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Register a new userSemigroups of Operators on Spaces of Fuzzy-Number-Valued Functions with Applications to Fuzzy Differential Equations
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In this paper we introduce and study semigroups of operators on spaces of fuzzy-number-valued functions, and various applications to fuzzy differential equations are presented. Starting from the space of fuzzy numbers, many new spaces sharing the same properties are introduced. We derive basic operator theory results on these spaces and new results in the theory of semigroups of linear operators on fuzzy-number kind spaces. The theory we develop is used to solve classical fuzzy systems of differential equations, including, for example, the fuzzy Cauchy problem and the fuzzy wave equation. These tools allow us to obtain explicit solutions to fuzzy initial value problems which bear explicit formulas similar to the crisp case, with some additional fuzzy terms which in the crisp case disappear. The semigroup method displays a clear advantage over other methods available in the literature (i.e., the level set method, the differential inclusions method and other fuzzification methods of the real-valued solution) in the sense that the solutions can be easily constructed, and that the method can be applied to a larger class of fuzzy differential equations that can be transformed into an abstract Cauchy problem.
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In this paper, a new concept, the fuzzy rate of an operator in linear spaces is proposed for the very first time. Some properties and basic principles of it are studied. Fuzzy rate of an operator $B$ which is specific in a plane is discussed. As its application, a new fixed point existence theorem is proved.
Let $(mathbb M, d,mu)$ be a metric measure space with upper and lower densities: $$ begin{cases} |||mu|||_{beta}:=sup_{(x,r)in mathbb Mtimes(0,infty)} mu(B(x,r))r^{-beta}<infty; |||mu|||_{beta^{star}}:=inf_{(x,r)in mathbb Mtimes(0,infty)} mu(B(x,r))r^{-beta^{star}}>0, end{cases} $$ where $beta, beta^{star}$ are two positive constants which are less than or equal to the Hausdorff dimension of $mathbb M$. Assume that $p_t(cdot,cdot)$ is a heat kernel on $mathbb M$ satisfying Gaussian upper estimates and $mathcal L$ is the generator of the semigroup associated with $p_t(cdot,cdot)$. In this paper, via a method independent of Fourier transform, we establish the decay estimates for the kernels of the fractional heat semigroup ${e^{-t mathcal{L}^{alpha}}}_{t>0}$ and the operators ${{mathcal{L}}^{theta/2} e^{-t mathcal{L}^{alpha}}}_{t>0}$, respectively. By these estimates, we obtain the regularity for the Cauchy problem of the fractional dissipative equation associated with $mathcal L$ on $(mathbb M, d,mu)$. Moreover, based on the geometric-measure-theoretic analysis of a new $L^p$-type capacity defined in $mathbb{M}times(0,infty)$, we also characterize a nonnegative Randon measure $ u$ on $mathbb Mtimes(0,infty)$ such that $R_alpha L^p(mathbb M)subseteq L^q(mathbb Mtimes(0,infty), u)$ under $(alpha,p,q)in (0,1)times(1,infty)times(1,infty)$, where $u=R_alpha f$ is the weak solution of the fractional diffusion equation $(partial_t+ mathcal{L}^alpha)u(t,x)=0$ in $mathbb Mtimes(0,infty)$ subject to $u(0,x)=f(x)$ in $mathbb M$.
We prove mapping properties of pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $a(x,eta)$ are elements of $C^{r}_{*}S^{m}_{1,delta}$ classes that have limited regularity in the $x$ variable. We show that the associated pseudodifferential operator $a(x,D)$ maps between Sobolev spaces $mathcal{H}^{s,p}_{FIO}(mathbb{R}^{n})$ and $mathcal{H}^{t,p}_{FIO}(mathbb{R}^{n})$ over the Hardy space for Fourier integral operators $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$. Our main result implies that for $m=0$, $delta=1/2$ and $r>n-1$, $a(x,D)$ acts boundedly on $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$ for all $pin(1,infty)$.
In this paper, the interval-valued intuitionistic fuzzy matrix (IVIFM) is introduced. The interval-valued intuitionistic fuzzy determinant is also defined. Some fundamental operations are also presented. The need of IVIFM is explain by an example.
We prove that a family of linear bounded evolution operators $({bf G}(t,s))_{tge sin I}$ can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators $bm{mathcal A}$ with unbounded coefficients defined in $Itimes Rd$ (where $I$ is a right-halfline or $I=R$) all having the same principal part. We establish some continuity and representation properties of $({bf G}(t,s))_{t ge sin I}$ and a sufficient condition for the evolution operator to be compact in $C_b(Rd;R^m)$. We prove also a uniform weighted gradient estimate and some of its more relevant consequence.
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