Differential Equations Of Sobolev Type Research Articles

Study of Uniqueness and Ulam-Type Stability of Abstract Hadamard Fractional Differential Equations of Sobolev Type via Resolvent Operators

This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach’s fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam’s stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives.

Inverse problem for an abstract neutral differential equation of Sobolev-type

In this manuscript, we consider an inverse problem for the neutral differential equation of Sobolev type with a deviated argument and identify the parameter using an overdetermined condition on a mild solution. A direct approach using Volterra integral equations for sufficiently regular data and an optimal control approach for less regular data are the main techniques to find the result. Under certain hypotheses, the characterization of the limit of the sequence of approximate solutions demonstrates that it is a solution to the original inverse problem under specific hypotheses. At last, an example is provided in the support of our results.

An analysis of approximate controllability for Hilfer fractional delay differential equations of Sobolev type without uniqueness

This study focused on the approximate controllability results for the Hilfer fractional delay evolution equations of the Sobolev type without uniqueness. Initially, the Lipschitz condition is derived from the hypothesis, which is represented by a measure of noncompactness, in particular, nonlinearity. We also examined the continuity of the solution map of the Sobolev type of Hilfer fractional delay evolution equation and the topological structure of the solution set. Furthermore, we prove the approximate controllability of the fractional evolution equation of the Sobolev type with delay. Finally, we provided an example to illustrate the theoretical results.

Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses

<abstract><p>This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion.</p></abstract>

Solvability to an initial-periodic problem for delay partial differential equations of Sobolev type

The initial-boundary value problem for higher-order delay partial differential equations is considered. We study the existence of its solution and also propose a method for finding approximate solutions. We are established a sufficient conditions for the existence and uniqueness of the solution to the problem under consideration. Introducing new unknown functions, we reduce the considered problem to an equivalent problem consisting of a family of problems for delay differential equation with functional parameters and integral relations. An algorithm for finding an approximate solution to the problem under study is proposed and its convergence is proved. Sufficient conditions for the existence of a unique solution to an equivalent problem with parameters are established. The conditions for the unique solvability of the initial-boundary value problem for delay partial differential equations are obtained in terms of the initial data. Unique solvability to the initial-boundary value problem for higher-order delay partial differential equations is interconnected with unique solvability to the family of problems with parameters for delay differential equations.

Results on neutral differential equation of sobolev type with nonlocal conditions

In this work, we analyse the study of neutral fractional differential equation in an arbitrary Hilbert space. An associated integral equation is studied and approximate integral equation is obtained. We demonstrate the existence and uniqueness of an approximate solution by using analytic semigroup theory and the Fixed point method. In the application part, we discuss the approximation and the convergence results for such an approximation.

Sobolev-type nonlinear Hilfer fractional stochastic differential equations with noninstantaneous impulsive

<abstract><p>The existence of a mild solution for nonlinear Hilfer fractional stochastic differential equations of the Sobolev type with non-instantaneous impulse in Hilbert space is investigated in this study. For nonlinear Hilfer fractional stochastic differential equations of Sobolev type with non-instantaneous impulsive conditions, sufficient criteria for controllability are established. Finally, an illustration of the acquired results is shown.</p></abstract>

Well-posedness and ill-posedness of boundary-value problems for one class of fourth-order differential equations of Sobolev type

Well-posedness and ill-posedness of boundary-value problems for one class of fourth-order differential equations of Sobolev type

The Cauchy Problem for the Nonlinear Long Longitudinal Waves Equation in a Viscoelastic Rod

We study the Cauchy problem in the space of continuous functions for some nonlinear differential equation of Sobolev type that simulates longitudinal waves in an infinite viscoelastic rod. Under consideration are the conditions for the existence of the global classical solution and the blow-up of the solution to the Cauchy problem on a finite time interval.

Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces

<p style='text-indent:20px;'>In this paper, we investigate the approximate controllability problems of certain Sobolev type differential equations. Here, we obtain sufficient conditions for the approximate controllability of a semilinear Sobolev type evolution system in Banach spaces. In order to establish the approximate controllability results of such a system, we have employed the resolvent operator condition and Schauder's fixed point theorem. Finally, we discuss a concrete example to illustrate the efficiency of the results obtained.</p>

Subordination principle for fractional diffusion-wave equations of Sobolev type

Abstract In this paper we study subordination principles for fractional differential equations of Sobolev type in Banach space. With the help of the theory of Sobolev type resolvent families (known also as propagation family) as well as these subordination principles, we obtain the existence of mild solutions for this kind of equations. We study simultaneously the case 0 < α < 1 and 1 < α < 2 for the Caputo and Riemann-Liouville fractional derivatives.

Solvability of Nonlocal Problems for Systems of Sobolev-Type Differential Equations with a Multipoint Condition

We consider a nonlocal problem for a system of loaded differential equations of the Sobolev type with a multipoint constraint. By introducing additional unknown functions, we reduce the problem under consideration to an equivalent problem consisting of a nonlocal multipoint problem for a system of loaded hyperbolic equations of the second order with functional parameters and integral correlations. We propose algorithms for solving the equivalent problem. Moreover, we establish conditions for the well-posedness of the nonlocal multipoint problem for the system of loaded hyperbolic equations of the second order and conditions for the existence of a unique classical solution to the nonlocal problem for the system of differential equations of the Sobolev type with a multipoint constraint.

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.

On the solvability of a nonlocal problem for the system of Sobolev-type differential equations with integral condition

Abstract Sufficient conditions for the existence and uniqueness of a classical solution to a nonlocal problem for a system of Sobolev-type differential equations with integral condition are established. By introducing a new unknown function, we reduce the considered problem to an equivalent problem consisting of a nonlocal problem for the system of hyperbolic equations of second order with a functional parameter and an integral relation. We propose the algorithm for finding an approximate solution to the investigated problem and prove its convergence.

О разрешимости нелокальной задачи для системы дифференциальных уравнений соболевского типа с многоточечным условием

О разрешимости нелокальной задачи для системы дифференциальных уравнений соболевского типа с многоточечным условием

BOUNDARY VALUE PROBLEMS FOR COMPOSITE TYPE EQUATIONS WITH A QUASIPARABOLIC OPERATOR IN THE LEADING PART HAVING THE VARIABLE DIRECTION OF EVOLUTION AND DISCONTINUOUS COEFFICIENTS

It is studied the solvability of boundary value problems for non-classical differential equations ofSobolev type with an alternating function, which has a discontinuity of the first kind at the point zero.Also, this function changes sign depending on the sign of the variable x. It is proved the existenceand uniqueness theorems for regular solutions, which has all generalizated derivatives including in thisequation. Presence of necessary a priori estimates for the solutions of the problems under study.

Existence results for Hilfer fractional evolution equations with boundary conditions

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of properties of Hilfer fractional calculus, the theory of propagation family as well as the theory of the measure of noncompactness and the fixed point methods, we obtain the existence results of mild solutions for Sobolev type fractional evolution differential equations involving Hilfer fractional derivative. Finally, two examples are presented to illustrate the main result.

Study on the mild solution of Sobolev type Hilfer fractional evolution equations with boundary conditions

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of properties of Hilfer fractional calculus, the theory of propagation family as well as the theory of the measure of noncompactness and the fixed point methods, we obtain the existence results of mild solutions for Sobolev type fractional evolution differential equations involving Hilfer fractional derivative. Finally, two examples are presented to illustrate the main result.

Existence of mild solutions for Sobolev-type Hilfer fractional evolution equations with boundary conditions

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of the properties of Hilfer fractional calculus, the theory of propagation families as well as the theory of the measure of noncompactness and fixed point methods, we obtain the existence results of mild solutions for Sobolev-type fractional evolution differential equations involving the Hilfer fractional derivative. Finally, an example is presented to illustrate the main result.