Existence of anti-periodic (differentiable) mild solutions to semilinear differential equations with nondense domain

In this paper, we investigate the existence of anti-periodic (or anti-periodic differentiable) mild solutions to the semilinear differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u'(t) = Au(t) + f (t, u(t))$$\end{document}u′(t)=Au(t)+f(t,u(t)) with nondense domain. Furthermore, an example is given to illustrate our results.

paper. To illustrate our abstract results, the existence and uniqueness of anti-periodic solutions to a partial differential equation is discussed.
The paper is organized as follows: In "Preliminaries" section, we give some definitions and fix notations which will be used in the sequel. In "Main results and proofs" section, the existence of anti-periodic (or anti-periodic differentiable) mild solution to some semilinear differential equations in Banach space are studied. In "An example" section, an example is given to illustrate our main results.

Preliminaries
In this section we recall some definitions and fix notations which will be used in the sequel. We assume that X is a Banach space endowed with the norm � · � and B(X) stands for the Banach space of all bounded linear operators from X to itself. R + denotes the set of nonnegative real numbers. C b (R, X) denotes the space of all bounded continuous functions from R → X. Moreover, we denote by C 1 (R, X) the space of all functions R → X which have a continuous derivative on R. C 1 b (R, X) is the subspace of C 1 (R, X) consists of such functions satisfying It is clear that C 1 b (R, X) turns out to be a Banach space with the norm We first recall some properties of Hille-Yosida operators and extrapolation spaces. For more details, we refer to Amir and Maniar (1999), Agarwal et al. (2011), Prato andGrisvard (1982), Engel and Nagel (2001), Hille and Philips (1975), Nagel and Sinestrari (1994) and the references therein.
Definition 1 (Agarwal et al. 2011) Let A be a linear operator with domain D(A). We say that (A, D(A)) is a Hille-Yosida operator on X if there exist ω ∈ R and a positive constant M ≥ 1 such that (ω, ∞) ⊆ ρ(A) and sup{( − ω) n �( − A)� −n } ≤ M. The infinimum of such a ω is called the type of A. If the constant ω can be chosen smaller than zero, A is said to be of negative type.
From the Hille-Yosida theorem (Engel and Nagel 2001, Theorem II.3.8) we have the following result.

Lemma 1 Let (A, D(A)) be a Hille-Yosida operator on X
Let ∈ ρ(A). we define a norm on space X 0 by The completion of (X 0 , � · � −1 ) will be called the extrapolation space of X 0 associated with A 0 and will be denoted by X −1 . One can show easily that, T 0 (t) has a unique bounded linear extension T −1 (t) to X −1 . The operator family (T −1 (t)) t≥0 is a C 0 semigroup on X −1 , called the extrapolated semigroup of (T 0 (t)) t≥0 . The domain of its generator A −1 is equal to A 0 . The following results have been established in Amir and Maniar (1999), Agarwal et al. (2011), Nagel andSinestrari (1994).
Lemma 2 Under the previous conditions, the following properties are verified. .
) be a Hille-Yosida operator of negative ω-type. Then the following properties are valid.

Now, we recall a useful compactness criterion.
Let h : R → R be a continuous function such that h(t) ≥ 1 for all t ∈ R, and h(t) → ∞ as |t| → ∞. We consider the space endowed with the norm Lemma 4 (Henríquez and Lizama 2009) A subset K ⊆ C h (X) is a relatively compact set if it verifies the following conditions: Remark 1 (Henríquez and Lizama 2009) It is clear that C h (X) is a Banach space isometrically isomorphic with the space C 0 (R, X) consisting of functions that vanish at infinity.
Also, we recall some notations about Stepanov bounded functions and anti-periodic functions.
Definition 2 (Pankov 1990 Definition 3 (Pankov 1990 ). This is a Banach space with the norm Definition 4 (Aizicovici et al. 2001 Denote by P TA (R, X) the set of all anti-periodic functions.
Lemma 5 (N'Guérékata and Valmorin 2012) Let f n ∈ P TA (R, X), such that f n → f uniformly on R. Then f ∈ P TA (R, X).
Lemma 6 (N'Guérékata and Valmorin 2012) The P TA (R, X) is a Banach space equipped with the supnorm.

Definition 5
A continuous function f is said to be anti-periodic differentiable if f ∈ P TA (R, X) and f ′ ∈ P TA (R, X).
for all t ≥ s > −∞ is called a mild solution of semilinear differential equation We give the famous Schauder's fixed point theorem as follows: Lemma 9 (Smart 1980) Let D be a nonempty, closed, bounded, convex subset of a Banach space X. Let F : D → D be a continuous and compact operator, then the operator equation Fu = u has a fixed point in D.

Main results and proofs
In this section, we study the existence of anti-periodic (or anti-periodic differentiable) mild solutions of Eq.
(1). The following are the main results of this work. First, we list some assumptions.
The following theorem is needed to establish our next results.
Theorem 1 Let (A, D(A)) be a Hille-Yosida operator of negative ω-type and f satisfy the condition (H 1 ). The Ŵ : C b (R, X) → C b (R, X 0 ) is a linear operator and Ŵu(t) is defined by for every t ∈ R.
Proof Firstly, it is easily to see that Thus Ŵ is well defined and Ŵu is bounded. Secondly, for any t ∈ R, h ∈ R is small enough Thus, �Ŵu(t + h) − Ŵu(t)� → 0 as h → 0, which proves that Ŵu is continuous. Finally, It follows from (H 1 ) that for any u ∈ P TA (R, X 0 ) and for each t ∈ R Therefore, Ŵu is anti-periodic. The proof is complete.

Theorem 2 Let (A, D(A)) be a Hille-Yosida operator of negative ω-type and f satisfy the conditions (H 1 ) and (H 2 ). Then Eq. (1) has a unique anti-periodic mild solution provided that
Proof Define a operator Ŵ as in Theorem 1 on P TA (R, X 0 ) by for every t ∈ R. By Theorem 1, the operator Ŵ is well defined and maps P TA (R, X 0 ) into itself. Next, we prove that the operator Ŵ has a unique fixed point in P TA (R, X 0 ). Let u, v ∈ P TA (R, X 0 ), then .
1−e ω �L� S P < 1, it follows from the Banach contraction mapping principle that Ŵ admits a unique fixed point u ∈ P TA (R, X 0 ).
Moreover, one can see easily that u(t) satisfies the variation of constants formula that is u(t) is a mild solution of Eq. (1). The proof is complete. (1) has a unique anti-periodic mild solution provided that 0 < M 1−e ω �L� S 1 < 1.
Proof Define a operator Ŵ as in Theorem 1 on P TA (R, X 0 ) by for every t ∈ R. By Theorem 1, the operator Ŵ is well defined and maps P TA (R, X 0 ) into itself. Next, we prove that the operator Ŵ has a unique fixed point in P TA (R, X 0 ). Let u, v ∈ P TA (R, X 0 ), then For 0 < M 1−e ω �L� S 1 < 1, it follows from the Banach contraction mapping principle that Ŵ admits a unique fixed point u ∈ P TA (R, X 0 ). The proof is complete.
Let L(·) ≡ L, then the following result is now immediate.

Theorem 4 Let (A, D(A)) be a Hille-Yosida operator of negative ω-type. The function f satisfies the condition (H 1 ) and the Lipschitz condition
or all t ∈ R, x, y ∈ X 0 , where L > 0 is a constant. If ML −ω < 1 and ω < 0, then the Eq.
(1) has a unique anti-periodic mild solution.
Proof Similar as the proof of Theorem 3, the proof is omitted.
Theorem 5 Let (A, D(A)) be a Hille-Yosida operator of negative ω-type. The function f ∈ C 1 b (R, X 0 ) satisfies the condition (H 1 ) and (1) has a unique anti-periodic differentiable mild solution.
Proof Consider the nonlinear operator V : P ′ TA (R, X 0 ) → C b (R, X 0 ) given by Firstly, similar as the proof of Theorem 1, V ∈ C b (R, X 0 ) is well defined.
Finally, we take u, v ∈ P ′ TA (R, X 0 ), then we have which prove that V is a contraction. Hence, by using Banach contraction mapping principle that V admits a unique fixed point u ∈ P ′ TA (R, X 0 ). The proof is complete.
We next study the existence of anti-periodic mild solutions of Eq. (1) when the function f is not Lipschitz continuous. To abridge the text, We assume that f : R × X 0 → X satisfies the following conditions: (A 1 ) There is a continuous nondecreasing function W : R + → R + , such that Proof Let D = {u ∈ P TA (R, X 0 ) ∩ C h (X 0 ) : �u� ≤ r}, and D(t) := {Ŵu : u ∈ D}. We define the operator Ŵ by We divide the proof in several steps.
Step 2. The map Ŵ is continuous. In fact, for ǫ > 0, we take δ involved in condition (A 3 ). If u, v ∈ C h (X 0 ) and �u − v� h ≤ δ, then which shows the assertion.
Step 3. We will show that Ŵ is a compact operator. We will prove that D(t) := {Ŵu : u ∈ D} is a relatively compact subset of X 0 for each t ∈ R.
Thus, D(t) := {Ŵu : u ∈ D} is a relatively compact subset of X 0 for each t ∈ R.