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This paper aims to discuss the mathematical details in Lewis’ model by considering the analyticity and integrability conditions of characteristic functions and payoff functions of contingent claims. In his seminal paper, Lewis shows that it is much easier to compute the option value in the Fourier space than computing in terminal security price space. He computes the option value as an integral in the Fourier space, the integrand being some elementary functions and the characteristic functions of a wide range of Levy processes. The model also illustrates how the residue calculus leads to several variations of option formulas through the contour integrals. In this paper, we provide with, to a reasonable extent, some rigor into the mathematical background of Lewis’ model and validate his results for particular Levy processes. We also simply give the analyticity conditions for the characteristic function of the Carr–Geman–Madan–Yor model and a simple derivation of the characteristic function of Kou’s double exponential model.

In this paper, we study the existence and multiplicity of solutions for a class of equations involving a nonlocal term and the biharmonic operator with critical exponent. By new techniques, multiplicity results are established together with variational method. It is worth noting that the method of this paper is obviously different from the existing literature.

In this paper, a general five-step discrete-time Zhang neural network is proposed for time-varying nonlinear optimization (TVNO). First, based on the bilinear transform and Routh–Hurwitz stability criterion, we propose a general five-step third-order Zhang dynamic (ZeaD) formula, which is shown to be convergent with the truncation error $$O(\tau ^3)$$ with $$\tau >0$$ the sampling gap. Second, based on the designed ZeaD formula and the continuous-time Zhang dynamic design formula, a general five-step discrete-time Zhang neural network (DTZNN) model with quartic steady-state error pattern is presented for TVNO, which includes many multi-step DTZNN models as special cases. Third, based on the Jury criterion, we derive the effective domain of step size in the DTZNN model, which determines its convergence and convergence speed. Fourth, a nonlinear programming is established to determine the optimal step size. Finally, simulation results and discussions are given to validate the theoretical results obtained in this paper.

International audience; In this paper, we first introduce the concept and properties of ω-periodic limit process. Then we apply specific criteria obtained to investigate asymptotically ω-periodic mild solutions of a Stochastic Differential Equation driven by a Brownian motion. Finally, we give an example to show usefulness of the theoritical results that we obtain in the paper. MSC : 34C25, 34C27, 60H30, 34 F05.

The purpose of this paper is by using a generalized forward–backward splitting method to propose an iterative algorithm for finding a common element of the set of solutions to a system of quasi-variational inclusions with accretive mappings and the set of fixed points for a $$\lambda $$ -strictly pseudo-contractive mapping in Banach spaces. Some strong convergence theorems of the sequence generated by the algorithm are proved. The results presented in the paper extend and improve some recent results. As applications, we utilize our results to study the approximation problem of solutions to a system of variational inequalities, accretive variational inequality problem and convex minimization problem in Banach spaces.

In this paper, we propose an iterative algorithm to find the common element of set of solutions of a split equilibrium problem and set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. The new method is used to prove the strong convergence for the result of this paper. The result extends the corresponding one in the literature.

In the present paper, we devote our aspiration to some initial and final value problems for a class of space-fractional diffusion equation with time-dependent diffusivity factor. For the initial value problem (IVP), we investigate the stability of the solution concerning the data and the fractional order. For the final value problem, we prove the ill-posedness and suggest a filter method to regularize the problem. Explicit convergence rate of Holder type is established. Finally, several numerical examples based on the finite difference approximation and the discrete Fourier transform are performed to demonstrate the effectiveness of the proposed method.

This paper investigates the nonlinear backward fractional diffusion equations (BFDEs) having a variable order and variable diffusion coefficient. It is well-known that the BFDEs are ill-posed in the sense of Hadamard. Moreover, we investigate the problem taking into account the disturbance both variable diffusion coefficient and variable order. This may lead to some common regularization strategy to failure. Therefore, we propose a regularization method for this kind of problem. Under some appropriate regularity assumptions of the exact solution, a convergence estimate of Holder-type is proved.

In this paper, an SEIRS epidemic model is proposed, incorporating appropriate compartments as isolated exposed class and diagnosed class, relevant to interventions. In the scenario of constant isolated proportion, the qualitative analyses are carried out in terms of the basic reproduction number $$\mathscr {R}_0$$ . The sensitivity of $$\mathscr {R}_0$$ with respect to model parameters is discussed. The Pontryagin’s Maximum Principle is applied to characterize the optimal control problem analytically, aiming at finding the optimal value of the control to minimize the cost of interventions. A general explicit expression for the optimal control is obtained. Numerical simulations are performed to illustrate analytical results.

The present paper deals with a generalization of the Baskakov type operators, which preserve an exponential function and approximate functions with arbitrary order. We give some direct results including error estimation and quantitative asymptotic formula. It is observed that the rate of approximation can be made smaller by arbitrarily chosen sequences.

The aim of this paper is to complement existing oscillation results for third-order neutral advanced differential equations under the condition of $$\gamma >0$$ ; in particular, the sufficient conditions are given in different way when $$\gamma =1$$ . Our main idea is by establishing sufficient conditions for nonexistence of so-called Kneser solutions. Then, combining with the results which guarantee the equation almost oscillation, we establish sufficient condition for oscillation of all solutions.

In this paper, we obtain error bound for pseudo-binomial and negative binomial approximations to weighted sums of locally dependent random variables, using Stein’s method. We also discuss approximation results for weighted sums of independent random variables. We demonstrate our results through some applications in finance and runs in statistics.

In this paper, we prove the global well-posedness of the 3-D generalized incompressible Navier–Stokes equations in critical Besov spaces under a polynomial smallness assumption on the initial data. Moreover, we construct a class of initial data with large vertical component, which satisfies that polynomial condition but cannot satisfy the exponential condition in Liu (2020).

In this paper, we study the Korovkin-type theorem for modified Meyer–Konig and Zeller operators via A-statistical convergence and power series summability method. The rate of convergence for this new summability methods is also obtained with the help of the modulus of continuity. Further, we establish Voronovskaya-type and Gruss–Voronovskaya-type theorems for A-statistical convergence.

In this paper, we concern with the data interpolation by using extended cubic uniform B-splines with shape parameters. Two iterative formats, namely the progressive iterative approximation (PIA) and the weighted progressive iterative approximation (WPIA), are proposed to interpolate given data points. We study the optimal shape parameter and the optimal weight for the proposed methods by solving the eigenvalues of the collocation matrix. The optimal shape parameter can make the iterative methods not only have the fastest convergence speed but also have smallest initial interpolation error. Numerical experiments are given to illustrate the effectiveness of the proposed methods.

In this paper, the minimum and maximum principle sufficiency properties for a nonsmooth variational inequality problem (NVIP) are studied. We discuss the relationship among the solution set of an NVIP and those defined by its dual problem and relevant gap functions. For a pseudomonotone NVIP, the weaker sharpness of its solution set has been shown to be sufficient for it to have minimum principle sufficiency property. As special cases, pseudomonotonicity $$ _{*} $$ and pseudomonotonicity $$ ^{+} $$ of the relevant bifunction have been characterized, from which the minimum and maximum principle sufficiency properties have also been characterized.

In this paper, we consider a nonlinear cancer invasion mathematical model with proliferation, growth and haptotaxis effects. We obtain lower bounds for the finite-time blow-up of solutions of the considered system with nonlinear diffusion operator when blow-up occurs. We have assumed both the Dirichlet and Neumann boundary conditions in $${\mathbb {R}}^n, n\ge 2$$ to attain the desire result.

Fourth-order tensors play a fundamental role in signal processing, wireless communication systems, image processing, data analysis and higher-order statistics. In this paper, we introduce a Z-identity tensor and establish two Z-eigenvalue inclusion sets for fourth-order tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness of fourth-order symmetric tensors. Further, we propose upper bounds on the Z-spectral radius of fourth-order nonnegative tensors and estimate the convergence rate of the greedy rank-one algorithms under suitable conditions.

In this paper, based on Alikhanov’s L2- $$1_\sigma $$ high-order approximation and anisotropic finite element methods, a fully discrete scheme for time fractional variable coefficient diffusion equations on anisotropic meshes is presented. Firstly, we prove that the discrete scheme is unconditionally stable in $$H^1$$ -norm, then the results of convergence in $$L_2$$ -norm and superclose in $$H^1$$ -norm are derived by combining interpolation with projection, and then, the superconvergence in $$H^1$$ -norm is obtained by using interpolation post-processing technique. In addition, it is worth mentioning the key technology of combining interpolation and projection. If interpolation or projection is used alone, the results of this article cannot be obtained. Finally, numerical examples are provided to verify the correctness of theoretical analysis.

In this paper, we study a Cucker–Smale-type flocking model with distributed time delays, in which individuals interact with each other through general communication weights, and delays are information processing and reactions of individuals. By constructing two dissipative differential inequalities on position and velocity together with a continuity argument, and using the Lyapunov functional approach, we present some sufficient conditions for the existence of asymptotic flocking solutions to two classes of different communication weights. All results are novel and can be illustrated by numerical simulations using some concrete influence functions.

In this paper, we consider a stochastic delayed two-species competition chemostat model with the Monod growth response function. First, we verify that there is a unique global positive solution for any given initial conditions for this stochastic delayed system. Second, we give the dynamical behavior of the solution of stochastic delay system around the washout equilibrium of deterministic system; moreover, we discuss the competition exclusion and coexistence of microorganisms $$x_1$$ and $$x_2$$ . Finally, computer simulations are carried out to illustrate the obtained results; in addition, results show that time delay has critical effects on the survival of the microorganisms.

The purpose of this paper is to obtain large deviations, moderate deviations and normal deviations for the convergence of martingale generated by a supercritical Galton–Watson process with a Poisson process as its time index. Harmonic moments and Shroder index play an important role in the proofs.

Existing works on approximate controllability often assume that the nonlinear item is uniformly bounded and the corresponding fractional linear system is approximate controllable, which is, however, too constrained. In this paper, we omit these two assumptions and investigate the approximate controllability of fractional stochastic differential equations driven by fractional Brownian motion. We also demonstrate that our results can be extended to fractional stochastic differential equations with bounded delay.

This paper is concerned with the global well-posedness of the n-dimensional generalized incompressible Navier–Stokes equations with initial data in the critical Besov space. By fully using the advantage of suitable weighted functions and the Fourier localization technique, the global well-posedness is obtained without the hypothesis of smallness on initial data, but for some nonlinear smallness of the first iterate. We also give an example of initial data satisfying the nonlinear smallness condition, but whose norm is arbitrarily large.

The goal of this paper is to introduce and to initiate a study of a predator–prey system with Holling type II functional response, which is disturbed by both Markov switching and white noise. Our main aim is to investigate the dynamical properties of this model. First, we establish sufficient conditions of the strong persistence in the mean and the extinction of the predator population and obtain the threshold between them. Then, we show the existence of a unique ergodic stationary distribution under some natural conditions. Finally, a numerical example and figures are presented to illustrate the results.

The main purpose of this paper is to introduce a method to handle some boundary value problems generated by fifth-order formally symmetric differential equation and separated, real-coupled and complex-coupled boundary conditions. Moreover, the continuity properties of the eigenvalues of these problems on some data are studied and some Frechet derivatives of the eigenvalues are introduced.

In this paper, the weak solutions to a class of nonlinear system of fractional boundary value problems are studied. Each equation of the system does have two kinds of nonlinear terms; one of them is in terms of gradient, and the other one is a nonlinear source term with respect to dependent variable. It is proved that there exists at least three distinct weak solutions by using the critical point theory and the variational method. To this aim, we apply the well-known theorem on the construction of the critical set of functionals with a weak compactness condition. Moreover, the main result is demonstrated by some examples to show its validity.

An entropy-stable residual distribution (RD) method is developed for the systems of two-dimensional shallow water equations (SWE). The construction of entropy stability for the residual distribution method is derived from finite volume method principles, albeit using a multidimensional approach. The paper delves into in-depth discussions on how finite volume methods achieve entropy stability in a “one-dimensional” sense for two-dimensional systems of SWE unlike the residual distribution methods. Results herein demonstrate the superiority of the entropy-stable RD methods relative to their finite volume counterparts, especially on highly irregular triangular grids. The comparative results with other established RD methods are also included, depicting similar performances with the Lax–Wendroff method for unsteady smooth flows but more accurate than the multidimensional upwind approaches (N, LDA) on both smooth and discontinuous test cases.

In this paper, the Cauchy problem associated with the biharmonic equation is investigated. We prove that in principle, the problem is severely ill-posed in the sense of Hadamard. Therefore, we propose a quasi-boundary value-type regularization method for stabilizing the ill-posed problem. Very sharp convergence estimates are established based on some a priori information on the exact solution. Finally, several numerical examples with random data are provided to show the effectiveness of the proposed method.

In this paper, a class of nonlinear viscoelastic Kirchhoff equation in a bounded domain with a time-varying delay in the weakly nonlinear internal feedback is considered, where the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo–Galerkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given.

In this paper, we consider the regularity criteria, where a condition is added only on the velocity field to guarantee the smoothness of whole unknowns, and the global existence of the solutions with small initial data for the 3D tropical climate model with diffusion.

In this paper, nonlinear Fredholm integral equation of the second kind is solved by using parameter continuation method. Then, we propose parameter continuation method to solve perturbed nonlinear Fredholm integral equation of the second kind, which appear as an extension of the method of contractive mapping and parameter continuation method for solving nonlinear Fredholm integral equation of the second kind. Illustrative examples are presented to show the effectiveness and convenience of parameter continuation method.

In this paper, we attempted to characterize the exponential q-distribution through the q-memorylessness property using the q-addition operator and Jackson integral. Moreover, an extended version of k-gamma q-distribution is introduced and the q-moments of this family is computed. Finally, we suggested a new q-inversion method to simulate data from a q-distribution.

In this paper, for the IBVP of a fourth-order nonlinear parabolic equation, which is related to image analysis, we studied the existence and uniqueness of weak solutions. Moreover, we also considered the asymptotic behavior and the regularity of solutions of such problem.

In this paper, we consider the Cauchy problem to the 2D generalized tropical climate model. We establish a new blow-up criterion only in terms of the first baroclinic mode of the vector velocity in Besov spaces with negative indices.

In this paper, we present a new fractional Tikhonov regularization method for solving an inverse problem for a time-fractional diffusion equation which is highly ill-posed in the two-dimensional setting. Fractional Tikhonov regularization method not only retains the advantage of classical Tikhonov method, but also overcomes the effect of over-smoothing of classical Tikhonov method. We give the selection of regularization parameters of the new method and the corresponding error estimation. Furthermore, numerical results show that the fractional Tikhonov method outperforms the classical one.

This paper deals with the Ulam stability of linear differential equations by using the method of variation of parameters, which provides a unified method to study the Ulam stability problem of linear differential equations of n-th order with constant and nonconstant coefficients. As an application of the main results, we also obtain the Hyers–Ulam stability of the Cauchy–Euler differential equations of second order, third order and n-th order. Our results make up to some deficiencies in the relevant literature.

In this paper, we study the well-posedness of mild solutions to stochastic parabolic partial functional differential equations with space–time white noise. Firstly, we establish an existence–uniqueness theorem under the global Lipschitz condition and the linear growth condition. Secondly, we show the existence–uniqueness property under the global/local Lipschitz condition but without assuming the linear growth condition. In particular, we consider the existence and uniqueness under the weaker condition than the Lipschitz condition. Finally, we obtain the nonnegativity and comparison theorems and utilize them to investigate the existence of nonnegative mild solutions under the linear growth condition without assuming the Lipschitz condition.

In this paper, we present a new optimal approach to the calculation of the twists for constructing bicubic Coons interpolating surfaces from given points. A new objective function called minimal oscillation is proposed to generate the bicubic Coons surface to approximate bilinear interpolation by optimizing twists of Coons surfaces. A linear system with block tridiagonal coefficient matrix is established to solve the optimal problem. The efficiency of the proposed method is illustrated by some numerical examples.

Our aim in this paper is to study variants and computational errors of the extragradient method for solving equilibrium problems. First, we consider convergence of the method when domains in the auxiliary subproblems of the extragradient algorithm are replaced by outer and inner approximation polyhedra. Then, computational errors are showed under the asymptotic optimality condition, but the bifunction must satisfy certain Lipschitz-type continuous conditions. Next, by using Armijo-type linesearch techniques commonly used in variational inequalities, we obtain an approximation linesearch algorithm without Lipschitz continuity. Convergence analysis of the algorithms is considered under mild conditions on the iterative parameters.

In this paper, we investigate the solutions of certain types of higher-order linear differential equations. By introducing a new concept of n-subnormal solution, we study the existence, growth, and numbers of solutions of this type, and we also estimate the growth of all other solutions.

This paper investigates the problem of finite-time stability and control for a class of nonlinear singular discrete-time neural networks with time-varying delays and disturbances. First, based on the implicit function theorem and singular value decomposition method, a sufficient condition for the existence of the solution of such systems is established in terms of a linear matrix inequality (LMI). Then, using the Lyapunov functional approach combined with LMI technique we provide new delay-dependent sufficient conditions for robust $$H_{\infty }$$ finite-time stability and control. Finally, some numerical examples are given to illustrate the efficiency of the proposed results.

In this paper, by using the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, we obtain sufficient conditions for the existence, uniqueness and global exponential stability of weighted pseudo-almost periodic solutions of a class of shunting inhibitory cellular neural networks with mixed delays on time scales. A numerical example is also presented to illustrate the feasibility of our results.

In the paper, we introduce the higher-order generalized radial epiderivative of set-valued maps. Then, its basic properties are studied. Finally, we apply this epiderivative to optimality conditions for set-valued optimization problems.

In this paper we consider a class of Kirchhoff-type problem with 2-superlinear potentials. The existence of one positive solution and one negative solution will be established by using iterative technique and the Mountain Pass theorem, and a sign changing solution will be obtained by combining iterative technique and the Nehari method.

In this paper, by applying the cone theory in ordered Banach spaces associated with the characters of increasing $$\varphi -(h,e)$$ -concave operators, we investigate the existence and uniqueness of nontrivial solutions for a nonlinear fractional q-difference equation boundary value problem. The main results show that we can construct an iterative scheme approximating the unique nontrivial solution. Relying on an example, we show the efficiency and applicability of the main result.

The Jensen’s inequality plays a crucial role to obtain inequalities for divergences between probability distributions. In this paper, we introduce a new functional, based on the f- divergence functional, and then, we obtain some estimates for the new functional, the f- divergence and the Rényi divergence by applying a cyclic refinement of the Jensen’s inequality. Some inequalities for Rényi and Shannon entropies are obtained too. Zipf–Mandelbrot law is used to illustrate the results.

Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two (possibly simpler) nonlinear operators. Most of the investigation on splitting methods is however carried out in the framework of Hilbert spaces. In this paper, we consider these methods in the setting of Banach spaces. We shall introduce a viscosity iterative forward–backward splitting method with errors to find zeros of the sum of two accretive operators in Banach spaces. We shall prove the strong convergence of the method under mild conditions. We also discuss applications of these methods to monotone variational inequalities, convex minimization problem and convexly constrained linear inverse problem.

In this paper, we study optimal control problems for a class of fractional stochastic functional differential equations of order \(\alpha \in (1, 2]\) in Hilbert spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence of mild solutions are investigated by using the fractional calculus, stochastic analysis theory, and fixed point theorems with the strongly continuous \(\alpha \)-order cosine family. Then, the existence conditions of optimal pairs of systems governed by a fractional stochastic partial differential equations of order \(\alpha \in (1, 2]\) with infinite delay are presented. The results are obtained under the mixed Lipschitz and Caratheodory conditions. Finally, an example is presented to illustrate the main results.

This paper establishes the existence-and-uniqueness theorem of a stochastic delayed predator–prey model with Hassell–Varley type functional response and examines stochastically ultimate boundedness, extinction and global asymptotic stability of this solution. It is interesting to note that the results are based on time-varying delay, which is different from the previous work (the results are delay-independent). Some numerical simulations are introduced to support the analytical findings.