集值优化导数型最优性条件

借助于相依上导数的概念,建立了锥次类凸集值映射的导数型择一性定理,并利用择一性定理获得了集值优化导数型的最优性必要条件和充分条件.     

Second-Derivative-Free Variant of the Chebyshev Method for Nonlinear Equations

In this paper, we introduce a numerical method for nonlinear equations, based on the Chebyshev third-order method, in which the second-derivative operator is replaced by a finite difference between first derivatives. We prove a semilocal convergence theorem which guarantees local convergence with R-order three under conditions similar to those of the Newton-Kantorovich theorem, assuming the Lipschitz continuity of the second derivative. In a subsequent theorem, the latter condition is replaced by the weaker assumption of Lipschitz continuity of the first derivative.     

Fractional Differential Equations with Anti-Periodic Boundary Conditions

We are concerned with the existence of solutions of a class of fractional differential equations with anti-periodic boundary conditions involving the Caputo fractional derivative. We give two results: the first is based on Banach's fixed-point theorem, and the second is based on Schauder's fixed-point theorem.     

单侧导数和对称导数混合方式下的微分中值定理

微分中值定理是分析中的一个重要定理,文[1-2]用对称导数讨论该定理,文[3-4]用单侧导数讨论该定理,而本文把两种导数结合起来以混合方式给出该定理的三种形式,且条件更弱.     

Existence Results for the Higher-Order Weighted Caputo-Fabrizio Fractional Derivative

By the definition of the higher-order fractional derivative, we explore the central properties of the higher-order Caputo-Fabrizio fractional derivative and integral with a weighted term. Furthermore, by dint of Schaefer''s fixed point theorem, $\alpha$-$\psi$-Contraction theorem, etc., we establish the existence of solutions for nonlinear equations. We also give three examples to make our main conclusion clear.     

无穷维空间的牛顿法收敛域定理

本文对无穷维空间的映象给出了广义导数的概念,利用这种导数替代光滑映象的Frechet导数,给出了无穷维空间非光滑算子方程的阻尼牛顿法收敛域的一个定理。     

On some properties of almost-analytic tensors in an *O-space

Summary We consider an *O-space which is a kind of almost-Hermitian space. There is stated a theorem, that is, in a compact *O-space a skew-symmetric covariant almost-analytic tensor is closed. From this theorem, we get many important properties of almost-analytic tensors in an almost-K?hlerian space or a K-space and an interesting result related to the Lie derivative.     

On the unbounded extendability of solutions to nonlinear differential algebraic equations

We consider a system of nonlinear ordinary differential equations that are unsolved for the derivative of the unknown vector function and identically degenerate on the domain. We prove a theorem on the unbounded extendability of solutions to this system, allowing arbitrarily high unsolvability index with respect to the derivative.     

Investigation of exponential dichotomy of linear Itô stochastic systems with random initial data by using quadratic forms

We study conditions for the mean-square exponential dichotomy of linear Itô stochastic systems. We prove that a sufficient condition for exponential dichotomy is the existence of a quadratic form whose derivative along the solutions of a system is negative definite. The converse theorem is also proved.     

Inverse spectral problem for normal matrices and the Gauss-Lucas theorem

We establish an analog of the Cauchy-Poincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss-Lucas theorem and prove the old conjecture of de Bruijn-Springer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle's theorem, conjectured by Schoenberg.

     

Mittag–Leffler Functions and the Truncated $${\mathcal {}}$$-fractional Derivative

In this paper, we introduce a new type of fractional derivative, which we called truncated \({\mathcal {V}}\)-fractional derivative, for \(\alpha \)-differentiable functions, by means of the six-parameter truncated Mittag–Leffler function. One remarkable characteristic of this new derivative is that it generalizes several different fractional derivatives, recently introduced: conformable fractional derivative, alternative fractional derivative, truncated alternative fractional derivative, M-fractional derivative and truncated M-fractional derivative. This new truncated \({\mathcal {V}}\)-fractional derivative satisfies several important properties of the classical derivatives of integer order calculus: linearity, product rule, quotient rule, function composition and the chain rule. Also, as in the case of the Caputo derivative, the derivative of a constant is zero. Since the six parameters Mittag–Leffler function is a generalization of Mittag–Leffler functions of one, two, three, four and five parameters, we were able to extend some of the classical results of the integer-order calculus, namely: Rolle’s theorem, the mean value theorem and its extension. In addition, we present a theorem on the law of exponents for derivatives and as an application we calculate the truncated \({\mathcal {V}}\)-fractional derivative of the two-parameter Mittag–Leffler function. Finally, we present the \({\mathcal {V}}\)-fractional integral from which, as a natural consequence, new results appear as applications. Specifically, we generalize the inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. We also calculate the \({\mathcal {V}}\)-fractional integral of the two-parameter Mittag–Leffler function. Further, we were able to establish the relation between the truncated \({\mathcal {V}}\)-fractional derivative and the truncated \({\mathcal {V}}\)-fractional integral and the fractional derivative and fractional integral in the Riemann–Liouville sense when the order parameter \(\alpha \) lies between 0 and 1 (\(0<\alpha <1\)).     

Boundary control problem for the first boundary value problem for a second-order system of hyperbolic type

We consider a boundary control problem for a system of second-order hyperbolic equations without the mixed derivative. The boundary functions are constructed. We state a theorem that gives existence conditions for boundary controls.     

Integro-differential equations with multivalued solutions

We prove a theorem on the unique existence of a classical solution of an integro-differential equation with Hukuhara derivative. We also justify an averaging scheme for equations of this type in the standard form.     

On behavior at infinity of the solutions to equations with dominating mixed derivative

We prove a theorem on the polynomial asymptotics at infinity for the solutions to differential equations with dominating mixed derivative with constant coefficients.     

非线性分数阶微分方程三点边值问题解的存在性

讨论了一类非线性分数阶微分方程三点边值问题解的存在性.微分算子是Riemann-Liouville导算子并且非线性项依赖于低阶分数阶导数.通过将所考虑的问题转化为等价的Fredholm型积分方程,利用Schauder不动点定理获得该三点边值问题至少存在一个解.     

An algebroid function and its derivative

In this paper, we investigate the value distribution of an algebroid function and its derivative, and obtain two inequations between Nevanlinna characteristic function of an algebroid function and that of its derivative. We extend Chuang Chitai's theorem of meromorphic functions to algebroid functions.     

Characterizing zero-derivative points

We study smooth functions in several variables with a Lipschitz derivative. It is shown that these functions have the “envelope property”: Around zero-derivative points, and only around such points, the functions are envelopes of a quadratic parabolloid. The property is used to reformulate Fermat’s extreme value theorem and the theorem of Lagrange under slightly more restrictive assumptions but without the derivatives.     

论泰勒中值定理“中间点”的性质

研究泰勒中值定理"中间点"的单调性、连续性及可导性.     

Existence results for fractional semilinear differential inclusions in Banach spaces

We investigate the existence of at least three positive solutions to a singular boundary value problem of fractional differential equation with first-order derivative. Our analysis relies on the Avery-Peterson fixed point theorem in a cone. An example is given to illustrate our results.     

一个广义的Cauchy型的Taylor公式

给出了一个高阶导数形式的、广义的C auchy型的T ay lor公式,它将数学分析中一阶微分形式的C auchy中值定理推广到高阶导数形式,同时它也是T ay lor中值定理的推广.