[a, \infty), a \geq 0

, for the first time. The results concerning the existence and uniqueness, along with the Ulam–Hyers, Ulam–Hyers–Rassias, and semi-Ulam–Hyers–Rassias stabilities, are established in an appropriate special Banach space according to fractional calculus, fixed point theory, and nonlinear analysis. At the end, a numerical example is presented for the interpretation of the main results. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

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19 pages, 805 KiB

Open AccessArticle

Solving Generalized Heat and Generalized Laplace Equations Using Fractional Fourier Transform

by Sri Sulasteri, Mawardi Bahri, Nasrullah Bachtiar, Jeffry Kusuma and Agustinus Ribal

Fractal Fract. 2023, 7(7), 557; https://doi.org/10.3390/fractalfract7070557 - 18 Jul 2023

Cited by 1 | Viewed by 1620

Abstract

In the present work, the main objective is to find the solution of the generalized heat and generalized Laplace equations using the fractional Fourier transform, which is a general form of the solution of the heat equation and Laplace equation using the classical Fourier transform. We also formulate its solution using a sampling formula related to the fractional Fourier transform. The fractional Fourier transform is introduced, and related theorems and essential properties are collected. Several results related to the sampling formula are derived. A few examples are presented to illustrate the effectiveness and powerfulness of the proposed method compared to the classical Fourier transform method. Full article

(This article belongs to the Special Issue Advances in Nonlinear Functional Analysis on Fractional Differential Equations)

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