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Fractal Fract
Special Issues
Advances in Fractional Initial and Boundary Value Problems
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Advances in Fractional Initial and Boundary Value Problems

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

Deadline for manuscript submissions: 20 August 2026 | Viewed by 5585

Special Issue Editors

Prof. Dr. Sotiris K. Ntouyas

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Guest Editor
Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Interests: initial value problems; boundary value problems; inequalities
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Bashir Ahmad

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Guest Editor
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
Interests: boundary value problems; ordinary & partial differential equations; fractional differential equations; analytical and numerical methods for nonlinear problems; methods of functional analysis; stability theory; applications in energy problems; ecology; fluid mechanics; acoustic scattering; disease models
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Jessada Tariboon

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Guest Editor
Department of Mathematics, King Mongkut's University of Technology North Bangkok, Bangkok, Thailand
Interests: differential equations; boundary value problems; nonlinear analysis applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The importance of initial and boundary value problems for different kinds of fractional differential equations (ordinary, partial, difference, functional, stochastic, integral, integro-differential, etc.) is well recognized in view of their extensive applications in applied sciences and engineering.

Single-valued and multi-valued initial and boundary value problems involving different kinds of boundary conditions have attracted significant attention during the last few decades. The literature on this topic is now more enriched and contains a variety of results ranging from existence theory to the methods of solutions for such problems. The techniques of functional analysis and fixed-point theory play a key role in proving the existence and uniqueness of solutions to these problems.

The aim of this Special Issue is to strengthen the available literature on the topic by publishing research and review articles on initial and boundary value problems of differential equations and inclusions in a broader sense.

Potential topics include but are not limited to the following:

Existence, uniqueness, and multiplicity results for initial and boundary value problems for differential equations and inclusions (ordinary, functional, fractional, partial, difference, stochastic, integral, etc.).

Also, please feel free to read and download all the published articles in our first volume:
https://www.mdpi.com/journal/fractalfract/special_issues/boundary_value

Prof. Dr. Sotiris K. Ntouyas
Prof. Dr. Bashir Ahmad
Prof. Dr. Jessada Tariboon
Guest Editors

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 250 words) can be sent to the Editorial Office for assessment.

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

  • qualitative properties of the solutions (positivity, oscillation, asymptotic behavior, stability, etc.)
  • topological methods in differential equations and inclusions
  • approximation of the solutions
  • eigenvalue problems
  • variational methods
  • fixed point theory
  • critical point theory
  • applications to real-world phenomena

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  • Reprint: MDPI Books provides the opportunity to republish successful Special Issues in book format, both online and in print.

Further information on MDPI's Special Issue policies can be found here.

Published Papers (7 papers)

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Research

21 pages, 343 KB  
Article
Results on Extremal Solutions for a Class of Boundary Value Problem of Nonlinear Fractional Order Differential Equations
by Yue Du, Yumei Zou and Yujun Cui
Fractal Fract. 2026, 10(5), 316; https://doi.org/10.3390/fractalfract10050316 - 7 May 2026
Viewed by 246
Abstract
This paper investigates a class of boundary value problems involving Caputo fractional derivatives of order ν∈(2,3]. We begin by establishing two novel comparison principles. Subsequently, by employing the monotone iterative technique coupled with upper and lower [...] Read more.
This paper investigates a class of boundary value problems involving Caputo fractional derivatives of order ν∈(2,3]. We begin by establishing two novel comparison principles. Subsequently, by employing the monotone iterative technique coupled with upper and lower solutions, we demonstrate the existence of extremal solutions for the corresponding fractional differential equations. Finally, an illustrative example is provided to validate our main findings. Full article
(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)
24 pages, 387 KB  
Article
Structural Analysis of Coupled ψ-Hilfer Pantograph Langevin Systems via Measure of Noncompactness
by Muath Awadalla and Dalal Alhwikem
Fractal Fract. 2026, 10(3), 201; https://doi.org/10.3390/fractalfract10030201 - 18 Mar 2026
Viewed by 310
Abstract
This paper investigates a class of coupled ψ-Hilfer fractional pantograph–Langevin equations with nonlocal integral boundary conditions. By reformulating the problem as an equivalent fixed point equation and employing Mönch’s fixed point theorem together with the Kuratowski measure of noncompactness, we establish sufficient [...] Read more.
This paper investigates a class of coupled ψ-Hilfer fractional pantograph–Langevin equations with nonlocal integral boundary conditions. By reformulating the problem as an equivalent fixed point equation and employing Mönch’s fixed point theorem together with the Kuratowski measure of noncompactness, we establish sufficient conditions for the existence of at least one solution. Under additional Lipschitz-type assumptions, we prove Ulam–Hyers stability on a suitable closed ball and derive explicit, computable stability constants. A concrete numerical example is presented in which all hypotheses are verified and the stability constants are explicitly computed (e.g., K1≈3.811, K2≈2.761), illustrating the applicability of the theoretical results. The study contributes additional qualitative results to the analysis of fractional pantograph–Langevin systems within the unified framework of ψ-Hilfer fractional derivatives. Full article
(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)
36 pages, 3788 KB  
Article
Mittag-Leffler Weighted Orthogonal Functions for the ABC Fractional Operator: A Formal Self-Adjointness Construction
by Muath Awadalla and Dalal Alhwikem
Fractal Fract. 2026, 10(3), 185; https://doi.org/10.3390/fractalfract10030185 - 11 Mar 2026
Viewed by 335
Abstract
This work constructs an orthogonal function system on bounded intervals [0,R] associated with the Atangana–Baleanu–Caputo (ABC) fractional derivative for α∈(1/2,1). Starting from a fractional Laguerre-type equation involving the ABC operator, [...] Read more.
This work constructs an orthogonal function system on bounded intervals [0,R] associated with the Atangana–Baleanu–Caputo (ABC) fractional derivative for α∈(1/2,1). Starting from a fractional Laguerre-type equation involving the ABC operator, solutions are obtained via a generalized Frobenius method, yielding series representations with characteristic exponent α−1. Rather than postulating a weight function by analogy with classical or Caputo settings, the weight is derived directly from the requirement that the underlying fractional operator be formally self-adjoint on a suitable admissible domain. This operator-theoretic approach leads to the explicit Mittag–Leffler weight wα(x)=x−(2α−1)Eα(−xα), which intrinsically reflects the nonlocal memory structure of the ABC kernel. A similarity transformation removes the universal singular factor and produces regularized eigenfunctions that are continuous on [0,R] and orthogonal in the weighted L2 space. The weight identity and formal self-adjointness are rigorously verified through a right-Volterra uniqueness argument. Numerical experiments confirm orthogonality up to machine precision, demonstrate spectral convergence for a model ABC differential equation, and illustrate consistency with classical Laguerre polynomials in the limit α→1−. The resulting framework provides a self-consistent orthogonal system suitable for spectral approximations of problems governed by the ABC operator on bounded domains. Full article
(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)
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16 pages, 330 KB  
Article
New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity
by Yuan Gao, Lishan Liu, Na Wei, Haibo Gu and Yonghong Wu
Fractal Fract. 2025, 9(10), 646; https://doi.org/10.3390/fractalfract9100646 - 4 Oct 2025
Cited by 1 | Viewed by 731
Abstract
By using the Ekeland variational principle and Nehari manifold, we study the following fractional p-Laplacian Kirchhoff equations: [...] Read more.
By using the Ekeland variational principle and Nehari manifold, we study the following fractional p-Laplacian Kirchhoff equations: M[u]s,pp+∫RNV(x)|u|pdx[(−Δ)psu+V(x)|u|p−2u]=λ|u|q−2uln|u|,x∈RN,(P). In these equations, λ∈R∖{0},p∈(1,+∞), s∈(0,1),sp<N,ps*=NpN−sp, M(τ)=a+bτθ−1, a,b∈R+,1<θ<ps*p, V(x)∈C(RN,R) is a potential function and (−Δ)ps is the fractional p-Laplacian operator. The existence of solutions is deeply influenced by the positive and negative signs of λ. More precisely, (i) Equation (P) has one ground state solution for λ>0 and pθ<q<ps*, with a positive corresponding energy value; and (ii) Equation (P) has at least two nontrivial solutions for λ<0 and p<q<ps*, with positive and negative corresponding energy values, respectively. Full article
(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)
12 pages, 267 KB  
Article
Extremal Solutions for a Caputo-Type Fractional-Order Initial Value Problem
by Keyu Zhang, Tian Wang, Donal O’Regan and Jiafa Xu
Fractal Fract. 2025, 9(5), 308; https://doi.org/10.3390/fractalfract9050308 - 10 May 2025
Viewed by 917
Abstract
In this paper, we study the existence of extremal solutions for a Caputo-type fractional-order initial value problem. By using the monotone iteration technique and the upper–lower solution method, we obtain our existence theorem when the nonlinearity satisfies a reverse-type Lipschitz condition. Note that [...] Read more.
In this paper, we study the existence of extremal solutions for a Caputo-type fractional-order initial value problem. By using the monotone iteration technique and the upper–lower solution method, we obtain our existence theorem when the nonlinearity satisfies a reverse-type Lipschitz condition. Note that our nonlinearity depends on the unknown function and its fractional-order derivative. Full article
(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)
21 pages, 370 KB  
Article
A Study of a Nonlocal Coupled Integral Boundary Value Problem for Nonlinear Hilfer–Hadamard-Type Fractional Langevin Equations
by Bashir Ahmad, Hafed A. Saeed and Sotiris K. Ntouyas
Fractal Fract. 2025, 9(4), 229; https://doi.org/10.3390/fractalfract9040229 - 4 Apr 2025
Cited by 4 | Viewed by 1087
Abstract
We discuss the existence criteria and Ulam–Hyers stability for solutions to a nonlocal integral boundary value problem of nonlinear coupled Hilfer–Hadamard-type fractional Langevin equations. Our results rely on the Leray–Schauder alternative and Banach’s fixed point theorem. Examples are included to illustrate the results [...] Read more.
We discuss the existence criteria and Ulam–Hyers stability for solutions to a nonlocal integral boundary value problem of nonlinear coupled Hilfer–Hadamard-type fractional Langevin equations. Our results rely on the Leray–Schauder alternative and Banach’s fixed point theorem. Examples are included to illustrate the results obtained. Full article
(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)
11 pages, 269 KB  
Article
A Fractional Dirac System with Eigenparameter-Dependent and Transmission Conditions
by Abdullah Kablan and Fulya Şahantürk
Fractal Fract. 2025, 9(4), 227; https://doi.org/10.3390/fractalfract9040227 - 3 Apr 2025
Viewed by 827
Abstract
This work investigates the fractional Dirac system that has transmission conditions, and its boundary condition contains an eigenparameter. Defining a convenient inner product space and a new operator that has the same eigenvalues as the considered problem, we demonstrate that the fractional Dirac [...] Read more.
This work investigates the fractional Dirac system that has transmission conditions, and its boundary condition contains an eigenparameter. Defining a convenient inner product space and a new operator that has the same eigenvalues as the considered problem, we demonstrate that the fractional Dirac system is symmetric in this space. Thus, we have reached some remarkable results for the spectral characteristics of the operator. Furthermore, in the next section of the study, the existence of solutions was examined. Full article
(This article belongs to the Special Issue Advances in Fractional Initial and Boundary Value Problems)
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