General methods for converting impulsive fractional differential equations to integral equations and applications

In this paper, we propose the concepts of Caputo fractional derivatives and Caputo type Hadamard fractional derivatives for piecewise continuous functions. We obtain general solutions of four classes of impulsive fractional differential equations (Theorem 3.1–Theorem 3.4) respectively. These results are applied to converting boundary value problems for impulsive fractional differential equations to integral equations. Some comments are made on recently published papers (see Section 4).     

On the Stability of a Retarded Functional Differential Equation Edgeworth Process

Following Smale's notations, let n and 1 denote respectively the number of agents and commodities in a pure exchange barter economy. Let u₁,u₂, ... u_n denote n real valued functions representing the individuals preferences and which depend on the past as well as present holdings. We assume that the mechanism which describes the adjustment in the quantities traded at any point in time be given by an Edgeworth process in the form of an autonomous retarded functional differential equation type. The aim of this study is first to develop necessary conditions for this type of mechanism to converge to the set of Pareto-points. Second, we examine the conditions under which positive results may be obtained as solutions of the Eggeworth dynamic approach the boundary of the smooth submanifold of feasible allocations W. Third, necessary conditions are developped to assure local stabiltyof a Pareto point. Finally, we investigate the conditions under which a Pareto point is unstable. The result on asymptotic behavior uses the ideas of LaSalle'Theorem and those on stability are proved by means of Liapunov'theory, using Hale'stability Theorem on retarded functional differential equations.     

Convolution integral equations with Gegenbauer function kernel

In this paper we develop a concise and transparent approach for solving Mellin convolution equations where the convolutor is the product of an algebraic function and a Gegenbauer function. Our method is primarily based on

1. the use of fractional integral/differential operators;

2. a formula for Gegenbauer functions which is a fractional extension of the Rodrigues formula for Gegenbauer polynomials (see Theorem 3);

3. an intertwining relation concerning fractional integral/differential operators (see Theorem 1), which in the integer case reads (d/dx)²ⁿ⁺¹ = (x⁻¹ d/dx)ⁿx²ⁿ⁺¹(x⁻¹ d/dx)ⁿ⁺¹.

Thus we cover most of the known results on this type of integral equations and obtain considerable extensions. As a special illustration we present the Gegenbauer transform pair associated to the Radon transformation.     

Ground state solution for a class fractional Hamiltonian systems

In this paper, we consider a class of Hamiltonian systems of the form $_tD_\infty^\alpha(_{-\infty} D_t^\alpha u(t))+L(t) u(t)-\nabla W(t,u(t))=0$ where $\alpha\in(\frac{1}{2},1)$, $_{-\infty}D_t^\alpha$ and $_{t}D_\infty^\alpha$ are left and right Liouville-Weyl fractional derivatives of order $\alpha$ on the whole axis $R$ respectively. Under weaker superquadratic conditions on the nonlinearity and asymptotically periodic assumptions, ground state solution is obtained by mainly using Local Mountain Pass Theorem, Concentration-Compactness Principle and a new form of Lions Lemma respect to fractional differential equations.     

Optimal control of systems governed by time-delayed,second-order,linear, parabolic partial differential equations with a first boundary condition

In this paper, we consider a class of systems governed by time-delayed, second-order, linear, parabolic partial differential equations with first boundary conditions. The existence and uniqueness of solutions of this class of systems are established in Theorem 3.2. A necessary condition for optimality for the corresponding controlled system is presented in Theorem 5.1. For the proof of this theorem, we develop several preparatory results in Sections 2, 3, and 4.     

Weak solutions to stochastic differential equations driven by fractional brownian motion

Existence of a weak solution to the n-dimensional system of stochastic differential equations driven by a fractional Brownian motion with the Hurst parameter H ∈ (0, 1) \ {1/2} is shown for a time-dependent but state-independent diffusion and a drift that may by split into a regular part and a singular one which, however, satisfies the hypotheses of the Girsanov Theorem. In particular, a stochastic nonlinear oscillator driven by a fractional noise is considered.     

Existence theorems and Hyers-Ulam stability for a class of Hybrid fractional differential equations with $p$-Laplacian operator

In this paper, we prove necessary conditions for existence and uniqueness of solution (EUS) as well Hyers-Ulam stability for a class of hybrid fractional differential equations (HFDEs) with $p$-Laplacian operator. For these aims, we take help from topological degree theory and Leray Schauder-type fixed point theorem. An example is provided to illustrate the results.     

Nonsmooth version of Fountain theorem and its application to a Dirichlet-type differential inclusion problem

In this paper we establish a nonsmooth version of Fountain theorem by adopting the framework of nonsmooth analysis theory, which is a generalization of Theorem 3.6 of Willem (1996) [2]. Then we present an application of this theorem to a Dirichlet-type differential inclusion problem.     

On Namias's Fractional Fourier Transforms

The objective of this paper is to make rigorous a formal study(Namias, 1980) of fractional powers for the Fourier transform.To make the theory unambiguous, it is found necessary to modifyNamias's fractional operators. Some theorems are then provedfor the modified operators and an operational calculus is developed.Finally, an application to an ordinary differential equationis considered.     

Boundary value problem for a first-order partial differential equation with a fractional discretely distributed differentiation operator

We solve a boundary value problem for a first-order partial differential equation in a rectangular domain with a fractional discretely distributed differentiation operator. The fractional differentiation is given by Dzhrbashyan–Nersesyan operators. We construct a representation of the solution and prove existence and uniqueness theorems. The results remain valid for the corresponding equations with Riemann–Liouville and Caputo derivatives. In terms of parameters defining the fractional differential operator, we derive necessary and sufficient conditions for the solvability of the problem.     

Families over special base manifolds and a conjecture of Campana

Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefold. The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces, as introduced by Campana in his theory of Orbifoldes Géométriques. We discuss a weak variant of the Harder?CNarasimhan Filtration and prove a version of the Bogomolov?CSommese Vanishing Theorem that take the additional fractional positivity along the boundary into account. A brief, but self-contained introduction to Campana??s theory is included for the reader??s convenience.     

On the maximum principle for time-optimal controls for a class of distributed-boundary control problems

In this note, we present the necessary conditions of optimality for time-optimal controls for a class of distributed-boundary control problems in general Banach spaces using the semigroup theory. Theorem 3.1 is based on a recent general maximum principle due to Barbu (Ref. 1), which was proved for strictly convex reflexive Banach spaces. Theorem 3.2 generalizes this result (for time-optimal control problems) by lifting the assumption.This work was supported by the National Science and Engineering Council of Canada under Grant No. 7109.     

Existence and Uniqueness of Positive Solutions for a System of Multi-order Fractional Differential Equations

In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equa-tions. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.     

A duality theorem for a homogeneous fractional programming problem

The usual theory of duality for linear fractional programs is extended by replacing the linear functions in the numerator and denominator by arbitrary positively homogeneous convex functions. In the constraints, the positive orthant inR ⁿ is replaced by an arbitrary cone. The resultant duality theorem contains a recent result of Chandra and Gulati as a special case.The authors wish to thank the referee for a number of valuable suggestions, particularly improvements in Theorem 3.4 and Corollary 3.1.     

Integral dispersion equation method to solve a nonlinear boundary eigenvalue problem

In this work a nonlinear eigenvalue problem for a nonlinear autonomous ordinary differential equation of the second order is considered. This problem describes the process of propagation of transverse-electric electromagnetic waves along a plane dielectric waveguide with nonlinear permittivity. We demonstrate, as far as we know, a new method that allows one to derive an equation w.r.t. spectral parameter (the dispersion equation) which contains all necessary information about the eigenvalues. The method is based on a simple idea that the distance between zeros of a periodic solution to the differential equation is the same for the adjacent zeros. This method has no connections with the perturbation theory or the notion of a bifurcation point. Theorem of equivalence between the eigenvalue problem and the dispersion equation is proved. Periodicity of the eigenfunctions is proved, a formula for the period is found, and zeros of the eigenfunctions are determined. The formula for the distance between adjacent zeros of any eigenfunction is given. Also theorems of existence and localization of the eigenvalues are proved.     

Inequalities of Eigenvalues for Uniformly Elliptic Operators with Higher Orders

Let (Ω) ⊂ R^m (m ≥ 2) be a bounded domain with piecewise smooth boundary ∂Ω. Let t be positive integer with t > 1. We consider the eigenvalue problems about (1.1) and (1.2), and obtain Theorem 2.1 and Theorem 2.2, which are generalizations of the results in [1-2]. This kind of problem is interesting and significant both in theory of partial differential equations and in applications to mechanics and physics.     

Differential Equations of Fractional Order: Methods,Results and Problems. II

The article, being a continuation of the first one [A.A. Kilbas and J.J. Trujillo (2001). Differential equations of fractional order. Methods, results and problems, I. Applicable Analysis , 78 (1-2), 153-192.], deals with the so-called differential equations of fractional order in which an unknown function is contained under the operation of a derivative of fractional order. The methods and the results in the theory of such fractional differential equations are presented including the Dirichlet-type problem for ordinary fractional differential equations, studying such equations in spaces of generalized functions, partial fractional differential equations and more general abstract equations, and treatment of numerical methods for ordinary and partial fractional differential equations. Problems and new trends of research are discussed.     

分数阶微分方程的一些新结果

第一部分,介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质．第二部分,利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性．第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性．第四部分,利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性．第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性（次线性）条件下C310,11正解存在的充分必要条件．最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性．     

On variational methods to non-instantaneous impulsive fractional differential equation

In this paper by using the variational methods for a class of impulsive differential equation of fractional order with non-instantaneous impulses, we setup sufficient conditions for the existence and uniqueness of weak solutions. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of a functional. Main results of the present work are established by using Lax–Milgram Theorem.     

Nonexistence for the Laplace equation with a dynamical boundary condition of fractional type

We consider the Laplace equation in ? ^d?1 × ?⁺ × (0,+∞) with a dynamical nonlinear boundary condition of order between 1 and 2. Namely, the boundary condition is a fractional differential inequality involving derivatives of noninteger order as well as a nonlinear source. Nonexistence results and necessary conditions are established for local and global existence. In particular, we show that the critical exponent depends only on the fractional derivative of the least order.