AN EXPLANATION ON FOUR NEW DEFINITIONS OF FRACTIONAL OPERATORS

Fractional calculus has drawn more attentions of mathematicians and engineers in recent years.A lot of new fractional operators were used to handle various practical problems.In this article,we mainly study four new fractional operators,namely the CaputoFabrizio operator,the Atangana-Baleanu operator,the Sun-Hao-Zhang-Baleanu operator and the generalized Caputo type operator under the frame of the k-Prabhakar fractional integral operator.Usually,the theory of the k-Prabhakar fractional integral is regarded as a much broader than classical fractional operator.Here,we firstly give a series expansion of the k-Prabhakar fractional integral by means of the k-Riemann-Liouville integral.Then,a connection between the k-Prabhakar fractional integral and the four new fractional operators of the above mentioned was shown,respectively.In terms of the above analysis,we can obtain this a basic fact that it only needs to consider the k-Prabhakar fractional integral to cover these results from the four new fractional operators.

A Set of Almost Automorphic Functions and Applications

For a set S of real numbers, we introduce the concept of S-almost automorphic functions valued in a Banach space. It generalizes in particular the space of Z-almost automorphic functions. Considering the space of S-almost automorphic functions, we give sufficient conditions of the existence and uniqueness of almost automorphic solutions of a differential equation with a piecewise constant argument of generalized type. This is done using the Banach fixed point theorem.

Performance investigation of plasma magnetohydrodynamic power generator

A magnetohydrodynamic （MHD） power generator system involves several subjects such as magnetohydrodynamics, plasma physics, material science, and structure mechanics. Therefore, the performance of the MHD power generator is affected by many factors, among which the load coefficient k is of great importance. This paper reveals the effect of some system parameters on the performance by three-dimensional （3D） numerical simulation for a Faraday type MHD power generator using He/Xe as working plasma. The results show that average electrical conductivity increases first and then decreases with the addition of magnetic field intensity. Electrical conductivity reaches the maximum value of 11.05 S/m, while the applied magnetic field strength is B = 1.75 T. When B 〉 3 T, the ionization rate along the midline well keeps stable, which indicates that the ionization rate and three-body recombination rate （three kinds of particles combining to two kinds of particles） are approximately equal, and the relatively stable plasma structure of the mainstream is preserved. Efficiency of power generation of the Faraday type channel increases with an increment of the load factor. However, enthalpy extraction first increases to a certain value, and then decreases with the load factor. The enthalpy extraction rate reaches the maximum when the load coefficient k equals 0.625, which is the best performance of the power generator channel with the maximum electricity production.

Solitons and Bifurcations for the Generalized Tzitzéica Type Equation in Nonlinear Fiber Optics

Solitons and bifurcations for the generalized Tzitzéica type equation are studied by using the theory of dynamical systems and Hamilton function. With the help of Maple and bifurcation theory of differential equations, the bifurcation parameter conditions and all the bifurcation phase portraits are obtained. Because the same energy value of the Hamiltonian function is corresponding to the same orbit, thus the periodic wave solutions, bright soliton and dark soliton solutions are defined.

Group of Weakly Continuous Operators Associated to a Generalized Schrödinger Type Homogeneous Model

In this work, we prove the existence and uniqueness of the solution of the generalized Schrödinger type homogeneous model in the periodic distributional space P’. Furthermore, we prove that the solution depends continuously respect to the initial data in P’. Introducing a family of weakly continuous operators, we prove that this family is a group of operators in P’. Then, with this family of operators, we get a fine version of the existence and dependency continuous theorem obtained. Finally, we give some remarks derived from this study.

Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for Lagrange systems

Based on the invariance of differential equations under infinitesimal transformations of group, Lie symmetries, exact invariants, perturbation to the symmetries and adiabatic invariants in form of non-Noether for a Lagrange system are presented. Firstly, the exact invariants of generalized Hojman type led directly by Lie symmetries for a Lagrange system without perturbations are given. Then, on the basis of the concepts of Lie symmetries and higher order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for the system with the action of small disturbance is investigated, the adiabatic invariants of generalized Hojman type for the system are directly obtained, the conditions for existence of the adiabatic invariants and their forms are proved. Finally an example is presented to illustrate these results.

Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Relativistic Birkhoffian Systems

For a relativistic Birkhoffian system, the Lie symmetrical perturbation and adiabatic invariants of generalized Bojman type are studied under general infinitesimal transformations. On the basis of the invariance of relativistic Birkhotfian equations under general infinitesimal transformations,Lie symmetrical transformations of the system are constructed, which only depend on the Birkhoffian variables. The exact invariants in the form of generalized Hojman conserved quantities led by the Lie symmetries of relativistic Birkhoffian system without perturbations are given. Based on the definition of higher-order adiabatic invariants of a mechanical system, the perturbation of Lie symmetries for relativistic Birkhoffian system with the action of small disturbance is investigated, and a new type of adiabatic invariants of the system is obtained. In the end of the paper, an example is given to illustrate the application of the results.

ASYMPTOTIC EQUIVALENCE OF ALTERNATELY ADVANCED AND DELAYED DIFFERENTIAL SYSTEMS WITH PIECEWISE CONSTANT GENERALIZED ARGUMENTS

In this paper, we investigate the existence, uniqueness and the asymptotic equiv- alence of a linear system and a perturbed system of differential equations with piecewise alternately advanced and retarded argument of generalized type （DEPCAG）. This is based in the study of an equivalent integral equation with Cauchy and Green matrices type and in a solution of a DEPCAG integral inequality of Gronwall type. Several examples are also given to show the feasibility of results.

UNIFORM APPROXIMATION OF ENTIRE FUNCTIONS OF SLOW GROWTH ON COMPACT SETS

In the present paper, we study the polynomial approximation of entire functions of several complex variables. The characterizations of generalized order and generalized type of entire functions of slow growth are obtained in terms of approximation and interpolation errors.

ON APPROXIMATION AND GENERALIZED TYPE OF ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES

In the present paper, we study the polynomial approximation of analytic functions of several complex variables. The characterizations of generalized type of analytic functions of several complex variables have been obtained in terms of approximation and interpolation errors.

New bounds of (s,S) policies in periodical review inventory systems

We study the stochastic inventory problem with optimal （s,S） policies.In a finite horizon model with lost sales,we establish new lower and upper bounds of s and S.These bounds have structural implications for the optimal solutions.Consequently,when demand has a generalized phase type distribution,there are no more than a pre-determined number of minima.Similar bounds can also be found for the system where unsatisfied demand is backordered instead of lost sales.

Unified treatment of one-range addition theorems for integer and non-integer n-STO, -GTO and -generalized exponential type orbitals with hyperbolic cosine in position, momentum and four-dimensional spaces

Simpler formulas are derived for one-range addition theorems for the integer and noninteger n generalized ex- ponential type orbitals, momentum space orbitals, and hyperspherical harmonics with hyperbolic cosine （GETO HC, GMSO HC, and GHSH HC） in position, momentum and four-dimensional spaces, respectively. The final results are expressed in terms of one-range addition theorems of complete orthonormal sets of Ca-exponential type orbitals, Ca- momentum space orbitals and za-hyperspherical harmonics. We notice that the one-range addition theorems for integer and noninteger n-Slater type orbitals and Gaussian type orbitals in position, momentum and four dimensional spaces are special cases of GETO HC, GMSO HC, and GHSH HC. The theorems presented can be useful in the accurate study of the electronic structure of atomic and molecular systems.

The Grow th of the Entire Func tion Represented by the Laplace-Stieltjes Transform in the Whole Plane

In the paper, the α-order of the Laplace-Stieltjes Transform is introducedfirstly, then we get the relationship between α-order represented by the maximum modulus and α-order represented by A^*n, λn. Lastly, we obtain the relationship between type τrepresented by the maximum modulus and type τ represented by A^*n, λn.

Fixed Point Theorem of{a, b, c}Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces

Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following： 1. If T is {a,b,c}-ntype mapping, then inf{ ｜｜ T（x）-x｜｜ ：x C C} =0, accordingly T has a unique fixed point. Moreover, any sequence {Xn}n∈NN in C with limn→∞｜｜T（xn） - Xn｜｜ = 0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any x∈C the sequence of iterates {Tn （x）}n∈N has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4, 7] and [13].

ELASTOSTATIC SOLUTOINS FOR SEMI-INFINITE ORTHOTROPIC CANTILEVERED STRIPS

The elastostatic solutions of semi-infinite orthotropic cantilevered strips with traction free edges and loading at infinity are governed by the differential equationφ,■+(2+δ_(0))φ,■+φ,■=0 withδ_(0)>-4 with Based on the work of[10]forδ_(0)>0 case,.this paper completes the caseδ_(0)=0 for isotropic materials and the case 0>δ_(0)>-4 for orthotropic materials.The solutions of the above problems have important application in the properly formulated boundary conditions of plate theories for prescribed displacement edge data.

BLOW－UP AND DIE－OUT OF SOLUTIONS OF NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS OF GENERALIZED NERVE CONDUCTION TYPE

This paper considers the initial boundary value problems with three types of the boundary conditions for nonlinear pseudo-hyperbolic equations of generalized nerve conduction type, using foe eigenfunction method, the conditions for which the solutions blow-up and die-out in the finile time are got.

Generalized S-KKM Theorems and Minimax Inequalities in FC-Spaces

The concept of FC-closed subset in FC-space without any linear structure was introduced. Then, the generalized KKM theorem is proved for FC-closed value mappings under some conditions. The FC-space in the theorem is the generalization of L-convex space and the condition of the mapping with finitely FC-closed value is weaker than that with finitely L-closed value.

DUAL FUNCTIONALS OF SAID-BZIER TYPE GENERALIZED BALL BASES OVER TRIANGLE DOMAIN AND THEIR APPLICATION

A family of Said-Bézier type generalized Ball （SBGB） bases and surfaces with a parameter H over triangular domain is introduced,which unifies Bézier surface and Said-Ball surface and includes several intermediate surfaces. To convert different bases and surfaces,the dual functionals of bases are presented. As an application of dual functionals,the subdivision formulas for surfaces are established.

SIMILARITY TRANSFORMATION, THE STRUCTURE OF THE TRAVELING WAVES SOLUTION AND THE EXISTENCE OF A GLOBAL SMOOTH SOLUTION TO GENERALIZED KURAMOTO SIVASHINSKY TYPE EQUATIONS

The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for the generalized KS equations are discussed and analysed by using the qualitative theory of ODE and Lie's infinitesimal transformation respectively.

On generalized impulsive piecewise constant delay differential equations

A variation of parameters formula with Green function type and Gronwall type integral inequality are proved for impulsive differential equations involving piecewise constant delay of generalized type. Some results for piecewise constant linear and nonlinear delay differential equations with impulsive effects are obtained.They include existence and uniqueness theorems, a variation of parameters formula, integral inequalities, the oscillation property and some applications.