Hyers-Ulam-Rassias Stability Of Several Types Of Operators And Equations

This doctoral dissertation includes the main results obtained by the author during the period he applied for the Ph.D.The author investigates the Hyers-Ulam stability of some operators on analytic function spaces by some techniques which are often applied to study the properties of the operators on function spaces;the author applies the Laplace transform method,the fixed point techniques and weighted spaces method to study the Hyers-Ulam-R.assias stability of a few kinds of fractional differential equations;the author also investigates the Hyers-Ulam-Rassias stability of two types of mixed type functional equations by the direct method.This dissertation consists of six chaptersIn Chapter 1,the author sketches the historical background,the significance of the research and the advance in the Hyers-Ulam-Rassias stability research.This chapter mainly contains the research overviews of the Hyers-Ulam-Rassias stability of the func-tional equations,the Hyers-Ulam-Rassias stability of the differential equations and the Hyers-Ulam stability of the operators on function spaces.The author also introduces emphatically the definition of the Hyers-Ulam stability of the operat,ors,the definitions of the Hyers-Ulam-Rassias stability of the equations and some important results.The author also presents the research contents of this dissertation and the innovation pointsIn Chapter 2,firstly,the author studies the Hyers-Ulam stability of the differentiation operator D and the composition operators C? on Hilbert spaces of entire functions E2(?)where?(z)= a,z+b,0<|a|?1.The author presents a sufficient and necessary condition for the Hyers-Ulam stability of the operator D by the comparison function,and proves that the composition operators C? are not Hyers-Ulam stable on E2(?).Secondly,the author investigates the Hyers-Ulam stability of the differential operator D on the weighted Hardy spaces with the reproducing kernel functions,and the author gets that the stability is closely related to the reproducing kernel functions on the spaces.Meanwhile,the author discusses the Hyers-Ulam stability of the operator T?.The author also studies the Hyers-Ulam stability of the partial differential operators on reproducing kernel function spaces Hf2(Bd)with several variables,and the author obtains that the partial differential operators are not Hyers-Ulam stable on the spaces.In Chapter 3,firstly,the author surveys the Hyers-Ulam-Rassias stability of the linear fractional differential equations with the Riemann-Liouville fractional derivative D?u(t)+du(t)=q(t),and some stability results are proved by using the Laplace transform method.Secondly.the author investigates the stability problems of two types of linear fractional differential equations with the Caputo fractional derivative(CD0?+y)(x)-?y(x)=f(x)and(CD0?+y)(x)-?(CD0?+y)(x)=g(x),and the author proves that the two kinds of differential equations are all Hyers-Ulam Rassias stable.In Chapter 4,the author discusses the Hyers-Ulam-Rassias stability of the following nonlinear fractional Cauchy type problem:(DTqy)(x)=f(x,y(x)),q>0,x ?[0,T],with initial conditions(DTqky)|x=T=bk,bk?R(k 1,...,n-1),bn =0,where(DT-ky)|x=T=limx?T(DTQ-ky)(x),1?k?n-1,bn=limx?T(ITnqy)(x),n=-[-q],DTq and ITq are the righti-sided Riemann-Liouville fractional derivative and the integral of order q respectively,and f(x,y)is bounded and continuous function.Some sufficient con-ditions for the Hyers-Ulam-Rassias stability of these equations are presented by applying the fixed point techniques and the weighted space method.In Chapter 5,the author discusses the Hyers-Ulam-Rassias stability of two kinds of mixed type functional equations.Firstly,the author investigates the general solution and the Hyers-Ulam-Rassias stability of a mixed type quartic-additive functional equation(k4-k)[f(kx + y)+ f+(kx-y)]=k2(k4-k)(f(x+y)+f(x-y))+2(1-k2)(f(ky)-kf(y))+2(k4-k)f(kx)-2k2(k4-k)f(x)on Banach spaces,where k?0,k?1,and some sufficient conditions for Hyers-Ulam Rassias stability of the equation are given.Secondly,the author establishes the general solution of a mixed type quadratic-additive functional equation with a parameter s 2k[f(x+ky)+f(kx+y)]=k(1-s+k+ks+2k2)f(x+y)+k(1-s-3k+ks+2k2)f(x-y)+ 2kf(kx)+2k(s+k-ks-2k2)f(x)+2(1-k-s)f(ky)+2ksf(y),and presents some sufficient conditions for Hyers-Ulam-Rassias stability of the equation on quasi-Banach spaces,where k>1 and s?1-2kThe sixth chapter summarizes the full results in the dissertation,and presents the future research work.