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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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  2. 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 Published on:

  3. 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 Published on:

  5. 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 Published on:

  6. 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 Published on:

  7. 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 Published on:

  8. 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 Published on:

  9. 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 Published on:

  10. 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 Published on:

  12. 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 Published on:

  15. 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 Published on:

  17. 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 Published on:

  18. 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 Published on:

  19. 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 Published on:

  20. In this paper we study fractional initial value problems with Caputo–Fabrizio derivative which involves nonsingular kernel. First we apply α-ℓ-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 Published on:

  21. In this work, we present new necessary and sufficient conditions for the oscillation of a class of second-order neutral delay impulsive differential equations. Our oscillation results complement, simplify and ...

    Authors: Shyam Sundar Santra, Apurba Ghosh, Omar Bazighifan, Khaled Mohamed Khedher and Taher A. Nofal
    Citation: Advances in Difference Equations 2021 2021:318
    Content type: Research Published on:

  22. In this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the...

    Authors: Raheel Kamal, Kamran, Gul Rahmat, Ali Ahmadian, Noreen Izza Arshad and Soheil Salahshour
    Citation: Advances in Difference Equations 2021 2021:317
    Content type: Research Published on:

  24. The objective of this article is to introduce function weighted L-R-complete dislocated quasi-metric spaces and to present fixed point results fulfilling generalized rational type F-contraction for a multivalued ...

    Authors: Abdullah Shoaib, Qasim Mahmood, Aqeel Shahzad, Mohd Salmi Md Noorani and Stojan Radenović
    Citation: Advances in Difference Equations 2021 2021:310
    Content type: Research Published on:

  26. In this work, we study the existence, uniqueness, and continuous dependence of solutions for a class of fractional differential equations by using a generalized Riesz fractional operator. One can view the resu...

    Authors: Muhammad Aleem, Mujeeb Ur Rehman, Jehad Alzabut, Sina Etemad and Shahram Rezapour
    Citation: Advances in Difference Equations 2021 2021:303
    Content type: Research Published on:

    The Correction to this article has been published in Advances in Continuous and Discrete Models 2023 2023:23

  27. In this paper, we mainly investigate the existence, continuous dependence, and the optimal control for nonlocal fractional differential evolution equations of order (1,2) in Banach spaces. We define a competen...

    Authors: Denghao Pang, Wei Jiang, Azmat Ullah Khan Niazi and Jiale Sheng
    Citation: Advances in Difference Equations 2021 2021:302
    Content type: Research Published on:

  28. This article proposes four distinct kinds of symmetric contraction in the framework of complete F-metric spaces. We examine the condition to guarantee the existence and uniqueness of a fixed point for these co...

    Authors: Aftab Hussain, Fahd Jarad and Erdal Karapinar
    Citation: Advances in Difference Equations 2021 2021:300
    Content type: Research Published on:

    The Correction to this article has been published in Advances in Difference Equations 2021 2021:335

  29. This study is aimed to investigate the sufficient conditions of the existence of unique solutions and the Ulam–Hyers–Mittag-Leffler (UHML) stability for a tripled system of weighted generalized Caputo fraction...

    Authors: Mohammed A. Almalahi, Satish K. Panchal, Fahd Jarad and Thabet Abdeljawad
    Citation: Advances in Difference Equations 2021 2021:299
    Content type: Research Published on:

  30. In this paper, we introduce a new integral transform, namely Aboodh transform, and we apply the transform to investigate the Hyers–Ulam stability, Hyers–Ulam–Rassias stability, Mittag-Leffler–Hyers–Ulam stabil...

    Authors: Ramdoss Murali, Arumugam Ponmana Selvan, Choonkil Park and Jung Rye Lee
    Citation: Advances in Difference Equations 2021 2021:296
    Content type: Research Published on:

  31. This article describes the corona virus spread in a population under certain assumptions with the help of a fractional order mathematical model. The fractional order derivative is the well-known fractal fracti...

    Authors: Hasib Khan, Razia Begum, Thabet Abdeljawad and M. Motawi Khashan
    Citation: Advances in Difference Equations 2021 2021:293
    Content type: Research Published on:

  32. In this article, we introduce two notions of interpolative F-contractions with shrink map and F-contractions with shrink map. We also study the existence of E-fixed points by using these notations on a metric spa...

    Authors: Monairah Alansari and Muhammad Usman Ali
    Citation: Advances in Difference Equations 2021 2021:282
    Content type: Research Published on:

  33. The forward–backward algorithm is a splitting method for solving convex minimization problems of the sum of two objective functions. It has a great attention in optimization due to its broad application to man...

    Authors: Suthep Suantai, Muhammad Aslam Noor, Kunrada Kankam and Prasit Cholamjiak
    Citation: Advances in Difference Equations 2021 2021:265
    Content type: Review Published on:

  35. This paper is devoted to finding out some realization of the concept of b-metric like space. First, we attain a fixed point for two fuzzy mappings satisfying a suitable requirement of contractiveness. Subsequentl...

    Authors: Tahair Rasham, Giuseppe Marino, Aqeel Shahzad, Choonkill Park and Abdullah Shoaib
    Citation: Advances in Difference Equations 2021 2021:259
    Content type: Research Published on:

  37. In this paper, we deal with Caputo-type fractional differential inequality where there is a low-order fractional derivative with the term polynomial source. We investigate the nonexistence of nontrivial global...

    Authors: Mohammed D. Kassim, Saeed M. Ali, Mohammed S. Abdo and Fahd Jarad
    Citation: Advances in Difference Equations 2021 2021:246
    Content type: Research Published on:

  38. In this article, we debate the existence of solutions for a nonlinear Hilfer fractional differential inclusion with nonlocal Erdélyi–Kober fractional integral boundary conditions (FIBC). Both cases of convex- ...

    Authors: Adel Lachouri, Mohammed S. Abdo, Abdelouaheb Ardjouni, Bahaaeldin Abdalla and Thabet Abdeljawad
    Citation: Advances in Difference Equations 2021 2021:244
    Content type: Research Published on:

  41. By using a nonlinear method, we try to solve partial fractional differential equations. In this way, we construct the Laguerre wavelets operational matrix of fractional integration. The method is proposed by u...

    Authors: Nasser Aghazadeh, Amir Mohammadi, Ghader Ahmadnezhad and Shahram Rezapour
    Citation: Advances in Difference Equations 2021 2021:231
    Content type: Research Published on:

  43. A class of the boundary value problem is investigated in this research work to prove the existence of solutions for the neutral fractional differential inclusions of Katugampola fractional derivative which inv...

    Authors: Sina Etemad, Mohammed Said Souid, Benoumran Telli, Mohammed K. A. Kaabar and Shahram Rezapour
    Citation: Advances in Difference Equations 2021 2021:214
    Content type: Research Published on:

  45. In this work, we consider a fractional diffusion equation with nonlocal integral condition. We give a form of the mild solution under the expression of Fourier series which contains some Mittag-Leffler functio...

    Authors: Nguyen Hoang Tuan, Nguyen Anh Triet, Nguyen Hoang Luc and Nguyen Duc Phuong
    Citation: Advances in Difference Equations 2021 2021:204
    Content type: Research Published on:

  46. In this research study, we are concerned with the existence and stability of solutions of a boundary value problem (BVP) of the fractional thermostat control model with ψ-Hilfer fractional operator. We verify the...

    Authors: Chatthai Thaiprayoon, Weerawat Sudsutad, Jehad Alzabut, Sina Etemad and Shahram Rezapour
    Citation: Advances in Difference Equations 2021 2021:201
    Content type: Research Published on:

  48. In this paper, we improve the Proinov theorem by adding certain rational expressions to the definition of the corresponding contractions. After that, we prove fixed point theorems for these modified Proinov co...

    Authors: Badr Alqahtani, Sara S. Alzaid, Andreea Fulga and Antonio Francisco Roldán López de Hierro
    Citation: Advances in Difference Equations 2021 2021:164
    Content type: Research Published on: