On a Neutral Itô and Arbitrary (Fractional) Orders Stochastic Differential Equation with Nonlocal Condition

In this paper, we are concerned with the combinations of the stochastic Itô-differential and the arbitrary (fractional) orders derivatives in a neutral differential equation with a stochastic, nonlinear, nonlocal integral condition. The existence of solutions will be proved. The sufficient conditions for the uniqueness of the solution will be given. The continuous dependence of the unique solution will be studied.

On a Boundary Value Problem of Hybrid Functional Differential Inclusion with Nonlocal Integral Condition

In this work, we present a boundary value problem of hybrid functional differential inclusion with nonlocal condition. The boundary conditions of integral and infinite points will be deduced. The existence of solutions and its maximal and minimal will be proved. A sufficient condition for uniqueness of the solution is given. The continuous dependence of the unique solution will be studied.

On a Coupled System of Random and Stochastic Nonlinear Differential Equations with Coupled Nonlocal Random and Stochastic Nonlinear Integral Conditions

It is well known that Stochastic equations had many useful applications in describing numerous events and problems of real world, and the nonlocal integral condition is important in physics, finance and engineering. Here we are concerned with two problems of a coupled system of random and stochastic nonlinear differential equations with two coupled systems of nonlinear nonlocal random and stochastic integral conditions. The existence of solutions will be studied. The sufficient condition for the uniqueness of the solution will be given. The continuous dependence of the unique solution on the nonlocal conditions will be proved.

On a time fractional diffusion with nonlocal in time conditions

In this work, we consider a fractional diffusion equation with nonlocal integral condition. We give a form of the mild solution under the expression of Fourier series which contains some Mittag-Leffler functions. We present two new results. Firstly, we show the well-posedness and regularity for our problem. Secondly, we show the ill-posedness of our problem in the sense of Hadamard. Using the Fourier truncation method, we construct a regularized solution and present the convergence rate between the regularized and exact solutions.

On a Kirchhoff diffusion equation with integral condition

This paper is devoted to Kirchhoff-type parabolic problem with nonlocal integral condition. Our problem has many applications in modeling physical and biological phenomena. The first part of our paper concerns the local existence of the mild solution in Hilbert scales. Our results can be studied into two cases: homogeneous case and inhomogeneous case. In order to overcome difficulties, we applied Banach fixed point theorem and some new techniques on Sobolev spaces. The second part of the paper is to derive the ill-posedness of the mild solution in the sense of Hadamard.

INITIAL-BOUNDARY VALUE PROBLEM FOR ONE-DIMENSION HYPERBOLIC EQUATION WITH INTEGRAL BOUNDARY CONDITION

In this paper, we study an initial-boundary value problem with nonlocal integral condition for a hyperbolic equation. The existence and uniqueness of a generalized solution of the problem is proved.

PROBLEM WITH NONLOCAL INTEGRAL CONDITION FOR PSEUDOHYPERBOLIC EQUATION OF THE FOURTH-ORDER

In this article, we study a problem for a multimensional pseudohyperbolic equation of the fourth-order with an integral condition. Existence and uniqueness of a generalized solution is proved.

A PROBLEM WITH SECOND KIND INTEGRAL CONDITIONS FOR HYPERBOLIC EQUATION

In this paper, we consider a problem for one-dimensional hyperbolic equation with second kind integral conditions and prove unique solvability. To prove this statement we suggest a new approach. The main idea of it is that given nonlocal integral condition is equivalent with a different condition, nonlocal as well but this new condition enables us to introduce a definition of a generalized solution bazed on an integral identity and derive a priori estimates of a required solution in Sobolev space. This approach shows that integral conditions are closely connected with dynamical conditions.