Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 68, pp. 1–25. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu MONOTONE ITERATIVE METHOD FOR RETARDED EVOLUTION EQUATIONS INVOLVING NONLOCAL AND IMPULSIVE CONDITIONS XUPING ZHANG, PENGYU CHEN, YONGXIANG LI Abstract. In this article, we apply the perturbation technique and monotone iterative method in the presence of the lower and the upper solutions to dis- cuss the existence of the minimal and maximal mild solutions to the retarded evolution equations involving nonlocal and impulsive conditions in an ordered Banach space X u′(t) +Au(t) = f(t, u(t), ut), t ∈ [0, a], t 6= tk, u(t+k ) = u(t−k ) + Ik(u(tk)), k = 1, 2, . . . ,m, u(s) = g(u)(s) + ϕ(s), s ∈ [−r, 0], where A : D(A) ⊂ X → X is a closed linear operator and −A generates a strongly continuous semigroup T (t) (t ≥ 0) on X, a, r > 0 are two constants, f : [0, a]×X×C0 → X is Carathéodory continuous, 0 < t1 < t2 < · · · < tm < a are pre-fixed numbers, Ik ∈ C(X,X) for k = 1, 2, . . . ,m, ϕ ∈ C0 is a priori given history, while the function g : Ca → C0 implicitly defines a comple- mentary history, chosen by the system itself. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the minimal and maximal mild solutions, the existence of at least one mild solutions as well as the uniqueness of mild solution between the lower and the upper solutions. An example is given to illustrate the feasibility of our theoretical results. 1. Introduction Let X be a real Banach space with norm ‖ · ‖, and let a, r > 0 be two constants. For every t ∈ [0, a], we denote by Ct := C([−r, t], X) the Banach space of all the continuous functions from [−r, t] into X endowed with the sup-norm ‖u‖Ct = sup−r≤s≤t ‖u(s)‖. In this article, we use the perturbation technique and monotone iterative method in the presence of the lower and the upper solutions to discuss the existence of the existence of the minimal and the maximal mild solutions, the existence of at least one mild solutions as well as the uniqueness of the mild solution to the retarded evo- lution equations involving nonlocal and impulsive conditions in an ordered Banach 2010 Mathematics Subject Classification. 35K90, 47D06, 47H08. Key words and phrases. Evolution equation; delay; impulsive function; nonlocal condition; iterative method; measure of noncompactness. c©2020 Texas State University. Submitted August 11, 2019. Published July 1, 2020. 1 2 X. ZHANG, P. CHEN, Y. LI EJDE-2020/68 space X, u′(t) +Au(t) = f(t, u(t), ut), t ∈ [0, a], t 6= tk, u(t+k ) = u(t−k ) + Ik(u(tk)), k = 1, 2, . . . ,m, u(s) = g(u)(s) + ϕ(s), s ∈ [−r, 0], (1.1) where state u(·) takes values in the Banach space X endowed with norm ‖ · ‖; A : D(A) ⊂ X → X is a closed linear operator and −A generates a strongly continuous semigroup T (t) (t ≥ 0) on X; f : [0, a]×X×C0 → X is a Carathéodory continuous nonlinear function; 0 < t1 < t2 < · · · < tm < a are pre-fixed numbers, Ik ∈ C(X,X) is an impulsive function, k = 1, 2, . . . ,m, u(t+k ) and u(t−k ) represent the right and the left limits of u(t) at t = tk, respectively; ϕ ∈ C0 is a priori given history, while the function g : Ca → C0 implicitly defines a complementary history, chosen by the system itself; ut denotes the function in C0 defined as ut(τ) = u(t+τ) for τ ∈ [−r, 0] and ut(·) represent the time history of the state from the time t− r up to the present time t. The study of abstract nonlocal Cauchy problem was initiated by Byszewski [11]. It is demonstrated that the nonlocal problems have better effects in applications than the traditional Cauchy problems, differential equations with nonlocal condi- tions were studied by many authors and some basic results on nonlocal problems have been obtained, see [3, 12, 16, 24, 26, 37, 45, 48, 49] and the references therein. It is a well-established idea to model the evolution of some physical, biological and economic systems using delay evolution equations, in which the response of the system depends not only on the current state of the system, but also on the past history of the system. The history function g, which defines the initial nonlocal condition, is in fact a feedback operator which adjusts a part of the past — if the local initial condition ϕ is present — or even the whole past — if ϕ is absent — according to some precise future requirements. Namely, if we wish that the system behavior in the future be described by a function u belonging to a certain class, then it is left at the system disposal to choose its past wherefrom to start its evolution, in the nonlocal implicit form u(t) = g(u)(t) + ϕ(t) for each t ∈ [−r, 0]. For more details on this topic, see for example, the books of Hale and Verduyn Lunel [30], Kolmanovskii and Myshkis [35] and the papers of Chuong and Ke [18], Kamaljeet and Bahuguna [34] and Travis and webb [43]. The dynamics of evolving processes is often subjected to abrupt changes at cer- tain moments such as shocks, harvesting and natural disasters. Often these short term perturbations are treated as having acted instantaneously or in the form of impulses. One of the emerging branches of the study associated with impulses is the theory of impulsive differential equations, which describes processes which ex- perience a sudden change in their states at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical, biological, population and dynamics, engineering and economics. The theory of impulsive differential equations has emerged as an important area of re- search in the previous decades. For more details on differential equations with impulses, one can see the monographs of Benchohra, Henderson and Ntouyas [9], Lakshmikantham, Bainov and Simeonov [36] and the papers of Benchohra and Hammouche [1], Ahmed [2], Abada, Barreira and Valls [8], Bonottoa et al. [10], Guo and Liu [29], Li and Liu [39] and Qian, Chen and Sun [42], where numer- ous properties of their solutions are studied and detailed bibliographies are given. Particularly, there has been a significant development in the theory of impulsive EJDE-2020/68 RETARDED EVOLUTION EQUATIONS 3 evolution equations with nonlocal conditions in Banach spaces. Liang, Liu and Xiao [37, 38] combined impulsive conditions and nonlocal conditions, and investigated the nonlocal impulsive evolution equation in Banach spaces. Later, Balachandran, Kiruthika and Trujillo [5], Chen, Li and Yang [17], Debbouche and Baleanu [19], Fan and Li [26], Wang and Wei [47], Yan [50] studied the impulsive evolution equa- tion with nonlocal conditions. Moreover, Abada, Benchohra and Hammouche [1], Cardindi and Rubbioni [13], Chang, Anguraj and Karthikeyan [14], Chuong and Ke [18], Fu and Cao [27], Ji and Li [33] studied the impulsive evolution inclusion with nonlocal conditions. We mention that in 2012, Chuong and Ke [18] studied the retarded evolution inclusions involving nonlocal and impulsive conditions u′(t) +Au(t) ∈ F (t, u(t), ut), t ∈ [0, a], t 6= tk, u(t+k ) = u(t−k ) + Ik(u(tk)), k = 1, 2, . . . ,m, u(s) + g(u)(s) = ϕ(s), s ∈ [−r, 0], (1.2) where X is a Banach space, F : [0, a] × X × C0 → P (X) is a multivalued map, P (X) stands for the collection of all nonempty subsets of X. A is a closed linear operator on X. By using the fixed point theory for multi-valued maps and the theory of differential inclusions, the authors obtain the existence of mild solutions for nonlocal problem (1.2). Furthermore, by applying corresponding measure of noncompactness estimates, they also proved the continuity of the solution map, which demonstrates that the solution set depends continuously on initial data. But so far we have not seen relevant papers that study delay evolution equations involving nonlocal and impulsive conditions by applying the iterative method, per- turbation technique and the method of lower and upper solutions. It is well known that the monotone iterative technique in the presence of the lower and the upper solutions is an important method for seeking solutions of differential equations in abstract spaces. The most advantage by using the iterative method based on lower and upper solutions is that it not only provides a method to obtain the existence of extremal mild solutions, but also yields iterative sequences of lower and upper approximate solutions that converge to the minimal and maximal mild solutions between the lower and upper solutions. The iterative sequences are very useful in numerical calculation, which provide a computing rule in computer simulation. Early on, Du and Lakshmikantham [21] investigated the existence of extremal solu- tions to the initial value problem of ordinary differential equations without impulse by using the method of the lower and the upper solutions coupled with the mono- tone iterative technique. Latter, Guo and Liu [29], Li and Liu [39] developed the iterative method for ordinary differential equations with instantaneous impulses in Banach spaces. Recently, the iterative method has been extended to evolution equations in ordered Banach spaces, we refer to the papers by EI-Gebeily, O’Regan and Nieto [23] and Wang and Wang [46] for evolution equations with classical initial vlaue conditions, and to the paper by Chen and Li [15] and Chen, Li and Yang [17] for evolution equations with impulses in Banach spaces. Inspired by the above-mentioned aspects, in the present paper we will combine these earlier works and extend the study to the retarded evolution equations in- volving nonlocal and impulsive conditions (1.1), which is more general than those in many previous publications. By combining the theory of semigroups of linear operators and the method of the lower and the upper solutions coupled with the 4 X. ZHANG, P. CHEN, Y. LI EJDE-2020/68 monotone iterative technique, we construct two monotone iterative sequences, and prove that the sequences monotonically converge to the minimal and the maximal mild solutions of problem (1.1), respectively, under the reasonably weak conditions on the semigroup T (t) (t ≥ 0), nonlinear function f , impulsive function Ik for k = 1, 2, . . . ,m and nonlocal term g. The work of this paper has three wedges: firstly, we will extend the study of impulsive Cauchy problems with nonlocal initial conditions to retarded evolution equations involving nonlocal and impulsive conditions. Secondly, we will obtain the existence theorems of minimal and the maximal mild solutions and the uniqueness theorems of mild solution to the nonlocal problem (1.1) under weaker conditions on the semigroup T (t) (t ≥ 0), nonlinear function f , impulsive function Ik for k = 1, 2, . . . ,m and nonlocal term g. Lastly, the perturbation technique and mono- tone iterative method are extended to study the retarded evolution equations with nonlocal and impulsive conditions in ordered Banach space. Moreover, even for cor- responding retarded impulsive Cauchy problems without nonlocal initial conditions, the results here are new. Remark 1.1. As the reader can see, the hypotheses on nonlinear function f , impulsive function Ik (k = 1, 2, . . . ,m) and nonlocal term g in our theorems are reasonably weak and different from those in many previous papers such as [5, 19, 26, 47, 50]. Furthermore, the techniques in the proofs of our theorems are essentially different from those used in [18]. The outline of this paper is as follows. In section 2, notation and preliminaries are introduced, which are used throughout this paper. In Section 3, we obtained the existence of extremal mild solutions as well as mild solutions for nonlocal problem (1.1) under the situation that the semigroup T (t) (t ≥ 0) and the nonlocal function g are compact. The existence of extremal mild solutions and the uniqueness of mild solution for nonlocal problem (1.1) are obtained under the situation that the semigroup T (t) (t ≥ 0) and nonlocal function g are not compact in Section 4. Finally, an concrete example is given to illustrate the feasibility of our abstract results. 2. Preliminaries We begin by giving some notation. Let X be an ordered Banach space with norm ‖ · ‖ and partial order “ ≤ ” with positive cone P = {u ∈ X | u ≥ θ} (θ is the zero element of X), which defines a partial ordering in X by u ≤ v if and only if v − u ∈ P . If u ≤ v and u 6= v, we say u < v. The cone P is called normal if there exists a positive constant N such that θ ≤ u ≤ v implies ‖u‖ ≤ N‖v‖, in which N is called normal constant. Evidently, for any t ∈ [0, a], Ct = ([−r, t], X) is also an ordered Banach space whose partial order “ ≤ ” induced by the positive cone PCt := {u ∈ Ct | u(s) ≥ θ, s ∈ [−r, t]}, and PCt is also a normal cone with the same normal constant N . Denote by PC([−r, a], X) = { u : [−r, a]→ X : u is continuous for t 6= tk, left continuous at t = tk and u(t+k ) exists for k = 1, 2, . . . ,m } EJDE-2020/68 RETARDED EVOLUTION EQUATIONS 5 be a piecewise continuous function space. It is easy to see that PC([−r, a], X) is a Banach space endowed with the supremum-norm ‖u‖PC = sup t∈[−r,a] ‖u(t)‖, ∀u ∈ PC([−r, a], X). (2.1) Evidently, PC([−r, a], X) is also an ordered Banach space with the partial order “ ≤ ” induced by the positive cone KPC = {u ∈ PC([−r, a], X) | u(t) ≥ θ, t ∈ [−r, a]}. KPC is also a normal cone with the same normal constant N . For v, w ∈ PC([−r, a], X) with v ≤ w, we use [v, w] to denote the order interval {u ∈ PC([−r, a], X) | v ≤ u ≤ w} in PC([−r, a], X), and [v(t), w(t)] to denote the order interval {u ∈ PC([−r, a], X) | v(t) ≤ u(t) ≤ w(t), t ∈ [−r, a]} in X. In the following, we denote J0 = [−r, 0], J1 = [0, t1], Jk = (tk−1, tk], k = 2, . . . , m + 1, tm+1 = a, I ′ = [−r, a]\{t1, t2, . . . , tm} and I ′′ = [−r, a]\{0, t1, t2, . . . , tm}, and useX1 to denote the Banach spaceD(A) with the graph norm ‖·‖1 = ‖·‖+‖A·‖. An abstract function u ∈ PC([−r, a], X)∩C1(I ′′, X)∩C(I ′, X1) is called a solution of nonlocal problem (1.1) if u(t) satisfies all the equalities in (1.1). Definition 2.1. If a function u ∈ PC([−r, a], X) ∩ C1(I ′′, X) ∩ C(I ′, X1) satisfies u′(t) +Au(t) ≤ f(t, u(t), ut), t ∈ [0, a], t 6= tk, u(t+k ) ≤ u(t−k ) + Ik(u(tk)), k = 1, 2, . . . ,m, u(s) ≤ g(u)(s) + ϕ(s), s ∈ [−r, 0], (2.2) we call it a lower solution of nonlocal problem (1.1); if all the inequalities in (2.2) are reversed, we call it an upper solution of nonlocal problem (1.1). Definition 2.2. A function f : [0, a] × X × C0 → X is said to be Carathéodory continuous provided that (i) for all (u, v) ∈ X × C0, f(·, u, v) : [0, a]→ X is measurable, (ii) for a.e. t ∈ [0, a], f(t, ·, ·) : X × C0 → X is continuous. Throughout this paper, let A : D(A) ⊂ X → X be a closed linear operator and let −A generate a strongly continuous semigroup T (t) (t ≥ 0) on ordered Banach space X. Then there exist constants C1 ≥ 1 and δ ∈ R such that ‖T (t)‖ ≤ C1e δt, t ≥ 0. (2.3) Denote L(X) be the Banach space of all bounded linear operators from X to X equipped with its natural topology. From (2.3) we know that C := sup t∈[0,a] ‖T (t)‖L(X) ≥ 1 (2.4) is a finite number. Definition 2.3. A function u ∈ PC([−r, a], X) is said to be a mild solution of nonlocal problem (1.1) if it satisfies the following equation u(t) =  g(u)(t) + ϕ(t), t ∈ [−r, 0]; T (t)[g(u)(0) + ϕ(0)] + ∫ t 0 T (t− s)f(s, u(s), us)ds, + ∑ 0 0. Definition 2.5. A strongly continuous semigroup T (t) (t ≥ 0) in X is said to be equicontinuous, if T (t) is continuous in the operator norm for every t > 0. Definition 2.6. A strongly continuous semigroup T (t) (t ≥ 0) in X is said to be positive, if order inequality T (t)u ≥ θ holds for each u ≥ θ, u ∈ X and t ≥ 0. One can easily to see that for any constant M ≥ 0, −(A+MI) also generates a strongly continuous semigroup S(t) = e−MtT (t) (t ≥ 0) in X, and sup t∈[0,a] ‖S(t)‖L(X) = sup t∈[0,a] ‖e−MtT (t)‖L(X) = C ≥ 1. (2.6) Therefore, S(t) (t ≥ 0) is a positive strongly continuous semigroup if T (t) (t ≥ 0) is a positive strongly continuous semigroup, S(t) (t ≥ 0) is a compact semigroup if T (t) (t ≥ 0) is a compact semigroup, S(t) (t ≥ 0) is an equicontinuous semigroup if T (t) (t ≥ 0) is an equicontinuous semigroup. For more details about the properties of the operator semigroups and positive strongly continuous semigroup, we refer to Banasiak and Arlotti [7], Henry [32], Pazy [41] and Vrabie [44]. Next, we recall some basic definitions and properties about Kuratowski measure of noncompactness that will be used in the proof of our main results. Definition 2.7 ([20, 6]). The Kuratowski measure of noncompactness α(·) defined on bounded set S of Banach space X is α(S) := inf{δ > 0 : S = ∪mi=1Si with diam(Si) ≤ δ for i = 1, 2, . . . ,m}. (2.7) It is easy to know from Definition 2.7 that 0 ≤ α(S) < ∞. The following properties about the Kuratowski measure of noncompactness are well known. Lemma 2.8 ([6, 20]). Let E be a Banach space and S, U ⊂ E be bounded. The following properties are satisfied: (i) α(S) = 0 if and only if S is compact, where S means the closure hull of S; (ii) α(S) = α(S) = α(convS), where convS means the convex hull of S; (iii) α(λS) = |λ|α(S) for any λ ∈ R; (iv) S ⊂ U implies α(S) ≤ α(U); (v) α(S ∪ U) = max{α(S), α(U)}; (vi) α(S + U) ≤ α(S) + α(U), where S + U = {x | x = y + z, y ∈ S, z ∈ U}; (vii) If the map Q : D(Q) ⊂ E → X is Lipschitz continuous with constant k, then α(Q(V )) ≤ kα(V ) for any bounded subset V ⊂ D(Q), where X is another Banach space. To introduce the useful lemmas which will be used in our argument, we use α(·) and αPC(·) to denote the Kuratowski measure of noncompactness on the bounded set of X and PC([−r, a], X), respectively. For any D ⊂ PC([−r, a], X) and t ∈ [0, a], set D(t) = {u(t) | u ∈ D} and Dt = {ut | u ∈ D}, then D(t) ⊂ X and Dt ⊂ C0. If D ⊂ PC([−r, a], X) is bounded, then for any t ∈ [0, a], D(t) ⊂ X and Dt ⊂ C0 are bounded, and α(D(t)) ≤ αPC(D). For details about the definition and properties of the measure of noncompactness, we refer to the monographs by Ayerbe, Domı́nguez and López [4], Banas̀ and Goebel [6], Deimling [20]. Now, we give the following lemmas about the measure of noncompactness which are used further in this paper. EJDE-2020/68 RETARDED EVOLUTION EQUATIONS 7 Lemma 2.9 ([29]). Let E be a Banach space. Assume that Ω ⊂ E is a bounded closed and convex set on E, the operator Q : Ω → Ω is k-set-contractive. Then Q has at least one fixed point in Ω. Lemma 2.10 ([31]). Let X be a Banach space, and D = {un}∞n=1 ⊂ PC([0, a], X) be a bounded and countable set. Then α(D(t)) is Lebesgue integrable on [0, a], and α ({∫ a 0 un(t)dt | n ∈ N }) ≤ 2 ∫ a 0 α(D(t))dt. Lemma 2.11 ([16]). Let X be a Banach space, and let D ⊂ X be bounded. Then there exists a countable set D∗ ⊂ D, such that α(D) ≤ 2α(D∗). Lemma 2.12 ([28]). Let P be a normal cone of the ordered Banach space X and v0, w0 ∈ X with v0 ≤ w0. Suppose that Q : [v0, w0] → X is a nondecreasing strict set-contraction operator such that v0 ≤ Qv0 and Qw0 ≤ w0. Then Q has a minimal fixed point u and a maximal fixed point u in [v0, w0]; moreover, vn → u and wn → u as n→∞, where vn = Qvn−1 and wn = Qwn−1 (n = 1, 2, . . . ) which satisfy v0 ≤ v1 ≤ · · · ≤ vn ≤ · · · ≤ u ≤ u ≤ · · · ≤ wn ≤ · · · ≤ w1 ≤ w0. 3. Case g compact In this section, we discuss the existence of extremal mild solutions as well as the existence of at least one mild solutions for nonlocal problem (1.1) under the situation that T (t) (t ≥ 0) is a compact strongly continuous semigroup and the nonlocal function g : PC([−r, a], X)→ C0 is also compact. Theorem 3.1. Let X be an ordered Banach space, whose positive cone P is normal, and let A : D(A) ⊂ X → X be a closed linear operator and the positive strongly continuous semigroup T (t) (t ≥ 0) generated by −A be compact on X. Assume that the nonlinear function f : [0, a] × X × C0 → X is Carathéodory continuous, Ik ∈ C(X,X) for k = 1, 2, . . . ,m and the nonlocal function g : PC([−r, a], X) → C0 is continuous and compact as well as the nonlocal problem (1.1) has a lower solution v(0) ∈ PC([−r, a], X)∩C1(I ′′, X)∩C(I ′, X1) and an upper solution w(0) ∈ PC([−r, a], X) ∩ C1(I ′′, X) ∩ C(I ′, X1) with v(0) ≤ w(0). Suppose also that the following conditions are satisfied: (H1) There exists a constant M > 0 such that f(t, u2, v2)− f(t, u1, v1) ≥ −M(u2 − u1), for ∀ t ∈ [0, a], u1, u2 ∈ X and v1, v2 ∈ C0 with v(0)(t) ≤ u1 ≤ u2 ≤ w(0)(t) and (v(0))t ≤ v1 ≤ v2 ≤ (w(0))t; (H2) The impulsive function Ik(·) satisfies Ik(u1) ≤ Ik(u2), k = 1, 2, . . . ,m, for any t ∈ [0, a], and v(0)(t) ≤ u1 ≤ u2 ≤ w(0)(t); (H3) The nonlocal function g(u) is increasing on order interval [v(0), w(0)]. Then nonlocal problem (1.1) has a minimal mild solution u and a maximal mild solution u between v(0) and w(0), which can be obtained by a monotone iterative procedure starting from v(0) and w(0), respectively. 8 X. ZHANG, P. CHEN, Y. LI EJDE-2020/68 Proof. It is easy to see that nonlocal problem (1.1) is equivalent to the following retarded evolution equations involving nonlocal and impulsive conditions u′(t) +Au(t) +Mu(t) = f(t, u(t), ut) +Mu(t), t ∈ [0, a], t 6= tk, u(t+k ) = u(t−k ) + Ik(u(tk)), k = 1, 2, . . . ,m, u(s) = g(u)(s) + ϕ(s), s ∈ [−r, 0], (3.1) for any constant M > 0. We consider the operatorQ : [v(0), w(0)]→ PC([−r, a], X) defined by (Qu)(t) =  g(u)(t) + ϕ(t), t ∈ [−r, 0]; S(t)[g(u)(0) + ϕ(0)] + ∫ t 0 S(t− s)[f(s, u(s), us) +Mu(s)]ds + ∑ 0 0, such that ‖f(s, u(s), us) +Mu(s)‖ ≤M1, s ∈ [0, t], t ∈ [0, a]. (3.6) EJDE-2020/68 RETARDED EVOLUTION EQUATIONS 9 Combining with (2.6), (3.2), (3.4)-(3.6) and Lebesgue’s dominated convergence the- orem, we know that for any t ∈ [0, a], ‖(Qu(n))(t)− (Qu)(t)‖ ≤ C‖g(u(n))(0)− g(u)(0)‖ + C ∫ t 0 ‖f(s, u(n)(s), (u(n))s) +Mu(n)(s)− f(s, u(s), us)−Mu(s)‖ds + C ∑ 0 0 such that for any v(n−1) ∈ B∗, ‖f(s, v(n−1)(s), (v(n−1))s) +Mv(n−1)(s)‖ ≤M2, s ∈ [0, t], t ∈ [0, a]. (3.13) Hence, for t ∈ (0, a] and 0 < ε < t, the operator (Qε1vn−1)(t) = S(t)[g(v(n−1))(0) + ϕ(0)] + ∫ t−ε 0 S(t− s)[f(s, v(n−1)(s), (v(n−1))s) +Mv(n−1)(s)]ds = S(t)[g(v(n−1))(0) + ϕ(0)] + S(ε) ∫ t−ε 0 S(t− s− ε)[f(s, v(n−1)(s), (v(n−1))s) +Mv(n−1)(s)]ds (3.14) is precompact in X since S(t) is compact for t > 0. By (2.6), (3.11), (3.13) and (3.14), we obtain ‖(Q1v (n−1))(t)−Qε1v(n−1)(t)‖ = ∫ t t−ε ‖S(t− s)[f(s, v(n−1)(s), v(n−1)s ) +Mv(n−1)(s)]‖ds ≤ CM2ε, which means that there exists precompact set {(Qε1v(n−1))(t) | v(n−1) ∈ B∗} suffi- ciently close to the set {(Q1v (n−1))(t) | v(n−1) ∈ B∗} for every t ∈ (0, a]. There- fore, by the total boundedness we know that for t ∈ (0, a], the set {(Q1v (n−1))(t) | v(n−1) ∈ B∗} is precompact in X. On the other hand, for any v(n−1) ∈ B∗ and k = 1, 2, . . . ,m, by the assumption (H2), we have Ik(v(0)(tk)) ≤ Ik(v(n−1)(tk)) ≤ Ik(w(0)(tk)). (3.15) EJDE-2020/68 RETARDED EVOLUTION EQUATIONS 11 By the normality of the cone P and (3.15), there exists a constant M3 > 0 such that ‖Ik(v(n−1)(tk))‖ ≤M3, v(n−1) ∈ B∗, k = 1, 2, . . . ,m. Therefore, {(Q2v (n−1))(t) | v(n−1) ∈ B∗} is precompact in X since the operator S(t) is compact for t > 0. Therefore, the set {v(n)(t)} = {(Qv(n−1))(t) | v(n−1) ∈ B∗} is precompact in X for any t ∈ [−r, a]. Hence, {v(n)(t)} has a convergent subse- quence. Combining this with the monotonicity (3.10), we easily prove that {v(n)(t)} itself is convergent, i.e., lim n→∞ v(n)(t) = u(t) for t ∈ [−r, a]. Similarly, we can prove lim n→∞ w(n)(t) = u(t) for t ∈ [−r, a]. Obviously, {vn(t)} ⊂ PC([−r, a], X), and u(t) is bounded integrable when t belongs to [−r, 0] and [0, a], respectively. For any t ∈ [−r, a], we know from (3.2) that v(n)(t) = (Qv(n−1))(t) =  g(v(n−1))(t) + ϕ(t), t ∈ [−r, 0], S(t)[g(v(n−1))(0) + ϕ(0)] + ∫ t 0 S(t− s)[f(s, v(n−1)(s), (v(n−1))s) +Mv(n−1)(s)]ds+ ∑ 0 0 as well as the compactness of nonlocal function g, we get that the set {S(t)[g(u)(0) + ϕ(0)] : u ∈ [v(0), w(0)]} EJDE-2020/68 RETARDED EVOLUTION EQUATIONS 13 is precompact in X. For t ∈ (0, a] and 0 < ε < t, the set{∫ t−ε 0 S(t− s)[f(s, u(s), us) +Mu(s)]ds : u ∈ [v(0), w(0)] } = { S(ε) ∫ t−ε 0 S(t− s− ε)[f(s, u(s), us) +Mu(s)]ds : u ∈ [v(0), w(0)] } (3.16) is precompact in X since S(t) is compact for t > 0. Furthermore, we know that for every u ∈ [v(0), w(0)],∫ t−ε 0 S(t− s)[f(s, u(s), us) +Mu(s)]ds → ∫ t 0 S(t− s)[f(s, u(s), us) +Mu(s)]ds as ε→ 0. (3.17) Combining (3.16) and (3.17) with the total boundedness, we know that the set{∫ t 0 S(t− s)[f(s, u(s), us) +Mu(s)]ds : u ∈ [v(0), w(0)] } is precompact in X. Therefore, for each t ∈ [0, a], Π2(t) is precompact in X. Next, we prove the equicontinuity of Π2. For any u ∈ [v(0), w(0)] and s ∈ [0, t], t ∈ [0, a], by the assumption (H1), we have f(s, v(0)(s), (v(0))s) +Mv(0)(s) ≤ f(s, u(s), us) +Mu(s) ≤ f(s, w(0)(s), (w(0))s) +Mw(0)(s). By the normality of the cone P , there exists M4 > 0 such that ‖f(s, u(s), us) +Mu(s)‖ ≤M4, s ∈ [0, t], t ∈ [0, a], u ∈ [v(0), w(0)]. (3.18) Therefore, by (2.6), (3.18) and the definition of the set Π2, we obtain that for 0 ≤ t′ < t′′ ≤ a and u ∈ [v(0), w(0)],∥∥∥S(t′′)[g(u)(0) + ϕ(0)]− S(t′)[g(u)(0) + ϕ(0)] + ∫ t′ 0 [S(t′′ − s)− S(t′ − s)] · [f(s, u(s), us) +Mu(s)]ds + ∫ t′′ t′ S(t′′ − s)[f(s, u(s), us) +Mu(s)]ds ∥∥∥ ≤ C‖S(t′′ − t′)[g(u)(0) + ϕ(0)]− [g(u)(0) + ϕ(0)]‖ +M4 ∫ t′ 0 ‖S(t′′ − s)− S(t′ − s)‖ds+ CM4(t′′ − t′) ≤ C‖S(t′′ − t′)[g(u)(0) + ϕ(0)]− [g(u)(0) + ϕ(0)]‖ +M4 ∫ t′ 0 ‖S(t′′ − t′ + s)− S(s)‖ds+ CM4(t′′ − t′). (3.19) Since the semigroup S(t) (t ≥ 0) is strongly continuous for t ≥ 0 and is continuous in the uniform operator topology for t > 0, then it is easy to see that the right hand of (3.19) tends to zero independently of u ∈ [v(0), w(0)] as t′′ − t′ → 0, which means that the functions in Π2 are equicontinuous. Therefore, by the Arzela-Ascoli theorem one can easily to justify that the set Π2 is precompact. 14 X. ZHANG, P. CHEN, Y. LI EJDE-2020/68 Now, we are in the position to prove the precompactness of Π3. From the definition of the set Π3 combined with the fact that the interval [0, a] is divided into finite subintervals by tk, k = 1, 2, . . . ,m, we only need to prove that the set {S(· − t1)I1(u(t1)) : · ∈ [t1, t2], u ∈ [v(0), w(0)]} is precompact in C([t1, t2], X), as the cases for other subintervals are the same. By the fact that the semigroup S(t) (t ≥ 0) is compact for every t > 0 we know that for each t ∈ [t1, t2], the set {S(t− t1)I1(u(t1)) : t ∈ [t1, t2], u ∈ [v(0), w(0)]} is precompact in X. For t1 ≤ t′ < t′′ ≤ t2 and u ∈ [v(0), w(0)], by (2.6) we obtain ‖S(t′′ − t1)I1(u(t1))− S(t′ − t1)I1(u(t1))‖ = ‖S(t′ − t1)[S(t′′ − t′)− S(0)]I1(u(t1))‖ ≤ C‖[S(t′′ − t′)− S(0)]I1(u(t1))‖. The above inequality combined with the strong continuity of the operator S(t) for t > 0 imply that the functions in {S(· − t1)I1(u(t1)) : · ∈ [t1, t2], u ∈ [v(0), w(0)]} are equicontinuous. Therefore, an application of the Arzela-Ascoli theorem justifies the precompactness of the set {S(· − t1)I1(u(t1)) : · ∈ [t1, t2], u ∈ [v(0), w(0)]} in C([t1, t2], X). Hence, using a completely similar method for the case k = 2, 3, . . . ,m, one can prove that the set Π3 is precompact. Hence, by the above discuss we know that Q : [v(0), w(0)] → [v(0), w(0)] is a compact operator, and therefore a completely continuous operator. Therefore, the famous Schauder’s fixed point theorem implies that the operator Q has at least one fixed point in ordered interval [v(0), w(0)], which gives rise to a mild solution of problem (1.1). This completes the proof. � Remark 3.4. In Theorems 3.1 and 3.2, we obtained the existence of minimal mild solution and maximal mild solution (two mild solutions) for nonlocal problem (1.1) by using the monotone iterative method. In Theorems 3.3, we obtained the existence of at least one mild solution by utilizing Schauder’s fixed point theorem. The results as well as the proof method used are all different. 4. Case g not compact In this section, we discuss the existence of extremal mild solutions and the uniqueness of mild solution for nonlocal problem (1.1) under the situation that the nonlocal function g : PC([−r, a], X) → C0 is not compact. Furthermore, the strongly continuous semigroup T (t) (t ≥ 0) generated by −A does not have to be compact. From the definition of regular cone in ordered Banach spaces and the proof of Theorem 3.1, we can easily obtain the following result in a general ordered Banach space whose positive cone is regular. EJDE-2020/68 RETARDED EVOLUTION EQUATIONS 15 Theorem 4.1. Let X be an ordered Banach space, whose positive cone P is regular, and let A : D(A) ⊂ X → X be a closed linear operator and −A generates a positive strongly continuous semigroup T (t) (t ≥ 0) on X. Assume that the nonlinear function f : [0, a] × X × C0 → X is Carathéodory continuous, Ik ∈ C(X,X) for k = 1, 2, . . . ,m and the nonlocal function g : PC([−r, a], X)→ C0 is continuous. If the nonlocal problem (1.1) has a lower solution v(0) ∈ PC([−r, a], X)∩C1(I ′′, X)∩ C(I ′, X1) and an upper solution w(0) ∈ PC([−r, a], X)∩C1(I ′′, X)∩C(I ′, X1) with v(0) ≤ w(0), and the conditions (H1)–(H3) are satisfied. Then nonlocal problem (1.1) has a minimal mild solution u and a maximal mild solution u between v(0) and w(0), which can be obtained by a monotone iterative procedure starting from v(0) and w(0), respectively. In the application of differential equations, we often choose Hilbert space, reflex- ive space and Lp(Ω) space for 1 ≤ p < ∞ as working spaces, which are all weakly sequentially complete spaces. Therefore, it is interesting to discuss the existence of extremal mild solutions for nonlocal problem (1.1) in weakly sequentially complete space. Theorem 4.2. Let X be an ordered Banach space, whose positive cone P is regular, and let A : D(A) ⊂ X → X be a closed linear operator and −A generates a positive strongly continuous semigroup T (t) (t ≥ 0) on X. Assume that the nonlinear function f : [0, a] × X × C0 → X is Carathéodory continuous, Ik ∈ C(X,X) for k = 1, 2, . . . ,m and the nonlocal function g : PC([−r, a], X)→ C0 is continuous. If the nonlocal problem (1.1) has a lower solution v(0) ∈ PC([−r, a], X)∩C1(I ′′, X)∩ C(I ′, X1) and an upper solution w(0) ∈ PC([−r, a], X)∩C1(I ′′, X)∩C(I ′, X1) with v(0) ≤ w(0), and the conditions (H1)–(H3) are satisfied. Then nonlocal problem (1.1) has a minimal mild solution u and a maximal mild solution u between v(0) and w(0), which can be obtained by a monotone iterative procedure starting from v(0) and w(0), respectively. Proof. By the proof of Theorem 3.1, we know that the operator Q defined by (3.2) maps ordered interval [v(0), w(0)] to [v(0), w(0)] is continuous and monotone increas- ing. Furthermore, if the conditions (H1)–(H3) are satisfied, then the sequences {v(n)} and {w(n)} defined by (3.9) satisfying the monotonicity (3.10). Therefore, for any t ∈ [−r, 0], the sequences {v(n)(t)} and {w(n)(t)} are monotone and order- bounded sequences in X. Noticing that X is a weakly sequentially complete Banach space, from [22, Theorem 2.2] we know that {v(n)(t)} and {w(n)(t)} are precompact in X. Combining this fact with the monotonicity (3.10), it follows that {v(n)(t)} and {w(n)(t)} are convergent in X. Denote lim n→∞ v(n)(t) = u(t) for t ∈ [−r, a]. Similarly, we can prove that lim n→∞ w(n)(t) = u(t) for t ∈ [−r, a]. Using a completely similar method to the one we used to prove Theorem 3.1 we can easily to prove that u and u are minimal and maximal mild solutions of nonlocal problem (1.1) in [v(0), w(0)], and u and u can be obtained by the iterative scheme (3.10) starting from v(0) and w(0), respectively. This completes the proof. � 16 X. ZHANG, P. CHEN, Y. LI EJDE-2020/68 Remark 4.3. In Theorems 4.1 and 4.2, we only assume that positive semigroup T (t) (t ≥ 0) is strongly continuous, the nonlinear function f , the impulsive function Ik and the nonlocal term g are continuous and satisfy some monotonicity conditions, which are more easily to satisfied. Therefore, Theorem 4.1 and Theorem 4.2 in this paper essentially extends the main results of the previous research in several ways, as far as the mild solution of the retarded evolution equations involving nonlocal and impulsive conditions (1.1) is concerned, by dropping the compactness and Lipschitz continuity of the impulsive item from the hypotheses. This distinguishes the present paper from earlier works on impulsive Cauchy problems. Theorem 4.4. Let X be an ordered Banach space, whose positive cone P is normal, and let A : D(A) ⊂ X → X be a closed linear operator and −A generates a positive and equicontinuous strongly continuous semigroup T (t) (t ≥ 0) on X. Assume that the nonlinear function f : [0, a] × X × C0 → X is Carathéodory continuous, Ik ∈ C(X,X) for k = 1, 2, . . . ,m and the nonlocal function g : PC([−r, a], X) → C0 is continuous. If the nonlocal problem (1.1) has a lower solution v(0) ∈ PC([−r, a], X)∩ C1(I ′′, X) ∩ C(I ′, X1) and an upper solution w(0) ∈ PC([−r, a], X) ∩ C1(I ′′, X) ∩ C(I ′, X1) with v(0) ≤ w(0), and the conditions (H1)–(H2) and the condition (H4) There exist positive constants Lf , Lg and Lk (k = 1, 2, . . . ,m) satisfying 4aC(2Lf +M) + C ( 2 M∑ k=1 Lk + Lg ) < 1 such that ‖g(u)− g(v)‖C0 ≤ Lg‖u− v‖PC , u, v ∈ [v(0), w(0)], α ({ f(t, u(n)(t), (u(n))t) }) ≤ Lf [ α ({ u(n)(t) }) + sup −r≤τ≤0 α ({ u(n)(t+ τ) })] , ∀t ∈ [0, a] α ({ Ik(u(n)(tk)) }) ≤ Lkα ({ u(n)(tk) }) , k = 1, 2, . . . ,m, where {u(n)} ⊂ [v(0), w(0)] is countable and increasing or decreasing mono- tonic set and {(u(n))t} ⊂ C0, are satisfied. Then nonlocal problem (1.1) has a minimal mild solution u and a maximal mild solution u between v(0) and w(0); moreover, v(n)(t)→ u(t), w(n)(t)→ u(t) uniformly for t ∈ [−r, a] as n→∞, where v(n)(t) = (Qv(n−1))(t), w(n)(t) = (Qw(n−1))(t), which satisfy v(0)(t) ≤ v(1)(t) ≤ · · · ≤ v(n)(t) ≤ · · · ≤ u(t) ≤ u(t) ≤ . . . ≤ w(n)(t) ≤ · · · ≤ w(1)(t) ≤ w(0)(t), ∀t ∈ [−r, a]. Proof. By the proof of Theorem 3.1, we know that the operator Q defined by (3.2) maps [v(0), w(0)] to [v(0), w(0)] is continuous and monotone increasing. In the following, we will prove that the operator Q : [v(0), w(0)] → [v(0), w(0)] is strict EJDE-2020/68 RETARDED EVOLUTION EQUATIONS 17 set-contraction. For this purpose, denote by (Q1u)(t) = { g(u)(t) + ϕ(t), t ∈ [−r, 0]; S(t)[g(u)(0) + ϕ(0)], t ∈ [0, a], (Q2u)(t) =  0, t ∈ [−r, 0];∫ t 0 S(t− s)[f(s, u(s), us) +Mu(s)]ds + ∑ 0 v. In accordance with (H3) and (H5), we arrive at θ ≤ g(u)(t)− g(v)(t) ≤ Lg(u(t)− v(t)), t ∈ [−r, 0]. By this and the normality of cone P , we have ‖g(u)(t)− g(v)(t)‖ ≤ NLg‖u(t)− v(t)‖ ≤ NLg‖u− v‖PC , t ∈ [−r, 0]. Hence ‖g(u)− g(v)‖C0 ≤ Lg‖u− v‖PC , where Lg = NLg. On the other hand, for t ∈ [0, a], let {u(n)} ⊂ [v(0), w(0)] be an increasing se- quences. For m,n ∈ N with m > n, from the conditions (H1), (H2) and (H5), we obtain θ ≤ f(t, u(m)(t), (u(m))t)− f(t, u(n)(t), (u(n))t) +M [u(m)(t)− u(n)(t)] ≤ (M +Mf )[u(m)(t)− u(n)(t)] + Lf [u(m)(t+ τ)− u(n)(t+ τ)], t ∈ [0, a], τ ∈ [−r, 0] and θ ≤ Ik(u(m)(tk))− Ik(u(n)(tk)) ≤ Lk(u(m)(tk)− u(n)(tk)), k = 1, 2, . . . ,m. From the above inequalities and the normality of cone P , it follows that for any t ∈ [0, a], τ ∈ [−r, 0], ‖f(t, u(m)(t), (u(m))t)− f(t, u(n)(t), (u(n))t)‖ ≤ N(M +Mf )‖u(m)(t)− u(n)(t)‖ +NLf‖u(m)(t+ τ)− u(n)(t+ τ)‖+M‖u(m)(t)− u(n)(t)‖ 20 X. ZHANG, P. CHEN, Y. LI EJDE-2020/68 ≤ [N(M +Mf ) +M ]‖u(m)(t)− u(n)(t)‖ +NLf sup −r≤τ≤0 ‖u(m)(t+ τ)− u(n)(t+ τ)‖ and ‖Ik(u(m)(tk))− Ik(u(n)(tk))‖ ≤ NLk‖u(m)(tk)− u(n)(tk)‖, k = 1, 2, . . . ,m. From these inequalities and the definition of the measure of noncompactness, we have α ({ f(t, u(n)(t), (u(n))t) }) ≤ Lf [ α ({ u(n)(t) }) + sup −r≤τ≤0 α ({ u(n)(t+ τ) })] , ∀t ∈ [0, a] and α ({ Ik(u(n)(tk)) }) ≤ Lkα ({ u(n)(tk) }) , k = 1, 2, . . . ,m, where Lf = max { N(M +Mf ) +M,NLf } , Lk = NLk, k = 1, 2, . . . ,m. If {u(n)} ⊂ [v(0), w(0)] is a decreasing sequence, then above inequality is also valid. Therefore, the condition (H4) holds. Hence, by Theorem 4.4, the nonlocal problem (1.1) has a minimal mild solution u and a maximal mild solution u between v(0) and w(0). By the proof Theorem 4.4, (3.2), (3.10) are valid. In what follows, we show that u = u. For t ∈ [−r, 0], by (3.2) and (3.10) and the assumption (H5), we have θ ≤ u(t)− u(t) = Au(t)−Au(t) = g(u)(t)− g(u)(t) ≤ Lg(u(t)− u(t)). From this and the normality of cone P it follows that ‖u(t)− u(t)‖ ≤ NLg‖u(t)− u(t)‖ ≤ NLg‖u− u‖PC , t ∈ [−r, 0]. (4.11) For t ∈ [0, a], by (3.2) and (3.10) and the assumption (H5), we obtain θ ≤ u(t)− u(t) = Au(t)−Au(t) = S(t)[g(u)(0)− g(u)(0)] + ∑ 0 0 are two constants, 0 < t1 < t2 < · · · < tm < a, ut(x, s) = u(t+ s, x) for s ∈ [−r, 0] and x ∈ [0, 1], γ ∈ L([−r, 0],R+), ρ(s, t) is a continuous function from [−r, 0]× [0, a] to R+, ϕ ∈ C([0, 1]× [−r, 0],R+). Let X = L2([0, 1],R) with the norm ‖·‖2, and let P = {u ∈ L2([0, 1],R) | u(x) ≥ 0a.e. x ∈ [0, 1]}. Then X is a Banach space and P is a regular cone of X. Define an operator A : D(A) ⊂ X → X by Au = − ∂2 ∂x2 u, u ∈ D(A), with domain D(A) = {u ∈ L2([0, 1],R) : u′, u′′ ∈ L2([0, 1],R), u(0) = u(1) = 0}. 22 X. ZHANG, P. CHEN, Y. LI EJDE-2020/68 It is well known that −A generates a uniformly bounded strongly continuous semi- group T (t) (t ≥ 0) which is positive, compact, and analytic. Furthermore, A has discrete spectrum with eigenvalues λn = n2π2 for n ∈ N associated normalized eigenvectors en(x) = √ 2 sin(nπx), the set {en : n ∈ N} is an orthonormal basis of X. For t ∈ [0, a] and s ∈ [−r, 0], we denote u(t) = u(·, t), f(t, u(t), ut) = sin u(·, t)√ 2 + ∫ 0 −r γ(s)ut(·, s) ds, Ik(u(tk)) = √ |u(·, tk)| 1 + |u(·, tk)| , k = 1, 2, . . . ,m, g(u)(s) = ∫ a 0 ρ(s, t) lg(1 + |u(·, t)|)dt, ϕ(s) = ϕ(·, s). Then the retarded parabolic partial differential equation involving nonlocal and impulsive conditions (5.1) can be transformed into the abstract form of nonlocal problem (1.1). Theorem 5.1. Assume that there exists a function w = w(x, t) ∈ PC([0, 1] × [−r, a],R) ∩ C1([0, 1]× I ′′,R) such that ∂ ∂t w(x, t)− ∂2 ∂x2 w(x, t) ≥ sin w(x, t)√ 2 + ∫ 0 −r γ(s)wt(x, s)ds, x ∈ [0, 1], t ∈ [0, a], t 6= tk, w(x, t+k ) ≥ w(x, t−k ) + √ |w(x, tk)| 1 + |w(x, tk)| , x ∈ [0, 1], k = 1, 2, . . . ,m, w(0, t) = w(1, t) = 0, t ∈ [0, a], w(x, s) ≥ ∫ a 0 ρ(s, t) lg(1 + |w(x, t)|)dt+ ϕ(x, s), x ∈ [0, 1], s ∈ [−r, 0]. Then the retarded parabolic partial differential equation involving nonlocal and im- pulsive conditions (5.1) exist a minimal mild solution and a maximal mild solution between 0 and w(x, t), which can be obtained by a monotone iterative procedure starting from 0 and w(x, t), respectively. Proof. From the assumption and the definition of nonlinear term f , impulsive func- tion Ik for k = 1, 2, . . . ,m and nonlocal function g, we can verify that that v(0) = 0 and w(0) = w(x, t) are the lower and the upper solutions of the the retarded para- bolic partial differential equation involving nonlocal and impulsive conditions (5.1) respectively, nonlocal term g is continuous and compact as well as the conditions (H1)–(H3) are satisfied with M = √ 2 2 . Therefore, our conclusion follows from Theorem 3.1. This completes the proof. � Acknowledgments. The authors would like to express sincere thanks to the anonymous referee for his/her carefully reading the manuscript and valuable com- ments and suggestions. This research was supported by Science Research Project for Colleges and Universities of Gansu Province (No. 2019B-047), Project of NWNU- LKQN2019-13 and Doctoral Research Fund of Northwest Normal University (No. 6014/0002020209). 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Xuping Zhang Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Email address: lanyu9986@126.com EJDE-2020/68 RETARDED EVOLUTION EQUATIONS 25 Pengyu Chen (corresponding author) Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Email address: chpengyu123@163.com Yongxiang Li Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Email address: liyx@nwnu.edu.cn 1. Introduction 2. Preliminaries 3. Case g compact 4. Case g not compact 5. An example Acknowledgments References