PRODUCTS OF RESOLVENTS AND MULTIVALUED HYBRID MAPPINGS IN C AT(0) SPACES

In this article, we introduce and investigate the concept of multivalued hybrid mappings in C AT（0） spaces by using the concept of quasilinearization. Also, we present a new iterative algorithm involving products of Moreau-Yosida resolvents for finding a common element of the set of minimizers of a finite family of convex functions and a common fixed point of two multivalued hybrid mappings in C AT（0） spaces.

A mathematical analysis of a tuberculosis epidemic model with two treatments and exogenous re-infection

In this paper,we propose a mathematical model of tuberculosis with two treatments and exogenous re-infection,in which the treatment is effective for number of infectious individuals and it fails for some other infectious individuals who are being treated.We show that the model exhibits the phenomenon of backward bifurcation,where a stable disease-free equilibrium coexists with a stable endemic equilibria when the related basic reproduction number is less than unity.Also,it is shown that under certain conditions the model cannot exhibit backward bifurcation.Furthermore,it is shown in the absence of re-infection,the backward bifurcation phenomenon does not exist,in which the disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity.The global asymptotic stability of the endemic equilibrium,when the associated reproduction number is greater than unity,is established using the geometric approach.Numerical simulations are presented to illustrate our main results.

Global stability and optim al control of a two-patch tuberculosis epidemic model using fractional-order derivatives

In this paper,we propose a fractional-order and two-patch model of tuberculosis(TB)epidemic,in which susceptible,slow latent,fast latent and infectious individuals can travel freely between the patches,but not under treatment infected individuals,due to medical reasons.We obtain the basic reproduction number Ro for the model and extend the classical LaSalle's invariance principle for fractional differential equations.We show that if R0<1,the disease-free equilibrium(DFE)is locally and globally asymptotically stable.If Ro>l,we obtain sufficient conditions under which the endernic equilibrium is unique and globally asymptotically stable.We extend the model by inclusion the time-dependent controls(effective treatment controls in both patches and controls of screening on travel of infectious individuals between patches),and formulate a fractional optimal control problem to reduce the spread of the disease.The numerical results show that the use of all controls has the most impact on disease control,and decreases the size of all infected compartments,but increases the size of susceptible compartment in both patches.We,also,investigate the impact of the fractional derivative order a on the values of the controls(0.7≤α≤1).The results show that the maximum levels of effective treatment controls in both patches increase when a is reduced from l,while the maximum level of the travel screening control of infectious individuals from patch 2 to patch 1 increases when o limits to 1.

Backward bifurcation in a fractional-order and two-patch model of tuberculosis epidemic with incomplete treatment

In this paper,we consider a tuberculosis model with incomplete treatment and extend the model to a Caputo fractional-order and two-patch version with exogenous re-infection among the treated individuals,in which only susceptible individuals can travel freely between the patches.The model has multiple equilibria.We determine conditions that lead to the appearance of a backward bifurcation.The results show that the TB model can have exogenous reinfection among the treated individuals and,at the same time,does not exhibit backward bifurcation.Also,conditions that lead to the global asymptotic stability of the disease-free equilibrium are obtained.In case without reinfection,the model has four equilibria.In this case,the global asymptotic stability of the equilibria is established using the Lyapunov function theory together with the LaSalle invariance principle for fractional differential equations(FDEs).Numerical simulations confirm the validity of the theoretical results.

Optimal control of a fractional-order model for the HIV/AIDS epidemic

In this paper,we present a general formulation for a fractional optimal control problem (FOCP),in which the state and co-state equations are given in terms of the left fractional derivatives.We develop the forward-backward sweep method (FBSM)using the Adamstype predictor-corrector method to solve the FOCP.We present a fractional model for transmission dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS)with treatment and incorporate three control efforts (effective use of condoms,ART treatment and behavioral change control)into the model aimed at controlling the spread of HIV/AIDS epidemic.The necessary conditions for fractional optimal control of the disease are derived and analyzed.The numerical results show that implementing all the control efforts increases the life time and the quality of life those living with HIV and decreases significantly the number of HIV-infected and AIDS people.Also,the maximum levels of the controls and the value of objective functional decrease when the derivative order a limits to 1(0.7≤a <1).In addition,the effect of the fractional derivative order a (0.7≤a <1)on the spread of HIV/AIDS epidemic and the treatment of HIV-infected population is investigated.The results show that the derivative order a can play the role of using ART treatment in the model.

A STRONG CONVERGENCE THEOREM FOR QUASI-EQUILIBRIUM PROBLEMS IN BANACH SPACES

In this paper,we study an extragradient algorithm for approximating solutions of quasi-equilibrium problems in Banach spaces.We prove strong convergence of the sequence generated by the extragradient method to a solution of the quasi-equilibrium problem.

Multi-Waves,Breathers,Periodic and Cross-Kink Solutions to the(2+1)-Dimensional Variable-Coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

The present article deals with multi-waves and breathers solution of the(2+1)-dimensional variable-coefficient CaudreyDodd-Gibbon-Kotera-Sawada equation under the Hirota bilinear operator method.The obtained solutions for solving the current equation represent some localized waves including soliton,solitary wave solutions,periodic and cross-kink solutions in which have been investigated by the approach of the bilinear method.Mainly,by choosing specific parameter constraints in the multi-waves and breathers,all cases the periodic and cross-kink solutions can be captured from the 1-and 2-soliton.The obtained solutions are extended with numerical simulation to analyze graphically,which results in 1-and 2-soliton solutions and also periodic and cross-kink solutions profiles.That will be extensively used to report many attractive physical phenomena in the fields of acoustics,heat transfer,fluid dynamics,classical mechanics,and so on.We have shown that the assigned method is further general,efficient,straightforward,and powerful and can be exerted to establish exact solutions of diverse kinds of fractional equations originated in mathematical physics and engineering.We have depicted the figures of the evaluated solutions in order to interpret the physical phenomena.

Application of G′/G-expansion Method to Kuramoto-Sivashinsky Equation

This paper obtains the solutions of the Kuramoto-Sivashinsky equation. The G′/G method is used to carry out the integration of this equation. Subsequently, its special case, will be integrated and topological 1- soliton solution will be obtained by the soliton ansatz method. The restrictions on the parameters and exponents are also identified.

Solving a class of boundary value problems and fractional Boussinesq-like equation with β-derivatives by fractional-order exponential trial functions

The nonlinear two-point boundary value problem(TPBVP)is converted to a new form of the problem including integral terms.Then the combination of weak-form integral equation method(WFIEM)and special functions create a robust and applicable numerical scheme to solve the problem.To display the accuracy of our method,some examples are investigated.Also,the fractional Boussinesq-like equation involving the β-derivative has been considered that describes the propagation of small amplitude long capillary-gravity waves on the surface of shallow water.

Randomly Weighted Averages on Order Statistics

We study a well-known problem concerning a random variable uniformly distributed between two independent random variables. Two different extensions, randomly weighted average on independent random variables and randomly weighted average on order statistics, have been introduced for this problem. For the second method, two-sided power random variables have been defined. By using classic method and power technical method, we study some properties for these random variables.

Estimation in the Complementary Exponential Geometric Distribution Based on Progressive Type-II Censored Data

Complementary exponential geometric distribution has many applications in survival and reliability analysis.Due to its importance,in this study,we are aiming to estimate the parameters of this model based on progressive type-II censored observations.To do this,we applied the stochastic expectation maximization method and Newton-Raphson techniques for obtaining the maximum likelihood estimates.We also considered the estimation based on Bayesian method using several approximate:MCMC samples,Lindely approximation and Metropolis-Hasting algorithm.In addition,we considered the shrinkage estimators based on Bayesian and maximum likelihood estimators.Then,the HPD intervals for the parameters are constructed based on the posterior samples from the Metropolis-Hasting algorithm.In the sequel,we obtained the performance of different estimators in terms of biases,estimated risks and Pitman closeness via Monte Carlo simulation study.This paper will be ended up with a real data set example for illustration of our purpose.

An analytical model for source code distributability verification

ne way to speed up the execution of sequential programs is to divide them into concurrent segments and execute such segments in a parallel manner over a distributed computing environment. We argue that the execution speedup primarily depends on the concurrency degree between the identified segments as well as communication overhead between the segments. To guar-antee the best speedup, we have to obtain the maximum possible concurrency degree between the identified segments, taking communication overhead into consideration. Existing code distributor and multi-threading approaches do not fulfill such re-quirements;hence, they cannot provide expected distributability gains in advance. To overcome such limitations, we propose a novel approach for verifying the distributability of sequential object-oriented programs. The proposed approach enables users to see the maximum speedup gains before the actual distributability implementations, as it computes an objective function which is used to measure different distribution values from the same program, taking into consideration both remote and sequential calls. Experimental results showed that the proposed approach successfully determines the distributability of different real-life software applications compared with their real-life sequential and distributed implementations.

A predictor-corrector scheme for solving nonlinear fractional differential equations with uniform and nonuniform meshes

In this paper,we develop two algorithms for solving linear and nonlinear fractional differential equations involving Caputo derivative.For designing new predictor–corrector(PC)schemes,we select the mesh points based on the two equal-height and equal-area distribution.Furthermore,the error bounds of PC schemes with uniform and equidistributing meshes are obtained.Finally,examples are constructed for illustrating the obtained PC schemes with uniform and equidistributing meshes.A comparative study is also presented.

Multiple rogue wave and solitary solutions for the generalized BK equation via Hirota bilinear and SIVP schemes arising in fluid mechanics

The multiple lump solutions method is employed for the purpose of obtaining multiple soliton solutions for the generalized Bogoyavlensky-Konopelchenko(BK)equation.The solutions obtained contain first-order,second-order,and third-order wave solutions.At the critical point,the second-order derivative and Hessian matrix for only one point is investigated,and the lump solution has one maximum value.He’s semi-inverse variational principle(SIVP)is also used for the generalized BK equation.Three major cases are studied,based on two different ansatzes using the SIVP.The physical phenomena of the multiple soliton solutions thus obtained are then analyzed and demonstrated in the figures below,using a selection of suitable parameter values.This method should prove extremely useful for further studies of attractive physical phenomena in the fields of heat transfer,fluid dynamics,etc.

Multiple-order line rogue wave,lump and its interaction,periodic,and cross-kink solutions for the generalized CHKP equation

The multiple-order line rogue wave solutions method is emp loyed for searching the multiple soliton solutions for the generalized(2+1)-dimensional Camassa-HolmKadomtsev-Petviashvili(CHKP)equation,which contains first-order,second-order,and third-order waves solutions.At the critical point,the second-order derivative and Hessian matrix for only one point will be investigated and the lump solution has one minimum value.For the case,the lump solution will be shown the bright-dark lump structure and for another case can be present the dark lump structure-two small peaks and one deep hole.Also,the interaction of lump with periodic waves and the interaction between lump and soliton can be obtained by introducing the Hirota forms.In the meanwhile,the cross-kink wave and periodic wave solutions can be gained by the Hirota operator.The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values.We alternative offer that the determining method is general,impressive,outspoken,and powerful and can be exerted to create exact solutions of various kinds of nonlinear models originated in mathematical physics and engineering.