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In this paper, we study the long-time asymptotic behavior of the solution of the Cauchy problem for the generalized Sasa-Satsuma equation. Starting with the 3 × 3 Lax pair related to the generalized Sasa-Satsuma equation, we construct a Rieman-Hilbert problem, by which the solution of the generalized Sasa-Satsuma equation is converted into the solution of the corresponding RiemanHilbert problem. Using the nonlinear steepest decent method for the Riemann-Hilbert problem, we obtain the leading-order asymptotics of the solution of the Cauchy problem for the generalized SasaSatsuma equation through several transformations of the Riemann-Hilbert problem and with the aid of the parabolic cylinder function.

In this paper, we first introduce the concept of mixed γ-proximally monotone property type mappings and investigate the existence of the coupled proximally coincidence point for such mappings in partially ordered complete metric spaces. Furthermore, we prove the existence and uniqueness of coupled common best proximity points. Our results extend, improve and generalize several known results in the literature.

In this paper, we discuss the extended generalized multi-index Bessel function by using the extended beta type function. Then we investigate its several properties including integral representation, derivatives, beta transform, Laplace transform, Mellin transforms, and some relations of extension of extended generalized multi-index Bessel function (E1GMBF) with the Laguerre polynomial and Whittaker functions. Further, we also discuss the composition of the generalized fractional integral operator having Appell function as a kernel with the extension of extended generalized multi-index Bessel function and establish these results in terms of Wright functions.

The main objective of this paper is to compute refinements of bounds of the generalized fractional integral operators containing an extended generalized Mittag-Leffler function in their kernels. The presented results also provide refinements of already known bounds of different fractional integral operators for convex, m- convex, s-convex and (s, m)-convex functions. Moreover, the refinements of some known fractional versions of the Hadamard inequality are given.

The central graph $C(G)$ of a graph $G$ is obtained by sub dividing each edge of $G$ exactly once and joining all the nonadjacent vertices in $G$. In this paper, we compute the adjacency, Laplacian and signless Laplacian spectra of central graph of a connected regular graph. Also, we define central vertex join and central edge join of two graphs and calculate their adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum. As an application, some new families of integral graphs and cospectral graphs are constructed. In addition to that the Kirchhoff index and number of spanning trees of the new joins are determined.

This paper investigates a patch structure Nicholson's blowflies model involving multiple pairs of different time-varying delays. Without assuming the uniform positiveness of the death rate and the boundedness of coefficients, we establish three novel criteria to check the global convergence, generalized exponential convergence and asymptotical stability on the zero equilibrium point of the addressed model, respectively. Our proofs make substantial use of differential inequality techniques and dynamical system approaches, and the obtained results improve and supplement some existing ones. Last but not least, a numerical example with its simulations is given to show the feasibility of the theoretical results.

In this paper, we investigate a class of third-order singular neutral differential equations with time-dependent delay. Applying Krasnoselskiios fixed point theorem, we prove the existence results of a positive periodic solution for this neutral equation.

In this paper, the Hermitian, skew-Hermitian, Re-nonnegative definite, Re-positive definite, Re-nonnegative definite least-rank and Re-positive definite least-rank solutions of the matrix equation $AXB = C$ are considered. The necessary and sufficient condition for the existence of such type of solution to the equation is provided and the explicit expression of the general solution is also given.

Let $G = (V, E)$ be a simple undirected graph. A Roman dominating function on $G$ is a function $f: V\to \{0, 1, 2\}$ satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to at least one vertex $v$ with $f(v) = 2$. The weight of a Roman dominating function is the value $f(G) = \sum_{u\in V} f(u)$. The Roman domination number of $G$ is the minimum weight of a Roman dominating function on $G$. The Roman bondage number of a nonempty graph $G$ is the minimum number of edges whose removal results in a graph with the Roman domination number larger than that of $G$. Rad and Volkmann [9] proposed a problem that is to determine the trees $T$ with Roman bondage number $1$. In this paper, we characterize trees with Roman bondage number $1$.

In this paper, a new kind of real-valued major Choquet integral, real-valued minor Choquet integral and interval-valued Choquet integrals for set-valued functions is introduced and investigated. The representations of the Choquet integral of set-valued functions with respect to a fuzzy measure are given. In particular, we focus on the case of the distorted Lebesgue measure as a fuzzy measure. Furthermore, the characteristics of the primitive of Choquet integral for set-valued functions are given as Radon-Nikodym property in some sense.

In this paper, we deal with a class of fractional Henon equation and by using the Lyapunov-Schmidt reduction method, under some suitable assumptions, we derive the existence of infinitely many solutions, whose energy can be made arbitrarily large. Compared to the previous works, we encounter some new challenges because of the nonlocal property for fractional Laplacian. But by doing some delicate estimates for the nonlocal term we overcome the difficulty and find infinitely many nonradial solutions.

The objective of this paper is to study the ordered $h$-regular semirings by the properties of their ordered $h$-ideals. It is proved that each $h$-regular ordered semiring is an ordered $h$-regular semiring but the converse does not follow. Important theorems relating to basic properties of the operator clousre and $h$-regular semirings are given. It is also proved that each regular ordered semiring is an ordered $h$-regular semiring but the converse does not hold. The classifications of the left and the right ordered $h$-regular semirings and the left and the right ordered $h$-weakly regular semirings are also presented.

Let $G(V, E)$ be a simple connected graph of order $n$. A graph of order $n$ is called pancyclic if it contains all the cycles $C_k$ for $k\in \{3, 4, \cdot\cdot\cdot, n\}$. In this paper, some new spectral sufficient conditions for the graph to be pancyclic are established in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph.

In this paper, we establish generalized fractional versions of Hermite-Hadamard inequalities for exponentially (α, h − m)-convex functions, exponentially (h − m)-convex functions and exponentially (α, m)-convex functions. These inequalities arise when using the generalized fractional integral operators containing Mittag-Leffler function via a monotonically increasing function. The presented results hold at the same time for various kinds of convexities and well-known fractional integral operators. Moreover, the established inequalities reproduce several known results which are part of the existing literature.

In this paper, the Ulam stability of two fuzzy number-valued functional equations in Banach spaces is investigated by using the metric defined on a fuzzy number space. Under some suitable conditions, some properties of the solutions for these equations such as existence and uniqueness are discussed.

In this paper, the new version of the celebrated Montgomery identity is determined via quantum integral operators. By using it, certain quantum integral inequalities of Ostrowski type are established. Moreover, the relevant connection of the obtained results of this work with the derived results in previously published works is discussed.

In this paper, we use the elementary methods, the properties of the Gauss sums and the estimate for character sums to study the calculating problems of a certain cubic residues modulo p, and give some interesting identities and asymptotic formulas for their counting functions.

In this paper, we present a multi-step hybrid iterative method. It is proven that under appropriate assumptions, the proposed iterative method converges strongly to a common element of fixed point of a finite family of nonexpansive mappings, the solution set of split monotone variational inclusion problem and the solution set of triple hierarchical variational inequality problem (THVI) in real Hilbert spaces. In addition, we give a numerical example of a triple hierarchical system derived from our generalization.

The aim of this paper is to establish a stability result regarding the inverse problem of retrieving the damping coefficient in Kawahara equation. We first establish an internal Carleman estimate for the linearized problem with the help of Dirichlet-Neumann type boundary conditions. Using the obtained Carleman estimate and the regularity of solutions for the Kawahara equation, we prove the Lipschitz type stability and uniqueness of the considered inverse problems.

This paper investigates the bounds of an integral operator for several kinds of convex functions. By applying definition of (h - m)-convex function upper bounds of left sided (1.12) and right sided (1.13) integral operators are formulated which particularly provide upper bounds of various known conformable and fractional integrals. Further a modulus inequality is investigated for differentiable functions whose derivative in absolute value are (h - m)-convex. Moreover a generalized Hadamard inequality for (h - m)-convex functions is proved by utilizing these operators. Also all the results are obtained for (α, m)-convex functions. Finally some applications of proved results are discussed.

Convex functions play a vital role in the derivation of inequalities. In this paper these functions are used to obtain certain bounds of a unified integral operator. A Hadamard inequality for these operators is established. Further bounds of several kinds of fractional and conformable integral operators are deduced in particular.

Let $(M, g, \varphi)$ be an $n$-dimensional locally decomposable Riemann manifold, that is, $g(\varphi X, Y) = g(X, \varphi Y)$ and $\nabla \varphi = 0$, where $\nabla $ is Riemann (Levi-Civita) connection of metric $g$. In this paper, we construct a new connection on locally decomposable Riemann manifold, whose name is statistical ($\alpha, \varphi)$-connection. A statistical $\alpha $-connection is a torsion-free connection such that $% \overline{\nabla }g = \alpha C$, where $C$ is a completely symmetric $(0, 3)$% -type cubic form. The aim of this article is to use connection $\overline{% \nabla }$ and product structure $\varphi $ in the same equation, which is possible by writing the cubic form $C$ in terms of the product structure $% \varphi $. We examine some curvature properties of the new connection and give examples of it.

Some characterizations for weighted composition operators to be invertible on Dirichlet type spaces $\mathfrak{D}_{\rho}$ are given in this paper when $\rho$ is finite lower type greater than $0$ and upper type less than $1$. In particular, the equivalence between invertible and preserve frames is established. Moreover, weighted composition operators that preserve tight frames and normalized tight frames on the Dirichlet type space $\mathfrak{D}_{\alpha}$ $(0 < \alpha < 1)$ are also investigated.

Various applications of degenerate polynomials in different areas call for the thoughtful study and research, and many extensions and variants can be found in the literature. In this paper, we introduce partially degenerate Laguerre-Genocchi polynomials and investigate their properties and identities. Furthermore, we introduce a generalized form of partially degenerate Laguerre-Genocchi polynomials and derive some interesting properties and identities. The results obtained are of general character and can be reduced to yield formulas and identities for relatively simple polynomials and numbers. © 2020 the Author(s), licensee AIMS Press.

This paper deals with a nonlinear implicit fractional differential equation with the anti-periodic boundary condition involving the Caputo-Katugampola type. The existence and uniqueness results are established by applying the fixed point theorems of Krasnoselskii and Banach. Further, by using generalized Gronwall inequality the Ulam-Hyers stability results are proved. To demonstrate the effectiveness of the main results, appropriate examples are granted.

In this paper, we are devoted to study asymptotic dynamics of the stochastic extended Brusselator system with a multiplicative noise. The stochastic extended Brusselator system is composed of three pairs of symmetrical coupling components. We firstly study the pullback absorbing property for the stochastic extended Brusselator system with a multiplicative noise. But coupling terms bring great difficulty on this problem, we use the scaling method and estimate groups to overcome this difficulty. Then, we apply the bootstrap pullback estimations to prove the pullback asymptotic compactness for the stochastic extended Brusselator system with a multiplicative noise. Finally, we show the existence of random attractors. In the study of the existence of random attractors for stochastic dynamics, we use the exponential transformation of the Ornstein-Uhlenbeck process to replace the exponential transformation of Brownian motion, which changes the structure of the original Brusselator equations and produces the non-autonomous terms. Based on this, we have to estimate groups to overcome the difficulties of coupling structure and make more complex estimates.

This paper provides a modified characteristics projection finite element method for the unsteady incompressible magnetohydrodynamics(MHD) equations. In this method, modified characteristics finite element method and the projection method will be combined for solving the unsteady incompressible MHD equations. Both the stability and the optimal error estimates both in L2 and H1 norms for the modified characteristics projection finite element method will be shown. In order to demonstrate the effectiveness of our method, we will present some numerical results at the end.

In this paper, by using of the martingale property and positive maximum principle, we investigate the stochastic invariance for a class of hybrid stochastic differential equations (HSDEs) and provide necessary and sufficient conditions for the invariance of closed sets of $\mathbb{R}^d$ with non-Lipschitz coefficients. Moreover, an example of the most probable phase portrait is given to illustrate the effectiveness of the main results.

In this paper, we obtain meromorphic exact solutions of the KdV-Sawada-Kotera equation via two different systematic methods. Applying the exp(-ψ(z))-expansion method, we achieve the trigonometric, exponential, hyperbolic and rational function solutions for the mentioned equation. It is more interesting that we firstly proposed the extended complex method based on the previous work of Yuan et al., and as an example we use it to search exact solutions to the KdV-Sawada-Kotera equation. Dynamic behaviors of solutions obtained by these two different systematic techniques are also shown by some graphs. The results show that these two methods are direct and efficient methods to deal with various differential equations in the applied sciences.

In this paper, we consider the following sublinear biharmonic equations $ \begin{equation*} \Delta^2 u + V\left( x \right)u = K(x)|u|^{p-1}u,\ x\in \mathbb{R}^N, \end{equation*} $ where $N\geq5, ~0 \lt p \lt 1$, and $K, V$ both change sign in $\mathbb{R}^N$. We prove that the problem has infinitely many solutions under appropriate assumptions on $K, V$. To our end, we firstly infer the boundedness of $PS$ sequence, and then prove that the $PS$ condition was satisfied. At last, we verify that the corresponding functional satisfies the conditions of the symmetric Mountain Pass Theorem.

Let $m$ be a positive integer and $C = \{1, 2, \dots, m\}$ be a set of $m$ colors. A polychromatic $m$-coloring of a hypergraph is a coloring of its vertices in such a way that every hyperedge contains at least one vertex of each color in $C$. This problem is a generalization of $2$-colorings of hypergraphs and has close relations with the longest lifetime problem for a wireless sensor network, cover decompositions problem of hypergraphs and vertex cover problem of hypergraphs. In this paper, a main work is to find the maximum $m$ that a hypergraph $H$, with $n$ hyperedges, admits a polychromatic $m$-coloring such that each color appears at least $k$ times on each hyperedge. A $2\ln n$ approximation to the number is obtained when $k$ is a fixed positive integer. For the case that $k = O(n\ln n)$, there exists an $O(\ln n)$ approximation algorithm; for the case that $k = \omega (n\ln n)$, there exists a $(2+\sqrt{3})k$ approximation algorithm.

In this paper, with the quaternion number and time-varying coefficients introduced into traditional BAMNNs, the model of quaternion-valued BAMNNs are formulated. For the first time, fixed-time stabilization of time-varying quaternion-valued BAMNNs is investigated. A novel fixedtime control method is adopted, in which the choice of the Lyapunov function is more general than in most previous results. To cope with the noncommutativity of the quaternion multiplication, two different fixed-time control methods are provided, a decomposition method and a non-decomposition method. Furthermore, to reduce the control strength and improve control efficiency, an adaptive fixed-time control strategy is proposed. Lastly, numerical examples are presented to demonstrate the effectiveness of the theoretical results.

In this paper, we aim to investigate the uniqueness of meromorphic functions that share small functions on annuli. As a matter of fact, we give several uniqueness theorems about meromorphic functions sharing four or three distinct small functions on the annulus $\mathbb{A}=\{z:\frac{1}{R_{0}}

In this paper we give some refinements and sharpness of the Huygens- and Wilker- type inequalities, and show a proof of the second conjecture by Chen and Chueng in [10].

In this paper, we study a class of nonlocal boundary value problems of fractional systems which involves left and right fractional derivatives at resonance. By using the coincidence degree theory, the solvability results for the problems are obtained under the resonant conditions. As an application of our results, we also deal with the existence result for the solution of fractional differential equation which involves both left and right fractional derivatives and satisfies certain boundary conditions under the resonant conditions. Finally, some examples are presented to illustrate our main results.

In this paper, we propose a new fractional derivative, which is based on a Caputo-type derivative with a smooth kernel. We show that the proposed fractional derivative reduces to the classical derivative and has a smoothing effect which is compatible with $\ell_{1}$ regularization. Moreover, it satisfies some classical properties.

In present paper, we introduced generalized iterative method to solve linear and nonlinear fractional differential equations with composite fractional derivative operator. Linear/nonlinear fractional diffusion-wave equations, time-fractional diffusion equation, time fractional Navier-Stokes equation have been solved by using generalized iterative method. The graphical representations of the approximate analytical solutions of the fractional differential equations were provided.

This paper deals with the generalization of null and non-null normal curves in Lorentzian n -space E1n. We reveal necessary and sufficient condition for a curve to be a normal curve in Lorentzian n -space E1n. We obtain the relationship between the curvatures for any arclength parametrized curve to be congruent to a normal curve in E1n. Moreover, we give differential equations by introducing a differentiable function f(s) which can be solved explicitly for a curve to be congruent to a normal curve.

In this article, we introduce the notion of $\mathscr{M}$-convex functions, $\log$-$\mathscr{M}$-convex functions and the notion of quasi $\mathscr{M}$-convex functions. We derive some new analogues of Hermite-Hadamard like inequalities associated with $\mathscr{M}$-convex functions by using the concepts of ordinary, fractional and quantum calculus. The main results of this paper may be useful where bounds for natural phenomena described by integrals such as mechanical work are frequently required. These results are also helpful in the field of numerical analysis where error analysis is required.

Topological indices are used to predict certain phsio-chemical properties of the chemical compounds. Among all indices, degree based indices are of vital importance. In this paper, we introduce generalized inverse sum indeg index and generalized inverse sum indeg energy of graphs. We study the generalized inverse sum indeg index and energy from an algebraic point of view. Extremal values of this index for some graph classes are determined. Some spectral properties of generalized inverse sum indeg matrix are studied. We also find Nordhaus-Gaddum-type results for generalized inverse sum indeg index spectral radius and energy.

The main purpose of this paper is using the analytic methods and the properties of the Legendre's symbol and quadratic residue mod p to study the computational problem of the fourth power mean of a sum analogous to Kloosterman sum, and give a sharp asymptotic formula for it.

The present paper deals with the problem of oscillation for a class of second order nonlinear neutral impulsive difference equations with fixed moments of impulse effect. The technique employed here is due to the classical impulsive inequalities. Some examples are given to illustrate our results.

In this paper, the metabolic regulation model of the body during weight loss is established according to the numerical variation law of the three major nutrients and the biological principles. The changing trend of the solution of the model is qualitatively analyzed by using the theory of time-varying differential equation. According to the normal values of the three major nutrients and the ketone body, the sufficient conditions for safe weight loss under the three ways of weight loss during dieting, exercise and drug are provided respectively. Combining with the actual data, the model parameters are identified by the numerical analysis method. The feasible range and simulation curve of the three major nutrients and the ketone body under three weight loss modes are given as well. Finally, the unsafe factors in the process of weight loss are analyzed, and the weight loss suggestions beneficial to health are given.

In the paper, the authors present some monotonicity properties of certain functions defined in terms of the complete elliptic integrals of the second kind and some elementary functions and, consequently, improve several known inequalities for the complete elliptic integrals of the second kind.

The present paper introduces and studies a subclass of analytic functions defined by using the concept of Bazilevic and Janowski functions. Various properties such as coefficient estimates, Fekete-Szego type inequalities, arc length problem and growth rate of coefficients are investigated for related functions.

In this study, we obtain the real representations of elliptic biquaternion matrices. Afterwards, with the aid of these representations, we develop a general method to solve the linear matrix equations over the elliptic biquaternion algebra. Also we apply this method to the well known matrix equations X - AXB = C and AX - XB = C over the elliptic biquaternion algebra. Then, we give some illustrative numerical examples to show how the aforementioned method and its results work. Furthermore, we provide numerical algorithms for all the problems considered in this paper. Elliptic biquaternions are generalized form of complex quaternions and so real quaternions. This relation is valid for their matrices, as well. Thus, the obtained results extend, generalize and complement some known results from the literature.

Topological indices are numerical parameters of a molecular graph, which characterize its topology and are usually graph invariant. In quantitative structure-activity relationship/quantitative structure-property relationship study, physico-chemical properties and topological indices such as Randic, atom-bond connectivity (ABC), and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we are taking Dominating David Derived networks, produced by honeycomb structure of dimension t and obtain analytical closed results of Multiplicative topological indices and acquire exact results of degree based indices.

In this paper, we introduce and study a new subclass of analytic functions defined by $\mathcal{D}^{k}\mathcal{L} _{a}^{\delta }f(z)$ differential operator in the unit disk. For this subclass, the Fekete-Szego type coefficient inequalities are derived.

In this paper, we introduce the concept of Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction by unifying the definitions of Suzuki type $\mathcal{Z}$-contraction and $\mathcal{Z_\mathcal{R}}$-contraction and also provide examples to highlight the genuineness of our newly introduced contraction over earlier mentioned ones. Chiefly, we prove an existence and corresponding uniqueness fixed point results for Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction employing an amorphous binary relation on metric spaces without completeness and also furnish an illustrative example to demonstrate the utility of our main results. Finally, we utilize our main results to discuss the existence and uniqueness of solutions of a family of nonlinear matrix equations.

Recently, Chen and Ma [21] constructed the generalized shift-splitting (GSS) preconditioner, and gave the corresponding theoretical analysis and numerical experiments. In this paper, based on the generalized shift-splitting (GSS) preconditioner, we generalize their algorithms and further study the two-sweep shift-splitting (TSSS) preconditioner for complex symmetric linear systems. Moreover, by similar theoretical analysis, we obtain that the two-sweep shift-splitting iterative method is unconditionally convergent. In finally, one example is provided to confirm the effectiveness.