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In this paper, we discuss the existence of solutions to a boundary value problem of differential equations of variable order, which is a piecewise constant function. Our results are based on the Schauder fixed point theorem. Then, under some assumptions on the nonlinear term, we obtain a generalized Lyapunov-type inequality to the two-point boundary value problem considered. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for boundary value problems in term of variable order. In addition, some examples of the obtained inequalities are given.

This paper aims to present a new pathwise approximation method, which gives approximate solutions of order $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals Iα by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals Iα. Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward&ndash, backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

The purpose of this paper is to introduce the split combination of variational inclusion problem which combines the concept of the modified variational inclusion problem introduced by Khuangsatung and Kangtunyakarn and the split variational inclusion problem introduced by Moudafi. Using a modified Halpern iterative method, we prove the strong convergence theorem for finding a common solution for the hierarchical fixed point problem and the split combination of variational inclusion problem. The result presented in this paper demonstrates the corresponding result for the split zero point problem and the split combination of variation inequality problem. Moreover, we discuss a numerical example for supporting our result and the numerical example shows that our result is not true if some conditions fail.

A mathematical chaos model for the dynamical behaviors of atmospheric acoustic-gravity waves is considered in this paper. Boundedness and globally attractive sets of this chaos model are studied by means of the generalized Lyapunov function method. The innovation of this paper is that it not only proves this system is globally bounded but also provides a series of global attraction sets of this system. The rate of trajectories entering from the exterior of the trapping domain to its interior is also obtained. Finally, the detailed numerical simulations are carried out to justify theoretical results. The results in this study can be used to study chaos control and chaos synchronization of this chaos system.

The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein&ndash, Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schr&ouml, dinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson&ndash, Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

In this paper, Normalized Weighted Bonferroni Mean (NWBM) and Normalized Weighted Bonferroni Harmonic Mean (NWBHM) aggregation operators are proposed. Besides, we check the properties thereof, which include idempotency, monotonicity, commutativity, and boundedness. As the intuitionistic fuzzy numbers are used as a basis for the decision making to effectively handle the real-life uncertainty, we extend the NWBM and NWBHM operators into the intuitionistic fuzzy environment. By further modifying the NWBHM, we propose additional aggregation operators, namely the Intuitionistic Fuzzy Normalized Weighted Bonferroni Harmonic Mean (IFNWBHM) and the Intuitionistic Fuzzy Ordered Normalized Weighted Bonferroni Harmonic Mean (IFNONWBHM). The paper winds up with an empirical example of multi-attribute group decision making (MAGDM) based on triangular intuitionistic fuzzy numbers. To serve this end, we apply the IFNWBHM aggregation operator.

The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation.

In this paper, a wind speed prediction method was proposed based on the maximum Lyapunov exponent (Le) and the fractional Levy stable motion (fLsm) iterative prediction model. First, the calculation of the maximum prediction steps was introduced based on the maximum Le. The maximum prediction steps could provide the prediction steps for subsequent prediction models. Secondly, the fLsm iterative prediction model was established by stochastic differential. Meanwhile, the parameters of the fLsm iterative prediction model were obtained by rescaled range analysis and novel characteristic function methods, thereby obtaining a wind speed prediction model. Finally, in order to reduce the error in the parameter estimation of the prediction model, we adopted the method of weighted wind speed data. The wind speed prediction model in this paper was compared with GA-BP neural network and the results of wind speed prediction proved the effectiveness of the method that is proposed in this paper. In particular, fLsm has long-range dependence (LRD) characteristics and identified LRD by estimating self-similarity index H and characteristic index &alpha, Compared with fractional Brownian motion, fLsm can describe the LRD process more flexibly. However, the two parameters are not independent because the LRD condition relates them by &alpha, H &gt, 1.

The aim of this paper is to study the Lagrange interpolation on the unit circle taking only into account the separation properties of the nodal points. The novelty of this paper is that we do not consider nodal systems connected with orthogonal or paraorthogonal polynomials, which is an interesting approach because in practical applications this connection may not exist. A detailed study of the properties satisfied by the nodal system and the corresponding nodal polynomial is presented. We obtain the relevant results of the convergence related to the process for continuous smooth functions as well as the rate of convergence. Analogous results for interpolation on the bounded interval are deduced and finally some numerical examples are presented.

In this paper, a mathematical model based on the one-parameter Mittag-Leffler function is proposed to be used for the first time to describe the relation between the unemployment rate and the inflation rate, also known as the Phillips curve. The Phillips curve is in the literature often represented by an exponential-like shape. On the other hand, Phillips in his fundamental paper used a power function in the model definition. Considering that the ordinary as well as generalised Mittag-Leffler function behave between a purely exponential function and a power function it is natural to implement it in the definition of the model used to describe the relation between the data representing the Phillips curve. For the modelling purposes the data of two different European economies, France and Switzerland, were used and an &ldquo, out-of-sample&rdquo, forecast was done to compare the performance of the Mittag-Leffler model to the performance of the power-type and exponential-type model. The results demonstrate that the ability of the Mittag-Leffler function to fit data that manifest signs of stretched exponentials, oscillations or even damped oscillations can be of use when describing economic relations and phenomenons, such as the Phillips curve.

Probabilistic evolution theory (PREVTH) forms a framework for the solution of explicit ODEs. The purpose of the paper is two-fold: (1) conversion of multinomial right-hand sides of the ODEs to purely second degree multinomial right-hand sides by space extension, (2) decrease the computational burden of probabilistic evolution theory by using the condensed Kronecker product. A first order ODE set with multinomial right-hand side functions may be converted to a first order ODE set with purely second degree multinomial right-hand side functions at the expense of an increase in the number of equations and unknowns. Obtaining purely second degree multinomial right-hand side functions is important because the solution of such equation set may be approximated by probabilistic evolution theory. A recent article by the authors states that the ODE set with the smallest number of unknowns can be found by searching. This paper gives the details of a way to search for the optimal space extension. As for the second purpose of the paper, the computational burden can be reduced by considering the properties of the Kronecker product of vectors and how the Kronecker product appears within the recursion of PREVTH: as a Cauchy product structure.

This paper focuses on the problem of finite-time stabilization of homogeneous, non-Lipschitz systems with dilations. A key contribution of this paper is the design of a virtual recursive Hölder, non-Lipschitz state feedback which renders the non-Lipschitz systems in the special case dominated by a lower-triangular nonlinear system, finite-time stable. The proof is based on a recursive design algorithm developed recently, to construct the virtual Hölder continuous, finite-time stabilizer as well as a C1 positive definite and proper Lyapunov function that guarantees finite-time stability of the non-Lipschitz non-linear systems.

This is a survey paper illuminating the distinguished role of the Mittag-Leffler function and its generalizations in fractional analysis and fractional modeling. The content of the paper is connected to the recently published monograph by Rudolf Gorenflo, Anatoly Kilbas, Francesco Mainardi and Sergei Rogosin.

This paper is concerned with the existence of stationary distribution and extinction for multispecies stochastic Lotka-Volterra predator-prey system. The contributions of this paper are as follows. (a) By using Lyapunov methods, the sufficient conditions on existence of stationary distribution and extinction are established. (b) By using the space decomposition technique and the continuity of probability, weaker conditions on extinction of the system are obtained. Finally, a numerical experiment is conducted to validate the theoretical findings.

The paper focuses on studying the Noether theorem for nonholonomic systems with time delay. Firstly, the differential equations of motion for nonholonomic systems with time delay are established, which is based on the Hamilton principle with time delay and the Lagrange multiplier rules. Secondly, based upon the generalized quasi-symmetric transformations for nonconservative systems with time delay, the Noether theorem for corresponding holonomic systems is given. Finally, we obtain the Noether theorem for the nonholonomic nonconservative systems with time delay. At the end of the paper, an example is given to illustrate the application of the results.

In this paper, we develop a rumor spreading model by introducing white noise into the model. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic Lyapunov function, which provides us a good description of persistence. Finally, we provide some numerical simulations to illustrate the analytical results.

To avoid singular generalized Jacobian matrix and further accelerate the convergence of the generalized Newton (GN) iteration method for solving generalized absolute value equations Ax - B|x| = b, in this paper we propose a new relaxed generalized Newton (RGN) iteration method by introducing a relaxation iteration parameter. The new RGN iteration method involves the well-known GN iteration method and the Picard iteration method as special cases. Theoretical analyses show that the RGN iteration method is well defined and globally linearly convergent under suitable conditions. In addition, a specific sufficient condition is studied when the coefficient matrix A is symmetric positive definite. Finally, two numerical experiments arising from the linear complementarity problems are used to illustrate the effectiveness of the new RGN iteration method.

This paper gropes the stability and Hopf bifurcation of a delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response. The critical point at which a Hopf bifurcation occurs can be figured out by using the escalating time delay of psychologically addicts as a bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are explored with aid of the central manifold theorem and normal form theory. Specially, global stability of the model is proved by constructing a suitable Lyapunov function. To underline effectiveness of the obtained results and analyze influence of some influential parameters on dynamics of the model, some numerical simulations are ultimately addressed.

In this paper, we investigate the global properties of two general models of pathogen infection with immune deficiency. Both pathogen-to-cell and cell-to-cell transmissions are considered. Latently infected cells are included in the second model. We show that the solutions are nonnegative and bounded. Lyapunov functions are organized to prove the global asymptotic stability for uninfected and infected steady states of the models. Analytical expressions for the basic reproduction number $\mathcal{R}_{0}$ and the necessary condition under which the uninfected and infected steady states are globally asymptotically stable are established. We prove that if $\mathcal{R}_{0}$ < 1 then the uninfected steady state is globally asymptotically stable (GAS), and if $\mathcal{R}_{0}$ > 1 then the infected steady state is GAS. Numerical simulations are performed and used to support the analytical results.

Recently, Hoang and Egbelowo (Boletin de la Sociedad Matematica Mexicana, 2020) proposed a nonstandard finite difference scheme (NSFD) to get a discrete SIR epidemic model with saturated incidence rate and constant vaccination. The discrete model was derived by discretizing the right-hand sides of the system locally and the first order derivative is approximated by the generalized forward difference method but with a restrictive denominator function. Their analysis showed that the NSFD scheme is dynamically-consistent only for relatively small time-step sizes. In this paper, we propose and analyze an alternative NSFD scheme by applying nonlocal approximation and choosing the denominator function such that the proposed scheme preserves the boundedness of solutions. It is verified that the proposed discrete model is dynamically-consistent with the corresponding continuous model for all time-step size. The analytical results have been confirmed by some numerical simulations. We also show numerically that the proposed NSFD scheme is superior to the Euler method and the NSFD method proposed by Hoang and Egbelowo (2020).

This paper aims to efficiently implement the maximum likelihood estimator (MLE) for Hurst exponent, a vital parameter embedded in the process of fractional Brownian motion (FBM) or fractional Gaussian noise (FGN), via a combination of the Levinson algorithm and Cholesky decomposition. Many natural and biomedical signals can often be modeled as one of these two processes. It is necessary for users to estimate the Hurst exponent to differentiate one physical signal from another. Among all estimators for estimating the Hurst exponent, the maximum likelihood estimator (MLE) is optimal, whereas its computational cost is also the highest. Consequently, a faster but slightly less accurate estimator is often adopted. Analysis discovers that the combination of the Levinson algorithm and Cholesky decomposition can avoid storing any matrix and performing any matrix multiplication and thus save a great deal of computer memory and computational time. In addition, the first proposed MLE for the Hurst exponent was based on the assumptions that the mean is known as zero and the variance is unknown. In this paper, all four possible situations are considered: known mean, unknown mean, known variance, and unknown variance. Experimental results show that the MLE through efficiently implementing numerical computation can greatly enhance the computational performance.

In recent years, there has been a renewed interest in finding methods to construct orthogonal transforms. This interest is driven by the large number of applications of the orthogonal transforms in image analysis and compression, especially for colour images. Inspired by this motivation, this paper first introduces a new orthogonal transform known as a discrete fractional COSHAD (FrCOSHAD) using the Kronecker product of eigenvectors and the eigenvalues of the COSHAD kernel functions. Next, this study discusses the properties of the FrCOSHAD kernel function, such as angle additivity. Using the algebra of quaternions, the study presents quaternion COSHAD/FrCOSHAD transforms to represent colour images in a holistic manner. This paper also develops an inverse polynomial reconstruction method (IPRM) in the discrete COSHAD/FrCOSHAD domains. This method can effectively recover a piecewise smooth signal from the finite set of its COSHAD/FrCOSHAD coefficients, with high accuracy. The convergence theorem has proved that the partial sum of COSHAD provides a spectrally accurate approximation to the underlying piecewise smooth signal. The experimental results verify the numerical stability and accuracy of the proposed methods.

In this paper, we analyze the dissipativity for descriptor systems with impulsive behavior based on image space analysis. First, a new image space is used to characterize state responses for descriptor systems. Based on such characterization and an integral property of delta function, a new necessary and sufficient condition for the dissipativity of descriptor systems is derived using the linear matrix inequality (LMI) approach. Also, some of the earlier related results on dissipativity for linear systems are investigated in the framework proposed in this paper. Finally, two examples are given to show the validity of the derived results.

This paper presents another new modified Lorenz system which is chaotic in a certain range of parameters. Besides that, this paper also presents explanations to solve the new modified Lorenz system. Furthermore, some of the dynamical properties of the system are shown and stated. Basically, this paper shows the finding that led to the discovery of fixed points for the system, dynamical analysis using complementary-cluster energy-barrier criterion (CCEBC), finding the Jacobian matrix, finding eigenvalues for stability, finding the Lyapunov functions, and finding the Lyapunov exponents to investigate some of the dynamical behaviours of the system. Pictures and diagrams will be shown for the chaotic systems using the aide of MAPLE in 2D and 3D views. Nevertheless, this paper is to introduce the new modified Lorenz system.

A nonzero residual intersymbol interference (ISI) causes the symbol error rate (SER) to increase where the achievable SER may not answer any more on the system’s requirements. In the literature, we may find for the single-input-single-output (SISO) case a closed-form approximated expression for the SER that takes into account the achievable performance of the chosen blind adaptive equalizer from the residual ISI point of view and a closed-form approximated expression for the residual ISI valid for the single-input-multiple-output (SIMO) case. Both expressions were obtained by assuming that the input noise is a white Gaussian process where the Hurst exponent (H) is equal to 0.5. In this paper, we derive a closed-form approximated expression for the residual ISI obtained by blind adaptive equalizers for the SIMO case, valid for fractional Gaussian noise (fGn) input where the Hurst exponent is in the region of . Based on this new expression for the residual ISI, a closed-form approximated expression is obtained for the SER valid for the SIMO and fGn case. In this paper, we show via simulation results that the SER might get improved for increasing values of H.

The eigenvalue problem over a polyhedral cone is studied in this paper. Based on the F-B NCP function, we reformulate this problem as a system of equations and propose a Jacobian-like method. The global convergence and local quadratic convergence of the proposed method are established under suitable assumptions. Preliminary numerical experiments for a special polyhedral cone are reported in this paper to show the validity of the proposed method.

In this paper, we study the existence of square-mean asymptotically almost periodic mild solutions for a class of second order nonautonomous stochastic evolution equations in Hilbert spaces. By using the principle of Banach contractive mapping principle, the existence and uniqueness of square-mean asymptotically almost periodic mild solutions of the equation are obtained. To illustrate the abstract result, a concrete example is given.

Finite time stability involving dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time have been studied for both continuous-time and discrete-time systems. For continuous-time systems, finite time stability is defined for equilibria of continuous but non-Lipschitzian nonlinear dynamics, whereas discrete-time systems can exhibit finite time stability even when the system dynamics are linear, and hence, Lipschitz continuous. Alternatively, for impulsive dynamical systems it may be possible to reset the system states to an equilibrium state achieving finite time stability without requiring a non-Lipschitz condition for the continuous-time part of the hybrid system dynamics. In this paper, we develop sufficient Lyapunov conditions for finite time stability of impulsive dynamical systems using both a scalar differential Lyapunov inequality on the continuous-time dynamics as well as a scalar difference Lyapunov inequality on the discrete-time resetting dynamics. Furthermore, using our proposed finite time stability results, we design universal hybrid finite time stabilizing control laws for impulsive dynamical systems. Finally, we present several numerical examples for finite time stabilization of network impulsive dynamical systems.

The paper presents a mathematical model of queueing tandem M|GI|∞ → GI|∞ with feedback. The service times at the first stage are independent and identically distributed (i.i.d.) with an arbitrary distribution function B1(x). Service times at the second stage are i.i.d. with an arbitrary distribution function B2(x). The problem is to determine the probability distribution of the number of repeated customers (r-flow) during fixed time period. To solve this problem, the Markov summation method was used, which is based on the consideration of Markov processes and the solution of the Kolmogorov equation. In the course of the solution, the so-called local r-flow was studied — the number of r-flow calls generated by one incoming customer received by the system. As a result, an expression is obtained for the characteristic probability distribution function of the number of calls in the local r-flow, which can be used to study queuing systems with a similar service discipline and non-Markov incoming flows. As a result of the study, an expression is obtained for the characteristic probability distribution function of the number of repeated calls to the system at a given time interval during non-stationary regime, which allows one to obtain the probability distribution of the number of calls in the flow under study, as well as its main probability characteristics.

The boundedness of singular and fractional integral operator on Lebesgue and Hardy spaces have been well studied. The theory of Herz space and Herz type Hardy space, as a local version of Lebesgue and Hardy space, have been developed. The main purpose of this paper is to establish the endpoint continuity properties of some multilinear operators related to certain non-convolution type fractional singular integral operators on Herz and Herz type Hardy spaces and the endpoint estimates for the multilinear operators on Herz and Herz type Hardy spaces are obtained.

Integral inequalities play a key role in applied and theoretical mathematics. The purpose of inequalities is to develop mathematical techniques in analysis. The goal of this paper is to develop a fractional integral operator having a non-singular function (generalized multi-index Bessel function) as a kernel and then to obtain some significant inequalities like Hermit Hadamard Mercer inequality, exponentially $ (s-m) $-preinvex inequalities, Polya-Szego and Chebyshev type integral inequalities with the newly developed fractional operator. These results describe in general behave and provide the extension of fractional operator theory (FOT) in inequalities.

In this paper, we establish new Lyapunov-type inequalities for a Hadamard fractional differential equation under Sturm-Liouville boundary conditions. Our conclusions cover many results in the literature.

In this paper, we establish a representation of mild solutions to fully nonlocal stochastic evolution problems. Through the iterative technique and energy estimates, we obtain the existence and uniqueness of mild solution. Furthermore, we prove the existence of optimal control for fully nonlocal stochastic control problems with a non-convex cost function. Two examples are given at the end.

In this paper, the reproducing kernel Hilbert space method had been extended to model a numerical solution with two-point temporal boundary conditions for the fractional derivative in the Caputo sense, convergent analysis and error bounds are discussed to verify the theoretical results. Numerical examples are given to illustrate the accuracy and efficiency of the presented approach.

In this paper, a Susceptible-Infected-Quarantined-Susceptible (SIQRS) computer virus propagation model with nonlinear infection rate and two-delay is formulated. The local stability of virus-free equilibrium without delay is examined. Furthermore, we also expound and prove that time-delay plays a crucial role in sufficient conditions for the local stability of the virus-existence equilibrium and the occurrence of Hopf bifurcation at the critical value. Especially, direction and stability of the Hopf bifurcation are demonstrated. Finally, some numerical simulations are presented in order to verify the theoretical results.

In this paper, we propose a predator-prey system with square root functional response, two delays and prey harvesting, in which an algebraic equation stands for the economic interest of the yield of the harvest effort. Firstly, the existence of the positive equilibrium is discussed. Then, by taking two delays as bifurcation parameters, the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained. Next, some explicit formulas determining the properties of Hopf bifurcation are analyzed based on the normal form method and center manifold theory. Furthermore, the control of Hopf bifurcation is discussed in theory. What's more, the optimal tax policy of system is given. Finally, simulations are given to check the theoretical results.

We consider in this paper the stochastic nonlinear Schrödinger equation forced by multiplicative noise in the Itô sense. We use two different methods as sine-cosine method and Riccati-Bernoulli sub-ODE method to obtain new rational, trigonometric and hyperbolic stochastic solutions. These stochastic solutions are of a qualitatively distinct nature based on the parameters. Moreover, the effect of the multiplicative noise on the solutions of nonlinear Schrödinger equation will be discussed. Finally, two and three-dimensional graphs for some solutions have been given to support our analysis.

The main purpose of this paper is using analytic methods to give a recurrence formula of the number of solutions of an equation over finite field. We use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of the Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid complicated calculations.

A counterexample to a compactness embedding result of Nagy is provided. Let V and H be real separable Hilbert spaces with V densely and continuously embedded in H. Identifying H with its dual we write V c H c V' algebraically and topologically, where V' is the dual space to V. Given T > 0, let Y = L2(0, T; V), * = L2(0, T; H) and 7' = L2(0, T; V') denote the spaces of the square summable functions defined on the interval (0, T) with values in V, H and V', respectively. We define 2 = {v E 7: v' E 7'},whichwiththeusualnorm Ilullr = (I1uII2+I1u'II2,)1/2 is a Hilbert space (cf. [7], Theorem 25.4). The following remarks about the space 2f are in order. In the definition the derivative u' = du is understood in the dt weak sense. Nevertheless, when we view u as a V'-valued function, we know (see [8], Proposition 23.23) that it is absolutely continuous, so its derivative exists in the strong sense almost everywhere on (0, T). It is known (see, e.g., [7], Theorem 25.5) that the embedding (1) 7C C(0 T; H) is continuous. Here C(0, T; H) denotes the space of continuous functions from [0, T] into H, endowed with the supremum norm. Therefore every function in /l can be, after modification on the set of measure zero, considered as an element of C(0, T; H). We also know that 2 c X compactly (see [8]). The following is the main result of Nagy (see [3], Theorem 2). Theorem. Let V and H be infinite-dimensional separable Hilbert spaces such that the embedding V c H is dense, continuous and compact. Then the embedding (1) is also compact. The aim of this paper is to exhibit a simple example which shows that the embedding (1) cannot be compact, i.e., the above theorem is not true. It should also be noted here that this theorem was exploited in several recent papers in Received by the editors December 1, 1993. 1991 Mathematics Subject Classification. Primary 46E35.

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.

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.

This paper deals with the amalgamated basic IS-LM business cycle model with Kaldor’s growth model to form an augmented model. Pertaining to substantial evidence, IS-LM model in paradigm with a specific economic extension (Kaldor-Kalecki Business cycle model in our case) provides an adept explanation of a developing but strong economy like that of our country. Occurring in the equation of capital accumulation, the two time delays are a result of the assumption in the investment function being both income and capital stock dependent in past period and maturity period. Investigating a model combined with capital accumulation is both interesting and important. From economist point of view, production without capital is impossible to even imagine. Moreover capital accumulation is impeccable to large-scale production, specialisation and creation of employment opportunities. In our model ‘I’ the investment function, ‘S’ the savings function and ‘L’ the demand for money are depending linearly on their arguments. We adhere to a linear model, contrary to the popular belief of non- linear models being the undisputed style for modern economics. The model is first shown to be mathematically and economically poised. The local stability of boundary and interior equilibrium points has been investigated. Three cases arise, pertaining to two time delays. System dynamics exhibits mutation under the influence of time delays and may clinch or discharge its local stability when subjected to the latter. Hopf bifurcation occurs when the delay parameter crosses a critical value.

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.

The main purpose of this paper is to present a new computational for approximate analytical solutions of nonlinear time-fractional wave-like equations with variable coefficients using the fractional residual power series method (FRPSM). The fractional derivative is considered in the Caputo sense. This method is based on the generalized Taylor series formula and residual error function. Unlike other analytical methods, FRPSM has a special advantage, that it solves the nonlinear problems without using linearization, discretization, perturbation or any other restrictions. By numerical examples, it is shown that the FRPSM is a simple, effective, and powerful method for finding approximate analytical solutions of nonlinear fractional partial differential equations.

In this paper, we present few Lyapunov-type inequalities for Hadamard fractional boundary value problems associated with different sets of boundary conditions. Further, we discuss two applications of the established results.

This paper deals with a Gause-type predator-prey model with Allee effect and Holling type III functional response. We also consider the influence of predator competition and the artificial harvesting on predator-prey system. The existence of multiple positive periodic solutions of the predator-prey model is established by using the Mawhin coincidence degree theory.

The periodic nonuniform sampling plays an important role in digital signal processing and other engineering fields. In this paper, we introduce the Gaussian regularization method to accelerate the convergence rate of periodic nonuniform sampling series. We prove that the truncation error of the Gaussian regularized periodic nonuniform sampling series decays exponentially. Numerical experiments are presented to demonstrate our result.

Using some of the extended fixed point results for Geraghty contractions in b-metric spaces given by Faraji, Savić and Radenović and their idea to apply these results to nonlinear integral equations, in this paper we present some existence and uniqueness conditions for the solution of a nonlinear Fredholm&ndash, Volterra integral equation with a modified argument.

There are some generalizations of the classical exponential distribution in the statistical literature that have proven to be helpful in numerous scenarios. Some of these distributions are the families of distributions that were proposed by Marshall and Olkin and Gupta. The disadvantage of these models is the impossibility of fitting data of a bimodal nature of incorporating covariates in the model in a simple way. Some empirical datasets with positive support, such as losses in insurance portfolios, show an excess of zero values and bimodality. For these cases, classical distributions, such as exponential, gamma, Weibull, or inverse Gaussian, to name a few, are unable to explain data of this nature. This paper attempts to fill this gap in the literature by introducing a family of distributions that can be unimodal or bimodal and nests the exponential distribution. Some of its more relevant properties, including moments, kurtosis, Fisher’s asymmetric coefficient, and several estimation methods, are illustrated. Different results that are related to finance and insurance, such as hazard rate function, limited expected value, and the integrated tail distribution, among other measures, are derived. Because of the simplicity of the mean of this distribution, a regression model is also derived. Finally, examples that are based on actuarial data are used to compare this new family with the exponential distribution.