The existence of solutions of a functional integro-differential equation with self-reference and state-dependence will be studied. The continuous dependence of the solution on the delay \(\phi(t)\), the functional \(g\) and initial data \(x_0\) will be proved.

In this paper, differential transform method (DTM) is presented to solve Tumor-immune system at two initial conditions where two different cases of the interaction between tumor cells and effector cells. The system is presented to show the ability of the method for non-linear systems of differential equations. By using small iteration, the results of DTM are near the results of Runge-Kutta fourth-fifth order method (ode45 solver in MATLAB) and better than the results of Runge-Kutta second-third order method (ode23 solver in MATLAB). Also, the residual error of DTM's solutions approach zero. Therefore, DTM's solutions approximate exact solutions. Finally, we conclude formulae that we can find DTM's solutions, better than the results of Runge-Kutta second-third order method, in any interval we need.

In this work, we obtain some fixed point and common fixed point theorems of comparable maps satisfying certain contractive conditions on partially ordered cone b-metric space over Banach algebras. Some examples are also provided to illustrate the main results presented in this paper, which extend and generalize several known results in cone b-metric spaces.

In this paper, we propose and study a new class of continuous distributions called the Marshall-Olkin exponentiated generalized G (MOEG-G) family which extends the Marshall-Olkin-G family introduced by Marshall and Olkin [A. W. Marshall, I. Olkin, Biometrika, \(\bf 84\) (1997), 641--652]. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, order statistics and probability weighted moments are derived. Some characterizations for the new family are presented. Maximum likelihood estimation for the model parameters under uncensored and censored data is addressed in Section 5 as well as a simulation study to assess the performance of the estimators. The importance and flexibility of the new family are illustrated by means of two applications to real data sets.

In this paper, we study optimal asset allocation strategy for a defined contribution (DC) pension fund with return of premium clause under Heston's volatility model in mean-variance utility frame work. In this model, members' next of kin are allowed to withdraw their family members' accumulated premium with predetermined interest. Also, investments in one risk free asset and one risky asset are considered to help increase the accumulated funds of the remaining members in order to meet their retirement needs. Using the actuarial symbol, we formulize the problem as a continuous time mean-variance stochastic optimal control problem. We establish an optimization problem from the extended Hamilton Jacobi Bellman equations using the game theoretic approach and solve the optimization problem to obtain the optimal allocation strategy for the two assets, the optimal fund size and also the efficient frontier of the pension members. We analyze numerically the effect of some parameters on the optimal allocation strategy and deduce that as the initial wealth, predetermined interest rate and risk averse level increases, the optimal allocation policy for the risky asset (equity) decreases. Furthermore, we give a theoretical comparison of our result with an existing result and observed that the optimal allocation policy whose return is with predetermined interest is higher compared to that without predetermined interest.

In this paper, we study the quenching behavior of semidiscretizations of the heat equation with nonlinear boundary conditions. We obtain some conditions under which the positive solution of the semidiscrete problem quenches in a finite time and estimate its semidiscrete quenching time. We also establish the convergence of the semidiscrete quenching time and obtain some results on numerical quenching rate. Finally we give some numerical results to illustrate our analysis.