Existence Results Of Mild Solutions Research Articles

Feedback control systems governed by evolution equations

ABSTRACTThe goal of this paper is to provide systematic approaches to study the feedback control systems governed by evolution equations in separable reflexive Banach spaces. We firstly give some existence results of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray–Schauder alternative fixed point theorem with Lipschitz conditions and some types of boundedness conditions. Next, by using the Filippove theorem and the Cesari property, a new set of sufficient assumptions are formulated to guarantee the existence of feasible pairs for the feedback control systems. Some existence results for an optimal control problem are given. Finally, we apply our main result to obtain a controllability result for semilinear evolution equations and existence results for a class of differential variational inequalities and Clarke's subdifferential inclusions.

Existence results for impulsive feedback control systems

The goal of this paper is to provide systematic approaches to study the feedback control systems governed by impulsive evolution equations in separable reflexive Banach spaces. We firstly give some existence results of mild solutions for the equations by applying the Banach’s fixed point theorem and the Leray–Schauder alternative fixed point theorem with Lipschitz conditions and different types of boundedness conditions. Then, by using the Filippove theorem and the Cesari property, a new set of sufficient assumptions are formulated to guarantee the existence results of feasible pairs for the feedback control systems. Finally, we apply our main result to obtain a controllability result for impulsive evolution equations and two existence results for a class of impulsive differential variational inequalities and impulsive Clarke’s subdifferential inclusions.

Existence of mild solutions for impulsive fractional evolution equations with periodic boundary conditions

In this paper, we concern on the periodic boundary value problem for a class of semilinear impulsive fractional evolution equations in an ordered Banach space E. First, we establish the existence results of mild solutions for the associated linear periodic boundary value problem. Next, we obtain the existence results of mild solutions by using the monotone iterative technique with L-quasi-upper and lower solutions, the results are new and extend some previously known results. Finally, two examples are also given to illustrate the main results.

Fractional non-autonomous evolution equation with nonlocal conditions

The aim of this paper is to discuss the existence of mild solutions for a class of time fractional non-autonomous evolution equations with nonlocal conditions and measure of noncompactness in infinite-dimensional Banach spaces. Combining the theory of fractional calculus and evolution families, the fixed point theorem with respect to k-set-contractive operator and a new estimation technique of the measure of noncompactness, we obtain the new existence results of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and noncompactness measure conditions. Our results generalize and improve some previous results on this topic by deleting the compactness condition on nonlocal function g and extending the study of fractional autonomous evolution equations in recent years to non-autonomous ones. Finally, as samples of applications, we consider a time fractional non-autonomous partial differential equation with homogeneous Dirichlet boundary condition and nonlocal conditions.

Existence of Mild Solutions for Sobolev-Type Hilfer Fractional Nonautonomous Evolution Equations with Delay

Abstract This paper treats the existence of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations with delay in Banach spaces. We first characterize the definition of mild solutions for the studied problem which was given based on an operator family generated by the operator pair (A,B) and probability density function. And then via Hilfer fractional derivative and combining the techniques of fractional calculus, measure of noncompactness and Sadovskii fixed-point theorem, we obtain new existence result of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations. Particularly, the existence or compactness of an operator B − 1 $B^{-1} $ is not necessarily needed in our results. Furthermore, our results obtained improve and extend some related conclusions on this topic. At last, an example is given to illustrate our main results.

Existence Results of Mild Solutions for Impulsive Fractional Integrodifferential Evolution Equations With Nonlocal Conditions

Abstract In this paper, we investigate the existence of mild solutions of impulsive fractional integrodifferential evolution equations with nonlocal conditions via the fixed point theorems and fractional cosine family combined with solutions operator theorems. Our results improve and generalize some classical results. Finally, an example is given to illustrate the main results.

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.

Study on Sobolev type Hilfer fractional integro-differential equations with delay

In this paper, we deal with a class of nonlinear Sobolev type fractional integro-differential equations with delay using Hilfer fractional derivative, which generalized the famous Riemann–Liouville fractional derivative. The definition of mild solutions for studied problem was given based on an operator family generated by the operator pair (A, B) and probability density function. Combining with the techniques of fractional calculus, measure of noncompactness and fixed point theorem, we obtain new existence result of mild solutions with two new characteristic solution operators and the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition. The results obtained improve and extend some related conclusions on this topic. At last, an example is given to illustrate our main results.

Study a class of nonlinear fractional non-autonomous evolution equations with delay

In this paper, we deal with a class of nonlinear fractional non-autonomous evolution equations with delay by using Hilfer fractional derivative, which generalized the famous Riemann–Liouville fractional derivative. Combining techniques of fractional calculus, measure of noncompactness and some fixed point theorem, we obtain new existence result of mild solutions when the associated semigroup is not compact. Furthermore, the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition. The results obtained improve and extend some related conclusions. Finally, two examples will be presented to illustrate the main results.

Solvability and Optimal Controls of a Fractional Impulsive Stochastic Partial Integro-Differential Equation with State-Dependent Delay

In this paper, a new class of fractional impulsive stochastic partial integro-differential control systems with state-dependent delay and their optimal controls in a Hilbert space is studied. We firstly prove an existence result of mild solutions for the control systems by using stochastic analysis, analytic \(\alpha \)-resolvent operator, fractional powers of closed operators and suitable fixed point theorems. Then we derive the existence conditions of optimal pairs to the fractional impulsive stochastic control systems. Finally, an example is given to illustrate the effectiveness of our main results.

Existence Results of Mild Solutions for Impulsive Fractional Evolution Equations with Periodic Boundary Condition

Abstract This paper discusses the existence of mild solutions for a class of fractional impulsive evolution equation with periodic boundary condition and noncompact semigroup. By using some fixed-point theorems, the existence theorems of mild solutions are obtained, our results are also more general than known results. Furthermore, as an application that illustrates the abstract results, two examples are given.

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.

Existence of solutions for delay evolution equations with nonlocal conditions

Abstract In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.

Study on fractional non-autonomous evolution equations with delay

In this paper, we deal with a class of nonlinear time fractional non-autonomous evolution equations with delay by introducing the operators ψ(t,s), φ(t,η) and U(t), which are generated by the operator −A(t) and probability density function. The definition of mild solutions for studied problem was given based on these operators. Combining the techniques of fractional calculus, operator semigroups, measure of noncompactness and fixed point theorem with respect to k-set-contractive, we obtain new existence result of mild solutions with the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition and the closed linear operator −A(t) generates an analytic semigroup for every t>0. The results obtained in this paper improve and extend some related conclusions on this topic. At last, by utilizing the abstract result obtained in this paper, the existence of mild solutions for a class of nonlinear time fractional reaction–diffusion equation introduced in Ouyang (2011) is obtained.

A study on existence of solutions for fractional functional differential equations

In this research article, we establish the existence results of mild solutions for semi-linear impulsive neutral fractional order integro-differential equations with state dependent delay subject to nonlocal initial condition by applying well known classical fixed point theorems. At last, we present an example of partial derivative to illuminate the results.

Approximate controllability of a multi-valued fractional impulsive stochastic partial integro-differential equation with infinite delay

In this paper, the approximate controllability of a new class of multi-valued fractional impulsive stochastic partial integro-differential equations with infinite delay in Hilbert spaces is studied. Firstly, by using stochastic analysis, analytic α-resolvent operator, fractional powers of closed operators and a fixed point theorem for multi-valued maps, we prove an existence result of mild solutions for the control systems under the mixed Lipschitz and Carathéodory conditions. Secondly, we discuss a new set of sufficient conditions for the approximate controllability of the systems. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is provided to illustrate the proposed theory.

The optimal control of a new class of impulsive stochastic neutral evolution integro-differential equations with infinite delay

ABSTRACTIn this paper, we introduce the optimal control problems governed by a new class of impulsive stochastic partial neutral evolution equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, the analytic semigroup theory, fractional powers of closed operators, and suitable fixed point theorems, we prove an existence result of mild solutions for the control systems in the α-norm without the assumptions of compactness. Next, we derive the existence conditions of optimal pairs of these systems. Finally, application to a nonlinear impulsive stochastic parabolic optimal control system is considered.

Fractional evolution equation nonlocal problems with noncompact semigroups

This paper is concerned with the existence results of mild solutions to the nonlocal problem of fractional semilinear integro-differential evolution equations. New existence theorems are obtained by means of the fixed point theorem for condensing maps. The results extend and improve some related results in this direction.