# 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. ### Applications of some new Krasnoselskii-type fixed-point results for generalized expansive and equiexpansive mappings

Authors: Niaz Ahmad, Nayyar Mehmood and Ali Akgül
Citation: Advances in Continuous and Discrete Models 2022 2022:30
2. ### On some accelerated optimization algorithms based on fixed point and linesearch techniques for convex minimization problems with applications

In this paper, we introduce and study a new accelerated algorithm based on forward–backward and SP-algorithm for solving a convex minimization problem of the sum of two convex and lower semicontinuous function...

Authors: Pornsak Yatakoat, Suthep Suantai and Adisak Hanjing
Citation: Advances in Continuous and Discrete Models 2022 2022:25
3. ### Convergence analysis of a novel iteration process with application to a fractional differential equation

The objective of this article is to study a three-step iteration process in the framework of Banach spaces and to obtain convergence results for Suzuki generalized nonexpansive mappings. We also provide numeri...

Authors: Izhar Uddin, Chanchal Garodia, Thabet Abdeljawad and Nabil Mlaiki
Citation: Advances in Continuous and Discrete Models 2022 2022:16
4. ### Global solution to the compressible non-isothermal nematic liquid crystal equations with constant heat conductivity and vacuum

This paper considers the initial-boundary value problem of the one-dimensional full compressible nematic liquid crystal flow problem. The initial density is allowed to touch vacuum, and the viscous and heat co...

Authors: Tariq Mahmood and Mei Sun
Citation: Advances in Difference Equations 2021 2021:517
5. ### Stability criteria for nonlinear Volterra integro-dynamic matrix Sylvester systems on measure chains

In this paper, we establish sufficient conditions for various stability aspects of a nonlinear Volterra integro-dynamic matrix Sylvester system on time scales. We convert the nonlinear Volterra integro-dynamic...

Authors: Sreenivasulu Ayyalappagari and Venkata Appa Rao Bhogapurapu
Citation: Advances in Difference Equations 2021 2021:514
6. ### On initial inverse problem for nonlinear couple heat with Kirchhoff type

The main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the...

Authors: Danh Hua Quoc Nam
Citation: Advances in Difference Equations 2021 2021:512
7. ### A new formulation of finite difference and finite volume methods for solving a space fractional convection–diffusion model with fewer error estimates

Convection and diffusion are two harmonious physical processes that transfer particles and physical quantities. This paper deals with a new aspect of solving the convection–diffusion equation in fractional ord...

Authors: Reem Edwan, Shrideh Al-Omari, Mohammed Al-Smadi, Shaher Momani and Andreea Fulga
Citation: Advances in Difference Equations 2021 2021:510
8. ### Common fixed point theorems for auxiliary functions with applications in fractional differential equation

In this work, we investigate h-ϕ contraction mappings with two metrics endowed with a directed graph which involve auxiliary functions. The achievement allows us to obtain applications for the existence of the so...

Authors: Ben Wongsaijai, Phakdi Charoensawan, Teeranush Suebcharoen and Watchareepan Atiponrat
Citation: Advances in Difference Equations 2021 2021:503
9. ### On the generalized fractional snap boundary problems via G-Caputo operators: existence and stability analysis

This research is conducted for studying some qualitative specifications of solution to a generalized fractional structure of the standard snap boundary problem. We first rewrite the mathematical model of the e...

Authors: Mohammad Esmael Samei, Mohammed M. Matar, Sina Etemad and Shahram Rezapour
Citation: Advances in Difference Equations 2021 2021:498
10. ### Existence of positive solutions for a system of nonlinear Caputo type fractional differential equations with two parameters

The main purpose of this paper is to prove the existence of positive solutions for a system of nonlinear Caputo-type fractional differential equations with two parameters. By using the Guo–Krasnosel’skii fixed...

Authors: Yang Chen and Hongyu Li
Citation: Advances in Difference Equations 2021 2021:497
11. ### Existence of solutions for a class of nonlinear boundary value problems on the hexasilinane graph

Chemical graph theory is a field of mathematics that studies ramifications of chemical network interactions. Using the concept of star graphs, several investigators have looked into the solutions to certain bo...

Authors: Ali Turab, Zoran D. Mitrović and Ana Savić
Citation: Advances in Difference Equations 2021 2021:494
12. ### An analysis on the controllability and stability to some fractional delay dynamical systems on time scales with impulsive effects

In this article, we establish a new class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. We investigate the qualitative properties of the considered systems. In fact,...

Citation: Advances in Difference Equations 2021 2021:491
13. ### Mathematical analysis of a fractional resource-consumer model with disease developed in consumer

The research presents a qualitative investigation of a fractional-order consumer-resource system with the hunting cooperation interaction functional and an infection developed in the resources population. The ...

Authors: Abdelheq Mezouaghi, Abdelkader Benali, Sunil Kumar, Salih Djilali, Anwar Zeb and Shahram Rezapour
Citation: Advances in Difference Equations 2021 2021:487
14. ### Some variants of Wardowski fixed point theorem

The purpose of this paper is to consider some F-contraction mappings in a dualistic partial metric space and to provide sufficient related conditions for the existence of a fixed point. The obtained results are e...

Citation: Advances in Difference Equations 2021 2021:485
15. ### A note on the approximate controllability of second-order integro-differential evolution control systems via resolvent operators

The approximate controllability of second-order integro-differential evolution control systems using resolvent operators is the focus of this work. We analyze approximate controllability outcomes by referring ...

Authors: Velusamy Vijayakumar, Anurag Shukla, Kottakkaran Sooppy Nisar, Wasim Jamshed and Shahram Rezapour
Citation: Advances in Difference Equations 2021 2021:484
16. ### A terrorism-based differential game: Nash differential game

In this paper, we investigate the problem of combating terrorism by the government, which is one of the most serious problems that direct governments and countries. We formulate the problem and use the Nash ap...

Authors: Abd El-Monem A. Megahed
Citation: Advances in Difference Equations 2021 2021:483
17. ### Well-posed conditions on a class of fractional q-differential equations by using the Schauder fixed point theorem

In this paper, we propose the conditions on which a class of boundary value problems, presented by fractional q-differential equations, is well-posed. First, under the suitable conditions, we will prove the exist...

Citation: Advances in Difference Equations 2021 2021:482
18. ### New discussion on nonlocal controllability for fractional evolution system of order \(1 < r < 2\)

Authors: M. Mohan Raja, Velusamy Vijayakumar, Anurag Shukla, Kottakkaran Sooppy Nisar and Shahram Rezapour
Citation: Advances in Difference Equations 2021 2021:481
19. ### Hyers–Ulam stability of impulsive Volterra delay integro-differential equations

This paper discusses different types of Ulam stability of first-order nonlinear Volterra delay integro-differential equations with impulses. Such types of equations allow the presence of two kinds of memory ef...

Authors: D. A. Refaai, M. M. A. El-Sheikh, Gamal A. F. Ismail, Bahaaeldin Abdalla and Thabet Abdeljawad
Citation: Advances in Difference Equations 2021 2021:477
20. ### Reconstructing the right-hand side of the Rayleigh–Stokes problem with nonlocal in time condition

In this paper, the problem of finding the source function for the Rayleigh–Stokes equation is considered. According to Hadamard’s definition, the sought solution of this problem is both unstable and independen...

Authors: Phuong Nguyen Duc, Ho Duy Binh, Le Dinh Long and Ho Thi Kim Van
Citation: Advances in Difference Equations 2021 2021:470
21. ### 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
22. ### 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
23. ### 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
24. ### 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
25. ### 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
26. ### 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
27. ### 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
28. ### 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
29. ### 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
30. ### 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
31. ### 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
32. ### 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
33. ### 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

34. ### 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
35. ### 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
36. ### 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
37. ### 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
38. ### 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
39. ### On solution of generalized proportional fractional integral via a new fixed point theorem

Citation: Advances in Difference Equations 2021 2021:427
40. ### 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
41. ### 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
42. ### 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
43. ### 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
44. ### 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
45. ### 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
46. ### 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
47. ### 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
48. ### 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
49. ### 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
50. ### 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