Evolution Equations Of Sobolev Type Research Articles

Controllability of a Class of Impulsive ψ-Caputo Fractional Evolution Equations of Sobolev Type

In this paper, we investigate the controllability of a class of impulsive ψ-Caputo fractional evolution equations of Sobolev type in Banach spaces. Sufficient conditions are presented by two new characteristic solution operators, fractional calculus, and Schauder fixed point theorem. Our works are generalizations and continuations of the recent results about controllability of a class of impulsive ψ-Caputo fractional evolution equations. Finally, an example is given to illustrate the effectiveness of the main results.

Existence Results of Mild Solutions for the Fractional Stochastic Evolution Equations of Sobolev Type

In this paper, by utilizing the resolvent operator theory, the stochastic analysis method and Picard type iterative technique, we first investigate the existence as well as the uniqueness of mild solutions for a class of α ∈ ( 1 , 2 ) -order Riemann–Liouville fractional stochastic evolution equations of Sobolev type in abstract spaces. Then the symmetrical technique is used to deal with the α ∈ ( 1 , 2 ) -order Caputo fractional stochastic evolution equations of Sobolev type in abstract spaces. Two examples are given as applications to the obtained results.

Approximate controllability of nonlocal non-autonomous Sobolev type evolution equations

The aim of this article is to investigate the existence of mild solutions as well as approximate controllability of non-autonomous Sobolev type differential equations with the nonlocal condition. To prove our results, we will take the help of Krasnoselskii fixed point technique, evolution system and controllability of the corresponding linear system.

Fractional differential equations of Sobolev type with sectorial operators

This paper treats the asymptotic behavior of resolvent operators of Sobolev type and its applications to the existence and uniqueness of mild solutions to fractional functional evolution equations of Sobolev type in Banach spaces. We first study the asymptotic decay of some resolvent operators (also called solution operators) and next, by using fixed point results, we obtain the existence and uniqueness of solutions to a class of Sobolev type fractional differential equation. We notice that, the existence or compactness of an operator $$E^{-1}$$ is not necessarily needed in our results.

Approximate Controllability Results for Second Order Neutral Impulsive Stochastic Evolution Equations of Sobolev Type with Unbounded Delay

Approximate Controllability Results for Second Order Neutral Impulsive Stochastic Evolution Equations of Sobolev Type with Unbounded Delay

Fully discrete approximation of general nonlinear Sobolev equations

We consider abstract quasilinear evolution equations of Sobolev type in a Hilbert setting. We propose two fully discrete schemes and prove some error estimates under minimal assumptions. Various examples that enter into our abstract framework are considered, for each of them our theoretical results are confirmed by several numerical experiments.

Existence and Optimal Controls for Fractional Stochastic Evolution Equations of Sobolev Type Via Fractional Resolvent Operators

This paper is mainly concerned with controlled stochastic evolution equations of Sobolev type for the Caputo and Riemann–Liouville fractional derivatives. Some sufficient conditions are established for the existence of mild solutions and optimal state-control pairs of the limited Lagrange optimal systems. The main results are investigated by compactness of fractional resolvent operator family, and the optimal control results are derived without uniqueness of solutions for controlled evolution equations.

Existence of mild solutions for fractional nonautonomous evolution equations of Sobolev type with delay

In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair (A,B) and probability density function. Combining the techniques of fractional calculus, measure of noncompactness, and fixed point theorem with respect to k-set-contractive, we obtain a new existence result of mild solutions. The results obtained improve and extend some related conclusions on this topic. At last, we present an application that illustrates the abstract results.

The Bounded Solutions on a Semiaxis for the Linearized Hoff Equation in Quasi-Sobolev Spaces

In this paper we investigate the properties of the linearized Hoff equation in quasi-Sobolev spaces. Hoff equation, specified on the interval, describes the buckling of the H-beam. Due to the fact that for certain values of the parameters in the equation may be missing the derivative with respect to time, this equation refers in a frame of class of nonclassical equations of mathematical physics. The article by relatively spectral theorem describes the morphology of the phase space and the existence of invariant spaces of solutions. Using these results, we prove the existence of bounded on the semiaxis solutions for homogeneous evolution equations of Sobolev type in quasi-Sobolev spaces. Apart of the introduction and bibliography the article contains three parts. The first one shows the results on the solvability of the investigated class of equations. The second part shows the existence of bounded on the semiaxis solutions for the homogeneous equations of the research class. Finally, the third part presents the results of the existence of solutions bounded on the semiaxis for analog linearized Hoff equation in quasi-Sobolev spaces.

Sobolev type fractional stochastic integro-differential evolution

In this paper, we prove the existence of $\alpha$-mild solutions for a class of fractional stochastic integrodifferential evolution equations of sobolev type with fractional sobolev stochastic nonlocal conditions in a real separable Hilbert space. To establish our main results, we use the Banach contraction mapping principle, fractional calculus, stochastic analysis and an analytic semigroup of linear operators. An example is given to illustrate the feasibility of our abstract result.

Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators

The paper is concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach spaces. With the help of two new characteristic solution operators and their properties, such as boundedness and compactness, we present the controllability results corresponding to two admissible control sets via the well-known Schauder fixed point theorem. Finally, an example is given to illustrate our theoretical results.

Difference schemes for the singularly perturbed Sobolev periodic boundary problem

A finite-difference method is suggested and analyzed for evolution equations of Sobolev type in a single space variable and with boundary layers. The convergence and error estimates for an exponentially fitted difference scheme in an equidistant mesh are obtained. The numerical methods discussed here are extremely robust, in the sense that they have good convergence properties not only for small values of the perturbation parameter, but also for moderate and large values.

On Some Evolution Inequalities in a Hilbert Space

In the present paper we establish some evolution inequalities related to Sobolev type evolution equations. The inequalities established here can be used to study the uniqueness and asymptotic growth of solutions of the corresponding evolution inequalities in a Hilbert space.

Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension

A semidiscrete Galerkin finite element method is defined and analyzed for nonlinear evolution equations of Sobolev type in a single space variable. Optimal orderL p error estimates are derived for 2?p??. And it is shown that the rates of convergence of the approximate solution and its derivative are one order better than the optimal order at certain spatial Jacobi and Gauss points, respectively. Also the standard nodal superconvergence results are established. Futher, it is considered that an a posteriori procedure provides superconvergent approximations at the knots for the spatial derivatives of the exact solution.

Approximation of point controls of second-order evolution equations of Sobolev type

Limit behavior of the optimal-point control associated with a Sobolev equation as inertial or viscous effect is allowed to vanish is studied. These results are also related to the approximation of a wave equation.