In this note we introduce a new class of abstract impulsive differential equations for which the impulses are not instantaneous. We introduce the concepts of mild and classical solution and we establish some results on the existence of these types of solutions. An example involving a partial differential equation is presented.

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On a new class of abstract impulsive differential equations

In this paper we discuss the existence of P C-mild solutions for Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving Caputo fractional derivative. By utilizing the theory of operators semigroup, probability density functions via impulsive conditions, a new concept on a P C-mild solution for our problem is introduced. Our main techniques based on fractional calculus and fixed point theorems. Some concrete applications to partial differential equations are considered.

https://www.intlpress.com/site/pub/files/_fulltext/journals/dpde/2011/0008/0004/DPDE-2011-0008-0004-a003.pdf

On the new concept of solutions and existence results for impulsive fractional evolution equations

Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses

One of the novelty of this paper is the study of a general class of impulsive differential equations, which is more reasonable to show dynamics of evolution processes in Pharmacotherapy. This fact reduces many difficulties in applying analysis methods and techniques in Bielecki's normed Banach spaces and thus makes the study of existence and uniqueness theorems interesting. The other novelties of this paper are new concepts of Ulam's type stability and Ulam-Hyers-Rassias stability results on compact and unbounded intervals.

/pdf/a-general-class-of-impulsive-evolution-equations-49oaf2gy9v.pdf

A General Class of Impulsive Evolution Equations

In this paper, we establish two sufficient conditions for nonlocal controllability for fractional evolution systems. Since there is no compactness of characteristic solution operators, our theorems guarantee the effectiveness of controllability results under some weakly compactness conditions.

Nonlocal Controllability of Semilinear Dynamic Systems with Fractional Derivative in Banach Spaces

Existence results for semilinear differential equations with nonlocal and impulsive conditions

In this paper, we discuss the exact solutions of linear homogeneous and nonhomogeneous fractional differential equations with double delays. Firstly, a new concept of double-delayed Mittag-Leffler type matrix function is introduced, which is the promotion of the double-delayed matrix exponential. Secondly, we apply the double-delayed Mittag-Leffler type matrix function and Laplace transform approach to obtain the exact solutions of fractional differential equations with double delays. Furthermore, the solution is used to investigate the Hyers-Ulam stability of the system. Lastly, we illustrate our techniques by an example. 

Exact solutions and Hyers-Ulam stability of fractional equations with double delays

The existence of mild solutions for Hilfer fractional evolution equations with nonlocal conditions in a Banach space is investigated in this manuscript. No assumptions about the compactness of a function or the Lipschitz continuity of a nonlinear function are imposed on the nonlocal item and the nonlinear function, respectively. However, we assumed that the nonlocal item is continuous, the nonlinear term is continuous and satisfies some specified assumptions, and the associated semigroup is compact. Our theorems are proved by means of approximate techniques, semigroup methods, and fixed point theorem. These methods are useful for fixing the noncompactness of operators caused by some specified given assumptions on this paper. The results obtained here improve some known results. Finlay, two examples are presented for illustration of our main results.

https://downloads.hindawi.com/journals/jfs/2023/8662375.pdf

Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

The approximate controllability of fractional linear evolution systems is
 considered in this paper. Firstly, the definitions of the mild solution and
 the approximate controllability of fractional linear evolution systems are
 obtained by using the theory of C?semigroups. Secondly, a new set of
 necessary and sufficient conditions are established to examine that linear
 system is approximately controllable with the help of symmetric operator.
 Moreover, the restricted condition of the state space is weakened by means
 of the dual mapping. Finally, as applications, the approximate
 controllability of nonlinear evolution systems are derived under the
 assumption that the corresponding linear system is approximately
 controllable. Our work essentially improves and generalized the
 corresponding results which are based on strongly continuous semigroups.

/pdf/necessary-and-sufficient-conditions-for-the-approximate-1y69lubi.pdf

Necessary and sufficient conditions for the approximate controllability of fractional linear systems via C−semigroups

In this paper, we investigate the concept of regional enlarged observability (ReEnOb) for fractional differential equations (FDEs) with the Hilfer derivative. To proceed this, we develop an approach based on the Hilbert uniqueness method (HUM). We mainly reconstruct the initial state ν01 on an internal subregion ω from the whole domain Ω with knowledge of the initial information of the system and some given measurements. This approach shows that it is possible to obtain the desired state between two profiles in some selective internal subregions. Our findings develop and generalize some known results. Finally, we give two examples to support our theoretical results.

/pdf/the-regional-enlarged-observability-for-hilfer-fractional-1552aqni.pdf

Zhenbin Fan

Papers

Existence results for semilinear differential equations with nonlocal and impulsive conditions

Exact solutions and Hyers-Ulam stability of fractional equations with double delays

Existence of Mild Solutions for Nonlocal Evolution Equations with the Hilfer Derivatives

Necessary and sufficient conditions for the approximate controllability of fractional linear systems via C−semigroups

The Regional Enlarged Observability for Hilfer Fractional Differential Equations