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This paper is devoted to the study of abstract time-fractional equations of the following form:

A great number of abstract time-fractional equations appearing in engineering, mathematical physics, and chemistry can be modeled through the abstract Cauchy problem

The aim of this paper is to develop some operator theoretical methods for solving the abstract time-fractional equations of the form (

In the second section, we continue the analysis from our recent paper [

On the other hand, the notions of

Throughout this paper, we will always assume that

Define

Given

Let

The following definition has been recently introduced in [

Suppose

Then it is said that

Let

Let

It will be convenient to remind us of the following definitions from [

(i) Let

(ii) Let

(i) Let

(ii) Let

For a global

The reader may consult [

Henceforth, we assume that

The following conditions will be used in the sequel:

(H1) ^{’}.

In this section, we will always assume that

A function

for every

for every

In the case of abstract Cauchy problem (

Given

Suppose

The above classes of propagation families can be defined by purely algebraic equations (cf. [

As indicated before, we will consider only nondegenerate

In case that

Let

Let

Let

Let

the equality

holds provided

any of the assumptions

the equality (

if

Let

The equations (

Suppose

If

If, additionally,

then (

The equality (

We will only prove the second part of proposition. Let

Let

Suppose now

The following definition also appears in [

Let

a strong solution of (

a mild solution of (

It is clear that every strong solution of (

Let

if

if

Suppose

if

for any

if

Therefore, there is at most one strong (mild) solution for (

We will only prove the second part of theorem. Let

If

The subsequent theorems can be shown by modifying the arguments given in the proof of [

Suppose

Suppose

The equality

holds provided

The equality

holds provided

Suppose

Suppose

Keeping in mind Theorem

The analytical properties of

Suppose

Assume

In this paper, we will not consider differential properties of

In the following theorem, which possesses several obvious consequences, we consider

(i) Suppose

(ii) Suppose

The proof is almost completely similar to that of [

In the second part of Theorem

Throughout this section, we will always assume that

Let

Following Xiao and Liang [

Suppose

A strongly continuous operator family

A strongly continuous operator family

A strongly continuous family

Suppose

In case

Integrating both sides of (

Let

Let

Suppose

In the first part of subsequent theorem (cf. also [

(i) Suppose

(ii) Suppose

A straightforward computation involving (

Before proceeding further, we would like to notice that the solution

The standard proof of following theorem is omitted (cf. also [

Suppose

(a) Let

Let the assumptions of (i) hold with

Suppose

The Hausdorff locally convex topology on

Now we focus our attention to the adjoint type theorems for (local)

(i) Suppose

(ii) Suppose

(iii) Suppose

Notice here that a similar theorem can be proved for the class of

Let

In the subsequent theorem, whose proof follows from a slight modification of the proof of [

Suppose

Suppose that all conditions quoted in the first part of the above theorem hold, and the family

The proof of following theorem can be derived by using Theorem

Suppose

Let

Define, for every

Let

for every

Suppose

(i) Consider the situation of Theorem

(ii) Let

(iii) Concerning the analytical properties of

The mapping

Let

The similar statements hold for the

The results on

We need the following definition.

Suppose

A strongly continuous operator family

A strongly continuous operator family

Notice also that one can introduce the classes of

The following facts are clear.

Suppose

Let

The proof of following subordination principle is standard and therefore omitted (cf. the proofs of [

(i) Suppose there is an exponentially equicontinuous

(ii) Suppose there is an exponentially equicontinuous

It is not difficult to reformulate Theorem

Although our analysis tends to be exhaustive, we cannot cover, in this limited space, many interested subjects. For example, the characterizations of some special classes of

We start this section with the following example.

Suppose

Suppose

Put

Due to the choice of

Therefore, we have the following: if the operator

Suppose

where

Suppose

Let

Arguing as in (a), we reveal that

Suppose