Semilinear Evolution Equations with Convex-Power Condensing Operators

i ) constitutes an impulsive condition. f and g areX-valued functions to be given later. As far as we know, the first paper dealing with abstract nonlocal initial value problems for semilinear differential equations is due to [1]. Because nonlocal conditions have better effect in the applications than the classical initial ones, many authors have studied the following type of semilinear differential equations under various conditions on S(t), f, and g:

As far as we know, the first paper dealing with abstract nonlocal initial value problems for semilinear differential equations is due to [1].Because nonlocal conditions have better effect in the applications than the classical initial ones, many authors have studied the following type of semilinear differential equations under various conditions on (), , and :   () =  () +  (,  ()) ,  ∈ [0, ] ,  (0) =  () . ( For instance, Byszewski and Lakshmikantham [2] proved the existence and uniqueness of mild solutions for nonlocal semilinear differential equations when  and  satisfy Lipschitz type conditions.In [3], Ntouyas and Tsamatos studied the case with compactness conditions.Byszewski and Akca [4] established the existence of solution to functionaldifferential equation when the semigroup is compact and  is convex and compact on a given ball.Subsequently, Benchohra and Ntouyas [5] discussed the second-order differential equation under compact conditions.The fully nonlinear case was considered by Aizicovici and McKibben [6], Aizicovici and Lee [7], Aizicovici and Staicu [8], García-Falset [9], Paicu and Vrabie [10], Obukhovski and Zecca [11], and Xue [12,13].
Recently, the theory of impulsive differential inclusions has become an important object of investigation because of its wide applicability in biology, medicine, mechanics, and control theory and in more and more fields.Cardinali and Rubbioni [14] studied the multivalued impulsive semilinear differential equation by means of the Hausdorff measure of noncompactness.Liang et al. [15] investigated the nonlocal impulsive problems under the assumptions that  is compact, Lipschitz, and  is not compact and not Lipschitz, respectively.All these studies are motivated by the practical interests of nonlocal impulsive Cauchy problems.For a more detailed bibliography and exposition on this subject, we refer to [14][15][16][17][18].
The present paper is motivated by the following facts.Firstly, the approach used in [9,12,13,19,20] relies on the assumption that the coefficient  of the function  about the measure of noncompactness satisfies a strong inequality, which is difficult to be verified in applications.Secondly, in Journal of Function Spaces and Applications [21], it seems that authors have considered the inequality restriction on coefficient function () of  may be relaxed for impulsive nonlocal differential equations.However, in fact, they only solve the classical initial value problems (0) =  0 rather than the nonlocal initial problems (0) =  0 + ().
For more details, one can refer to the proof of Theorem 3.1 in [21] (see the inequalities (3.3) and (3.4) in page 5 and the estimations of the measure of noncompactness in page 6 and page 7 of [21]).
Therefore, we will continue to discuss the impulsive nonlocal differential equations under more general assumptions.Throughout this work, we mainly use the property of convexpower condensing operators and fixed point theorems to obtain the main result (Theorem 10).Indeed, the fixed point theorem about the convex-power condensing operators is an extension for Darbo-Sadovskii's fixed point theorem.But the former seems more effective than the latter at times for some problems.For example, in [22] we ever applied the former to study the nonlocal Cauchy problem and obtained more general and interesting existence results.Based on the results obtained, we discuss the impulsive nonlocal differential equations.Fortunately, applying the techniques of convexpower condensing operators and fixed point theorems solves the difficulty involved by coefficient restriction that is, the constraint condition for the coefficient function () of  is unnecessary (see Theorem 10).Therefore, our results generalize and improve many previous ones in this field, such as [9,12,13,19,20].
The outline of this paper is as follows.In Section 2, we recall some concepts and facts about the measure of noncompactness, fixed point theorems, and impulsive semilinear differential equations.In Section 3, we obtain the existence results of (1) when  is compact in ([0, ]; ).In Section 4, we discuss the existence result of (1) when  is Lipschitz continuous, while Section 5 contains two illustrating examples.
The map  :  ⊂  →  is said to be -condensing if for every bounded and not relatively compact  ⊂ , we have () < () (see [23]).
Lemma 2 (see [9]: Darbo-Sadovskii).If  ⊂  is bounded closed and convex, the continuous map  :  →  is condensing, then the map  has at least one fixed point in .
In the sequel, we will continue to generalize the definition of condensing operator.First of all, we give some notations.
Let  ⊂  be bounded closed and convex, the map  :  → , and  0 ∈  for every  ⊂ , set where co means the closure of the convex hull.Now we give the definition of a kind of new operator.
Definition 3. Let  ⊂  be bounded closed and convex, the map  :  →  is said to be -convex-power condensing if there exist  0 ∈ ,  0 ∈  and for every bounded and not relatively compact  ⊂ , we have From this definition, if ( ( 0 , 0 ) ()) = (), one obtains  ⊂  as relatively compact.
Subsequently, we give the fixed point theorem about the convex-power condensing operator.
Lemma 4 (see [23]).If  ⊂  is bounded closed and convex, the continuous map  :  →  is -convex-power condensing, then the map  has at least one fixed point in .
For further information about the theory of semigroup of operators, we may refer to some classic books, such as [24][25][26].
To discuss the problem (1), we also need the following lemma.

𝑔 Is Compact
In this section, we state and prove the existence theorems for the nonlocal impulsive problem (1).First, we give the following hypotheses: To prove the above theorem, we need the following lemma.
Lemma 11.If the condition (  ) holds, then for arbitrary bounded set  ⊂   , we have This proof is quite similar to that of Lemma 3.1 in [20]; we omit it.
Proof of Theorem 10.We consider the operator  : ([0, ]; ) → ([0, ]; ) defined by It is easy to see that the fixed points of  are the mild solutions of nonlocal impulsive semilinear differential equation (1).Subsequently, we shall prove that  has a fixed point by using Lemma 4.
Set  = co(  ).It is obvious that  is equicontinuous on [0, ] and  maps  into itself.
Next, we shall prove that  :  →  is a convexpower condensing operator.Take  0 ∈ ; by the definition of convex-power condensing operator, we shall show that there exists a positive integral  0 such that   ( ( 0 , 0 ) ) <   () (23) if  ⊂  is not relatively compact.In fact, by using the conditions (  ) and (  ), we get from Lemma 11 that Since () ∈  1 (0, ;  + ), there exists a continuous function  : [0, ] →  1 such that for any 0 <  < 1, where Thus, and hence, by the method of mathematical induction, for any positive integer  and  ∈ [0, ], we obtain 6 Journal of Function Spaces and Applications Therefore, for any positive integer , we have Since lim  → +∞ [ −1 (/( − 1)) −1 ] 1/ = , it follows from the Stirling Formula (see [28]) that and hence, there exists sufficiently large positive integer  0 such that which shows that  :  →  is a convex-power condensing operator.From Lemma 4, we get that  has at least one fixed point in ; that is, (1) has at least one mild solution  ∈ .This completes the proof.
Remark 12.The technique of constructing convex-power condensing operator plays a key role in the proof of Theorem 10, which enables us to get rid of the strict inequality restriction on the coefficient function () of .However, in many previous articles, such as [9,12,13,19,20], the authors had to impose a strong inequality condition on the integrable function (), as they used Darbo-Sadovskii's fixed point theorem only.Thus, our result extends and complements those obtained in [9,12,13,19,20] and has more broad applications.
Therefore, we can get the following consequence.

𝑔 Is Lipschitz Continuous
In this section, by applying the proof of Theorem 10 and Darbo-Sadovskii's fixed point theorem, we give the existence of mild solutions of the problem (1) when the nonlocal condition  is Lipscitz continuous in ([0, ]; ).We give the following hypotheses: (39) From the proof of Theorem 10, we can easily see that there exists at least one mild solution to (39).Define  :  →  by that  is the mild solution to (39).Then which implies that In addition, since ( + ∑  =1   + 4‖‖  1 ) < 1, it follows that the mapping  is a   -condensing operator on .In view of Lemma 2, the mapping  has at least one fixed point in , which produces a mild solution for the nonlocal impulsive problem (1).Remark 16.Similarly, one can show that the conclusion of Theorem 15 remains valid provided that hypothesis (  )(3) is replaced by condition (  )(3  ).
Remark 17.In Theorem 15, we do not assume the compactness of nonlocal item .Under the Lipschitz assumption, we make full use of the conclusion of Theorem 10, the properties of noncompact measure and the technique of fixed point to deal with the solution operator .
Remark 18.Recently, the existence results for fractional differential equations have been widely studied in many papers.For more details on this theory one can refer to [29,30] and references therein.It should be pointed out that the techniques and ideas in this paper can also be used to study fractional equations.In the future, we will also try to investigate to nonlocal controllability of impulsive differential equations by applying the similar techniques, methods, and compactness conditions.Further discussions on this topic will be in our consequent papers.

Examples
In this section, we shall give two examples to illustrate Theorems 10 and 15.
Let us observe that the problem (43) may be reformulated as the abstract problem (1) under the above conditions.By using Theorem 10, the problem (43) has at least one mild solution  ∈ ([0, ];  2 (Ω)) provided that the hypothesis (  ) holds.As is known to all, the operator  is an infinitesimal generator of the semigroup () defined by ()() = ( + ) for each  ∈ .Here, () is equicontinuous but is not compact.
We now suppose the following.