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Fixed Point Theory and Applications to Fractional Ordinary and Partial Difference and Differential Equations

An important concept in mathematics, differential and integral calculus appears naturally in numerous scientific problems, which have been widely applied in physics, chemical technology, optimal control, finance, signal processing, etc. and are modeled by ordinary or partial difference and differential equations.

In recent years, it was observed that many real-world phenomena cannot be modeled by ordinary or partial differential equations or standard difference equations defined via the classical derivatives and integrals. In fact, these problems followed the appearance of fractional calculus (fractional derivatives and integrals), intended to handle the problems for which the classical calculus was insufficient. Together with the development and progress in fractional calculus, the theory and applications of ordinary and partial differential equations with fractional derivatives became one of the most studied topics in applied mathematics. The wide application potential of fractional differential equations in many fields of science has been underlined by a huge number of articles, books, and scientific events on the subject.

Fixed point theory on the other hand, is a very strong mathematical tool to establish the existence and uniqueness of almost all problems modeled by nonlinear relations. Consequently, existence and uniqueness problems of fractional differential equations are studied by means of fixed point theory. For about a century, fixed point theory has begun to take shape, and developed rapidly. Due to its applications, fixed point theory is highly appreciated and continues to be explored. Besides, this theory can be applied in many types of spaces, such as abstract spaces, metric spaces, and Sobolev spaces. This feature of fixed point theory makes it very valuable in studying numerous problems of practical sciences modeled by fractional ordinary and partial differential and difference equations. 

This special issue presents ideas for theoretical advances on fixed point theory and applications to fractional ordinary and partial difference and differential equations.

Edited by:  Erdal Karapinar, Tomás Caraballo, Inci Erhan, Nguyen Huy Tuan

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  1. This paper deals with the generalized Bagley–Torvik equation based on the concept of the Caputo–Fabrizio fractional derivative using a modified reproducing kernel Hilbert space treatment. The generalized Bagle...

    Authors: Shatha Hasan, Nadir Djeddi, Mohammed Al-Smadi, Shrideh Al-Omari, Shaher Momani and Andreea Fulga

    Citation: Advances in Difference Equations 2021 2021:469

    Content type: Research

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  2. The goal of this study is to propose the existence results for the Sobolev-type Hilfer fractional integro-differential systems with infinite delay. We intend to implement the outcomes and realities of fraction...

    Authors: K. Kavitha, Kottakkaran Sooppy Nisar, Anurag Shukla, Velusamy Vijayakumar and Shahram Rezapour

    Citation: Advances in Difference Equations 2021 2021:467

    Content type: Research

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  3. In this work, we study a q-differential inclusion with doubled integral boundary conditions under the Caputo derivative. To achieve the desired result, we use the endpoint property introduced by Amini-Harandi and...

    Authors: Mehdi Shabibi, Mohammad Esmael Samei, Mehran Ghaderi and Shahram Rezapour

    Citation: Advances in Difference Equations 2021 2021:466

    Content type: Research

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  4. In this paper, we introduce a new structure of the generalized multi-point thermostat control model motivated by its standard model. By presenting integral solution of this boundary problem, the existence prop...

    Authors: Sina Etemad, Brahim Tellab, Chernet Tuge Deressa, Jehad Alzabut, Yongkun Li and Shahram Rezapour

    Citation: Advances in Difference Equations 2021 2021:458

    Content type: Research

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  5. In our manuscript, we extend the controllability outcomes given by Bashirov (Math. Methods Appl. Sci. 44(9):7455–7462, 2021) for a family of second-order semilinear control system by formulating a sequence of pie...

    Authors: Urvashi Arora, V. Vijayakumar, Anurag Shukla, Kottakkaran Sooppy Nisar, Shahram Rezapour and Wasim Jamshed

    Citation: Advances in Difference Equations 2021 2021:455

    Content type: Research

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  6. In this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch’s fixed point theorem along with a t...

    Authors: Abdelatif Boutiara, Maamar Benbachir, Sina Etemad and Shahram Rezapour

    Citation: Advances in Difference Equations 2021 2021:454

    Content type: Research

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  7. For finding a common fixed point of a finite family of G-nonexpansive mappings, we implement a new parallel algorithm based on the Ishikawa iteration process with the inertial technique. We obtain the weak conver...

    Authors: Phakdi Charoensawan, Damrongsak Yambangwai, Watcharaporn Cholamjiak and Raweerote Suparatulatorn

    Citation: Advances in Difference Equations 2021 2021:453

    Content type: Research

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  8. Some complicated events can be modeled by systems of differential equations. On the other hand, inclusion systems can describe complex phenomena having some shocks better than the system of differential equati...

    Authors: Seher Melike Aydogan, Fethiye Muge Sakar, Mostafa Fatehi, Shahram Rezapour and Hashem Parvaneh Masiha

    Citation: Advances in Difference Equations 2021 2021:449

    Content type: Research

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  9. The aim of this paper is to investigate the interpolative contractions involving rational forms in the framework of b-metric spaces. We prove the existence of a fixed point of such a mapping with different combin...

    Authors: Andreea Fulga

    Citation: Advances in Difference Equations 2021 2021:448

    Content type: Research

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  10. The main target of this paper is to study a problem of recovering a spherically symmetric domain with fractional derivative from observed data of nonlocal type. This problem can be established as a new boundar...

    Authors: Le Dinh Long, Ho Thi Kim Van, Ho Duy Binh and Reza Saadati

    Citation: Advances in Difference Equations 2021 2021:445

    Content type: Research

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  11. The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankt...

    Authors: Muhammad Salman Khan, Maria Samreen, Hassen Aydi and Manuel De la Sen

    Citation: Advances in Difference Equations 2021 2021:443

    Content type: Research

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  12. In this work, we establish some necessary results about existence theory to a class of boundary value problems (BVPs) of hybrid fractional differential equations (HFDEs) in the frame of Atangana–Baleanu–Caputo (A...

    Authors: Rozi Gul, Kamal Shah, Zareen A. Khan and Fahd Jarad

    Citation: Advances in Difference Equations 2021 2021:437

    Content type: Research

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  13. In this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in th...

    Authors: H. Jafari, S. Nemati and R. M. Ganji

    Citation: Advances in Difference Equations 2021 2021:435

    Content type: Research

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  14. In this paper, we consider the biparabolic problem under nonlocal conditions with both linear and nonlinear source terms. We derive the regularity property of the mild solution for the linear source term while...

    Authors: Le Dinh Long, Ho Duy Binh, Kim Van Ho Thi and Van Thinh Nguyen

    Citation: Advances in Difference Equations 2021 2021:434

    Content type: Research

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  15. HIV patients are vulnerable to developing active visceral leishmaniasis (VL). To understand this complication, we studied a mathematical model for HIV and visceral leishmaniasis coinfection. In this approach, ...

    Authors: Zinabu Teka Melese and Haileyesus Tessema Alemneh

    Citation: Advances in Difference Equations 2021 2021:429

    Content type: Research

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  16. In this paper, we consider a new coupled system of fractional boundary value problems based on the thermostat control model. With the help of fixed point theory, we investigate the existence criterion of the s...

    Authors: Sina Etemad, Brahim Tellab, Jehad Alzabut, Shahram Rezapour and Mohamed Ibrahim Abbas

    Citation: Advances in Difference Equations 2021 2021:428

    Content type: Research

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  17. The main objective of this paper is to investigate the existence, uniqueness, and Ulam–Hyers stability of positive solutions for fractional integro-differential boundary values problem. Uniqueness result is ob...

    Authors: Mohammed M. Matar, Manar abu Jarad, Manzoor Ahmad, Akbar Zada, Sina Etemad and Shahram Rezapour

    Citation: Advances in Difference Equations 2021 2021:423

    Content type: Research

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  18. The main purpose of this paper is to present some fixed-point results for a pair of fuzzy dominated mappings which are generalized V-contractions in modular-like metric spaces. Some theorems using a partial order...

    Authors: Tahair Rasham, Awais Asif, Hassen Aydi and Manuel De La Sen

    Citation: Advances in Difference Equations 2021 2021:417

    Content type: Research

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  19. In this work, we solve the system of integro-differential equations (in terms of Caputo–Fabrizio calculus) using the concepts of the best proximity pair (point) and measure of noncompactness. We first introduc...

    Authors: Anupam Das, Hemant Kumar Nashine, Rabha W. Ibrahim and Manuel De la Sen

    Citation: Advances in Difference Equations 2021 2021:414

    Content type: Research

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  20. In this work, we modify the inertial hybrid algorithm with Armijo line search using a parallel method to approximate a common solution of nonmonotone equilibrium problems in Hilbert spaces. A weak convergence ...

    Authors: Suthep Suantai, Damrongsak Yambangwai and Watcharaporn Cholamjiak

    Citation: Advances in Difference Equations 2021 2021:410

    Content type: Research

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  21. In this paper, we propose two new contractions via simulation function that involves rational expression in the setting of partial b-metric space. The obtained results not only extend, but also generalize and ...

    Authors: Erdal Karapınar, Chi-Ming Chen, Maryam A. Alghamdi and Andreea Fulga

    Citation: Advances in Difference Equations 2021 2021:409

    Content type: Research

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  22. In this paper, we study the existence of solutions for a generalized sequential Caputo-type fractional neutral differential inclusion with generalized integral conditions. The used fractional operator has the ...

    Authors: Adel Lachouri, Mohammed S. Abdo, Abdelouaheb Ardjouni, Sina Etemad and Shahram Rezapour

    Citation: Advances in Difference Equations 2021 2021:404

    Content type: Research

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  23. In this paper we propose a solution to the nonlinear Fredholm integral equations in the context of w-distance. For this purpose, we also provide a fixed point result in the same setting. In addition, we provide b...

    Authors: P. Dhivya, M. Marudai, Vladimir Rakočević and Andreea Fulga

    Citation: Advances in Difference Equations 2021 2021:398

    Content type: Research

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  24. Some fundamental conditions and hypotheses are established to ensure the existence, uniqueness, and stability to a class of implicit boundary value problems (BVPs) with Atangana–Baleanu–Caputo type derivative ...

    Authors: Asma, Sana Shabbir, Kamal Shah and Thabet Abdeljawad

    Citation: Advances in Difference Equations 2021 2021:395

    Content type: Research

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  25. In this paper, we study the fractional nonlinear Rayleigh–Stokes equation under nonlocal integral conditions, and the existence and uniqueness of the mild solution to our problem are considered. The ill-posedn...

    Authors: Nguyen Hoang Luc, Le Dinh Long, Ho Thi Kim Van and Van Thinh Nguyen

    Citation: Advances in Difference Equations 2021 2021:388

    Content type: Research

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  26. The current paper is about the investigation of a new integral transform introduced recently by Jafari. Specifically, we explore the applicability of this integral transform on Atangana–Baleanu derivative and ...

    Authors: M. Meddahi, H. Jafari and M. N. Ncube

    Citation: Advances in Difference Equations 2021 2021:385

    Content type: Research

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  27. The aim of this article is to discuss the uniqueness and Ulam–Hyers stability of solutions for a nonlinear fractional integro-differential equation involving a generalized Caputo fractional operator. The used ...

    Authors: Mohammed D. Kassim, Thabet Abdeljawad, Wasfi Shatanawi, Saeed M. Ali and Mohammed S. Abdo

    Citation: Advances in Difference Equations 2021 2021:375

    Content type: Research

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  28. In this manuscript, we study a certain classical second-order fully nonlinear coupled system with generalized nonlinear coupled boundary conditions satisfying the monotone assumptions. Our new results unify th...

    Authors: Imran Talib, Thabet Abdeljawad, Manar A. Alqudah, Cemil Tunc and Rabia Ameen

    Citation: Advances in Difference Equations 2021 2021:368

    Content type: Research

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  29. In this paper, we consider a nonlinear sequential q-difference equation based on the Caputo fractional quantum derivatives with nonlocal boundary value conditions containing Riemann–Liouville fractional quantum i...

    Authors: Abdelatif Boutiara, Sina Etemad, Jehad Alzabut, Azhar Hussain, Muthaiah Subramanian and Shahram Rezapour

    Citation: Advances in Difference Equations 2021 2021:367

    Content type: Research

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  30. We introduce double and triple F-expanding mappings. We prove related fixed point theorems. Based on our obtained results, we also prove the existence of a solution for fractional type differential equations by u...

    Authors: Shahid Bashir, Naeem Saleem, Hassen Aydi, Syed Muhammad Husnine and Asma Al Rwaily

    Citation: Advances in Difference Equations 2021 2021:359

    Content type: Research

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  31. In this article we make an improvement in the Banach contraction using a controlled function in controlled metric like spaces, which generalizes many results in the literature. Moreover, we present an applicat...

    Authors: Wasfi Shatanawi, Nabil Mlaiki, Doaa Rizk and Enyinda Onunwor

    Citation: Advances in Difference Equations 2021 2021:358

    Content type: Research

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  32. This work is concerned with the problem for stochastic genetic regulatory networks (GRNs) subject to mixed time delays via passivity control in which mixed time delays consist of leakage, discrete, and distrib...

    Authors: S. Senthilraj, T. Saravanakumar, R. Raja and J. Alzabut

    Citation: Advances in Difference Equations 2021 2021:353

    Content type: Research

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  33. In the existing study, we investigate the criteria of existence of solution for relatively new categories of φ-Caputo fractional differential equations and inclusions problems equipped with nonlocal φ-integral bo...

    Authors: Muhammad Qamar Iqbal and Azhar Hussain

    Citation: Advances in Difference Equations 2021 2021:350

    Content type: Research

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  34. In this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves R...

    Authors: Ahmed Nouara, Abdelkader Amara, Eva Kaslik, Sina Etemad, Shahram Rezapour, Francisco Martinez and Mohammed K. A. Kaabar

    Citation: Advances in Difference Equations 2021 2021:343

    Content type: Research

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  35. In this paper, we aim to discuss the common fixed point of Proinov type mapping via simulation function. The presented results not only generalize, but also unify the corresponding results in this direction. W...

    Authors: Badr Alqahtani, Sara Salem Alzaid, Andreea Fulga and Seher Sultan Yeşilkaya

    Citation: Advances in Difference Equations 2021 2021:328

    Content type: Research

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  36. In this paper, we study classes of linear and nonlinear multi-term fractional differential equations involving a fractional derivative with generalized Mittag-Leffler kernel. Estimates of fractional derivative...

    Authors: Mohammed Al-Refai, Abdalla Aljarrah and Thabet Abdeljawad

    Citation: Advances in Difference Equations 2021 2021:325

    Content type: Research

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  37. The Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville...

    Authors: Zohreh Zeinalabedini Charandabi, Hakimeh Mohammadi, Shahram Rezapour and Hashem Parvaneh Masiha

    Citation: Advances in Difference Equations 2021 2021:323

    Content type: Research

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  38. In this paper we study fractional initial value problems with Caputo–Fabrizio derivative which involves nonsingular kernel. First we apply α--contraction and α-type F-contraction mappings to study the existence ...

    Authors: Hojjat Afshari, Hossein Hosseinpour and H. R. Marasi

    Citation: Advances in Difference Equations 2021 2021:321

    Content type: Research

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