# 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. ### Numerical solvability of generalized Bagley–Torvik fractional models under Caputo–Fabrizio derivative

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. ### A discussion concerning the existence results for the Sobolev-type Hilfer fractional delay integro-differential systems

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. ### Some analytical and numerical results for a fractional q-differential inclusion problem with double integral boundary conditions

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...

Citation: Advances in Difference Equations 2021 2021:466

Content type: Research

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4. ### Existence and nonexistence of entire k-convex radial solutions to Hessian type system

In this paper, a Hessian type system is studied. After converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, the existence of entire k-convex radial solutions i...

Authors: Jixian Cui

Citation: Advances in Difference Equations 2021 2021:462

Content type: Research

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5. ### On a generalized fractional boundary value problem based on the thermostat model and its numerical solutions via Bernstein polynomials

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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6. ### Infinite Geraghty type extensions and their applications on integral equations

In this article, we discuss about a series of infinite dimensional extensions of some theorems given in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018), (Fisher in Math. Mag. 48(4):223–225, 1975), and ...

Authors: R. Bardhan, C. Ozel, L. Guran, H. Aydi and Choonkil Park

Citation: Advances in Difference Equations 2021 2021:456

Content type: Research

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7. ### Results on exact controllability of second-order semilinear control system in Hilbert spaces

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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8. ### Kuratowski MNC method on a generalized fractional Caputo Sturm–Liouville–Langevin q-difference problem with generalized Ulam–Hyers stability

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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9. ### An inertial parallel algorithm for a finite family of G-nonexpansive mappings with application to the diffusion problem

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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10. ### Two hybrid and non-hybrid k-dimensional inclusion systems via sequential fractional derivatives

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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11. ### On interpolative contractions that involve rational forms

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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12. ### On backward problem for fractional spherically symmetric diffusion equation with observation data of nonlocal type

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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13. ### Qualitative analysis of a discrete-time phytoplankton–zooplankton model with Holling type-II response and toxicity

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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14. ### On a class of boundary value problems under ABC fractional derivative

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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15. ### Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

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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16. ### Well-posed results for nonlocal biparabolic equation with linear and nonlinear source terms

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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17. ### Enhancing reservoir control in the co-dynamics of HIV-VL: from mathematical modeling perspective

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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18. ### Approximate solutions and Hyers–Ulam stability for a system of the coupled fractional thermostat control model via the generalized differential transform

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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19. ### On solution of generalized proportional fractional integral via a new fixed point theorem

Citation: Advances in Difference Equations 2021 2021:427

Content type: Research

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20. ### On the existence and stability of two positive solutions of a hybrid differential system of arbitrary fractional order via Avery–Anderson–Henderson criterion on cones

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...

Citation: Advances in Difference Equations 2021 2021:423

Content type: Research

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21. ### Computation of Fourier transform representations involving the generalized Bessel matrix polynomials

Motivated by the recent studies and developments of the integral transforms with various special matrix functions, including the matrix orthogonal polynomials as kernels, in this article we derive the formulas...

Authors: M. Abdalla and M. Akel

Citation: Advances in Difference Equations 2021 2021:418

Content type: Research

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22. ### On pairs of fuzzy dominated mappings and applications

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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23. ### Best proximity point results and application to a system of integro-differential equations

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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24. ### Revising the Hardy–Rogers–Suzuki-type Z-contractions

The aim of this study is to introduce a new interpolative contractive mapping combining the Hardy–Rogers contractive mapping of Suzuki type and Authors: Maha Noorwali

Citation: Advances in Difference Equations 2021 2021:413

Content type: Research

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25. ### Solving common nonmonotone equilibrium problems using an inertial parallel hybrid algorithm with Armijo line search with applications to image recovery

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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26. ### Advances on the fixed point results via simulation function involving rational terms

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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27. ### A generalized neutral-type inclusion problem in the frame of the generalized Caputo fractional derivatives

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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28. ### Mathematical model of survival of fractional calculus, critics and their impact: How singular is our world?

Fractional calculus as was predicted by Leibniz to be a paradox, has nowadays evolved to become a centre of interest for many researchers from various backgrounds. As a result, multiple innovative ideas had em...

Authors: Abdon Atangana

Citation: Advances in Difference Equations 2021 2021:403

Content type: Research

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29. ### A solution to nonlinear Fredholm integral equations in the context of w-distances

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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30. ### Stability analysis for a class of implicit fractional differential equations involving Atangana–Baleanu fractional derivative

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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31. ### Refinements of some fractional integral inequalities for refined $$(\alpha ,h-m)$$-convex function

Authors: Chahn Yong Jung, Ghulam Farid, Hafsa Yasmeen, Yu-Pei Lv and Josip Pečarić

Citation: Advances in Difference Equations 2021 2021:391

Content type: Research

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32. ### A nonlinear fractional Rayleigh–Stokes equation under nonlocal integral conditions

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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33. ### New general integral transform via Atangana–Baleanu derivatives

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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34. ### On Caputo–Hadamard type coupled systems of nonconvex fractional differential inclusions

This research article is mainly concerned with the existence of solutions for a coupled Caputo–Hadamard of nonconvex fractional differential inclusions equipped with boundary conditions. We derive our main res...

Citation: Advances in Difference Equations 2021 2021:377

Content type: Research

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35. ### A qualitative study on generalized Caputo fractional integro-differential equations

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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36. ### New results and applications on the existence results for nonlinear coupled systems

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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37. ### On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria

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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38. ### Developments of some new results that weaken certain conditions of fractional type differential equations

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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39. ### Fredholm-type integral equation in controlled metric-like spaces

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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40. ### Delay-dependent passivity analysis of nondeterministic genetic regulatory networks with leakage and distributed delays against impulsive perturbations

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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41. ### System of fractional boundary value problem with p-Laplacian and advanced arguments

In this paper, we discuss the existence and multiplicity of positive solutions for a system of fractional differential equations with boundary condition and advanced arguments. The existence result is proved v...

Authors: Amina Mahdjouba, Juan J. Nieto and Abdelghani Ouahab

Citation: Advances in Difference Equations 2021 2021:352

Content type: Research

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42. ### Existence criteria via α–ψ-contractive mappings of φ-fractional differential nonlocal boundary value problems

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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43. ### Results on fixed circles and discs for $$L_{ (\omega,C ) }$$-contractions and related applications

Authors: Eskandar Ameer, Hassen Aydi, Muhammad Nazam and Manuel De la Sen

Citation: Advances in Difference Equations 2021 2021:349

Content type: Research

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44. ### Finite-time stability of linear stochastic fractional-order systems with time delay

Authors: Lassaad Mchiri, Abdellatif Ben Makhlouf, Dumitru Baleanu and Mohamed Rhaima

Citation: Advances in Difference Equations 2021 2021:345

Content type: Research

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45. ### A study on multiterm hybrid multi-order fractional boundary value problem coupled with its stability analysis of Ulam–Hyers type

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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46. ### Fixed point theory in the setting of $$(\alpha,\beta,\psi,\phi)$$-interpolative contractions

Authors: Erdal Karapınar, Andreea Fulga and Antonio Francisco Roldán López de Hierro

Citation: Advances in Difference Equations 2021 2021:339

Content type: Research

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47. ### Common fixed point theorem on Proinov type mappings via simulation function

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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48. ### Analysis of fractional differential equations with fractional derivative of generalized Mittag-Leffler kernel

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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49. ### On partial fractional Sturm–Liouville equation and inclusion

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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50. ### Application of some new contractions for existence and uniqueness of differential equations involving Caputo–Fabrizio derivative

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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