On the nonlocal Cauchy problem for semilinear fractional order evolution equations

In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.     

On a nonlocal problem for nonlinear pseudoparabolic equations

For a nonlinear pseudoparabolic equation with one space dimension we consider its initial boundary value problem on an interval. The boundary condition on the left end is of Dirichlet type, the right end condition is replaced by a nonlocal one. Because it is given by an integral, the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet counterpart. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in a weighted Sobolev space.     

On a problem for operator-differential second-order equations with nonlocal boundary condition

An operator-differential second-order equation with nonlocal boundary condition at zero is considered on the semiaxis. Here we give sufficient conditions on the operator coefficients for the regular solvability of the boundary-value problem. Moreover, we obtain conditions for the completeness andminimality of the derivative of the chain of eigen- and associated vectors generated by the boundary-value problem under study and establish the completeness and minimality of the decreasing elementary solutions of the operator-differential equation under consideration.     

On some nonlocal boundary value problems for evolution equations

In the Sobolev-Besov spaces, we examine the question on solvability of nonlocal boundary value problems for operator-differential equations of the form

Existence-uniqueness and long time behavior for a class of nonlocal nonlinear parabolic evolution equations

We establish existence and uniqueness of solutions for a general class of nonlocal nonlinear evolution equations. An application of this theory to a class of nonlinear reaction-diffusion problems that arise in population dynamics is presented. Furthermore, conditions on the initial population density for this class of problems that result in finite time extinction or persistence of the population is discussed. Numerical evidence corroborating our theoretical results is given.

     

On some nonlocal evolution equations in Banach spaces

This note is concerned with the initial value problem for the abstract nonlocal equation where A is a maximal monotone operator from a reflexive Banach space E to its dual E^*, while B is a nonlocal maximal monotone operator from . Under proper boundedness and coercivity assumptions on the operators, a solution is achieved by means of a discretization argument. Uniqueness and continuous dependence are also discussed and we prove some estimates for the discretization error. Finally, we deal with the approximation of linear Volterra integrodifferential operators.     

A problem nonlocal in time for the equations of the dynamics of a barotropic ocean

The research was supported by the Russian Foundation for Fundamental Research (Grant 93-013-16384) and the International Science Foundation (Grant NR 5000).     

Perturbation method for nonlocal impulsive evolution equations

This paper deals with the existence of mild solutions for a class of semilinear nonlocal impulsive evolution equations in ordered Banach spaces. The existence and uniqueness theorem of mild solution for the associated linear nonlocal impulsive evolution equation is established. With the aid of the theorem, the existence of mild solutions for nonlinear nonlocal impulsive evolution equation is obtained by using perturbation method and monotone iterative technique. The theorems proved in this paper improve and extend some related results in ordinary differential equations and partial differential equations. Moreover, we present two examples to illustrate the feasibility of our abstract results.     

A problem with a nonlocal,with respect to time,condition for multidimensional hyperbolic equations

We study a boundary-value problem for a hyperbolic equation with a nonlocal with respect to time-variable integral condition. We obtain sufficient conditions for unique solvability of the nonlocal problem. The proof is based on reduction of the nonlocal first-type condition to the second-type one. This allows to reduce the nonlocal problem to an operator equation. We show that unique solvability of the operator equation implies the existence of a unique solution to the problem.     

Existence results for a class of semilinear nonlocal evolution equations

In this paper, we study a class of semilinear evolution equations with nonlocal initial conditions and give some new results on the existence of mild solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

On blow-up results for solutions of inhomogeneous evolution equations and inequalities

The main purpose of this work is to obtain blow-up results for solutions of the inequality

     

On a nonlocal problem for a parabolic equation

We study a nonlocal boundary-value problem for a parabolic equation in a two-dimensional domain, establish ana priori estimate in the energy norm, prove the existence and uniqueness of a generalized solution from the classW ₂ ^1,0 (Q _T ), and construct a difference scheme for the second-order approximation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 790–800, June, 1995.     

An inhomogeneous nonlocal diffusion problem with unbounded steps

We consider the following nonlocal equation

$$\int J\left(\frac{x-y}{g(y)}\right) \frac{u(y)}{g(y)} dy -u(x)=0\qquad x\in \mathbb{R},$$

where J is an even, compactly supported, Hölder continuous kernel with unit integral and g is a continuous positive function. Our main concern will be with unbounded functions g, contrary to previous works. More precisely, we study the influence of the growth of g at infinity on the integrability of positive solutions of this equation, therefore determining the asymptotic behavior as \({t\to +\infty}\) of the solutions to the associated evolution problem in terms of the growth of g.

     

On solutions of neutral nonlocal evolution equations with nondense domain

In this paper, with Sadovskii's and Banach's fixed point theorems applied, we establish some results on the existence of integral solutions, strong solutions, and strict solutions for a class of nondensely defined neutral evolution equations with nonlocal conditions. Also, an example is given in the end to show the applications of the obtained results.     

The Neumann problem for nonlocal nonlinear diffusion equations

We study nonlocal diffusion models of the form

A problem with nonlocal conditions for mixed-type equations

We study a problem with the Bitsadze-Samarskii conditions on the ellipticity boundary and on a segment of degeneration line and with the displacement condition on parts of boundary characteristics of the Gellerstedt equation with singular coefficient. With the help of the maximum principle we prove the uniqueness of a solution to the problem, and with the help of the method of integral equations we prove the existence of a solution to the problem.     

An inverse problem for a family of two parameters time fractional diffusion equations with nonlocal boundary conditions

The determination of a space‐dependent source term along with the solution for a 1‐dimensional time fractional diffusion equation with nonlocal boundary conditions involving a parameter β>0 is considered. The fractional derivative is generalization of the Riemann‐Liouville and Caputo fractional derivatives usually known as Hilfer fractional derivative. We proved existence and uniqueness results for the solution of the inverse problem while over‐specified datum at 2 different time is given. The over‐specified datum at 2 time allows us to avoid initial condition in terms of fractional integral associated with Hilfer fractional derivative.     

On a singular nonlocal time fractional order mixed problem with a memory term

We use the priori estimate method to prove the existence and uniqueness of a solution as well as its dependence on the given data of a singular time fractional mixed problem having a memory term. The considered fractional equation is associated with a nonlocal condition of integral type and a Neuman condition. Our results develop and show the efficiency and effectiveness of the energy inequalities method for the time fractional order differential equations with a nonlocal condition.     

On a certain nonlocal problem for a hyperbolic equation

The author proves the existence and uniqueness of a generalized solution of a nonlocal problem with an integral condition for a hyperbolic equation with n spatial variables. This work is a continuation of the studies started in [3–5], where the solvability problem of nonlocal problems with an integral condition was studied for hyperbolic equations on the plane. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.