Search

We study the existence of bounded asymptotic mild solutions to evolution equations of the form $u'(t)=Au(t)+f(t), t\ge 0$ in a Banach space $\X$, where $A$ generates an (analytic) $C_0$-semigroup and $f$ is bounded. We find spectral conditions on $A$ and $f$ for the existence and uniqueness of asymptotic mild solutions with the same "profile" as that of $f$. In the resonance case, a sufficient condition of Massera type theorem is found for the existence of bounded solutions with the same profile as $f$. The obtained results are stated in terms of spectral properties of $A$ and $f$, and they are analogs of classical results of Katznelson-Tzafriri and Massera for the evolution equations on the half line. Applications from PDE are given.

In this paper we study the well-posedness of the evolution equation of the form $u'(t)=Au(t)+Cu(t)$, $t\ge 0$, where $A$ is the generator of a $C_0$- semigroup and $C$ is a (possibly unbounded) linear operator in a Banach space $\mathbb{X}$. We prove that if $A$ generates a $C_0$-semigroup $\left (T_A(t)\right )_{t \geq 0}$ with $\|T(t)\| \le Me^{\omega t}$ in a Banach space $\mathbb{X}$ and $C$ is a linear operator in $\mathbb{X}$ such that $D(A)\subset D(C)$ and $\| CR(\mu ,A)\| \le K/(\mu -\omega)$ for each $\mu>\omega$, then, the above-mentioned evolution equation is well-posed, that is, $A+C$ generates a $C_0$-semigroup $\left (T_{A+C}(t)\right )_{t \geq 0}$ satisfying $\| T_{A+C}(t)\| \le Me^{(\omega +MK)t}$. Our approach is to use the Hille-Yosida Theorem. Discussions on the persistence of asymptotic behavior of the perturbed equations such as the roughness of exponential dichotomy are also given. The obtained results seem to be new., Comment: 12 pages

This paper is concerned with perturbation of evolution equation with delay $u'(t)=Au(t)+Bu_t \ (*)$ where $A,B$ are linear operators in a Banach space $\mathbb{X}$ and from $C([-r,0],\mathbb{X})$ to $\mathbb{X}$, respectively; $r>0$ is a given constant. To measure the size of unbounded perturbation we introduce the so-called \textit{Yosida distance} of linear operators $U$ and $V$ in a Banach space $\mathbb{X}$ as $d_Y(U,V):=\limsup_{\mu\to +\infty} \| U_\mu-V_\mu\|$, where $U_\mu$ and $V_\mu$ are the Yosida approximations of $U$ and $V$, respectively. The obtained results states that if $d_Y(A, A_1)$ and $d_Y(B, B_1)$ are sufficiently small, then the equation $u'(t)=A_1u(t)+B_1u_t \ (**)$ also has an exponential dichotomy, if (*) has it. The proofs are based on estimates of the Yosida distance of the generators of the solution semigroups generated by (*) and (**) in the phase space $C([-r,0],\mathbb{X})$ whose domains have no relations at all. The obtained results seem to be new., Comment: arXiv admin note: text overlap with arXiv:2301.12080

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $ D^{\alpha}_Cu(t)=Au(t)+f(t), u(0)=x, 0<\alpha\le1, ( *) $ where $D^{\alpha}_Cu(t)$ is the derivative of the function $u$ in the Caputo's sense, $A$ is a linear operator in a Banach space $\X$ that may be unbounded and $f$ satisfies the property that $\lim_{t\to \infty} (f(t+1)-f(t))=0$ which we will call asymptotic $1$-periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator $A$ for all asymptotic mild solutions of Eq. (*) to be asymptotic $1$-periodic, or there exists an asymptotic mild solution that is asymptotic $1$-periodic., Comment: 13 pages. arXiv admin note: text overlap with arXiv:1910.08609

We introduce a new concept of Yosida distance between two (unbounded) linear operators $A$ and $B$ in a Banach space $\mathbb{X}$ defined as $d_Y(A,B):=\limsup_{\mu\to +\infty} \| A_\mu-B_\mu\|$, where $A_\mu$ and $B_\mu$ are the Yosida approximations of $A$ and $B$, respectively, and then study the persistence of evolution equations under small Yosida perturbation. This new concept of distance is also used to define the continuity of the proto-derivative of the operator $F$ in the equation $u'(t)=Fu(t)$, where $F \colon D(F)\subset \mathbb{X} \rightarrow \mathbb{X}$ is a nonlinear operator. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of $F$ is continuous. The Yosida distance approach to perturbation theory allows us to free the requirement on the domains of the perturbation operators. Finally, the obtained results seem to be new.

We study the existence of traveling waves of reaction-diffusion systems with delays in both diffusion and reaction terms of the form $\partial u(x,t)/\partial t = \Delta u(x,t-\tau_1)+f(u(x,t),u(x,t-\tau_2))$, where $\tau_1,\tau_2$ are positive constants. We extend the monotone iteration method to systems that satisfy typical monotone conditions by thoroughly studying the sign of the Green function associated with a linear functional differential equation. Namely, we show that for small positive $r$ the functional equation $x''(t)-ax'(t+r)-bx(t+r)=f(t)$, where $a\not=0, b>0$ has a unique bounded solution for each given bounded and continuous $f(t)$. Moreover, if $r>0$ is sufficiently small, $f(t)\ge 0$ for $t\in {\mathbb R}$, then the unique bounded solution $x_f(t)\le 0$ for all $t\in {\mathbb R}$. In the framework of the monotone iteration method that is developed based on this result, upper and lower solutions are found for Fisher-KPP and Belousov-Zhabotinski equations to show that traveling waves exist for these equations when delays are small in both diffusion and reaction terms. The obtained results appear to be new., Comment: 36 pages

In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by $n^\nu$, where $\nu$ is a natural number. We apply this spectral theory to study the asymptotic behavior of solutions of fractional difference equations of the form $\Delta^\alpha x(n)=Tx(n)+y(n)$, $n\in \mathbb{N}$, where $0<\alpha\le 1$. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that if the $\alpha$-resolvent operator $S_\alpha$ satisfies $\sup_{n\in\mathbb{N}} \| S_\alpha (n)\| /n^\nu <\infty$ and for all $z_0\in \{z\in \mathbb{C}: \ |z|=1\}$, but $z_0=1$, the complex function $(z^{1-\alpha}(z-1)^\alpha -T)^{-1}$ \ exists and is holomorphic in a neighborhood of $z_0$, then \begin{align*} \lim_{n\to \infty} \frac{1}{n^\nu} \sum_{k=0}^{\nu+1} \frac{(\nu+1)!}{k!(\nu+1-k)!} (-1)^{\nu+1+k} S_\alpha (n+k) =0. \end{align*} Three concrete examples are also included to illustrate the obtained results., Comment: 20 pages

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $D^{\alpha}_Cu(t)=Au(t)+f(t)$ on the half line, where $D^{\alpha}_Cu(t)$ is the derivative of the function $u$ in Caputo's sense, $A$ is generally an unbounded closed operator, $f$ is polynomially bounded. To this end we develop a spectral theory for functions of polynomial growth on the half line. Our main result claims that if $u$ is mild solution of the Cauchy problem such that $\lim_{h\downarrow 0} \sup_{t\ge 0} \| u(t+h)-u(t)\|/(1+t)^n=0$, and $\sup_{t\ge 0} \| u(t)\| /(1+t)^n <\infty$, then, $\lim_{t\to\infty} u(t)/(1+t)^n =0$ provided that the spectral set $\Sigma (A,\alpha )\cap i\R$ is countable, where $\Sigma (A,\alpha )$ is defined to be the set of complex numbers $\xi$ such that $\lambda^{\alpha -1} (\lambda^\alpha -A)^{-1}$ is analytic in a neighborhood of $\xi$, and $u$ satisfies some ergodic, Comment: 16 pages

We study conditions for the abstract periodic linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t)$ to have almost periodic with the same structure of frequencies as $f$. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the frequencies of the forcing term $f$. The obtained results extend recent results on the subject. A discussion on how the results could be extended to the case when $A$ depends on $t$ is given., Comment: 17 pages

We study conditions for the abstract linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t), t\ge 0$ to have asymptotic almost periodic solutions, where $F(\cdot )$ is periodic, $f$ is asymptotic almost periodic. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the circular spectrum of the forcing term $f$. The obtained results extend recent results on the subject., Comment: arXiv admin note: text overlap with arXiv:1807.03826

We analyze the procedure providing quantum cheques of S. R. Moulick and P. K. Panigrahi to produce quantum e-cheques, based on multiparty quantum telecommunication between customer and cooperated branches of bank., Comment: 6 pages, 3 figures

Neutrosophic set has the ability to handle uncertain, incomplete, inconsistent, indeterminate information in a more accurate way. In this paper, we proposed a neutrosophic recommender system to predict the diseases based on neutrosophic set which includes single-criterion neutrosophic recommender system (SC-NRS) and multi-criterion neutrosophic recommender system (MC-NRS). Further, we investigated some algebraic operations of neutrosophic recommender system such as union, complement, intersection, probabilistic sum, bold sum, bold intersection, bounded difference, symmetric difference, convex linear sum of min and max operators, Cartesian product, associativity, commutativity and distributive. Based on these operations, we studied the algebraic structures such as lattices, Kleen algebra, de Morgan algebra, Brouwerian algebra, BCK algebra, Stone algebra and MV algebra. In addition, we introduced several types of similarity measures based on these algebraic operations and studied some of their theoretic properties. Moreover, we accomplished a prediction formula using the proposed algebraic similarity measure. We also proposed a new algorithm for medical diagnosis based on neutrosophic recommender system. Finally to check the validity of the proposed methodology, we made experiments on the datasets Heart, RHC, Breast cancer, Diabetes and DMD. At the end, we presented the MSE and computational time by comparing the proposed algorithm with the relevant ones such as ICSM, DSM, CARE, CFMD, as well as other variants namely Variant 67, Variant 69, and Varian 71 both in tabular and graphical form to analyze the efficiency and accuracy. Finally we analyzed the strength of all 8 algorithms by ANOVA statistical tool., Comment: Keywords: Medical diagnosis, neutrosophic set, neutrosophic recommender system, non-linear regression model

Motility is characteristic of life, but a common basis for movement has remained to be identified. Diverse systems in motion shift between two states depending on interactions that turnover at the rate of an applied cycle of force. Although one phase of the force cycle terminates the decay of the most recent state, continuation of the cycle of force regenerates the original decay process in a recursive cycle. By completing a cycle, kinetic energy is transformed into probability of sustaining the most recent state and the system gains a frame of reference for discrete transitions having static rather than time-dependent probability. The probability of completing a recursive cycle is computed with a Markov chain comprised of two equilibrium states and a kinetic intermediate. Given rate constants for the reactions, a random walk reproduces bias and recurrence times of walking motor molecules and bacterial flagellar switching with unrivaled fidelity., Comment: 15 pages, 5 figures, 1 table, supplemental materials

Fuzzy Geographically Weighted Clustering (FGWC) is considered as a suitable tool for the analysis of geo-demographic data that assists the provision and planning of products and services to local people. Context variables were attached to FGWC in order to accelerate the computing speed of the algorithm and to focus the results on the domain of interests. Nonetheless, the determination of exact, crisp values of the context variable is a hard task. In this paper, we propose two novel methods using fuzzy approaches for that determination. A numerical example is given to illustrate the uses of the proposed methods., Comment: 11 pages