Recent Development on Nonlinear Methods in Function Spaces and Applications in Nonlinear Fractional Differential Equations

This special issue mainly focuses on the theme: recent development on nonlinear methods in function spaces and applications in nonlinear fractional differential equations. Nonlinear fractional differential equations arise from scientific research, modeling of nonlinear phenomena, and optimal control of complex systems. Function space theory has played an important role in the study of real world nonlinear fractional differential equations and the development of new technologies. Key research areas in this field include wellposedness of fractional mathematical models, development on nonlinear methods in function space, approximation theories and operator theory in function space, numerical computational methods, and control for nonlinear fractional differential equations. In general terms, studies of nonlinear fractional differential equations are very important nowadays and thus this special issue aims to present some of the recent developments in this field. Here, we would like to appraise highly the excellent contributions of all authors and reviewers. The efforts made by members of guest editors are also greatly acknowledged. This issue contains 25 papers selected through a rigorous peer-reviewed process, covering most of the fields of nonlinear fractional differential equations. Recent developments in nonlinear methods in function spaces and applications are addressed in most of the papers in this issue and many innovative ideas and in-depth scientific approaches are given. These contributions will attract attention from the research community of this field and promote further research in this field. In the following, we briefly summarize the main contributions of each paper.

In the following, we briefly summarize the main contributions of each paper.
(1) In the paper titled "Approximation of Functions on a Square by Interpolation Polynomials at Vertices and Few Fourier Coefficients," the author developed a new approximation scheme to overcome the weakness of Fourier approximation.In detail, the author used Lagrange interpolation and linear interpolation on the boundary of the square to derive a new approximation scheme such that one can use the values of the target function at vertices of the square and few Fourier coefficients to reconstruct the target function with very small error.
(2) In the paper titled "Approximate Controllability for Functional Equations with Riemann-Liouville Derivative by Iterative and Approximate Method," the authors discussed the functional control systems governed by differential equations with Riemann-Liouville fractional derivative in general Banach spaces.By the approach of fixed point and fractional resolvent under more general settings, the uniqueness and existence of mild solutions for functional differential equations were derived.Moreover, by means of the iterative and approximate method, some new sufficient conditions for approximate controllability of functional control systems were also presented.(3) In the paper titled "Positive Solutions for Singular Semipositone Fractional Differential Equation Subject to Multipoint Boundary Conditions," the existence results together with multiplicity result of positive solutions of higher-order fractional multipoint boundary value problems were established by considering the integrations of height functions on some special bounded sets.The nonlinearity of equation in this paper may be sign-changing and may possess singularities on the time and the space variables at the same time.
(4) In the paper titled "Existence of Solutions for a Class of Coupled Fractional Differential Systems with Nonlocal Boundary Conditions," by applying Schauder fixed-point theorem and Leray-Schauder nonlinear alternative theory, the authors were concerned with the existence of solutions to coupled fractional differential systems with fractional integral boundary value conditions.Two examples were given to illustrate the application of the main results.
(5) In the paper titled "On the Existence of Positive Solutions for a Fourth-Order Boundary Value Problem," by using the method of order reduction and the fixedpoint index, the existence of positive solutions for a fourth-order boundary value problem was studied and the conditions guaranteeing the existence of positive solutions were given which are related to the first eigenvalue corresponding to the relevant linear differential equation with dependence on the derivatives of the unknown function.
(6) In the paper titled "Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities," the authors were concerned with the existence of periodic solutions for the fully second-order ordinary differential equation, where the nonlinearity is continuous and (, , ) is 2-periodic in .Under certain inequality conditions that (, , ) may be superlinear growth on (, ), an existence result of odd 2-periodic solutions was obtained via the Leray-Schauder fixed-point theorem.
(7) In the paper titled "A Regularity Criterion for the 3D Incompressible Magnetohydrodynamics Equations in the Multiplier Spaces," the authors were concerned with the regularity criterion for weak solutions to the 3D incompressible MHD equations.They showed that if some partial derivatives of the velocity components and magnetic components belong to the multiplier spaces, then the solution actually is smooth on (0, ).
(8) In the paper titled "Rogue Wave Solutions and Generalized Darboux Transformation for an Inhomogeneous Fifth-Order Nonlinear Schrödinger Equation," the rogue wave solutions were discussed for an inhomogeneous fifth-order nonlinear Schrödinger equation, which describes the dynamics of a sitedependent Heisenberg ferromagnetic spin chain.
Using the Darboux matrix, the generalized Darboux transformation was constructed and a recursive formula was derived.Based on the transformation, the first-order to the third-order rogue wave solutions were obtained.Then, the nonlinear dynamics of the first-order to the third-order rogue waves were also studied on the basis of some free parameters.
Several new structures of the rogue waves were found by using numerical simulation.The results give a supportive tool to study the rogue waves.
(9) In the paper titled "Oscillation Criteria for Nonlinear Third-Order Neutral Dynamic Equations with Damping on Time Scales," the authors established several oscillation criteria for a class of third-order nonlinear dynamic equations with a damping term and a nonpositive neutral coefficient by using the Riccati transformation.Two illustrative examples were presented to show the significance of the results obtained.
(10) In the paper titled "Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels," the authors developed a generalized Jacobi-Galerkin method for the second kind Volterra integral equations with weakly singular kernels.In this method, some known singular nonpolynomial functions in the approximation space of the conventional Jacobi-Galerkin method were first introduced and then the Gauss-Jacobi quadrature rules to approximate the integral term in the resulting equation were used to obtain high-order accuracy for the approximation.In the end, the author established that the approximate equation has a unique solution and the approximate solution arrives at an optimal convergence order.A numerical example was presented to demonstrate the effectiveness of the proposed method.
(11) The paper titled "Existence of Mild Solutions and Controllability of Fractional Impulsive Integrodifferential Systems with Nonlocal Conditions" was concerned with the existence results of nonlocal problems for a class of fractional impulsive integrodifferential equations in Banach spaces.The authors defined a piecewise continuous control function to obtain the results on controllability of the corresponding fractional impulsive integrodifferential control systems.The results were obtained by means of fixed-point methods.An example to illustrate the applications of the main results was given.
(12) In the paper titled "Existence and Multiplicity of Nontrivial Solutions for a Class of Semilinear Fractional Schrödinger Equations," the authors were concerned with the existence of solutions to a class of fractional Schrödinger type equations where the primitive of the nonlinearity  is of superquadratic growth near infinity in  and the potential  is allowed to be sign-changing.By using variant fountain theorems, a sufficient condition was obtained for the existence of infinitely many nontrivial high energy solutions.
(13) In the paper titled "New Result on the Critical Exponent for Solution of an Ordinary Fractional Differential Problem," the authors gave some background of fractional Cauchy problems and corrected an existing result of Cauchy problem for the fractional differential inequality, and they also gave an example to illustrate the statement.
(14) In the paper titled "Ulam Type Stability for a Coupled System of Boundary Value Problems of Nonlinear Fractional Differential Equations," the authors discussed existence, uniqueness, and Hyers-Ulam stability of solutions for coupled nonlinear fractional order differential equations (FODEs) with boundary conditions.Using generalized metric space, they obtained some relaxed conditions for uniqueness of positive solutions for the mentioned problem by using Perov's fixed-point theorem.Moreover, necessary and sufficient conditions were obtained for existence of at least one solution by the Leray-Schauder-type fixed-point theorem.Further, the authors also developed some conditions for Hyers-Ulam stability.To demonstrate the main result, a proper example was provided.
(15) In the paper titled "Multiple Positive Solutions for Quadratic Integral Equations of Fractional Order," the existence of multiple positive solutions for a class of quadratic integral equation of fractional order was obtained by utilizing Avery-Henderson and Leggett-Williams multiple fixed-point theorems on cones.An example was given to illustrate the applicability of the results.
(16) In the paper titled "Global Structure of Positive Solutions for Some Second-Order Multipoint Boundary Value Problems," the authors investigated a class of second-order multipoint boundary value problems.Under some conditions, they obtained global structure of positive solution set of the boundary value problem and the behavior of positive solutions with respect to parameter  by using the global bifurcation method.The authors also obtained the infinite interval of parameter  about the existence of positive solution.
(17) In the paper titled "A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay," a linearized compact finite difference scheme was constructed for solving the fractional neutral parabolic differential equation with proportional delay.By the energy method, the unconditional stability of the scheme was proved, and the convergence order of the scheme was proved to be ( 2− + ℎ 4 ).A numerical test was also conducted to validate the accuracy and efficiency of the numerical algorithm.
(18) In the paper titled "The Existence of Solutions to Integral Boundary Value Problems of Fractional Differential Equations at Resonance," the authors dealt with the integral boundary value problems of fractional differential equations at resonance.By Mawhin's coincidence degree theory, they presented some new results on the existence of solutions for a class of differential equations of fractional order with integral boundary conditions at resonance.An example was also included to illustrate the main results.
(19) In the paper titled "Exact Solutions of the Vakhnenko-Parkes Equation with Complex Method," the authors derived the exact solutions to the Vakhnenko-Parkes equation by means of the complex method, and the main results are illustrated by some computer simulations.
(20) In the paper titled "Rapid Convergence for Telegraph Systems with Periodic Boundary Conditions," the generalized quasilinearization method was applied to a telegraph system with periodic boundary conditions where the forcing function satisfies some conditions.The authors developed nonlinear iterates of order −1 which are different with  being even or odd.Finally, they developed two sequences which converge to the solution of the telegraph system and the convergence is of order .Theory" was concerned with the existence of multiple solutions for a class of nonlinear fractional boundary value problems.The existence of infinitely many nontrivial high or small energy solutions was obtained by using variant fountain theorems.
(23) In the paper titled "Approximating Solution of Fabrizio-Caputo Volterra's Model for Population Growth in a Closed System by Homotopy Analysis Method," the authors studied the existence, uniqueness, and approximating for a Caputo-Fabrizio Volterra's model for population growth in a closed system subject to some initial condition.The mechanism for approximating the solution was Homotopy Analysis Method which is a semianalytical technique to solve nonlinear ordinary and partial differential equations.Furthermore, the authors used the same method to analyze a similar closed system by considering classical Caputo's fractional derivative.Comparison between the results for these two factional derivatives was also included.
(24) In the paper titled "Periodicities of a System of Difference Equations," the authors studied the periodicities of a system of difference equations with some initial values.They showed that if sequences are two periodic sequences, then every solution of the above system is eventually periodic with period 2. If  is even, there must be one sequence that converges to period-two solution.
(25) In the paper titled "Weak and Strong Convergence Theorems for the Multiple-Set Split Equality Common Fixed-Point Problems of Demicontractive Mappings," the authors considered mixed parallel and cyclic iterative algorithms to solve the multiple-set split equality common fixed-point problem which is a generalization of the split equality problem and the split feasibility problem for the demicontractive mappings without prior knowledge of operator norms in real Hilbert spaces.Some weak and strong convergence results were established.The results obtained in this paper generalize and improve the recent ones obtained by many others.
Through the special issue, we also hope to open the opportunity for the journal readers to make comments on the work presented.

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In the paper titled "Positive Solutions of Fractional Differential Equations with -Laplacian," the multiplicity of positive solution for a new class of four-point boundary value problems of fractional differential equations with -Laplacian operator was investigated.By the use of the Leggett-Williams fixed-point theorem, the multiplicity results of positive solutions are obtained.An example was given to illustrate the main results.(22) The paper titled "Multiple Solutions for a Nonlinear Fractional Boundary Value Problem via Critical Point