The Second-Order Differential Equation System with the Controlled Process for Variational Inequality with Constraints

In this paper, the variational inequality with constraints can be viewed as an optimization problem. Using Lagrange function and projection operator, the equivalent operator equations for the variational inequality with constraints under the certain conditions are obtained. Then, the second-order differential equation system with the controlled process is established for solving the variational inequality with constraints. We prove that any accumulation point of the trajectory of the second-order differential equation system is a solution to the variational inequality with constraints. In the end, one example with three kinds of different cases by using this differential equation system is solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the controlled process for solving the variational inequality with constraints.

New Aspects for Oscillation of Differential Systems with Mixed Delays and Impulses

Oscillation and symmetry play an important role in many applications such as engineering, physics, medicine, and vibration in flight. In this work, we obtain sufficient and necessary conditions for the oscillation of the solutions to a second-order differential equation with impulses and mixed delays when the neutral coefficient lies within [0,1). Furthermore, an examination of the validity of the proposed criteria has been demonstrated via particular examples.

Oscillation Results of Emden–Fowler-Type Differential Equations

In this article, we obtain oscillation conditions for second-order differential equation with neutral term. Our results extend, improve, and simplify some known results for neutral delay differential equations. Several effective and illustrative implementations are provided.

Fixed point for F⊥-weak contraction

In this paper, we establish some fixed point results for F⊥-weak contraction in orthogonal metric space and we give an application for the solution of second order differential equation.

Dirichlet problem for a second-order ordinary differential equation with a distributed differentiation operator

For an ordinary second-order differential equation with an operator of continuously distributed differentiation with variable coefficients, a solution to the Dirichlet problem is constructed using the Green’s function method.

Some new results on the stability and boundedness of solutions of certain class of second order vector differential equations

n this paper, we provide certain conditions that guarantee the stability of the zero solution when P(t, X, Y) = 0 and boundedness of all solutions when P(t, X, Y)# 0 of a certain system of second order differential equation using a suitable Lyapunov function. The results in this paper are quite new and complement those in the literature. Examples are given to demonstrate the correctness of the established results.

Generalized boundary value problem for a second order differential equation with fractional derivative

A generalized boundary value problem for a second-order differential equation with a fractional derivative is solved. An explicit representation of the solution of the problem under study is constructed, a condition for unique solvability is found, and a uniqueness theorem for the solution is proved.