Existence Results Of Mild Solutions Research Articles

Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions

Abstract In this article, we investigate the existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions. Using the theory of resolvent operators, some fixed point theorems, and an estimation technique of Kuratowski measure of noncompactness, we first establish some existence results of mild solutions for the proposed equation. Subsequently, we show by applying a newly established lemma that these solutions have regularity property under some conditions. Finally, as a sample of application, the obtained results are applied to a class of non-autonomous nonlocal partial integrodifferential equations.

An analysis on time optimal feedback control for Caputo fractional neutral systems with delay

ABSTRACT This paper aims to provide a systematic approach to investigate the optimal feedback control systems governed by the fractional neutral evolution system with fixed delay in the sense of Caputo fractional derivatives in separable reflexive Banach spaces. Initially, we study the existence result of mild solutions for the system by applying fixed point theorem for a condensing map due to Martelli. The existence of feasible pairs is obtained by using the Filippov theorem and the Cesari property. Furthermore, the existence of optimal feedback control and time optimal control are studied for the corresponding control problem. Finally, an example is given to validate our findings.

A class of time-optimal feedback control for fractional neutral evolution hemivariational inequalities with fixed delay

ABSTRACT In this paper, we study a class of optimal feedback control for a neutral fractional system with hemivariational inequalities. Initially, we derive some existence results of mild solutions for the system by applying the Marteli's fixed point theorem of multivalued maps, properties of generalized Clarke's subdifferential and semigroup operators. Then, by using the Filippov theorem and the Cesari property, a new set of sufficient conditions are formulated to ensure the existence results of a feasible pair for the feedback control systems. Also, we apply our main results to obtain the optimal feedback control pair. Further, we investigate the time-optimal feedback control for our control system. In the end, an example is given to illustrate our theory.

Nonlocal partial fractional evolution equations with state dependent delay

In this work, we propose sufficient conditions guaranteeing an existence result of mild solutions by using the nonlinear Leray-Schauder alternative in Banach spaces combined with the semigroup theory for the class of Caputo partial semilinear fractional evolution equations with finite state-dependent delay and nonlocal conditions.

Controllability of fractional measure evolution systems with state-dependent delay and nonlocal condition

<p style='text-indent:20px;'>This paper is concerned with the existence of mild solutions and exact controllability for a class of fractional measure evolution systems with state-dependent delay and nonlocal conditions. We first establish an existence result of mild solutions for the concerned problem by applying an integral equation which is given in terms of probability density and semigroup theory. Then, the exact controllability are obtained by using the fractional calculus theory, Kuratowski measure of noncompactness and Mönch fixed point theorem, without imposing the Lipschitz continuity on nonlinear term. Finally, we give two applications to support the validity of the study.</p>

Existence results of mild solutions for some stochastic integrodifferential equations with state-dependent delay and noninstantaneous impulses in Hilbert spaces

The primary objective of this work is to investigate the existence of mild solutions for some neutral stochastic integrodifferential systems in Hilbert spaces. These systems have a state-dependent delay and noninstantaneous impulses. In order to achieve this objective, the neutral stochastic integrodifferential system that has been proposed with state-dependent delay and noninstantaneous impulses is converted into an equivalent fixed point problem by means of an appropriate integral operator.Together with the theory of stochastic analysis, we develop several existence results that are founded on the theory of resolvent operators in the sense of Grimmer, Hausdorff’s measures of noncompactness, and fixed point theorems.In the last part of this paper, specific examples are used to show how the general conclusions reached in this paper are made.

A Cauchy problem for fractional evolution equations with Hilfer’s fractional derivative on semi-infinite interval

In this paper, we consider a Cauchy problem for fractional evolution equations with Hilfer’s fractional derivative on semi-infinite interval. An elementary fact shows that semi-infinite interval is not compact, the classical Ascoli-Arzelà theorem is not valid. In order to establish the global existence criteria, we first generalize Ascoli-Arzelà theorem into the semi-infinite interval. Next, we introduce a new concept of mild solutions based on cosine/sine family and probability density function and obtain several existence results of mild solutions on semi-infinite interval.

Existence results of mild solutions for nonlocal fractional delay integro-differential evolution equations via Caputo conformable fractional derivative

<abstract><p>In this paper, we investigate the existence of mild solutions for nonlocal delay fractional Cauchy problem with Caputo conformable derivative in Banach spaces. We establish a representation of a mild solution using a fractional Laplace transform. The existence of such solutions is proved under certain conditions, using the Mönch fixed point theorem and a general version of Gronwall's inequality under weaker conditions in the sense of Kuratowski measure of non compactness. Applications illustrating our main abstract results and showing the applicability of the presented theory are also given.</p></abstract>

Existence and Stability Results for Second-Order Neutral Stochastic Differential Equations With Random Impulses and Poisson Jumps

The objective of this paper is to investigate the existence and stability results of secondorder neutral stochastic functional differential equations (NSFDEs) in Hilbert space. Initially, we establish the existence results of mild solutions of the aforementioned system using the Banach contraction principle. The results are formulated using stochastic analysis techniques. In the later part, we investigate the stability results through the continuous dependence of solutions on initial conditions.

Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type

The basic motivation of this research paper is the approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type which generalized the Riemann-Liouville fractional derivative. Combining with the techniques of fractional calculus, fixed point theorem for multi-value dmaps and Sobolev-type and Clarke subdifferential to the approximate controllability system we obtain new existence result of mild solutions. Next, we demonstrate linear and semilinear systems for approximate controllability. Finally, to verify our theoretical results, an illustration is also included.

Global Attractor for Second-Order Nonlinear Evolution Differential Inclusions

In this paper, we address the model of global attractor formulated in the form of evolution differential inclusions with second order in Banach spaces. Firstly, based on the fixed point theorem, the existence result of mild solutions is deduced. Then, by implementing the measure of noncompactness, the existence of global attractor associated with m -semiflow is validated. Finally, a concrete application of the main result is demonstrated to enhance the practical signification.

Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives

<p style='text-indent:20px;'>The goal of this paper is to provide systematic approaches to study the feedback control systems governed by fractional impulsive delay evolution equations involving Caputo fractional derivatives in separable reflexive Banach spaces. This work is a continuation of previous work. We firstly give an existence result of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray-Schauder alternative fixed point theorem. Next, by using the Filippove theorem and the Cesari property, we obtain the existence result of feasible pairs for the feedback control system. Finally, some applications are given to illustrate our main results.</p>

Existence results on impulsive stochastic semilinear differential inclusions

In this paper, we present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive stochastic semilinear differential inclusions driven by Poisson jumps with periodic boundary conditions.We consider the cases in which the right hand side can be either convex . The results are obtained by using fixed point theorems for multivalued mappings, more precisely, the technique is based on fixed point theorem a nonlinear alternative of Leray-Schauder's fixed point theorem in generalised metric and Banach spaces.

Existence and uniqueness of mild solutions for fractional impulsive integro-differentialevolution equations of order bm$1<\beta\leq2$with nonlocal 条件

This paper investigates the existence and uniqueness ofmild solutions for fractional impulsive integro-differentialevolution equations with nonlocal 条件 of order$1<\\beta\\leq2$. Under two cases where the 解 operator iscompact and noncompact respectively, the existence results of mild solutions areestablished by the (generalized) Darbo fixed point 定理 andSchauder fixed point 定理. Meanwhile, uniqueness of mildsolutions is also given by the (generalized) Banach contractionmapping principle. These results are obtained by the semigroup定理, 解 operator theorems, measures of noncompactness andfixed point 定理. At last, an 例 is presented to illustratethe effectiveness of the main results.

Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families

<p style='text-indent:20px;'>In this paper, we investigate the non-autonomous stochastic evolution equations of parabolic type with nonlinear noise and nonlocal initial conditions in Hilbert spaces, where the operators in linear part depend on time <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula> and generate an noncompact evolution family. New existence result of mild solutions is established under some weaker growth and measure of noncompactness conditions on nonlinear functions and nonlocal term. The discussions are based on Sadovskii's fixed-point theorem as well as the theory of evolution family. At last, as a sample of application, the obtained abstract result is applied to a class of non-autonomous stochastic partial differential equations of parabolic type with nonlocal initial conditions. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions

In this paper, we study the non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions in Hilbert spaces, where the operators in linear part (possibly unbounded) depend on time \begin{document}$ t $\end{document} and generate an evolution family. New existence result of mild solutions is established under more weaker conditions by introducing a new Green's function. The discussions are based on Schauder's fixed-point theorem as well as the theory of evolution family. At last, an example is also given to illustrate the feasibility of our theoretical results. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families

This paper investigates the Cauchy problem to a class of stochastic non-autonomous evolution equations of parabolic type governed by noncompact evolution families in Hilbert spaces. Combining the theory of evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we established some new existence results of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the strong restriction on the constants in the condition of noncompactness measure is completely deleted, and also the condition of uniformly continuity of the nonlinearity is not required. At last, as samples of applications, we consider the Cauchy problem to a class of stochastic non-autonomous partial differential equation of parabolic type.

The nonlinear Rayleigh‐Stokes problem with Riemann‐Liouville fractional derivative

The Rayleigh‐Stokes problem has gained much attention with the further study of non‐Newtonain fluids. In this paper, we are interested in discussing the existence of solutions for nonlinear Rayleigh‐Stokes problem for a generalized second grade fluid with Riemann‐Liouville fractional derivative. We firstly show that the solution operator of the problem is compact and continuous in the uniform operator topology. Furtherly, we give an existence result of mild solutions for the nonlinear problem.

A Study on Impulsive Hilfer Fractional Evolution Equations with Nonlocal Conditions

Abstract In this paper, we concern with the existence of mild solution to nonlocal initial value problem for nonlinear Sobolev-type impulsive evolution equations with Hilfer fractional derivative which generalized the Riemann–Liouville fractional derivative. At first, we establish an equivalent integral equation for our main problem. Second, by means of the properties of Hilfer fractional calculus, combining measure of noncompactness with the fixed-point methods, we obtain the existence results of mild solutions with two new characteristic solution operators. The results we obtained are new and more general to known results. At last, an example is provided to illustrate the results.

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.