Narrow Results

In this paper, we propose two methods for solving general mixed variational inequalities. In the first method, the new iterate is obtained by using a descent direction while the second method can be viewed as an extension of the first one by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. It is proved theoretically that the lower-bound of the progress obtained by the second method is greater than that by the first one. Under certain conditions, the global convergence of the both methods is proved. The comparison of these methods with other methods for solving the mixed genera variational inequalities is an open interesting problem.

In this paper, we study the existence and continuous dependence on coefficients of mild solutions for first-order McKean–Vlasov integrodifferential equations with delay driven by a cylindrical Wiener process using resolvent operator theory and Wasserstein distance. Under the situation that the nonlinear term depends on the probability distribution of the state, the existence and uniqueness of solutions are established. An example illustrating the general results is included.

In this paper, we suggest and analyze a new three-step iterative method for solving mixed variational inequalities. The new iterate is obtained by using a descent direction. We prove that the new method is globally convergent under suitable mild conditions. Our results can be viewed as significant extensions of the previously known results for mixed variational inequalities. Since mixed variational inequalities include variational inequalities as special cases, our method appears to be a new one for solving variational inequalities. Preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method.

In this work, we introduce a modified inertial algorithm for solving the split common null point problem without the prior knowledge of the operator norms in Banach spaces. The strong convergence theorem of our method is proved under suitable assumptions. We apply our result to the split feasibility problem, split equilibrium problem and split minimization problem. Finally, we provide some numerical experiments including compressed sensing to illustrate the performances of the proposed method. The result presented in this paper improves and generalizes many recent important results in the literature.

In this paper, a new class of general nonlinear quasi-variational inclusions involving m–η – accretive operators in Banach spaces is introduced and studied. And a new iterative algorithm for this class of general nonlinear quasi-variational inclusions and its convergence result are established.

This paper considers strategic entry decisions in a duopoly market when the underlying state variable follows a diffusion with volatility that depends on the current state variable. The extension to this case is more than marginal, since empirical studies have suggested that the volatility is indeed non-constant in real options practices. It is shown that, even in the extended model, three types of equilibria exist in the case of strategic substitution, as for the geometric Brownian case, when the revenue functions are linear. Also, the presence of strategic interactions may push a firm with cost advantage to invest earlier, and the firm value as well as the optimal threshold for the investment decision increases as the market uncertainty increases.