We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L(H ; ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under bo...

We introduce and study a new class of generalized nonlinear mixed variational-like inequalities in reflexive Banach spaces. By applying Ding’s technique we prove several existence and uniqueness theorems of solutions for the generalized nonlinear mixed variational-like inequality, extend the auxiliary problem technique to suggest and analyze an iterative method to compute the approximate soluti...

In this paper, we propose a new method for the numerical solution of two-dimensional linear and nonlinear Volterra integral equations of the first and second kinds, which avoids from using starting values. An existence and uniqueness theorem is proved and convergence isverified by using an appropriate variety of the Gronwall inequality. Application of the method is demonstrated for solving the ...

In the present research paper we derive results about existence and uniqueness of solutions and Ulam--Hyers and Rassias stabilities of nonlinear Volterra--Fredholm delay integrodifferential equations. Pachpatte's inequality and Picard operator theory are the main tools that are used to obtain our main results. We concluded this work with applications of ob...

The correspondence principle suggests that quantum systems grow classical when large. Classical cannot violate Bell inequalities. Yet agents given substantial control can inequalities proven for large-scale systems. We consider who have little control, implementing only general operations suited to macroscopic experimentalists: preparing small-scale entanglement and measuring properties while s...

H, control theory for nonlinear systems has been developed, which is based on the concept of the energy dissipation. A nonlinear H, controller using the energy dissipation is designed in the sense of Lagain attenuation from a disturbance to performance and it is essential to find the solution of the Hamilton Jacobi (HJ) equation (or inequality) for application However, it is dificult to obtain ...

When solving the general smooth nonlinear optimization problem involving equality and/or inequality constraints, an approximate first-order critical point of accuracy ǫ can be obtained by a second-order method using cubic regularization in at most O(ǫ) problem-functions evaluations, the same order bound as in the unconstrained case. This result is obtained by first showing that the same result ...

In this paper we present a penalty method for solving a complementarity problem involving 2nd-order nonlinear parabolic differential operators. In this work we first rewrite the complementarity problem as a nonlinear variational inequality. Then, we define a nonlinear parabolic partial differential equation (PDE) approximating the variational inequality using a power penalty term with a penalty...