Optimal control of impulsive systems with nonlocal boundary conditions

In this paper we study an optimal control problem, where states of a control system are described by impulsive differential equations with nonlocal boundary conditions. With the help of the contraction principle we prove the existence and uniqueness of a solution to the corresponding boundary value problem with fixed admissible controls. We calculate the first and second variation of the functional. Using the variation of controls, we establish various necessary optimality conditions of the second order.     

Existence of Global Solutions for a Class of Abstract Second-Order Nonlocal Cauchy Problem with Impulsive Conditions in Banach Spaces

In this paper, we are concerned with a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. First, we study the existence of mild solutions for a class of second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces on an interval [0,a]. Later, we study a couple of cases where we can establish the existence of global solutions for a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. We apply our theory to study the existence of solutions for impulsive partial differential equations.     

Nonlinear impulsive Langevin equation with mixed derivatives

In this paper, we consider a nonlocal boundary value problem of nonlinear impulsive Langevin equation with mixed derivatives. Some sufficient conditions are constructed to observe the existence, uniqueness, and generalized Ulam-Hyers-Rassias stability of our proposed model, with the help of Diaz-Margolis' fixed-point approach over generalized complete metric space. We give an example that supports our main result.     

Mathematical analysis of impulsive fractional differential inclusion of pantograph type

In this article, we formulate fractional differential inclusion of pantograph type (IFDIP), incorporating impulsive behavior of the solution. The boundary conditions taken into account are nonlocal in nature. We will consider the convex problem and prove the Filippov–Wazewski-type theorem. Moreover, existence of solution, uniqueness of a solution, and the topological properties of the solution's set will be examined for the problem under consideration. In the second part, the study will be confined to the second-order impulsive fractional differential equation of pantograph type. For certain geometric characteristics of the solution's set, Aronszajn–Browder–Gupta-type results will be explored for the newly introduced differential equation. Also, it will prove the existence of solution for the first-order fractional differential equation of pantograph type having impulsive behavior of the solution.     

A general study on random integro-differential equations of arbitrary order

Here the broad study is depending on random integro-differential equations (RIDE) of arbitrary order. The fractional order is in terms of $\psi$-Hilfer fractional operator. This work reveals the dynamical behaviour such as existence, uniqueness and stability solutions for RIDE involving fractional order. Thus initial value problem (IVP), boundary value problem (BVP), impulsive effect and nonlocal conditions are taken in account to prove the results.     

On a mixed problem with integral conditions for one-dimensional parabolic equation

We consider an initial-boundary value problem for a one-dimensional parabolic equation with nonlocal boundary conditions. These nonlocal conditions are given in terms of integrals. Based on solution of the Dirichlet problem for the parabolic equation, we constructively establish the well-posedness for the nonlocal problem.     

Generalized Cauchy problems involving nonlocal and impulsive conditions

We study the abstract Cauchy problem involving a class of nonlinear differential inclusions, with impulsive and nonlocal conditions. By using MNC estimates, the existence result and continuous dependence on initial data of the solution set are proved.     

On well-posedness of generalized neural field equations with impulsive control

We consider nonlinear nonlocal integral equation generalizing equations typically used in mathematical neuroscience. We investigate solutions tending to zero at any fixed moment with unbounded growth of the spatial variable (these solutions correspond to normal brain functioning). We consider an impulsive control problem, which models electrical stimulation used in the presence of various diseases of central nervous system. We define suitable complete metric space, where we obtain conditions for existence, uniqueness and extendability of solution to the problem as well as continuous dependence of this solution on the impulsive control.     

非局部条件下的脉冲中立型泛函微分方程

利用Sadovskii不动点定理研究了一类脉冲中立型泛函微分方程，证明了适度解的存在性．最后，给出了上述问题在偏微分方程方面的一个应用．     

On the unique solvability of a nonlocal boundary value problem with data on intersecting lines for systems of hyperbolic equations

We consider a nonlocal boundary value problem for a system of hyperbolic equations with two independent variables with data on intersecting lines one of which is a characteristic. In terms of the data of the nonlocal boundary value problem, we obtain sufficient coefficient conditions for its unique solvability.     

A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions

We study a nonlocal boundary value problem of Hadamard type coupled sequential fractional differential equations supplemented with coupled strip conditions (nonlocal Riemann-Liouville integral boundary conditions). The nonlinearities in the coupled system of equations depend on the unknown functions as well as their lower order fractional derivatives. We apply Leray-Schauder alternative and Banach’s contraction mapping principle to obtain the existence and uniqueness results for the given problem. An illustrative example is also discussed.     

Locally one-dimensional difference scheme for a pseudoparabolic equation with nonlocal conditions

We study a two-dimensional linear pseudoparabolic equation with nonlocal integral boundary conditions in one coordinate direction and use a locally one-dimensional method for solving this problem. We prove the stability of a finite-difference scheme based on the structure of spectrum of the difference operator with nonlocal conditions.     

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions

We present a model for nonlocal diffusion with Dirichlet boundary conditions in a bounded smooth domain. We prove that solutions of properly rescaled nonlocal problems approximate uniformly the solution of the corresponding Dirichlet problem for the classical heat equation.     

On Global Existence of Solutions of Initial Boundary Value Problem for a System of Semilinear Parabolic Equations with Nonlinear Nonlocal Neumann Boundary Conditions

We establish conditions for the existence and nonexistence of global solutions of initial boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal Neumann boundary conditions. We show that these conditions are determined by the behavior of the problem coefficients as t→∞.     

Nonlocal problem for complete second-order hyperbolic operator-differential equations with variable domains

We prove the well-posed solvability (in the strong sense) of complete second-order hyperbolic operator-differential equations with variable domains of unbounded operator coefficients under nonlocal initial conditions. We are the first to establish the well-posed solvability of the mixed problem for the complete string vibration equation with nonstationary boundary conditions and nonlocal initial conditions.     

Green’s functions for stationary problems with nonlocal boundary conditions

In this paper, we investigate Green’s functions for various stationary problems with nonlocal boundary conditions. We express the Green’s function per Green’s function for a problem with classical boundary conditions. This property is illustrated by various examples. Properties of Green’s functions with nonlocal boundary conditions are compared with those for classical problems. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-73/09.     

Existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions

In this paper, we consider the existence of solutions for a class of nonlinear impulsive problems with Dirichlet boundary conditions. We obtain some new existence theorems of solutions for the nonlinear impulsive problem by using critical point theory. We extend and improve some recent results.     

Simultaneous Determination of Two Coefficients of a Parabolic Equation in the Case of Nonlocal and Integral Conditions

We establish conditions for the existence and uniqueness of a solution of the inverse problem for a parabolic equation with two unknown time-dependent coefficients in the case of nonlocal boundary conditions and integral overdetermination conditions.     

Mixed Problem with an Integral Condition for the One-Dimensional Biwave Equation

We consider a mixed problem for the one-dimensional biwave equation with boundary conditions and a nonlocal integral condition. We prove the existence and uniqueness of the classical solution of the problem and obtain an analytic representation of the solution.