Initial and Boundary Value Problems for Differential Equations

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 (30 June 2023) | Viewed by 28416

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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
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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 Issue Information

Dear Colleagues,

The importance of initial and boundary value problems of different kinds of differential equations (ordinary, functional, fractional, partial, difference, stochastic, integral, 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 much enriched and contains a variety of results ranging from the existence theory to the methods of solution 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:

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

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

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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

Published Papers (22 papers)

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Research

Jump to: Review, Other

10 pages, 326 KiB  
Article
Precise Conditions on the Unique Solvability of the Linear Fractional Functional Differential Equations Related to the ς-Nonpositive Operators
by Natalia Dilna
Fractal Fract. 2023, 7(10), 720; https://doi.org/10.3390/fractalfract7100720 - 29 Sep 2023
Viewed by 561
Abstract
Exact conditions for the existence of the unique solution of a boundary value problem for linear fractional functional differential equations related to ς-nonpositive operators are established. The exact solvability conditions are based on the a priori estimation method. All theoretical investigations are [...] Read more.
Exact conditions for the existence of the unique solution of a boundary value problem for linear fractional functional differential equations related to ς-nonpositive operators are established. The exact solvability conditions are based on the a priori estimation method. All theoretical investigations are illustrated by an example of the pantograph-type model from electrodynamics. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
17 pages, 356 KiB  
Article
Nonlinear Integral Inequalities Involving Tempered Ψ-Hilfer Fractional Integral and Fractional Equations with Tempered Ψ-Caputo Fractional Derivative
by Milan Medveď, Michal Pospíšil and Eva Brestovanská
Fractal Fract. 2023, 7(8), 611; https://doi.org/10.3390/fractalfract7080611 - 8 Aug 2023
Cited by 1 | Viewed by 814
Abstract
In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered Ψ-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered Ψ-Hilfer fractional integrals, are [...] Read more.
In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered Ψ-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered Ψ-Hilfer fractional integrals, are presented as corollaries. To illustrate the results, the problem of the nonexistence of blowing-up solutions of initial value problems for fractional differential equations with tempered Ψ-Caputo fractional derivative of order 0<α<1, where the right side may depend on time, the solution, or its tempered Ψ-Caputo fractional derivative of lower order, is investigated. As another application of the integral inequalities, sufficient conditions for the Ψ-exponential stability of trivial solutions are proven for these kinds of differential equations. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
14 pages, 380 KiB  
Article
On Pantograph Problems Involving Weighted Caputo Fractional Operators with Respect to Another Function
by Saeed M. Ali
Fractal Fract. 2023, 7(7), 559; https://doi.org/10.3390/fractalfract7070559 - 19 Jul 2023
Cited by 2 | Viewed by 986
Abstract
In this investigation, weighted psi-Caputo fractional derivatives are applied to analyze the solution of fractional pantograph problems with boundary conditions. We establish the existence of solutions to the indicated pantograph equations as well as their uniqueness. The study also takes [...] Read more.
In this investigation, weighted psi-Caputo fractional derivatives are applied to analyze the solution of fractional pantograph problems with boundary conditions. We establish the existence of solutions to the indicated pantograph equations as well as their uniqueness. The study also takes into account the situation where ψ(x)=x. With the aid of Banach’s and Krasnoselskii’s classic fixed point results, we have established a the qualitative study. Different values of ψ(x) and w(x) are discussed as special cases that are relevant to our current results. Additionally, in light of our findings, we provide a more general fractional system with the weighted ψ-Caputo-type that takes into account both the new problems and certain previously existing, related problems. Finally, we give two illustrations to support and validate the outcomes. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
21 pages, 886 KiB  
Article
The Hölder Regularity for Abstract Fractional Differential Equation with Applications to Rayleigh–Stokes Problems
by Jiawei He and Guangmeng Wu
Fractal Fract. 2023, 7(7), 549; https://doi.org/10.3390/fractalfract7070549 - 16 Jul 2023
Cited by 1 | Viewed by 798
Abstract
In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this [...] Read more.
In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this article was to establish the Hölder regularities of mild solutions, classical solutions, and strict solutions. We introduced an interpolation space in terms of an analytic resolvent to lower the spatial regularity of initial value data. By virtue of the properties of analytic resolvent and the interpolation space, the Hölder regularities were obtained. As applications, the main conclusions were applied to the regularities of fractional Rayleigh–Stokes problems. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
23 pages, 385 KiB  
Article
The Existence of Mild Solutions for Hilfer Fractional Stochastic Evolution Equations with Order μ∈(1,2)
by Qien Li and Yong Zhou
Fractal Fract. 2023, 7(7), 525; https://doi.org/10.3390/fractalfract7070525 - 2 Jul 2023
Cited by 3 | Viewed by 831
Abstract
In this study, we investigate the existence of mild solutions for a class of Hilfer fractional stochastic evolution equations with order μ(1,2) and type ν[0,1]. We prove the existence of [...] Read more.
In this study, we investigate the existence of mild solutions for a class of Hilfer fractional stochastic evolution equations with order μ(1,2) and type ν[0,1]. We prove the existence of mild solutions of Hilfer fractional stochastic evolution equations when the semigroup is compact as well as noncompact. Our approach is based on the Schauder fixed point theorem, the Ascoli–Arzelà theorem and the Kuratowski measure of noncompactness. An example is also provided, to demonstrate the efficacy of this method. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
32 pages, 422 KiB  
Article
New Variant of Hermite–Hadamard, Fejér and Pachpatte-Type Inequality and Its Refinements Pertaining to Fractional Integral Operator
by Muhammad Tariq, Sotiris K. Ntouyas and Asif Ali Shaikh
Fractal Fract. 2023, 7(5), 405; https://doi.org/10.3390/fractalfract7050405 - 16 May 2023
Cited by 3 | Viewed by 1355
Abstract
In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, [...] Read more.
In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the (m,tgs)-type s-preinvex function, Godunova–Levin (s,m)-preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite–Hadamard (H–H), Fejér, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H–H and Fejér-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
14 pages, 1216 KiB  
Article
Qualitatively Stable Schemes for the Black–Scholes Equation
by Mohammad Mehdizadeh Khalsaraei, Ali Shokri, Yuanheng Wang, Sohrab Bazm, Giti Navidifar and Pari Khakzad
Fractal Fract. 2023, 7(2), 154; https://doi.org/10.3390/fractalfract7020154 - 4 Feb 2023
Cited by 2 | Viewed by 1042
Abstract
In this paper, the Black–Scholes equation is solved using a new technique. This scheme is derived by combining the Laplace transform method and the nonstandard finite difference (NSFD) strategy. The qualitative properties of the method are discussed, and it is shown that the [...] Read more.
In this paper, the Black–Scholes equation is solved using a new technique. This scheme is derived by combining the Laplace transform method and the nonstandard finite difference (NSFD) strategy. The qualitative properties of the method are discussed, and it is shown that the new method is positive, stable, and consistent when low volatility is assumed. The efficiency of the new method is demonstrated by a numerical example. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
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11 pages, 310 KiB  
Article
Existence Results for a Differential Equation Involving the Right Caputo Fractional Derivative and Mixed Nonlinearities with Nonlocal Closed Boundary Conditions
by Bashir Ahmad, Manal Alnahdi and Sotiris K. Ntouyas
Fractal Fract. 2023, 7(2), 129; https://doi.org/10.3390/fractalfract7020129 - 30 Jan 2023
Cited by 4 | Viewed by 1374
Abstract
In this study, we present a new notion of nonlocal closed boundary conditions. Equipped with these conditions, we discuss the existence of solutions for a mixed nonlinear differential equation involving a right Caputo fractional derivative operator, and left and right Riemann–Liouville fractional integral [...] Read more.
In this study, we present a new notion of nonlocal closed boundary conditions. Equipped with these conditions, we discuss the existence of solutions for a mixed nonlinear differential equation involving a right Caputo fractional derivative operator, and left and right Riemann–Liouville fractional integral operators of different orders. We apply a decent and fruitful approach of fixed point theory to establish the desired results. Examples are given for illustration of the main results. The paper concludes with some interesting observations. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
17 pages, 343 KiB  
Article
Nonlocal Problems for Hilfer Fractional q-Difference Equations
by Chunping Tian, Haibo Gu and Zunkai Yang
Fractal Fract. 2023, 7(2), 101; https://doi.org/10.3390/fractalfract7020101 - 17 Jan 2023
Viewed by 970
Abstract
In the paper, we investigate a kind of Hilfer fractional q-difference equations with nonlocal condition. Firstly, the existence and uniqueness results of solutions are obtained by using topological degree theory and Banach fixed point theorem. Subsequently, the existence of extremal solutions in an [...] Read more.
In the paper, we investigate a kind of Hilfer fractional q-difference equations with nonlocal condition. Firstly, the existence and uniqueness results of solutions are obtained by using topological degree theory and Banach fixed point theorem. Subsequently, the existence of extremal solutions in an ordered Banach space is discussed by monotone iterative method. In that following, we consider the Ulam stability results for equations. Finally, two examples are given to illustrate the effectiveness of theory results. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
10 pages, 314 KiB  
Article
Unique Solvability of the Initial-Value Problem for Fractional Functional Differential Equations—Pantograph-Type Model
by Natalia Dilna
Fractal Fract. 2023, 7(1), 65; https://doi.org/10.3390/fractalfract7010065 - 5 Jan 2023
Cited by 1 | Viewed by 916
Abstract
Contrary to the initial-value problem for ordinary differential equations, where the classical theory of establishing the exact unique solvability conditions exists, the situation with the initial-value problem for linear functional differential equations of the fractional order is usually non-trivial. Here we establish the [...] Read more.
Contrary to the initial-value problem for ordinary differential equations, where the classical theory of establishing the exact unique solvability conditions exists, the situation with the initial-value problem for linear functional differential equations of the fractional order is usually non-trivial. Here we establish the unique solvability conditions for the initial-value problem for linear functional differential equations of the fractional order. The advantage is the lack of the calculation of fractional derivatives, which is a complicated task. The unique solution is represented by the Neumann series. In addition, as examples, the model with a discrete memory effect and a pantograph-type model from electrodynamics are studied. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
20 pages, 380 KiB  
Article
A Three-Field Variational Formulation for a Frictional Contact Problem with Prescribed Normal Stress
by Andaluzia Cristina Matei
Fractal Fract. 2022, 6(11), 651; https://doi.org/10.3390/fractalfract6110651 - 4 Nov 2022
Cited by 3 | Viewed by 1013
Abstract
In the present work, we address a nonlinear boundary value problem that models frictional contact with prescribed normal stress between a deformable body and a foundation. The body is nonlinearly elastic, the constitutive law being a subdifferential inclusion. We deliver a three-field variational [...] Read more.
In the present work, we address a nonlinear boundary value problem that models frictional contact with prescribed normal stress between a deformable body and a foundation. The body is nonlinearly elastic, the constitutive law being a subdifferential inclusion. We deliver a three-field variational formulation by means of a new variational approach governed by the theory of bipotentials combined with a Lagrange-multipliers technique. In this new approach, the unknown of the mechanical model is a triple consisting of the displacement field, a Lagrange multiplier related to the friction force and the Cauchy stress tensor. We obtain existence, uniqueness, boundedness and convergence results. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
11 pages, 308 KiB  
Article
Qualitative Properties of Positive Solutions of a Kind for Fractional Pantograph Problems using Technique Fixed Point Theory
by Hamid Boulares, Abbes Benchaabane, Nuttapol Pakkaranang, Ramsha Shafqat and Bancha Panyanak
Fractal Fract. 2022, 6(10), 593; https://doi.org/10.3390/fractalfract6100593 - 14 Oct 2022
Cited by 12 | Viewed by 1390
Abstract
The current paper intends to report the existence and uniqueness of positive solutions for nonlinear pantograph Caputo–Hadamard fractional differential equations. As part of a procedure, we transform the specified pantograph fractional differential equation into an equivalent integral equation. We show that this equation [...] Read more.
The current paper intends to report the existence and uniqueness of positive solutions for nonlinear pantograph Caputo–Hadamard fractional differential equations. As part of a procedure, we transform the specified pantograph fractional differential equation into an equivalent integral equation. We show that this equation has a positive solution by utilising the Schauder fixed point theorem (SFPT) and the upper and lower solutions method. Another method for proving the existence of a singular positive solution is the Banach fixed point theorem (BFPT). Finally, we provide an example that illustrates and explains our conclusions. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
12 pages, 285 KiB  
Article
Results for Fuzzy Mappings and Stability of Fuzzy Sets with Applications
by Aqeel Shahzad, Abdullah Shoaib, Nabil Mlaiki and Suhad Subhi Aiadi
Fractal Fract. 2022, 6(10), 556; https://doi.org/10.3390/fractalfract6100556 - 30 Sep 2022
Cited by 3 | Viewed by 990
Abstract
The purpose of this paper is to develop some fuzzy fixed point results for the sequence of locally fuzzy mappings satisfying rational type almost contractions in complete dislocated metric spaces. We apply our results to obtain new results for set-valued and single-valued mappings. [...] Read more.
The purpose of this paper is to develop some fuzzy fixed point results for the sequence of locally fuzzy mappings satisfying rational type almost contractions in complete dislocated metric spaces. We apply our results to obtain new results for set-valued and single-valued mappings. We also study the stability of fuzzy fixed point γ-level sets. An example is presented in favor of these results. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
21 pages, 550 KiB  
Article
Hermite Fitted Block Integrator for Solving Second-Order Anisotropic Elliptic Type PDEs
by Emmanuel Oluseye Adeyefa, Ezekiel Olaoluwa Omole, Ali Shokri and Shao-Wen Yao
Fractal Fract. 2022, 6(9), 497; https://doi.org/10.3390/fractalfract6090497 - 5 Sep 2022
Cited by 3 | Viewed by 1324
Abstract
A Hermite fitted block integrator (HFBI) for numerically solving second-order anisotropic elliptic partial differential equations (PDEs) was developed, analyzed, and implemented in this study. The method was derived through collocation and interpolation techniques using the Hermite polynomial as the basis function. The Hermite [...] Read more.
A Hermite fitted block integrator (HFBI) for numerically solving second-order anisotropic elliptic partial differential equations (PDEs) was developed, analyzed, and implemented in this study. The method was derived through collocation and interpolation techniques using the Hermite polynomial as the basis function. The Hermite polynomial was interpolated at the first two successive points, while the collocation occurred at all the suitably chosen points. The major scheme and its complementary scheme were united together to form the HFBI. The analysis of the HFBI showed that it had a convergence order of eight with small error constants, was zero-stable, absolutely-stable, and satisfied the condition for convergence. In order to confirm the usefulness, accuracy, and efficiency of the HFBI, the method of lines approach was applied to discretize the second-order anisotropic elliptic partial differential equation PDE into a system of second-order ODEs and consequently used the derived HFBI to obtain the approximate solutions for the PDEs. The computed solution generated by using the HFBI was compared to the exact solutions of the problems and other existing methods in the literature. The proposed method compared favorably with other existing methods, which were validated through test problems whose solutions are presented in tabular form, and the comparisons are illustrated in the curves. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
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11 pages, 294 KiB  
Article
Solution of the Ill-Posed Cauchy Problem for Systems of Elliptic Type of the First Order
by Davron Aslonqulovich Juraev, Ali Shokri and Daniela Marian
Fractal Fract. 2022, 6(7), 358; https://doi.org/10.3390/fractalfract6070358 - 26 Jun 2022
Cited by 4 | Viewed by 1046
Abstract
We study, in this paper, the Cauchy problem for matrix factorizations of the Helmholtz equation in the space Rm. Based on the constructed Carleman matrix, we find an explicit form of the approximate solution of this problem and prove the stability [...] Read more.
We study, in this paper, the Cauchy problem for matrix factorizations of the Helmholtz equation in the space Rm. Based on the constructed Carleman matrix, we find an explicit form of the approximate solution of this problem and prove the stability of the solutions. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
16 pages, 330 KiB  
Article
Radially Symmetric Solution for Fractional Laplacian Systems with Different Negative Powers
by Haiyong Xu, Bashir Ahmad, Guotao Wang and Lihong Zhang
Fractal Fract. 2022, 6(7), 352; https://doi.org/10.3390/fractalfract6070352 - 23 Jun 2022
Cited by 1 | Viewed by 1150
Abstract
By developing the direct method of moving planes, we study the radial symmetry of nonnegative solutions for a fractional Laplacian system with different negative powers: [...] Read more.
By developing the direct method of moving planes, we study the radial symmetry of nonnegative solutions for a fractional Laplacian system with different negative powers: (Δ)α2u(x)+uγ(x)+vq(x)=0,xRN, (Δ)β2v(x)+vσ(x)+up(x)=0,xRN, u(x)|x|a,v(x)|x|bas|x|, where α,β(0,2), and a,b>0 are constants. We study the decay at infinity and narrow region principle for the fractional Laplacian system with different negative powers. The same results hold for nonlinear Hénon-type fractional Laplacian systems with different negative powers. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
28 pages, 413 KiB  
Article
Existence and Stability Results for a Tripled System of the Caputo Type with Multi-Point and Integral Boundary Conditions
by Murugesan Manigandan, Muthaiah Subramanian, Thangaraj Nandha Gopal and Bundit Unyong
Fractal Fract. 2022, 6(6), 285; https://doi.org/10.3390/fractalfract6060285 - 26 May 2022
Cited by 8 | Viewed by 1536
Abstract
In this paper, we introduce and investigate the existence and stability of a tripled system of sequential fractional differential equations (SFDEs) with multi-point and integral boundary conditions. The existence and uniqueness of the solutions are established by the principle of Banach’s contraction and [...] Read more.
In this paper, we introduce and investigate the existence and stability of a tripled system of sequential fractional differential equations (SFDEs) with multi-point and integral boundary conditions. The existence and uniqueness of the solutions are established by the principle of Banach’s contraction and the alternative of Leray–Schauder. The stability of the Hyer–Ulam solutions are investigated. A few examples are provided to identify the major results. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
17 pages, 356 KiB  
Article
Nonlocal Coupled System for (k,φ)-Hilfer Fractional Differential Equations
by Ayub Samadi, Sotiris K. Ntouyas and Jessada Tariboon
Fractal Fract. 2022, 6(5), 234; https://doi.org/10.3390/fractalfract6050234 - 23 Apr 2022
Cited by 8 | Viewed by 1568
Abstract
In this paper, we study a coupled system consisting of (k,φ)-Hilfer fractional differential equations of the order (1,2], supplemented with nonlocal coupled multi-point boundary conditions. The existence and uniqueness of the results are [...] Read more.
In this paper, we study a coupled system consisting of (k,φ)-Hilfer fractional differential equations of the order (1,2], supplemented with nonlocal coupled multi-point boundary conditions. The existence and uniqueness of the results are established via Banach’s contraction mapping principle, the Leray–Schauder alternative and Krasnosel’skiĭ’s fixed-point theorem. Numerical examples are constructed to illustrate the obtained results. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
13 pages, 301 KiB  
Article
Some Fixed-Disc Results in Double Controlled Quasi-Metric Type Spaces
by Nabil Mlaiki, Nihal Taş, Salma Haque and Doaa Rizk
Fractal Fract. 2022, 6(2), 107; https://doi.org/10.3390/fractalfract6020107 - 12 Feb 2022
Cited by 1 | Viewed by 1498
Abstract
In this paper, we introduce new types of general contractions for self mapping on double controlled quasi-metric type spaces, where we prove the existence and uniqueness of fixed disc and circle for such mappings. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
16 pages, 296 KiB  
Article
Common Fixed Point Theorems for Two Mappings in Complete b-Metric Spaces
by Lili Chen, Xin Xia, Yanfeng Zhao and Xin Liu
Fractal Fract. 2022, 6(2), 103; https://doi.org/10.3390/fractalfract6020103 - 11 Feb 2022
Cited by 1 | Viewed by 2061
Abstract
Our paper is devoted to the issue of the existence and uniqueness of common fixed points for two mappings in complete b-metric spaces by virtue of the new functions F and θ, respectively. Moreover, two specific examples to indicate the validity [...] Read more.
Our paper is devoted to the issue of the existence and uniqueness of common fixed points for two mappings in complete b-metric spaces by virtue of the new functions F and θ, respectively. Moreover, two specific examples to indicate the validity of our results are also given. Eventually, the generalized forms of Jungck fixed point theorem in the above spaces is investigated. Different from related literature, the conditions that the function F needs to satisfy are weakened, and F only needs to be non-decreasing in this paper. To some extent, our conclusions and methods improve the results of previous literature. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

Review

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35 pages, 424 KiB  
Review
A Survey on Recent Results on Lyapunov-Type Inequalities for Fractional Differential Equations
by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon
Fractal Fract. 2022, 6(5), 273; https://doi.org/10.3390/fractalfract6050273 - 18 May 2022
Cited by 7 | Viewed by 1434
Abstract
This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville, ψ-Caputo, ψ-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, [...] Read more.
This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville, ψ-Caputo, ψ-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, and proportional fractional derivative operators. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

Other

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10 pages, 315 KiB  
Brief Report
Fractional p-Laplacian Equations with Sandwich Pairs
by Jose Vanterler da C. Sousa
Fractal Fract. 2023, 7(6), 419; https://doi.org/10.3390/fractalfract7060419 - 23 May 2023
Cited by 1 | Viewed by 936
Abstract
The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational methods in ψ-fractional space [...] Read more.
The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational methods in ψ-fractional space Hpα,β;ψ(Ω). The results obtained in this paper are the first to make use of the theory of ψ-Hilfer fractional operators with p-Laplacian. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
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