Monotone iterative technique for non-autonomous semilinear differential equations with nonlocal condition

where A(t) : D(A(t)) ⊂ X → X is a linear operator, J = [0, b], f is a given function from J × X to X satisfying certain assumptions, ti ∈ (0, b), for i < j, ti < tj , 0 = ̸ ξi ∈ R, i = 1, 2, . . . , k ∈ N, x0 ∈ X. Nonlocal condition is a generalization of classical initial conditionwhich ismore effective to produce better results in the application of physical problems rather than classical initial condition. The existence results for nonlocal Cauchy problem was first studied by Byszewski [1]. To describe the diffusion phenomenon of a small amount of gas in a transparent tube, Deng [2] used the nonlocal condition. In [3], authors discussed the existence of mild solutions for a class of fractional evolution equations with nonlocal integral conditions with the help of compact semigroup, approximation technique and Schauder fixed point theorem. Monotone iterative technique is an effective method to find the existence and uniqueness of mild solutions. Using this method, we get monotone sequences of approximate solutions that converge to maximal and minimal mild solutions. Du [4], first used this technique to find extremal mild solutions for a differential equation. The technique has been used for autonomous system till now. Chen and Li [5] used monotone iterative technique to establish the existence and uniqueness of mild solutions for a semilinear differential equation with nonlocal condition. In [6], Chen and Mu discussed the existence and uniqueness of mild solutions for a semilinear impulsive integro-differential equation by using monotone iterative method. Mu [7] studied the existence and uniqueness of mild solutions for a class of fractional evolution equations with the


Introduction
In this work, we will apply monotone iterative method to find the existence and uniqueness of mild solutions for the following non-autonomous system in an ordered Banach space X: x ′ (t) + A(t)x(t) = f (t, x(t)), t ∈ (0, b], (1.1) where A(t) : D(A(t)) ⊂ X → X is a linear operator, J = [0, b], f is a given function from J × X to X satisfying certain assumptions, t i ∈ (0, b), for i < j, t i < t j , 0 ≠ ξ i ∈ R, i = 1, 2, . . . , k ∈ N, x 0 ∈ X. Nonlocal condition is a generalization of classical initial condition which is more effective to produce better results in the application of physical problems rather than classical initial condition. The existence results for nonlocal Cauchy problem was first studied by Byszewski [1]. To describe the diffusion phenomenon of a small amount of gas in a transparent tube, Deng [2] used the nonlocal condition. In [3], authors discussed the existence of mild solutions for a class of fractional evolution equations with nonlocal integral conditions with the help of compact semigroup, approximation technique and Schauder fixed point theorem.
Monotone iterative technique is an effective method to find the existence and uniqueness of mild solutions. Using this method, we get monotone sequences of approximate solutions that converge to maximal and minimal mild solutions. Du [4], first used this technique to find extremal mild solutions for a differential equation. The technique has been used for autonomous system till now. Chen and Li [5] used monotone iterative technique to establish the existence and uniqueness of mild solutions for a semilinear differential equation with nonlocal condition. In [6], Chen and Mu discussed the existence and uniqueness of mild solutions for a semilinear impulsive integro-differential equation by using monotone iterative method. Mu [7] studied the existence and uniqueness of mild solutions for a class of fractional evolution equations with the help of monotone iterative method. In [8], Mu and Li investigated the existence and uniqueness of mild solutions for an impulsive fractional differential equation by using monotone iterative technique. Chen et al. [9] studied the existence of mild solutions for a nonlocal impulsive semilinear evolution equation by using perturbation method and monotone iterative technique. In [10], authors considered a fractional differential equation with mixed monotone nonlocal conditions to investigate the existence of mild solutions with the help of coupled lower and upper mild L-quasi solutions and monotone iterative technique. Kamaljeet [11] and Renu [12] used monotone iterative method to discuss the existence and uniqueness of extremal mild solutions for nonlocal fractional differential equations with finite delay and for neutral fractional differential equations with infinite delay, respectively. Chen et al. [13] used perturbation method and iterative technique to establish the existence of mild solutions for a class of evolution equations with non-instantaneous impulses.
The existence of mild solutions of a non-autonomous integro-differential equations with nonlocal conditions by using the theory of evolution families, Banach contraction principle and Schauder's fixed point theorem is studied by Yan [14]. Haloi et al. [15] studied existence, uniqueness and asymptotic stability of non-autonomous differential equations with deviated arguments via Banach fixed point theorem and theory of analytic semigroup. In [16], Alka et al. established the existence and uniquenss of mild solutions for non-autonomous instantaneous impulsive differential equations with iterated deviating arguments by using analytic semigroup theory and Banach fixed point theorem. Borai [17] first derived the sufficient conditions for the existence of solution for a fractional non-autonomous Cauchy problem. Fang Li [18] discussed the existence of mild solutions for fractional non-autonomous integro-differential equations with nonlocal conditions. Chen et al. [19] studied the existence of mild solutions for a class of fractional non-autonomous evolution equations with delay by using analytic semigroup, measure of noncompactness and fixed point theorem with respect to k-set contractive operator. Nieto [20] established the sufficient conditions for the existence of mild solutions for a second order non-autonomous evolution equation having nonlocal conditions by using measure of noncompactness and Tikhonoff fixed point theorem. In [21], Chen et al. established the existence of mild solutions for a non-autonomous fractional differential equation by using the theory of evolution families and fixed point theorem with respect to k-set contractive operator. In [22], authors first studied the local existence of mild solutions, then obtained a blowup alternative result for fractional non-autonomous integrodifferential equation of Volterra type.
To the best of our knowledge, there is no work yet reported on the existence and uniqueness of mild solutions for non-autonomous differential equations by using monotone iterative method. Motivated by this fact, in this article, we consider a non-autonomous differential system with nonlocal condition to investigate the sufficient conditions for the existence and uniqueness of mild solutions with the help of monotone iterative technique.
We organize the article as following: In section 2, we will recall some basic definitions, notations and theorems. In section 3, we will study the existence of extremal mild solutions for the system (1.1)-(1.2), and also we will show the uniqueness of extremal mild solutions. Finally, in section 4, we will present an example to illustrate our results.

Preliminaries
Now, we recall some basic theory which is required for our main results.
Let (X, ‖ · ‖, ) be a partially ordered complete norm space, P = {x ∈ X| x 0} (0 is the zero of X) is a positive cone of X. The cone P is called normal if there is a real number N > 0 such that 0 x 1 x 2 ⇒ ‖x 1 ‖ N‖x 2 ‖, for all x 1 , x 2 ∈ X, the smallest value of such N is called normal constant. Let C(J, X) be space of all continuous maps from J to X, with sup norm. For x 1 , Definition 2.1. [23] Let X be a Banach space. A two parameter family of bounded linear operators U(t, s), 0 s t b on X is called an evolution system, if the following conditions are satisfied: For the family of linear operators {A(t) : t ∈ J} on X, we impose the following assumptions: The above assumptions imply that, −A(t) generates a unique evolution system {U(t, s) : 0 s t b} of bounded linear operators on X, which satisfies the following properties (see [23,24]): the space of bounded linear operators on X, and is continuous strongly for 0 exists in strong operator topology for 0 s < τ b, and strongly continuous in 3. An evolution system U(t, s) is called positive if U(t, s)y 0, for y ∈ P and 0 s t b.
If the inequalities of (2.1) are opposite, solution is known as upper solution.
Theorem 2.5. [23,24] Suppose that the assumptions (A1)-(A3) hold and f satisfies uniform Hölder continuity on J with exponent β ∈ (0, 1], then unique solution of the following linear Cauchy problem is Now, we state the definition and some properties of Kuratowski measure of noncompactness. For more details, we refer [25] and [26]. Definition 2.6. If Y is a Banach space and M(Y) is a collection of subsets of Y, which are bounded, then the function µ : M(Y) → [0, ∞) as following is known as Kuratowski measure of noncompactness.
Lemma 2.7. If X 1 and X 2 be Banach spaces and C, D ⊂ X 1 be bounded, then the following properties are satisfied:

Main results
First, we will derive the expression of mild solution for the problem (1.1)-(1.2), then the existence as well as uniqueness of extremal mild solutions will be discussed. Let us introduce our basic assumptions: (H0) The evolution system U(t, s) is positive.
by operator spectrum theorem, we know that exists, bounded and can be expressed as Since U(t, s) is positive, so is B and X) is said to be a mild solution of the problem (1 .1)-(1.2), if x satisfies the following integral equation: 2) is given as Proof. By Theorem 2.5, the mild solution of problem (1.1)-(1.2) can be expressed as So, for i = 1, 2, 3, . . . , k. By using (3.7) in (1.2), we get Now, assumption (H1) and (3.1) imply that Hence (3.6) and (3.8) imply that (3.5) is satisfied.
Let us define Q : C(J, X) → C(J, X) as: To prove that the system (1.1)-(1.2) has a mild solution, we need to show the operator Q has a fixed point.
Therefore, from the normality of P we get a constant c > 0, such that (3.10) First, we will prove that the map Q : D → C(J, X) is continuous. Let {xn} be a sequence in D such that xn → x ∈ D. Since f is continuous, so f (ϱ, xn(ϱ)) → f (ϱ, x(ϱ)) for ϱ ∈ J, and from (3.10) we get that ‖f (ϱ, xn(ϱ)) − f (ϱ, x(ϱ))‖ 2c. So, by Lebesgue dominated convergence theorem, we estimate Thus Q is continuous map on D.
From the above assumptions it is clear that (H1) holds, and ω 0 = 0 and ν 0 = v(t, z) are the lower and upper solutions for the system (4.1). So, by Corollary 3.4, we conclude that the minimal and maximal mild solutions for (4.1) exist between the lower solution 0 and upper solution v.