Boundedness of higher order commutators of generalized fractional integral operators on Hardy spaces

Let Tμ,b,m be the higher order commutator generated by a generalized fractional integral operator Tμ and a BMO function b. In this paper, we will study the boundedness of Tμ,b,m on classical Hardy spaces and Herz-type Hardy spaces.     

BOUNDEDNESS OF GENERALIZED FRACTIONAL INTEGRAL OPERATORS

The authors introduce a new kind of fractional integral operators,namely,the so called (θ,N)-type fractional integral operators,and discuss their boundedness on the Hardy spaces,the weak Hardy spaces and the Herz-type Hardy spaces.     

Boundedness of generalized fractional integral operators

The authors introduce a new kind of fractional integral operators, namely, the so called (θ,N)-type fractional integral operators, and discuss their boundedness on the Hardy spaces, the weak Hardy spaces and the Herz-type Hardy spaces.     

A formula for the powers of fractional order operators

We derive none some explicit formula for the power of fractional order (differential and integral) operators.     

Approximation with arbitrary order by certain linear positive operators

This paper aims to highlight classes of linear positive operators of discrete and integral type for which the rates in approximation of continuous functions and in quantitative estimates in Voronovskaya type results are of an arbitrarily small order. The operators act on functions defined on unbounded intervals and we achieve the intended purpose by using a strictly decreasing positive sequence \((\lambda _n)_{n\ge 1}\) such that \(\lim \limits _{n\rightarrow \infty }\lambda _n=0\), how fast we want. Particular cases are presented.     

A generalized moment problem for self-adjoint operators

Let the sequence {λ _i } (i≧0) satisfy condition (1.1) and let {A _n} (n≧0) be a sequence of bounded self-adjoint operators over a complex Hilbert spaceH. We give a necessary and sufficient condition in order that {A _n} (n≧0) should possess the representation (1.2).     

Scattering theory for elliptic operators of arbitrary order

We prove the main theorems of scattering theory for selfadjoint elliptic partial differential operators of arbitrary order. Under various hypotheses we show that the wave operators exist and are complete, that the intertwining relations hold, and that the invariance principle holds. One of our main hypotheses is that each lower order coefficientq(x) satisfies. $$(1 + \left| x \right|)^\alpha \int\limits_{\left| {x - y} \right|< a} {\left| {q(y)} \right|dy \in L^p (E^n )}$$ for some α≥0,a>0 and forp≤∞ such that $$\alpha > 1 - \frac{{2n}}{{(n + 1)p}}$$      

Variational principle of fractional order generalized thermoelasticity

Recently, Youssef constructed a new theory of fractional order generalized thermoelasticity by taking into account the theory of heat conduction in deformable bodies, which depends upon the idea of the Riemann–Liouville fractional integral operator. In this paper, the variational theorem is obtained for the generalized thermoelasticity model for a homogeneous and isotropic body.     

Initial value problems for arbitrary order fractional differential equations with delay

By fixed point theory the nonlinear alternative of Leray–Schauder type, and the properties of absolutely continuous functions space, we study the existence and uniqueness of initial value problems for nonlinear higher fractional equations with delay, and obtain some new results involving local and global solutions.     

On the generalized Hankel-Clifford transformation of arbitrary order

Two generalized Hankel-Clifford integral transformations verifying a mixed Parseval relation are investigated on certain spaces of generalized functions for any real value of their orders (α-β).     

A problem with operators of fractional differentiation in boundary condition for mixed-type equation

We consider a question on unique solvability of a boundary-value problem with fractional derivatives for a mixed-type equation of second order. We prove first a uniqueness theorem. The existence theorem is proved by means of reduction to Fredholm equation of the second kind, and its unconditional solvability follows from the uniqueness of solution.     

Second order duality in minmax fractional programming with generalized univexity

In this paper, the concept of second order generalized α-type I univexity is introduced. Based on the new definitions, we derive weak, strong and strict converse duality results for two second order duals of a minmax fractional programming problem.     

The generalized order complementarity problem

Given an ordered Banach Space (E,K) andm functionsf ₁,f ₂,...,f _m:EE, the generalized order complementarity problem associated with {f _i} andK is to findx ₀K such thatf _i(x ₀)K,i=1,...,m, and (x ₀,f ₁(x ₀),...,f _m(x ₀))=0. The problem is shown to be equivalent to several fixed-point problems and equivalent to the order complementarity problem studied by Borwein and Dempster and by Isac. Existence and uniqueness of solutions and least-element theory are shown in the spacesC(, ) andL _p(, ). For general locally convex spaces, least-element theory is derived, existence is proved, and an algorithm for computing a solution is presented. Applications to the mixed lubrication theory of fluid mechanics are described.     

Theory of fractional order in generalized thermoelectric MHD

A new mathematical model of thermoelectric MHD theory has been constructed in the context of a new consideration of heat conduction with fractional orders. This model is applied to Stokes’ first problem for a conducting fluid with heat sources. Laplace transforms and state-space techniques [1] will be used to obtain the general solution for any set of boundary conditions. According to the numerical results and its graphs, conclusion about the new theory has been constructed. Some comparisons have been shown in figures to estimate the effects of the fractional order parameters on all the studied fields.     

Spectral asymptotics of elliptic strongly degenerate operators of arbitrary order

We compute the principal term of the spectral asymptotics for elliptic operators of an arbitrary order which is degenerate at the boundary of the domain. The degree of the degeneracy is such that the order of the decrease of the eigenvalues of the boundary-value problems is different from the classical one and the asymptotic coefficient depends on the form of the boundary conditions (strong degeneracy).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 25–30, 1976.In conclusion, the author expresses his deep gratitude to M. Z. Solomyak for his guidance in the preparation of this paper.     

Duality in nondifferentiable multiobjective fractional programming problem with generalized invexity

In this paper, a new class of higher-order (V,α,ρ,θ)-invex function is introduced. Conditions are obtained under which a fractional function is higher-order (V,α,ρ,θ)-invex. Sufficiency of Karush-Kuhn-Tucker conditions is shown under this class of function. We then consider a nondifferentiable multiobjective fractional programming problem and derive the duality theorems.     

Two-weighted estimates for generalized fractional maximal operators on non-homogeneous spaces

Let μ be a nonnegative Borel measure on R ^d satisfying that μ(Q) ? l(Q)ⁿ for every cube Q ? R ⁿ , where l(Q) is the side length of the cube Q and 0 < n ? d.We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function B in the context of non-homogeneous spaces related to the measure μ. Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result for certain fractional maximal operator proved in W.Wang, C. Tan, Z. Lou (2012).