Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

On a general degenerate/singular parabolic equation with a nonlocal space term

In this paper we study the null controllability for the problems associated to the operators y_t-Ay - \lambda/b(x) y+\int_0^1 K(t,x,\tau)y(t, \tau) d\tau, (t,x) \in (0,T)\times (0,1) where Ay := ay_{xx} or Ay := (ay_x)_x and the functions a and b degenerate at an interior point x0 Ë .0; 1/. To this aim, as a first step we study the well posedness, the Carleman estimates and the null controllability for the associated nonhomogeneous degenerate and singular heat equations. Then,using the Kakutani’s fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.

On the Nonlocal Problems in Time for Time-fractional Subdiffusion Equations

The nonlocal boundary value problem, dt&rho;u(t)+Au(t)=f(t) (0&amp;lt;&rho;&amp;lt;1, 0&amp;lt;t&le;T), u(&xi;)=&alpha;u(0)+&phi; (&alpha; is a constant and 0&amp;lt;&xi;&le;T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann-Liouville derivative; naturally, in the case of the Riemann - Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant &alpha; on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function &phi; in the boundary conditions are investigated.

On the role of numerical viscosity in the study of the local limit of nonlocal conservation laws

We deal with the numerical investigation of the local limit of nonlocal conservation laws. Previous numerical experiments seem to suggest that the solutions of the nonlocal problems converge to the entropy admissible solution of the conservation law in the singular local limit. However, recent analytical results state that (i) in general convergence does not hold because one can exhibit counterexamples; (ii)~convergence can be recovered provided viscosity is added to both the local and the nonlocal equations. Motivated by these analytical results, we investigate the role of numerical viscosity in the numerical study of the local limit of nonlocal conservation laws. In particular, we show that Lax-Friedrichs type schemes may provide the wrong intuition and erroneously suggest that the solutions of the nonlocal problems converge to the entropy admissible solution of the conservation law in cases where this is ruled out by analytical results. We also test Godunov type schemes, less affected by numerical viscosity, and show that in some cases they provide an intuition more in accordance with the analytical results.

On a Coupled System of Stochastic Ito^-Differential and the Arbitrary (Fractional) Order Differential Equations with Nonlocal Random and Stochastic Integral Conditions

The fractional stochastic differential equations had many applications in interpreting many events and phenomena of life, and the nonlocal conditions describe numerous problems in physics and finance. Here, we are concerned with the combination between the three senses of derivatives, the stochastic Ito^-differential and the fractional and integer orders derivative for the second order stochastic process in two nonlocal problems of a coupled system of two random and stochastic differential equations with two nonlocal stochastic and random integral conditions and a coupled system of two stochastic and random integral conditions. We study the existence of mean square continuous solutions of these two nonlocal problems by using the Schauder fixed point theorem. We discuss the sufficient conditions and the continuous dependence for the unique solution.