PC-mild Solutions Research Articles

Approximate Controllability for a Class of Semi-Linear Fractional Integro-Differential Impulsive Evolution Equations of Order 1 < α < 2 with Delay

This article is mainly concerned with the approximate controllability for some semi-linear fractional integro-differential impulsive evolution equations of order 1<α<2 with delay in Banach spaces. Firstly, we study the existence of the PC-mild solution for our objective system via some characteristic solution operators related to the Mainardi’s Wright function. Secondly, by using the spatial decomposition techniques and the range condition of control operator B, some new results of approximate controllability for the fractional delay system with impulsive effects are obtained. The results cover and extend some relevant outcomes in many related papers. The main tools utilized in this paper are the theory of cosine families, fixed-point strategy, and the Grönwall-Bellman inequality. At last, an example is given to demonstrate the effectiveness of our research results.

On the periodic boundary value problems for fractional nonautonomous differential equations with non-instantaneous impulses

In this paper, we investigate periodic boundary value problems for Caputo type fractional semilinear nonautonomous differential equations with non-instantaneous impulses. By using semigroup theory combined with the measure of noncompactness and some fixed point theorems, the existence of PC-mild solutions for the equations is established. At the end, an example is presented to illustrate the application of our main results.

Impulsive Fractional Semilinear Integrodifferential Equations with Nonlocal Conditions

This paper is devoted to a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions. Based on the semigroup theory and some fixed point theorems, the existence theory of PC-mild solutions is established under the condition of compact resolvent operator. Furthermore, the uniqueness of PC-mild solutions is proved in the case of the noncompact resolvent operator.

New Study of the Existence and Dimension of the Set of Solutions for Nonlocal Impulsive Differential Inclusions with a Sectorial Operator

In this article, we are interested in a new generic class of nonlocal fractional impulsive differential inclusions with linear sectorial operator and Lipschitz multivalued function in the setting of finite dimensional Banach spaces. By modifying the definition of PC-mild solutions initiated by Shu, we succeeded to determine new conditions that sufficiently guarantee the existence of the solutions. The results are obtained by combining techniques of fractional calculus and the fixed point theorem for contraction maps. We also characterize the topological structure of the set of solutions. Finally, we provide a demonstration to address the applicability of our theoretical results.

On the fractional partial integro-differential equations of mixed type with non-instantaneous impulses

In this paper, we consider the initial boundary value problem for a class of nonlinear fractional partial integro-differential equations of mixed type with non-instantaneous impulses in Banach spaces. Sufficient conditions of existence and uniqueness of PC-mild solutions for the equations are obtained via general Banach contraction mapping principle, Krasnoselskii’s fixed point theorem, and α-order solution operator.

A study on impulsive fractional hybrid evolution equations using sequence method

In this paper, we introduce a new concept called $$\alpha $$ -order cosine-resolvent family, by using the theory of fractional calculus, the concepts of measure of noncompactness and Hybrid fixed point theorem, we consider the existence of PC-mild solutions for a class of impulsive fractional hybrid evolution equation in a Banach space . Furthermore, we obtain some sufficient conditions for approximate controllability of our concern problem. At the end, an example is given to illustrate the feasibility of our results.

PC-Mild Solutions to Sobolev-Type Fractional Differential Equations with Non-instantaneous Impulses

In this paper, we study the existence and uniqueness of piecewise-continuous (in brief, PC)-mild solutions for Sobolev-type fractional impulsive differential equations (in brief, STFIDEs) with state-dependent delay by virtue of Kuratowski measure of noncompactness in a Banach space. We establish a general framework to find PC-mild solutions for STFIDEs with compact and noncompact semigroups, which will provide an effective way to deal with such problems. Finally, an application is given to illustrate that our results are valuable.

Existence and controllability of fractional evolution equation with sectorial operator and impulse

This paper deals with the existence and uniqueness of PC-mild solutions for fractional impulsive evolution equation involving nonlocal conditions and sectorial operators. We also study the nonlocal controllability of the control system governed by fractional impulsive evolution equation.

Existence of mild solution for nonlocal impulsive fractional semilinear differential inclusion in Banach space

In this paper, we deal with the existence of PC-mild solutions of nonlocal impulsive differential inclusions in Banach space when the values of the orient field is convex (P). By using methods and results of semilinear differential inclusions, and techniques of fixed point theorems, we establish sufficient conditions that guarantee the existence of PC-mild solutions of (P). Our results develop and extend various results proved recently.

A general class of noninstantaneous impulsive fractional differential inclusions in Banach spaces

In this paper we introduce the concept of a PC-mild solution to a general new class of noninstantaneous impulsive fractional differential inclusions involving the generalized Caputo derivative with the lower bound at zero in infinite dimensional Banach spaces. Using the formula of a PC-mild solution, we give two classes of sufficient conditions to guarantee the existence of PC-mild solutions via fixed point theorems for multivalued functions. Also we characterize the compactness of the solution set. We introduce the concept of generalized Ulam-Hyers stability and present a generalized Ulam-Hyers stability result using multivalued weakly Picard operator theory. Examples are given to illustrate the theoretical results.

Hybrid fixed point theorems with application to fractional evolution equations

In this paper, we first prove some hybrid fixed point theorems of Krasnoselskii type via the measure of noncompactness in Banach spaces. Then some coupled fixed point theorems are established using the obtained hybrid fixed point theorems. As an application, we consider the existence of PC-mild solutions for the fractional evolution equations involving sectorial operators and impulsive effects.

English

In this paper, for the impulsive fractional integro-differential equations involving Caputo fractional derivative in Banach space, we investigate the existence and uniqueness of a pseudo almost periodic PC-mild solution. The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. Some known results are improved and generalized. Finally, existence and uniqueness of a pseudo almost periodic PC-mild solution of a two-dimensional impulsive fractional predator-prey system with diffusion are investigated.

Nonlinear impulsive evolution equations with nonlocal conditions and optimal controls

This paper is concerned with controlled nonlinear impulsive evolution equations with nonlocal conditions. The existence of PC-mild solutions is proved, but the uniqueness cannot be obtained. By constructing approximating minimizing sequences of functions, the existence of optimal controls of systems governed by nonlinear impulsive evolution equations is also presented.

Some results for impulsive fractional differential inclusions with infinite delay and sectorial operators in Banach spaces

This paper is concerned with impulsive fractional differential inclusions with sectorial operators, nonlocal conditions and infinite delay in Banach spaces. Using tools involving the measure of noncompactness and multi-valued fixed point theory, we derive some existence results of PC-mild solutions when the multivalued map F is convex as well as nonconvex. Furthermore, the compactness of the set of solutions and continuous dependence results are also obtained. As an application, an impulsive neutral partial functional differential equation is also investigated at last.

On a new class of impulsive fractional evolution equations

This paper is concerned with the existence of PC-mild solutions for Cauchy and nonlocal problems of impulsive fractional evolution equations for which the impulses are not instantaneous. By using the theory of operator semigroups, probability density functions, and some suitable fixed point theorems, we establish some existence results for these types of problems.

Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces

This paper investigates existence of PC-mild solutions of impulsive fractional differential inclusions with nonlocal conditions when the linear part is a fractional sectorial operators like in Bajlekova (2001) [1] on Banach spaces. We derive two existence results of PC-mild solutions when the values of the semilinear term F is convex as well as another existence result when its values are nonconvex. Further, the compactness of the set of solutions is characterized.

Existence and Controllability Results for Fractional Impulsive Integrodifferential Systems in Banach Spaces

We firstly study the existence of PC-mild solutions for impulsive fractional semilinear integrodifferential equations and then present controllability results for fractional impulsive integrodifferential systems in Banach spaces. The method we adopt is based on fixed point theorem, semigroup theory, and generalized Bellman inequality. The results obtained in this paper improve and extend some known results. At last, an example is presented to demonstrate the applications of our main results.

A class of nonlinear integrodifferential impulsive periodic systems of mixed type and optimal controls on Banach space

A class of nonlinear integrodifferential impulsive periodic systems of mixed type on Banach space is considered. Existence of periodic PC-mild solutions is proved. Existence of periodic optimal pairs of systems governed by nonlinear impulsive integrodifferential equations of mixed type is also presented. An example is given for demonstration.

A class of differential equations with impulses at variable times on Banach spaces

In this paper, two typical classes of semilinear evolution equation with impulses at variable times on infinite dimensional spaces are considered. Introducing the reasonable PC-mild solutions, we discuss the local existence, maximal existence interval, global existence of PC-mild solution for two models. In addition to the pulse phenomena, an example is given for demonstration.

Existence and Global Asymptotical Stability of Periodic Solution for theT-Periodic Logistic System with Time-Varying Generating Operators andT0-Periodic Impulsive Perturbations on Banach Spaces

This paper studies the existence and global asymptotical stability of periodic PC-mild solution for theT-periodic Logistic system with time-varying generating operators andT0-periodic impulsive perturbations on Banach spaces. Two sufficient conditions that guarantee the exponential stability of the impulsive evolution operator corresponding to homogenous well-posedT-periodic system with time-varying generating operators andT0-periodic impulsive perturbations are given. It is shown that the system have a unique periodic PC-mild solution which is globally asymptotically stable whenTandT0are rational dependent and its period must benT0for somen∈N. At last, an example is given for demonstration.