Search Results (27)

It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem. Full article

Frustrated magnetic systems arising in geometrically constrained lattices represent rich platforms for exploring unconventional phases of matter, including fractional magnetization plateaus, incommensurate orders and complex domain dynamics. However, determining the microscopic spin configurations that stabilize such phases is a key challenge, especially when in-plane and out-of-plane spin components coexist and compete. Here, we combine neutron scattering and magnetic susceptibility experiments with simulations to investigate the emergence of field-induced fractional plateaus and the related criticality in a frustrated magnet holmium tetraboride (HoB₄) that represents the family of rare earth tetraborides that crystalize in a Shastry–Sutherland lattice in the $ab$ plane. We focus on the interplay between classical and quantum criticality near phase boundaries, as well as the role of material defects in the stabilization of the ordered phases. We find that simulations using classical annealing can explain certain observed features in the experimental Laue diffraction and the origin of multiple magnetization plateaus. Our results show that defects and out-of-plane interactions play an important role and can guide the route towards resolving microscopic spin textures in highly frustrated magnets. Full article

In the qualitative theory of differential equations in Banach spaces, the resolving families of operators of such equations play an important role. We obtained necessary and sufficient conditions for the existence of strongly continuous resolving families of operators for a linear homogeneous equation resolved with respect to the Hilfer derivative. These conditions have the form of estimates on derivatives of the resolvent of a linear closed operator from the equation and generalize the Hille–Yosida conditions for infinitesimal generators of $C_0$-semigroups of operators. Unique solvability theorems are proved for the corresponding inhomogeneous equations. Illustrative examples of the operators from the considered classes are constructed. Full article

An algebraic graph is defined in terms of graph theory as a graph with related algebraic structures or characteristics. If the vertex set of a graph $\mathcal{G}$ is a group, a ring, or a field, then $\mathcal{G}$ is called an algebraic structure graph. This work uses an algebraic structure graph based on the modular ring ${\mathbb{Z}}_n$, known as a hyper-chordal ring network. The lower and upper bounds of the local fractional metric dimension are computed for certain families of hyper-chordal ring networks. Utilizing the cardinalities of local fractional resolving sets, local fractional resolving (LFR)M-polynomials are computed for hyper-chordal ring networks. Further, new topological indices based on (LFR)M-polynomials are established for the proposed networks. The local fraction entropies are developed by modifying the first three kinds of Zagreb entropies, which are calculated for the chosen hyper-chordal ring networks. Furthermore, numerical and graphical comparisons are discussed to observe the order between newly computed topological indices. Full article

We primarily investigate the existence of solutions for fractional neutral integro-differential equations with nonlocal initial conditions, which are crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by using fixed-point theorems, we establish the existence of mild solutions under more relaxed conditions. Full article

Fractional integrodifferential diffusion equations play a significant role in describing anomalous diffusion phenomena. In this paper, we study the existence and uniqueness of mild solutions to these equations. Firstly, we construct an appropriate resolvent family, through which the related equicontinuity, strong continuity, and compactness properties are studied using the convolution theorem of Laplace transform, the probability density function, the Cauchy integral formula, and the Fubini theorem. Then, we construct a reasonable mild solution for the considered equations. Finally, we obtain some sufficient conditions for the existence and uniqueness of mild solutions to the considered equations by some fixed point theorems. Full article

Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders. Full article

The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph’s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications in many different domains, including graph theory, network architecture, and facility location problems. A graph G with order n is known as a Toeplitz graph over the subset S of consecutive collections of integers from one to n, and two vertices will be adjacent to each other if their absolute difference is a member of S. A graph $G\left({\mathbb{Z}}_n\right)$ is called a zero-divisor graph over the zero divisors of a commutative ring ${\mathbb{Z}}_n$, in which two vertices will be adjacent to each other if their product will leave the remainder zero under modulo n. Since the local fractional metric dimension problem is NP-hard, it is computationally difficult to identify an optimal solution or to precisely determine the minimal size of a local resolving set; in the worst case, the process takes exponential time. Different upper bound sequences of local fractional metric dimension are suggested in this article, along with a comparison analysis for certain families of Toeplitz and zero-divisor graphs. Furthermore, we note that the analyzed local fractional metric dimension upper bounds fall into three metric families: constant, limited, and unbounded. Full article

In this paper, we introduce and investigate several new classes of $(F,G,C)$-regularized resolvent operator families subgenerated by multivalued linear operators in locally convex spaces. The known classes of $(a,k)$-regularized C-resolvent operator-type families are special cases of the classes introduced in this paper. We provide certain applications of $(F,G,C)$-regularized resolvent operator families and $(a,k)$-regularized C-resolvent families to abstract fractional differential–difference inclusions and abstract Volterra integro-difference inclusions. Full article

In this study, we consider fractional-in-time Venttsel’ problems in fractal domains of the Koch type. Well-posedness and regularity results are given. In view of numerical approximation, we consider the associated approximating pre-fractal problems. Our main result is the convergence of the solutions of such problems towards the solution of the fractional-in-time Venttsel’ problem in the corresponding fractal domain. This is achieved via the convergence (in the Mosco–Kuwae–Shioya sense) of the approximating energy forms in varying Hilbert spaces. Full article

The fractional powers of generators for analytic operator semigroups are used for the proof of the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. Here, we use an analogous construction of fractional powers $A^{\gamma }$ for an operator A such that $-A$ generates analytic resolving families of operators for a fractional order equation. Under the condition of local Lipschitz continuity with respect to the graph norm of $A^{\gamma }$ for some $\gamma \in (0,1)$ of a nonlinear operator, we prove the local unique solvability of the Cauchy problem to a fractional order quasilinear equation in a Banach space with several Gerasimov–Caputo fractional derivatives in the nonlinear part. An analogous nonlocal Lipschitz condition is used to obtain a theorem of the nonlocal unique solvability of the Cauchy problem. Abstract results are applied to study an initial-boundary value problem for a time-fractional order nonlinear diffusion equation. Full article

The metric dimension has various applications in several fields, such as computer science, image processing, pattern recognition, integer programming problems, drug discovery, and the production of various chemical compounds. The lowest number of vertices in a set with the condition that any vertex can be uniquely identified by the list of distances from other vertices in the set is the metric dimension of a graph. A resolving function of the graph G is a map $\vartheta :V\left(G\right)\to [0,1]$ such that ${\sum }_{u\in \mathcal{R}\{v,w\}}\vartheta \left(u\right)\ge 1,$ for every pair of adjacent distinct vertices $v,w\in V\left(G\right)$. The local fractional metric dimension of the graph G is defined as ${\mathrm{ldim}}_{\mathrm{f}}\left(G\right)$ = $min\{{\sum }_{v\in V\left(G\right)}\vartheta \left(v\right)$, where $\vartheta$ is a local resolving function of $G\}$. This paper presents a new family of planar networks namely, rotationally heptagonal symmetrical graphs by means of up to four cords in the heptagonal structure, and then find their upper-bound sequences for the local fractional metric dimension. Moreover, the comparison of the upper-bound sequence for the local fractional metric dimension is elaborated both numerically and graphically. Furthermore, the asymptotic behavior of the investigated sequences for the local fractional metric dimension is addressed. Full article

We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter $\widehat{\mathcal{H}}\in (1/2,1)$. Using fixed point techniques, a q-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system. Moreover, under appropriate conditions, we investigate the approximate controllability of the considered system. Finally, the main results are demonstrated with an illustrative example. Full article

In this work, we provide several applications of (a, k)-regularized C-resolvent families to the abstract impulsive Volterra integro-differential inclusions. The resolvent operator families under our consideration are subgenerated by multivalued linear operators, which can degenerate in the time variable. The use of regularizing operator C seems to be completely new within the theory of the abstract impulsive Volterra integro-differential equations. Full article

In traditional Chinese medicine (TCM), insects from the family Blattidae have a long history of application, and their related active compounds have excellent pharmacological properties, making them a prominent concern with significant potential for medicinal and healthcare purposes. However, the medicinal potential of the family Blattidae has not been fully exploited, and many problems must be resolved urgently. Therefore, a comprehensive review of its chemical composition, pharmacological activities, current research status, and existing problems is necessary. In order to make the review clearer and more systematic, all the contents were independently elaborated and summarized in a certain sequence. Each part started with introducing the current situation or a framework and then was illustrated with concrete examples. Several pertinent conclusions and outlooks were provided after discussing relevant key issues that emerged in each section. This review focuses on analyzing the current studies and utilization of medicinal insects in the family Blattidae, which is expected to provide meaningful and valuable relevant information for researchers, thereby promoting further exploration and development of lead compounds or bioactive fractions for new drugs from the insects. Full article