Existence of Some Semilinear Nonlocal Functional Differential Equations of Neutral Type

and Applied Analysis 3 3. Mild Solutions Definition 2. A continuous function u : [0, T] → X is called a mild solution of (1) on [0, T] if, for each t ∈ [0, T], the function s 󳨃→ AT(t − s)F(s, u(s)) is integrable on [0, t), and the following equation is satisfied: u (t) = T (t) [u0 + F (0, u (0)) − g (u)] − F (t, u (t)) + ∫ t 0 AT (t − s) F (s, u (s)) ds


Introduction
The purpose of this paper is to study the existence of mild and strong solutions for the following neutral evolution problem with nonlocal initial conditions: [ () +  (,  ())] = − () +  (,  ()) ,  ∈ [0, ] , in a Banach space (, ‖ ⋅ ‖), where  > 0 and − generates an analytic compact semigroup (⋅) on .The functions , , and  will be specified later.The Cauchy problem with the nonlocal condition (0) + () =  0 was first considered by Byszewski [1], and since it reflects physical phenomena more precisely than the classical initial condition (0) =  0 does, this issue has gained enormous attention in the past several years.For more detailed information about the importance of nonlocal initial conditions in applications, we refer to the works of Byszewski [2], Byszewski and Lakshmikantham [3], and to many other authors [4][5][6][7] and the references therein.Equation (1) has been studied by many authors under various assumptions on the linear part , the nonlinear terms , , and the nonlocal condition  see, for example, [8][9][10][11][12][13][14].
A basic approach to this problem is to define the solution operator Φ : ([0, ], ) → ([0, ], ) by and to use various fixed-point theorems, including Schauder fixed-point theorem, Banach contraction principle, Leray-Schauder alternative, and Sadovskii fixed-point theorem, to show that Φ has a fixed point, which is the mild solution of (1).When using fixed-point theorems, it is necessary that the semigroup () generated by the linear part of (1) be compact; that is, () is a compact operator, for all  > 0, so that the norm continuity of (), for  > 0, becomes a key point in the study of the existence of mild solutions.Thus, because of the absence of compactness of the solution operator at  = 0, most of the papers on the relevant topics (e.g., [8][9][10]14]) assume complete continuity on the nonlocal term .However, it is too restrictive in terms of applications.
where   's are given constants, and in this case, we have measurements at  = 0 ≤  0 <  1 < ⋅ ⋅ ⋅ <   ≤  rather than just at  = 0. Thus, by assuming that there is a  ∈ (0, ) such that the authors utilize fixed-point theorem twice to deduce the existence results.More recently, Liu and Yuan [16] gave existence results using Schauder fixed-point theorem and a limiting process under the following hypothesis.
Motivated by the works in [15,16], we drop the compactness assumption on the nonlocal condition  and discuss the existence of solutions for (1).The obtained results generalize recent conclusions on this topic.
The present work is organized as follows.Section 1 is devoted the introduction of the problem we studied.Section 2, we explain some known notations and results we will use.The basic hypotheses on (1) are also given in this section.In Section 3, we study the existence of mild solutions to (1) and in Section 4, we investigate some conditions for (1) to come up with strong solutions.In Section 5, an example is given to illustrate the existence results.
Let   be the Banach space (  ) endowed with the norm ‖ ⋅ ‖  .Then, we denote by   the operator norm of  − , that is,   := ‖ − ‖, and let  be the Banach space ([0, ], ) endowed with the supnorm given by and, for any  ∈ (0, ), set   := ([, ], ).Moreover, let   be the Banach space ([0, ],   ) endowed with the supnorm given by The following hypotheses are the basic assumptions of this paper.
Abstract and Applied Analysis 3
To see the existence of mild solution of nonlocal problem (1), we will, in view of (12), locate the fixed point of a mapping Φ defined on  by For this, we first observe the following result, where for all  ∈ N, we let   = { ∈  : ‖()‖ ≤ ,  ∈ [0, ]}.Lemma 3. Assume that hypotheses (H1)-(H3) are satisfied, and, in addition, there holds the following inequality: (H4) Then, Φ  ⊂   , for some  ∈ N.
Proof.Suppose, on the contrary, that, for each  > 0, there exist Dividing the two sides by  and taking the lower limit as  → +∞, we have which is a contradiction.This completes the proof.
By Lemma 3, we see that the mapping Φ :  →  defined by (13) maps   into itself.We will show that Φ has a fixed point in   .To see this, note first that Φ is continuous by the continuity of ,  and .We decompose Φ as Φ = Φ 1 + Φ 2 , where We show that Φ 1 is a contraction in   and Φ 2 is a compact operator in   .

Lemma 5. Assume that hypotheses (H1)-(H4) are satisfied, and, in addition, the following is given.
(H5) There exists a  ∈ (0, ) Then the problem (1) has at least one mild solution in   for some  ∈ N.
Proof.Let  be given by (H5), and let For any  ∈   (), let  ∈   be defined by With a similar argument as in the proof of Lemma 4, one sees that Φ 1, is a contraction on   ().
By the fixed-point theorem of Sadovskiȋ [18], this shows that Φ  has a fixed point in   (); that is, there is a  ∈   () such that (34) That is,  is a mild solution of (1).
For the main results in this section, we introduce a family of nonlocal neutral problems as follows.Firstly, we define, for each  ∈ (0, ), an operator B  on  by It is clear that V ∞ is strongly measurable, V ∞ ∈   , and It therefore follows from Lebesgue's dominated convergence theorem that there is a subsequence {  } ∈N of {  0 }  0 ∈N such that This shows that the sequence {B     } ∈N is relatively compact on , and, hence, by the continuity of , it follows that By ( 44) and (49), we see the relative compactness of {  } ∈N on .Thus, there is a subsequence of {  } ∈N denoted by then ( 50) and the uniform continuity of  ∞ imply that lim ] → ∞ ‖B  ]  ] −  ∞ ‖  = 0.By taking limits in (39), we see that  ∞ is a mild solution of ( 1) and this completes the proof.
We will consider the case more generally; that is, the nonlocal condition  is defined on  rather than  1 ([0, ], ).Theorem 8. Suppose that, hypotheses (H1) and (H2) are satisfied, and, in addition, there hold the following hypotheses.
Proof.Let {  } ∈N and {  } ∈N be the sequences defined as in the proof of Theorem 7.With the same arguments as in the proof of Lemma 4, we see that Φ  ⊂   , for some  ∈ N.Moreover, it follows from the same arguments as in the proof of Theorem 7 that (44), and (45) also hold, and, for every subsequence {  0 }  0 ∈N of {  } ∈N , there exist a subsequence {V  } ∞ =1 and a function V ∞ : (0, ] →  such that V ∞ is continuous on (0, ] and, for every   , lim Let  > 0 be given.It follows from (H7) and (53) that there is a  > 0 such that and that, for every  ∈ N, there is an   such that  >   implies that Choose  that is large enough so that   < , and define  : [0, ] →  by Thus, (H7), (54), and (55) insure that And, hence, by the continuity of  and the compactness of (), for  > 0, (49) is also valid in this case.Therefore, a similar argument as in the last paragraph of the proof of Theorem 7 shows the existence of a mild solution for (1).
where  0 is a constant independent of .Since (H11) implies that then whenever Therefore, Φ has a fixed point  which is a mild solution of (1).By the above calculation, we see that, for this (⋅), all of the functions  () =  (,  ()) , are Lipschitz continuous, respectively.Since  is Lipschitz continuous on [0, ] and the space  is reflexive, then a result of [19] asserts that (⋅) is a.e.differentiable on (0, ] and   (⋅) ∈  1 ([0, ], ).A similar argument shows that (⋅), (⋅), (⋅), and (⋅) also have this property.Furthermore, with a standard argument as in [17] (Theorem 4. This shows that (⋅) is also a strong solution to the nonlocal Cauchy problem (1), and the proof is completed.
The following result is an immediate corollary of Theorems 8 and 10.