Stability and monotony dependence of second-order fuzzy functional systems in partially ordered metric spaces

Abstract

In this paper, we consider Cauchy problems for second order fuzzy functional differential equations (DEs) with generalized Hukuhara (gH) derivatives. We study the solvability of the problem by using Perov fixed point theorem in ordered partial metric spaces. The data monotony, continuity, diferentiability dependence of mild solutions with respect to parameters are investigated via weak Picard operators. Moreover, the stability of mild solutions is addressed in sense of Ulam-Hyers stability related to the technique of coefficient matrix converges to zero. Some examples are presented to demonstrate for theoretical results.

Keywords

Fuzzy functional DEs gH-derivatives ordered partial metric spaces

1 Introduction

Many technical systems are modeled by differential equations (DEs) with delay. However, these models are often impacted by some types of uncertainties. Fuzzy functional DEs are proposed as one of the natural ways to model the real world system, such as the propagations of epistemic uncertainty or vague in a dynamical environment. Subjects related to fuzzy DEs were interested many mathematicians as [4 , 19–22]. Arshad and Lupulescu [3, 18] used the methods of successive approximations to establish the global existence and uniqueness for a class of fuzzy functional DEs. Long et al. studied the existence, uniqueness, and boundedness of solutions for some classes of fuzzy partial differential equations in [15, 16]. Some recent researches of dynamical systems related to extended fuzzy sets were presented in [23 –25]. Fuzzy DE with granular derivatives were investigated in [26, 28]. For more references on the recent applications of fuzzy theory and advanced fuzzy theory in real world, the reader is kindly requested to go through [1 , 27].

When we fuzzify the models containing uncertain phenomena, which have lag time in the independent variables, it turns out fuzzy functional DEs. Khastan and Nieto [13] considered a large class of fuzzy two-point boundary value problems under the Hukuhara differentiability or the gH differentiability for fuzzy DEs in order 2. Hasan et al [8] investigated approximate solutions of fuzzy functional DEs in order 2 by using the reproducing kernel Hilbert space method. Guo et al. [7] used different formulation for the fuzzy initial value problems based on a family of differential inclusions at each α-level set. They discussed the oscillation properties of a class of fuzzy delay DEs of second order. Meawhile, in some cases, conditions to permit that family of differential inclusions at each α-level with 0 ≤ α ≤ 1 will be fuzzy solutions of initial problems are not known. Therefore, it is clear that all existed results on fuzzy DEs in order 2 based on reducing to real-cases and only at the first stage of considering the existence and some basic properties of solutions. It lacks literatures in researching qualitative properties of fuzzy solutions for these equations.

In this paper, by using the technique of ordered partial metric spaces and matrix converges to zero, we consider Cauchy problems for second-order fuzzy functional DEs reduced to the canonical form on [t₀, T] $D^{2} u (t) = f (t, u (t), D u (t), u (t - d), D u (t - d)),$ (1) with initial condition $u (t) = η (t), t \in [t_{0} - d, t_{0}],$ where t₀ < T and d > 0, where $D u, D^{2} u$ are gH-derivatives of fuzzy-valued function u in order 1 and order 2.

The main aim of paper can be highlighted as follows.

- To study qualitative properties of Cauchy problems for the second-ordered DEs with indeterminacy and delay such as the solvability of the problem; the data monotony, continuity and the diferentiability dependence of mild solutions with respect to parameter.

- To study the stability and continuous differentiability with respect to parameters of fuzzy solutions under suitable conditions on ordered partial metric spaces by using the idea fuzzy transform method and successive approximations.

The rest of the paper is organized as follows. Section 2 recalls some necessary concepts on generalized metric space and the state of problem is presented in Section 3. Section 4 studies the existence and uniqueness of solutions for considered fuzzy functional DEs with initial conditions. Section 5 presents some results on the continuity and monotony of mild solutions while the smoothness of solutions is exhibited in Section 6. The Ulam-Hyers of systems is investigated in Section 7. Finally, some conclusions and future works are discussed in section 8.

2 Preliminaries

Let $ℝ_{F}$ be the space of fuzzy sets on $ℝ$ . For $u \in ℝ_{F}$ , the α-cuts or level sets of u are defined by $[u]^{α} = {x \in ℝ : u (x) \geq α} = [u_{l α}, u_{r α}] for 0 < α \leq 1$ and $[u]^{0} = \bar{{x \in ℝ : u (x) > 0}}$ is denoted for the support of u. $(ℝ_{F}, ρ)$ is a complete metric space, where $ρ (u, v) = sup_{0 \leq α \leq 1} max {sup_{a \in {[u]}^{α}} inf_{b \in {[v]}^{α}} | a - b |} .$

The Hukuhara difference of v and w in $ℝ_{F}$ , denoted by e = v ⊖ w, if v = w + e. And we denote by v_⊝gHw-the gH-difference of v and w, if there exists $e \in ℝ_{F}$ such that v = w + e or w = v + (-1) ⊙ e.

For $\tilde{a}, \tilde{b} \in ℝ_{F}$ , $[\tilde{a}]^{α} = [{\tilde{a}}_{l α}, {\tilde{a}}_{r α}], [\tilde{b}]^{α} = [{\tilde{b}}_{l α}, {\tilde{b}}_{r α}] .$ We have $\tilde{a} \leq \tilde{b}$ if and only if ${\tilde{a}}_{l α} \leq {\tilde{b}}_{l α}$ and ${\tilde{a}}_{r α} \leq {\tilde{b}}_{r α}$ for all α ∈ [0, 1]. For $I \subset ℝ$ , $f, g \in C (I, ℝ_{F})$ , we say that f ≤ g if and only if f (x) ≤ g (x) for all x ∈ I.

Denote $C (I, R_{F})$ by the space of all continuous mappings $f : I \to ℝ_{F}$ and $C^{2} (I, R_{F})$ by the space of all continuously differentiable up to order two $f : I \to ℝ_{F} .$ The first order gH-derivatives of a fuzzy valued mapping $f : I \subset ℝ \to ℝ_{F}$ at x₀ ∈ I is a fuzzy number $D f (x_{0})$ , which satisfied $D f (x_{0}) = lim_{h \to 0} \frac{1}{h} (f (x_{0} + h) ⊝_{gH} f (x_{0})),$ provided that the gH-differences f (x₀ + h) _⊝gHf (x₀) and the limit in the right hand side exist for h close enough to 0.

Analogously, we define for higher order gH-derivatives. Denote by $C^{2} (I, R_{F})$ the set of all functions $f \in C (I, ℝ_{F})$ which have gH-derivatives up to order 2, denote by $D^{2} f,$ which is continuous on I .

Definition 2.1. [2] Let $f : I \to ℝ_{F}$ be gH-differentiable at x₀ ∈ I. We say that f is (i)-gH differentiable at x₀ ∈ I if $[D f (x_{0})]^{α} = [f_{l α}^{'} (x_{0}), f_{r α}^{'} (x_{0})] \forall α \in [0, 1]$ and that f is (ii)-gH differentiable at x₀ ∈ I if $[D f (x_{0})]^{α} = [f_{r α}^{'} (x_{0}), f_{l α}^{'} (x_{0})], \forall α \in [0, 1],$ where [f (x)] ^α = [f_lα (x) , f_rα (x)] , x ∈ I, α ∈ [0, 1] and g′ (x₀) is the derivative of real function $g : I \to ℝ$ at x₀.

3 State the problem

We consider following functional DEs on [t₀, T]

$D^{2} u (t) = f (t, u (t), D u (t), u (t - d), D u (t - d)),$ (2) with initial condition

$u (t) = η (t), t \in [t_{0} - d, t_{0}],$ (3) where d is the delay factor, $u : [t_{0}, T] \to ℝ_{F}$ is a fuzzy-valued function and $f : I \times (C ([t_{0} - d, T], ℝ_{F}))^{4} \to R_{F}$ , $η : [t_{0} - d, t_{0}] \to ℝ_{F}$ are given functions. Set $v : = D u$ , then equation (2) is transformed to following system for t ∈ [t₀, T] ${\begin{matrix} D u (t) = v (t), \\ D v (t) = f (t, u (t), v (t), u (t - d), v (t - d)) \end{matrix}$ (4) with initial conditions $u (t) = η (t), v (t) = D η (t), t \in [t_{0} - d, t_{0}] .$ (5) Denote $W_{d} ([t_{0}, T]) = (C ([t_{0} - d, T], ℝ_{F}) \cap C^{1} ([t_{0}, T], ℝ_{F}))^{2} .$ By a solution of the problem (4)-(5) we recall a vector function (u, v) ^T∈ W_d ([t₀, T]) , which satisfies system (4) and initial condition (5).

We have $\int_{t_{0}}^{t} D u (s) ds = \int_{t_{0}}^{t} v (s) ds$ . It follows $u (t) \underset{gH}{⊖} u (t_{0}) = \int_{t_{0}}^{t} v (s) ds .$ That leads to two case

- If u (.) is (i)-gH differentiable, then $u (t) = η (t_{0}) \oplus \int_{t_{0}}^{t} v (s) ds$ .

- If u (.) is (ii)-gH differentiable, then $u (t) = η (t_{0}) ⊖ (- 1) ⊙ \int_{t_{0}}^{t} v (s) ds$ .

Similarly, from $D v (t) = f (t, u (t), v (t), u (t - d), v (t - d))$ , if v (.) is (i)-gH differentiable then $\begin{matrix} v (t) = & D η (t_{0}) \oplus \int_{t_{0}}^{t} f (s, u (s), \\ v (s), u (s - d), v (s - d)) ds \end{matrix}$ and if v (.) is (ii)-gH differentiable then $\begin{matrix} v (t) = & D η (t_{0}) ⊖ (- 1) ⊙ \int_{t_{0}}^{t} f (s, u (s), \\ v (s), u (s - d), v (s - d)) ds . \end{matrix}$

Denote F (s, u, v) = f (s, u (s) , v (s) , u (s – d) , v (s – d)). If (u, v) ^T ∈ W_d ([t₀, T]) is a solution of the problem (4) – (5) and u, v are (i)-gH differentiable, then (u, v) ^T is a solution of following integral system $\begin{matrix} (u, v)^{T} (t) \\ = {\begin{matrix} (η, D η)^{T} (t), t \in [t_{0} - d, t_{0}] \\ {(η (t_{0}) \oplus \int_{t_{0}}^{t} v (s) ds, D η (t_{0}) \oplus \int_{t_{0}}^{t} F (s, u, v) ds)}^{T}, t \in [t_{0}, T] . \end{matrix} \end{matrix}$ (6)

If u, v are (ii)-gH differentiable then (u, v) ^T is a solution of following integral system $\begin{matrix} (u, v)^{T} (t) \\ = {\begin{matrix} (η, D η)^{T} (t), t \in [t_{0} - d, t_{0}] \\ {(η (t_{0}) ⊖ (- 1) ⊙ \int_{t_{0}}^{t} v (s) ds, D η (t_{0}) ⊖ (- 1) ⊙ \int_{t_{0}}^{t} F (s, u, v) ds)}^{T}, t \in [t_{0}, T] . \end{matrix} \end{matrix}$ (7)

By using analogous method, we can see that if u is (i)-gH differentiable and v is (ii)-gH differentiable then (u, v) ^T satisfies following integral system

$\begin{matrix} (u, v)^{T} (t) \\ = {\begin{matrix} (η, D η)^{T} (t), t \in [t_{0} - d, t_{0}] \\ {(η (t_{0}) + \int_{t_{0}}^{t} v (s) ds, D η (t_{0}) ⊖ (- 1) ⊙ \int_{t_{0}}^{t} F (s, u, v) ds)}^{T}, t \in [t_{0}, T] \end{matrix} \end{matrix}$ (8) and if u is (ii)-gH differentiable and v is (i)-gH differentiable then $\begin{matrix} (u, v)^{T} (t) \\ = {\begin{matrix} (η, D η)^{T} (t), t \in [t_{0} - d, t_{0}] \\ {(η (t_{0}) ⊖ (- 1) ⊙ \int_{t_{0}}^{t} v (s) ds, D η (t_{0}) + \int_{t_{0}}^{t} F (s, u, v) ds)}^{T}, t \in [t_{0}, T] . \end{matrix} \end{matrix}$ (9)

Definition 3.1.

(1) A couple mappings (u, v) ∈ W_d ([t₀, T]) is called a mild solution in type 1 of differential system (4)-(5) if it satisfies integral system (6). In this cases, we call u is a mild solution in type 1 of Cauchy problem (2)-(3).

(2) A couple mappings $(u, v) \in {\hat{W}}_{d} ([t_{0}, T])$ is called a mild solution in type 2 (type 3, type 4) of problem (4)-(5) if it satisfies integral system (7) (system (8), system (9) respectively), where ${\hat{W}}_{d} ([t_{0}, T])$ is the space of all functions (u, v) ∈ W_d ([t₀, T]) such that the gH-differences in the right hand side of (7) exist. And then u is called a mild solution in type 2 (type 3, type 4) of Cauchy problem (2)-(3), respectively.

4 The well-posedness of the problem

In W_d ([t₀, T]), we consider the metric D (U₁, U₂) = (H (u₁, u₂) , H (v₁, v₂)) ^T, where U₁ = (u₁, v₁) ^T, U₂ = (u₂, v₂) ^T, and $H (u, w) = sup_{t \in [t_{0} - d, T]} {ρ (u (t), w (t)}$ , $u, w \in C ([t_{0} - d, T], ℝ_{F})$ .

Set $\begin{matrix} X_{(η, D η)} = & {(u, v)^{T} \in W_{d} ([t_{0}, T]) : (u (t), v (t))^{T} \\ = (η (t), D η (t))^{T} for t \in [t_{0} - d, t_{0}]}, \end{matrix}$ where $(η, D η)^{T} \in (C ([t_{0} - d, t_{0}], ℝ_{F}))^{2}$ . Then $W_{d} ([t_{0}, T]) = \cup_{(η, D η)^{T} \in (C ([t_{0} - d, t_{0}], ℝ_{F}))^{2}} X_{(η, D η)}$ .

Theorem 4.1. Suppose that problem (2) – (3) satisfies following conditions

$f \in C ([t_{0}, T] \times ℝ_{F}^{4}, ℝ_{F}), η \in C^{1} ([t_{0} - d, t_{0}], ℝ_{F})$ ;

there exist positive constants L₁ and L₂ such that for all $t \in [t_{0}, T], u_{i}, v_{i}, \tilde{u_{i}}, \tilde{v_{i}} \in C ([t_{0}, T] \times ℝ_{F}), (i = 1, 2)$ , we have $\begin{matrix} ρ (f (t, u_{1}, v_{1}, u_{2}, v_{2}), f (t, \tilde{u_{1}}, \tilde{v_{1}}, \tilde{u_{2}}, \tilde{v_{2}})) \\ \leq L_{1} max_{[t_{0}, T]} {ρ (u_{1} (t), \tilde{u_{1}} (t)), ρ (u_{2} (t), \tilde{u_{2}} (t))} \\ + L_{2} max_{[t_{0}, T]} {ρ (v_{1} (t), \tilde{v_{1}} (t)), ρ (v_{2} (t), \tilde{v_{2}} (t))} . \end{matrix}$

By setting $Q = (T - t_{0}) (\begin{matrix} 0 1 \\ L_{1} L_{2} \end{matrix})$ . If Q converges to zero matrix, then problem (2) – (3) has a unique mild solution in type 1.

Proof. Consider $B_{(η, D η)}^{f} : W_{d} ([t_{0}, T]) \to W_{d} ([t_{0}, T])$ , $B_{(η, D η)}^{f} (t) = {(B_{1} (u, v) (t), B_{2} (u, v) (t))}^{T}$ defined by the right hand side of (6) for t ∈ [t₀ – d, T].

Since $B_{(η, D η)}^{f} (u, v) |_{[t_{0} - d, t_{0}]} = (η, D η)^{T}$ for $(u, v)^{T} \in X_{(η, D η)},$ then $B_{(η, D η)}^{f} (u, v) |_{[t_{0} - d, t_{0}]} \in X_{(η, D η)}$ . This implies $B_{(η, D η)}^{f} (X_{(η, D η)}) \subset X_{(η, D η)}$ . For t ∈ [t₀ – d, t₀], we have $\begin{matrix} D (B_{(η, D η)}^{f} (u_{1}, v_{1}), B_{(η, D η)}^{f} (u_{2}, v_{2})) \\ = D ((η, D η), (η, D η)) = 0 . \end{matrix}$

When t ∈ [t₀, T], we have

$\begin{matrix} D (B_{(η, D η)}^{f} (u_{1}, v_{1}), B_{(η, D η)}^{f} (u_{2}, v_{2})) \\ = D ((B_{1} (u_{1}, v_{1}), B_{2} (u_{1}, v_{1})), (B_{1} (u_{2}, v_{2}), B_{2} (u_{2}, v_{2}))) \\ = D ((η (t_{0}) \oplus \int_{t_{0}}^{t} v_{1} (s) ds, D η (t_{0}) \\ \oplus \int_{t_{0}}^{t} F (s, u_{1}, v_{1}) ds), (η (t_{0}) \\ \oplus \int_{t_{0}}^{t} v_{2} (s) ds, D η (t_{0}) \oplus \int_{t_{0}}^{t} F (s, u_{2}, v_{2}) ds)) . \end{matrix}$

We have $\begin{matrix} ρ (η (t_{0}) \oplus \int_{t_{0}}^{t} v_{1} (s) ds, η (t_{0}) \oplus \int_{t_{0}}^{t} v_{2} (s) ds) \\ = ρ (\int_{t_{0}}^{t} v_{1} (s) ds, \int_{t_{0}}^{t} v_{2} (s) ds) \\ \leq \int_{t_{0}}^{t} ρ (v_{1} (s), v_{2} (s)) ds \leq (t - t_{0}) H (v_{1}, v_{2}) . \end{matrix}$ On the other hand,

$\begin{matrix} ρ (D η (t_{0}) \oplus \int_{t_{0}}^{t} & F (s, u_{1}, v_{1}) ds, D η (t_{0}) \oplus \int_{t_{0}}^{t} F (s, u_{2}, v_{2}) ds) \\ = ρ (\int_{t_{0}}^{t} F (s, u_{1}, v_{1}) ds, \int_{t_{0}}^{t} F (s, u_{2}, v_{2}) ds) \\ \leq \int_{t_{0}}^{t} ρ (F (s, u_{1}, v_{1}), F (s, u_{2}, v_{2})) ds \\ \leq (t - t_{0}) [L_{1} H (u_{1}, u_{2}) + L_{2} H (v_{1}, v_{2})] . \end{matrix}$ It follows

$\begin{matrix} D & (B_{(η, D η)}^{f} (u_{1}, v_{1}), B_{(η, D η)}^{f} (u_{2}, v_{2})) \\ \leq (H (\int_{t_{0}}^{t} v_{1} (s) ds, \int_{t_{0}}^{t} v_{2} (s) ds), \\ H (\int_{t_{0}}^{t} F (s, u_{1}, v_{1}) ds, \int_{t_{0}}^{t} F (s, u_{2}, v_{2}) ds))^{T} \\ \leq ((t - t_{0}) H (v_{1}, v_{2}), (t - t_{0}) [L_{1} H (u_{1}, u_{2}) \\ + L_{2} H (v_{1}, v_{2})])^{T} \\ \leq (T - t_{0}) (H (v_{1}, v_{2}), L_{1} H (u_{1}, u_{2}) + L_{2} H (v_{1}, v_{2}))^{T} \\ = (T - t_{0}) (\begin{matrix} 0 1 \\ L_{1} L_{2} \end{matrix}) (H (u_{1}, u_{2}), H (v_{1}, v_{2}))^{T} \\ = Q . D ((u_{1}, v_{1}), (u_{2}, v_{2})) . \end{matrix}$

Because Q converges to zero matrix, by applying Perov fixed point theorem in W_d ([t₀, T]), we conclude that there exists a unique mild solution in type 1 of problem (4)-(5).□

Proposition 4.1. Suppose that all the hypotheses of Theorem 4.1 are fulfilled. And, assume that the space ${\hat{W}}_{d} ([t_{0}, T])$ is not empty. Then problem (4) – (5) has a unique mild solution in type 2.

Proof. Consider ${\tilde{B}}_{(η, D η)}^{f} : {\hat{W}}_{d} ([t_{0}, T]) \to {\hat{W}}_{d} ([t_{0}, T])$ which is defined by ${\tilde{B}}_{(η, D η)}^{f} (t) = ({\tilde{B}}_{1} (u, v) (t), {{\tilde{B}}_{2} (u, v) (t))}^{T}$ - the right hand side of (7) for t ∈ [t₀ – d, T].

Similar to the proof of Theorem 4.1, we have ${\tilde{B}}_{(η, D η)}^{f} (X_{(η, D η)}) \subset X_{(η, D η)}$ and ${\tilde{B}}_{(η, D η)}^{f} |_{X_{(η, D η)}} = \tilde{E} |_{X_{(η, D η)}}$ .

For t ∈ [t₀ – d, t₀], we have $D ({\tilde{B}}_{(η, D η)}^{f} (u_{1}, v_{1}), {\tilde{B}}_{(η, D η)}^{f} (u_{2}, v_{2})) = 0 .$

For t ∈ [t₀, T], using the fact that for all $u, v, w, e \in ℝ_{F}$ , if the H-differences u ⊖ v, w ⊖ e exist, then $ρ (u ⊖ v, w ⊖ e) \leq ρ (u, w) + ρ (v, e),$ one gets $\begin{matrix} D ({\tilde{B}}_{(η, D η)}^{f} (u_{1}, v_{1}), {\tilde{B}}_{(η, D η)}^{f} (u_{2}, v_{2})) \\ \leq (H (\int_{t_{0}}^{t} v_{1} (s) ds, \int_{t_{0}}^{t} v_{2} (s) ds), H (\int_{t_{0}}^{t} F (s, u_{1}, v_{1}) ds, \\ \int_{t_{0}}^{t} F (s, u_{2}, v_{2}) ds))^{T} \\ \leq ((t - t_{0}) H (v_{1}, v_{2}), (t - t_{0}) [L_{1} H (u_{1}, u_{2}) \\ + L_{2} H (v_{1}, v_{2})])^{T} \\ \leq Q . D ((u_{1}, v_{1})^{T}, (u_{2}, v_{2})^{T}) . \end{matrix}$ Then ${\tilde{B}}_{(η, D η)}^{f}$ is a contraction. The unique fixed point of ${\tilde{B}}_{(η, D η)}^{f}$ is the unique mild solution of Cauchy problem (4) – (5) in type 2.

Using analogous arguments, we receive the unique existence of mild solvability in type 3 or type 4 of Cauchy problem (4) – (5). It completes the proof.

□

Remark 4.1. From now on, with the assumption that the W_d ([t₀, T]) is not empty, by

$ρ (x ⊖ y, e ⊖ r) \leq ρ (x, e) + ρ (y, r)$ (10) for all $x, y, e, r \in ℝ_{F}$ , we will receive similar results for mild solutions in type 2, type 3, type 4. Therefore, without loss of generalization, we present only results for mild solutions in type 1 (we will call mild solutions in general). The similar results for mild solutions in type 2, 3 or 4 will be obtained by using analogous method and inequality (10).

Definition 4.1.

An operator A : X → X is a weakly Picard operator if the sequence ${A^{n} (x)}_{n \in ℕ}$ converges for all x ∈ X and its limit is a fixed point of A. Denote Fix (A) = {x ∈ X| A (x) = x} by the set of all fixed points of A. Then denote A^∞ by A^∞ : X → X, $A^{\infty} (x) : = lim_{n \to \infty} A^{n} (x)$ . It is clear that A^∞ (X) = Fix (A).

An operator A : X → X is a Picard operator if there exists x^* ∈ X such that Fix (A) = {x^*} and for all x₀ ∈ X, the sequence ${A^{n} (x_{0})}_{n \in ℕ}$ converges to x^*.

Consider self mapping E : W_d ([t₀, T]) → W_d ([t₀, T]), defined by

$E (u, v) (t) = {\begin{matrix} (u (t), v (t))^{T}, t \in [t_{0} - d, t_{0}] \\ {(u (t_{0}) \oplus \int_{t_{0}}^{t} v (s) ds, v (t_{0}) \oplus \int_{t_{0}}^{t} F (s, u, v) ds)}^{T}, t \in [t_{0}, T] . \end{matrix}$ (11) For short, we denote E (u, v) (t) = (E₁ (u, v) (t) , E₂ (u, v) (t)) ^T.

Denote $\tilde{E} : \hat{W} ([t_{0} - d, T], ℝ_{F}^{2}) \to \hat{W} ([t_{0} - d, T], ℝ_{F}^{2})$ by

$\tilde{E} (u, v) (t) = {\begin{matrix} (u (t), v (t))^{T}, t \in [t_{0} - d, t_{0}] \\ {(u (t_{0}) ⊖ (- 1) ⊙ \int_{t_{0}}^{t} v (s) ds, v (t_{0}) ⊖ (- 1) ⊙ \int_{t_{0}}^{t} F (s, u, v) ds)}^{T}, t \in [t_{0}, T] . \end{matrix}$ (12)

Corollary 4.1. Assume that hypotheses (H₁) – (H₂) are satisfied and Q converges to zero, then the operator $B_{(η, D η)}^{f}$ is a Picard operator and the operator E is a weakly Picard operator in W_d ([t₀, T]); ${\tilde{B}}_{(η, D_{η})}^{f}$ is a Picard operator and $\tilde{E}$ is a weakly Picard operator in W_d ([t₀, T]).

Proof. Let $(u, v)^{T} \in W_{d} ([t_{0}, T]) = \cup_{(η, D η) \in (C ([t_{0} - d, t_{0}], ℝ_{F}))^{2}} X_{(η, D η)} .$ There exists $(η_{0}, η_{0}^{'}) \in (C ([t_{0} - d, t_{0}], ℝ_{F}))^{2}$ such that $(u, v)^{T} \in X_{(η_{0}, η_{0}^{'})}$ .

Consider the problem (2) with the initial condition $u (t) = η_{0} (t); v (t) = η_{0}^{'} (t), t \in [t_{0} - d, t_{0}] .$ We have $E (u, v) |_{[t_{0} - d, t_{0}]} = B_{(η_{0}, η_{0}^{'})}^{f} (u, v) |_{[t_{0} - d, t_{0}]}$ and $B_{(η_{0}, η_{0}^{'})}^{f}$ is a Picard operator. Thus $(B_{(η_{0}, η_{0}^{'})}^{f})^{n} (u, v)$ converges to $(u^{*}, v^{*})^{T} = Fix (B_{(η_{0}, η_{0}^{'})}^{f}) .$ Then Eⁿ (u, v) → (u^*, v^*) ^T uniquely for each $(u, v)^{T} \in X_{(η_{0}, η_{0}^{'})}$ . However this is NOT true in the whole space X = W_d ([t₀, T]). Therefore, E is a weakly Picard operator.

Using the analogous arguments as in the proof of Theorem 4.1, ${\tilde{B}}_{(η_{0}, η_{0}^{'})}^{f}$ is a Picard operator and $\tilde{E}$ is a weakly Picard operator on ${\hat{W}}_{d} ([t_{0}, T]) \neq \emptyset$ .□

5 The continuity and monotony of mild solutions

Theorem 5.1. [Comparison theorem] One supposes that

all conditions in Theorem 4.1 are satisfied;

for $u_{i}, v_{i}, \tilde{u_{i}}, \tilde{v_{i}} \in ℝ_{F}, (u_{i}, v_{i})^{T} \leq (\tilde{u_{i}}, \tilde{v_{i}})^{T}$ , i.e.; $u_{i} \leq \tilde{u_{i}}$ and $v_{i} \leq \tilde{v_{i}}$ for all i = 1, 2, we have $\begin{matrix} f (t, u_{1}, v_{1}, u_{2}, v_{2}) \\ \leq f (t, \tilde{u_{1}}, \tilde{v_{1}}, \tilde{u_{2}}, \tilde{v_{2}}), t \in [t_{0}, T] . \end{matrix}$

Let (x^*, y^*) ^T be a mild solution of (4) – (5) and (u^*, w^*) ^T satisfies the following system ${\begin{matrix} D u (t) \leq w (t) \\ D w (t) \leq f (t, u (t), w (t), u (t - d), w (t - d)) \end{matrix} t \in [t_{0}, T]$ and u (t) , w (t) are defined on [t₀ – d, T] . Then from (u^*, w^*) ^T|_{[t₀-d,t₀]} ≤ (x^*, y^*) ^T|_{[t₀-d,t₀]}, we have (u^*, w^*) ^T ≤ (x^*, y^*) ^T in [t₀, T] .

Proof. Since, (x^*, y^*) ^T is a mild solution of (4) – (5), then (x^*, y^*) ^T = E (x^*, y^*). We will show that (u^*, w^*) ^T ≤ E (u^*, w^*).

From $D u^{*} (t) \leq w^{*} (t)$ , we have $\int_{t_{0}}^{t} D u (s) ds \leq \int_{t_{0}}^{t} w^{*} (s) ds$ . Then $\int_{t_{0}}^{t} [u_{l α}^{*} (s)]^{'} ds \leq \int_{t_{0}}^{t} w_{l α}^{*} (s) ds$ and $\int_{t_{0}}^{t} [u_{r α}^{*} (s)]^{'} ds \leq \int_{t_{0}}^{t} w_{r α}^{*} (s) ds$ . It implies $u_{l α}^{*} (t) - u_{l α}^{*} (t_{0}) \leq \int_{t_{0}}^{t} w_{l α}^{*} (s) ds$ and $u_{r α}^{*} (t) - u_{r α}^{*} (t_{0}) \leq \int_{t_{0}}^{t} w_{r α}^{*} (s) ds .$ Then $u^{*} (t) \leq u^{*} (t_{0}) + \int_{t_{0}}^{t} w^{*} (s) ds$ . Similarly, we have $w^{*} (t) \leq w^{*} (t_{0}) + \int_{t_{0}}^{t} f (s, y^{*} (s), w^{*} (s), y^{*} (s - d), w^{*} (s - d)) ds$ . Hence, (u^*, w^*) ^T ≤ E (u^*, w^*) in [t₀, T].

Now we will prove that E is increasing. Let (u₁ (t) , v₁ (t)) ^T ≤ (u₂ (t) , v₂ (t)) ^T and (u₁ (t – d) , v₁ (t – d)) ^T ≤ (u₂ (t – d) , v₂ (t – d)) ^T. Then by assumptions, we get

$\begin{matrix} f (t, u_{1} (t), v_{1} (t), u_{1} (t - d), v_{1} (t - d)) \\ \leq f (t, u_{2} (t), v_{2} (t), u_{2} (t - d), v_{2} (t - d)), t \in [t_{0}, T] . \end{matrix}$ (13) For t ∈ [t₀ – d, t₀], it follows E (u₁, v₁) (t) ≤ E (u₂, v₂) (t).

For t ∈ [t₀, T] , u₁ (t₀) ≤ u₂ (t₀) and $\int_{t_{0}}^{t} v_{1} (s) ds \leq \int_{t_{0}}^{t} v_{2} (s) ds$ . It follows $u_{1} (t_{0}) \oplus \int_{t_{0}}^{t} v_{1} (s) ds \leq u_{2} (t_{0}) \oplus$ $\int_{t_{0}}^{t} v_{2} (s) ds$ . Moreover from v₁ (t₀) ≤ v₂ (t₀) and (13), then

$v_{1} (t_{0}) \oplus \int_{t_{0}}^{t} F (s, u_{1}, v_{1}) ds \leq v_{2} (t_{0}) \oplus \int_{t_{0}}^{t} F (s, u_{2}, v_{2}) ds .$

Thus E (u₁, v₁) (t) ≤ E (u₂, v₂) (t). Similarly, $\tilde{E}$ can be proved to be increasing by the same steps.

By induction, Eⁿ (u₁, v₁) ≤ Eⁿ (u₂, v₂) for all n. It follows $lim_{n \to \infty} E^{n} (u_{1}, v_{1}) \leq lim_{n \to \infty} E^{n} (u_{2}, v_{2})$ or E^∞ (u₁, v₁) ≤ E^∞ (u₂, v₂) . Hence E^∞ is increasing. It is similar to the proof of ${\tilde{E}}^{\infty}$ .

Let (u^*, w^*) ^T ∈ X_{(η_o,χ_o)}. E is a weak Picard operator, and $E |_{X_{(η_{o}, χ_{o})}} = B_{(η_{o}, χ_{o})}^{f} |_{X_{(η_{o}, χ_{o})}}$ . It implies $E^{\infty} (u^{*}, w^{*}) = ({\bar{u}}^{*}, {\bar{w}}^{*})^{T}$ . For $({\tilde{u}}^{*}, {\tilde{w}}^{*})^{T} \in X_{(η_{o}, χ_{o})}$ , we also get that $E^{\infty} ({\tilde{u}}^{*}, {\tilde{w}}^{*}) (t) = ({\bar{u}}^{*}, {\bar{w}}^{*})^{T}$ . Then $E^{\infty} (u^{*}, w^{*}) = E^{\infty} ({\tilde{u}}^{*}, {\tilde{w}}^{*})$ .

Next, we consider (x^*, y^*) ^T ∈ X_{(η₁,χ₁)}, that is, (x^*, y^*) ^T |_{[t₀-d,t₀]} = (η₁, χ₁) ^T. Then (u^*, w^*) ^T |_{[t₀-d,t₀]} ≤ (x^*, y^*) ^T |_{[t₀-d,t₀]}, or (η_o, χ_o) ^T (t) ≤ (η₁, χ₁) ^T (t). Then we shall choose $({\tilde{u}}^{*}, {\tilde{w}}^{*})^{T} \in X_{(η_{0}, χ_{0})}$ and $({\tilde{x}}^{*}, {\tilde{y}}^{*})^{T} \in X_{(η_{1}, χ_{1}})$ such that for t ∈ [t₀ – d, T], $({\tilde{u}}^{*}, {\tilde{w}}^{*})^{T} \leq ({\tilde{x}}^{*}, {\tilde{y}}^{*})^{T}$ . It follows $E^{\infty} ({\tilde{u}}^{*}, {\tilde{w}}^{*}) \leq E^{\infty} ({\tilde{x}}^{*}, {\tilde{y}}^{*})$ . Moreover, $E^{\infty} ({\tilde{x}}^{*}, {\tilde{y}}^{*}) = E^{\infty} (x^{*}, y^{*}) = (x^{*}, y^{*})^{T}$ . And hence, $(u^{*}, w^{*})^{T} \leq E (u^{*}, w^{*}) = E^{\infty} ({\tilde{u}}^{*}, {\tilde{w}}^{*}) \leq E^{\infty} ({\tilde{x}}^{*}, {\tilde{y}}^{*}) = (x^{*}, y^{*})^{T}$ . Thus (u^*, w^*) ^T |_[t₀-d,T] ≤ (x^*, y^*) ^T |_[t₀-d,T]. It completes the proof.□

Theorem 5.2. [Comparison theorem] Let $f_{i} \in C ([t_{0}, T] \times ℝ_{F}^{4}, ℝ_{F}), i = 1, 2, 3,$ be satisfied conditions (H₁) , (H₂) and f₁ ≤ f₂ ≤ f₃; $f_{2} (t, ., ., ., .) : ℝ_{F}^{4} ⟶ ℝ_{F}$ is increasing for all t ∈ [t₀, T]. Assume that $(u_{i}^{*}, v_{i}^{*})^{T}$ are mild solutions in type 1 of the systems ${\begin{matrix} D u (t) = v (t) \\ D v (t) = f_{i} (t, u (t), v (t), u (t - d), v (t - d)) \end{matrix}$ (14)defined on [t₀ – d, T] and $\begin{matrix} (u_{1}^{*}, v_{1}^{*})^{T} |_{[t_{0} - d, t_{0}]} & \leq (u_{2}^{*}, v_{2}^{*})^{T} |_{[t_{0} - d, t_{0}]} \\ \leq (u_{3}^{*}, v_{3}^{*})^{T} |_{[t_{0} - d, t_{0}]} . \end{matrix}$ (15)

Then inequality (15) still hold in the whole domain [t₀ – d, T].

Proof. Consider the operators E_i corresponding to each system (14). The operators E_i, i = 1, 2, 3, are weakly Picard operators. Taking into the condition that f₂ is increasing, it implies that E₂ is increasing.

•When t ∈ [t₀ – d, t₀], we have E₁ (u, v) (t) = E₂ (u, v) (t) = E₃ (u, v) (t).

•When t ∈ [t₀, T], since f₁ ≤ f₂ ≤ f₃, one gets $\begin{matrix} f_{1} (t, u (t), v (t), u (t - d), v (t - d)) \\ \leq f_{2} (t, u (t), v (t), u (t - d), v (t - d)) \\ \leq f_{3} (t, u (t), v (t), u (t - d), v (t - d)) . \end{matrix}$ Hence E₁ (u, v) (t) ≤ E₂ (u, v) (t) ≤ E₃ (u, v) (t) for t ∈ [t₀ – d, T] .

Assume that $(u_{j}^{*}, v_{j}^{*})^{T}$ is a fixed point of E_j|_{X
_{(η_j,χ_j)}}, $(\tilde{u_{j}^{*}}, \tilde{v_{j}^{*}})^{T} \in X_{(η_{j}, χ_{j})}$ such that $(u_{j}^{*}, v_{j}^{*})^{T} = E^{\infty} (u_{j}^{*}, v_{j}^{*}) = E^{\infty} (\tilde{u_{j}^{*}}, \tilde{v_{j}^{*}})$ , j = 1, 2, 3.

Assume that $(u_{1}^{*}, v_{1}^{*})^{T} |_{[t_{0} - d, t_{0}]} \leq (u_{2}^{*}, v_{2}^{*})^{T} |_{[t_{0} - d, t_{0}]} \leq (u_{3}^{*}, v_{3}^{*})^{T} |_{[t_{0} - d, t_{0}]}$ . It implies that $\begin{matrix} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}})^{T} |_{[t_{0} - d, t_{0}]} \\ \leq (\tilde{u_{2}^{*}}, \tilde{v_{2}^{*}})^{T} |_{[t_{0} - d, t_{0}]} \leq (\tilde{u_{3}^{*}}, \tilde{v_{3}^{*}})^{T} |_{[t_{0} - d, t_{0}]} . \end{matrix}$

For t ∈ [t₀, T], we choose $(\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}})^{T} \leq (\tilde{u_{2}^{*}}, \tilde{v_{2}^{*}})^{T} \leq (\tilde{u_{3}^{*}}, \tilde{v_{3}^{*}})^{T}$ . Then $\tilde{u_{1}^{*}} (t_{0}) = η_{1} \leq \tilde{u_{2}^{*}} (t_{0}) = η_{2} \leq \tilde{u_{3}^{*}} (t_{0}) = η_{3}$ and $\tilde{u_{1}^{*}} (t) \leq \tilde{u_{2}^{*}} (t) = \frac{1}{2} (\tilde{u_{1}^{*}} (t) \oplus \tilde{u_{3}^{*}} (t)) \oplus η_{2} \underset{gH}{⊖} \frac{12}{(} η_{1} \oplus η_{3}) \leq \tilde{u_{3}^{*}} (t),$ where $\tilde{u_{1}^{*}} (t), \tilde{u_{3}^{*}} (t)$ are continuous. So does $\tilde{u_{2}^{*}} (t)$ . Moreover, $\tilde{v_{1}^{*}} (t_{0}) = χ_{1} \leq \tilde{v_{2}^{*}} (t_{0}) = χ_{2} \leq \tilde{v_{3}^{*}} (t_{0}) = χ_{3}$ and $\tilde{v_{1}^{*}} (t) \leq \tilde{v_{2}^{*}} (t) = \frac{12}{(} \tilde{v_{1}^{*}} (t) \oplus \tilde{v_{3}^{*}} (t)) \oplus χ_{2} \underset{gH}{⊖} \frac{12}{(} χ_{1} \oplus χ_{3}) \leq \tilde{v_{3}^{*}} (t)$ , where $\tilde{v_{1}^{*}} (t), \tilde{v_{3}^{*}} (t)$ are continuous. So does $\tilde{v_{2}^{*}} (t)$ .

Since E₂ is increasing, then $E_{2}^{\infty} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}}) \leq E_{2}^{\infty} (\tilde{u_{2}^{*}}, \tilde{v_{2}^{*}}) \leq E_{2}^{\infty} (\tilde{u_{3}^{*}}, \tilde{v_{3}^{*}})$ .

The next step is to prove that $E_{1}^{\infty} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}}) \leq E_{2}^{\infty} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}})$ . It has been known that $E_{1} |_{X_{(η_{1}, χ_{1})}} = B_{(η_{1}, χ_{1})}^{f} |_{X_{(η_{1}, χ_{1})}}$ and $E_{2} |_{X_{(η_{2}, χ_{2})}} = B_{(η_{1}, χ_{1})}^{f} |_{X_{(η_{1}, χ_{1})}}$ .

Suppose that $Fix (B_{(η_{1}, χ_{1})}^{f}) = {(\bar{u_{1}^{*}}, \bar{v_{1}^{*}})^{T}}$ and $Fix (B_{(η_{1}, χ_{1})}^{f}) = {(\bar{u_{1}^{*}}, \bar{v_{1}^{*}})^{T}}$ . Then $lim_{n \to \infty} {(B_{(η_{1}, χ_{1})}^{f})}^{n} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}})^{T} = ({\bar{u}}_{1}^{*}, {\bar{v}}_{1}^{*})^{T}$ and $lim_{n \to \infty} {(B_{(η_{1}, χ_{1})}^{f})}^{n} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}})^{T} = ({\bar{u}}_{1}^{*}, {\bar{v}}_{1}^{*})^{T} .$ Moreover, $\begin{matrix} lim_{n \to \infty} {(B_{(η_{1}, χ_{1})}^{f})}^{n} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}})^{T} \\ \leq lim_{n \to \infty} {(B_{(η_{1}, χ_{1})}^{f})}^{n} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}})^{T} . \end{matrix}$ Thus $(\bar{u_{1}^{*}}, \bar{v_{1}^{*}})^{T} \leq ({\bar{u}}_{1}^{*}, {\bar{v}}_{1}^{*})^{T}$ . Thus, we deduce that $E_{1}^{\infty} |_{X_{(η_{1}, χ_{1})}} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}}) \leq E_{2}^{\infty} |_{X_{(η_{1}, χ_{1})}} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}})$ and so $E_{1}^{\infty} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}}) \leq E_{2}^{\infty} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}})$ .

By the analogous argument, we obtain that $E_{2}^{\infty} (\tilde{u_{3}^{*}}, \tilde{v_{3}^{*}}) \leq E_{3}^{\infty} (\tilde{u_{3}^{*}}, \tilde{v_{3}^{*}}) .$ It follows $E_{1}^{\infty} (\tilde{u_{1}^{*}}, \tilde{v_{1}^{*}}) \leq E_{2}^{\infty} (\tilde{u_{2}^{*}}, \tilde{v_{2}^{*}}) \leq E_{3}^{\infty} (\tilde{u_{3}^{*}}, \tilde{v_{3}^{*}}) .$ And hence $(u_{1}^{*}, v_{1}^{*})^{T} \leq (u_{2}^{*}, v_{2}^{*})^{T} \leq (u_{3}^{*}, v_{3}^{*})^{T}$ .□

Now we consider the continuity of mild solutions of (4) – (5) with the dates $f_{i} \in C ([t_{0}, T] \times R_{F}^{4}, R_{F})$ and suppose that f_i satisfy the Lipschitz conditions with the same constants, i = 1, 2.

Theorem 5.3. Assume that $f_{i} \in C ([t_{0}, T] \times ℝ_{F}^{4}, ℝ_{F}), η_{i} \in C ([t_{0} - d, t_{0}], ℝ_{F}), i = 1, 2,$ satisfy (H₁) – (H₂) and there exists γ₃ > 0 such that $\begin{matrix} ρ (f_{1} (t, u_{1}, u_{2}, u_{3}, u_{4}), f_{2} (t, u_{1}, u_{2}, u_{3}, u_{4})) \\ \leq γ_{3} \forall t \in [t_{0}, T]; u_{i} \in E, i = 1, 2, 3, 4 . \end{matrix}$ If there exists γ₁, γ₂ > 0 such that $\begin{matrix} D ((η_{1}, D η_{1})^{T} (t), (η_{2}, D η_{2})^{T} (t)) \\ \leq (γ_{1}, γ_{2})^{T} \forall t \in [t_{0} - d, t_{0}], \end{matrix}$ then

$\begin{matrix} D ((u_{1}^{*}, v_{1}^{*})^{T}, (u_{2}^{*}, v_{2}^{*})^{T}) \\ \leq (I - Q)^{- 1} (γ_{1}, γ_{2} + (T - t_{0}) γ_{3})^{T}, \end{matrix}$ (16) where $(u_{i}^{*}, v_{i}^{*})^{T}$ are the unique mild solution in type 1 of (4) – (5) with f_i and η_i, i = 1, 2.

Proof. From Theorem 4.1, it follows that $\begin{matrix} D (B_{η_{1}, f_{1}} (u_{1}, v_{1}), B_{η_{1}, f_{1}} (u_{2}, v_{2})) \\ \leq QD ((u_{1}, v_{1})^{T}, (u_{2}, v_{2})^{T}), \\ \forall (u_{i}, v_{i})^{T} \in X, i = 1, 2 . \end{matrix}$ (17)

Additionally, $\begin{matrix} D (B_{η_{1}, f_{1}} (u, v), B_{η_{2}, f_{2}} (u, v)) \\ = D ((\begin{matrix} B_{1 η_{1}, f_{1}} (u, v) \\ B_{2 η_{1}, f_{1}} (u, v) \end{matrix}), (\begin{matrix} B_{1 η_{2}, f_{2}} (u, v) \\ B_{2 η_{2}, f_{2}} (u, v) \end{matrix})) . \end{matrix}$

For t ∈ [t₀ – d, t₀], we have $\begin{matrix} D ((\begin{matrix} B_{1 η_{1}, f_{1}} (u, v) \\ B_{2 η_{1}, f_{1}} (u, v) \end{matrix}), (\begin{matrix} B_{1 η_{2}, f_{2}} (u, v) \\ B_{2 η_{2}, f_{2}} (u, v) \end{matrix})) \\ = D (η_{1}, D (η_{1})^{T} (t), (η_{2}, D η_{2})^{T} (t)) \\ \leq (\begin{matrix} γ_{1} \\ γ_{2} \end{matrix}) \leq (\begin{matrix} γ_{1} \\ γ_{2} + (T - t_{0}) γ_{3} . \end{matrix}) \end{matrix}$

For t ∈ [t₀, T], by putting F_i (s, u, v) : = f_i (s, u (s) , v (s) , u (s – d) , v (s – d)), i = 1, 2, we have $\begin{matrix} D ((\begin{matrix} B_{1 η_{1}, f_{1}} (u, v) \\ B_{2 η_{1}, f_{1}} (u, v) \end{matrix}), (\begin{matrix} B_{1 η_{2}, f_{2}} (u, v) \\ B_{2 η_{2}, f_{2}} (u, v) \end{matrix})) \\ \leq (\begin{matrix} H (η_{1} (t_{0}) \oplus \int_{t_{0}}^{t} v (s) ds, η_{2} (t_{0}) \oplus \int_{t_{0}}^{t} v (s) ds) \\ H (η_{1}^{'} (t_{0}) \oplus F_{1} (s, u, v)) ds, η_{2}^{'} (t_{0}) \oplus \int_{t_{0}}^{t} F_{2} (s, u, v) ds) \end{matrix}) \\ = (\begin{matrix} H (η_{1} (t_{0}), η_{2} (t_{0})) \\ H (D η_{1} (t_{0}), D η_{2} (t_{0})) + H (\int_{t_{0}}^{t} F_{1} (s, u, v) ds, \int_{t_{0}}^{t} F_{2} (s, u, v) ds) \end{matrix}) \\ \leq (γ_{1}, γ_{2} + (T - t_{0}) γ_{3})^{T} . \end{matrix}$

From (17) and these inequation, we have $\begin{matrix} D & ((u_{1}^{*}, v_{1}^{*})^{T}, (u_{2}^{*}, v_{2}^{*})^{T}) \\ = D (B_{η_{1}, f_{1}} (u_{1}^{*}, v_{1}^{*}), B_{η_{2}, f_{2}} (u_{2}^{*}, v_{2}^{*})) \\ \leq D (B_{η_{1}, f_{1}} (u_{1}^{*}, v_{1}^{*}), B_{η_{1}, f_{1}} (u_{2}^{*}, v_{2}^{*})) \\ + D (B_{η_{1}, f_{1}} (u_{2}^{*}, v_{2}^{*}), B_{η_{2}, f_{2}} (u_{2}^{*}, v_{2}^{*})) \\ \leq QD ((u_{1}^{*}, v_{1}^{*})^{T}, (u_{2}^{*}, v_{2}^{*})^{T}) \\ + (γ_{1}, γ_{2} + (T - t_{0}) γ_{3})^{T} . \end{matrix}$ Since Qⁿ → 0 as n→ ∞, then there exists (I – Q) ^-1. Then we receive (16). It completes the proof.

□

6 The smoothness of solutions

Let $σ \in J \subset ℝ$ , consider following system in [t₀, T]

${\begin{matrix} D u (t; σ) = v (t; σ) \\ D v (t; σ) = f (t, u (t; σ), v (t; σ), u (t - d; σ), v (t - d; σ); σ) . \end{matrix}$ (18)

Theorem 6.1. Assume that $f \in C ([t_{0}, T] \times ℝ_{F}^{4} \times J, ℝ_{F})$ and $η \in C^{1} ([t_{0} - d, t_{0}], ℝ_{F})$ . Moreover, the conditions (C₁) , (C₂) hold, where

there exists L₁, L₂ > 0 such that $\begin{matrix} ρ (\frac{\partial f (t, u_{1}, u_{2}, u_{3}, u_{4}; σ)}{\partial u_{i}}, \hat{0}) \leq \frac{L_{1}}{2}, \\ u_{1}, u_{3} \in ℝ_{F}, i = 1, 3, σ \in J; \\ ρ (\frac{\partial f (t, u_{1}, u_{2}, u_{3}, u_{4}; σ)}{\partial u_{i}}, \hat{0}) \leq \frac{L_{2}}{2}, \\ u_{2}, u_{4} \in ℝ_{F}, i = 2, 4, σ \in J; \end{matrix}$

for $Q = (T - t_{0}) (\begin{matrix} 0 1 \\ L_{1} L_{2} \end{matrix})$ , we have Qⁿ ⟶ 0 as n→ ∞.

Then, problem (18) has a unique mild solution (u^*, v^*) ^T (. , σ) in $C ([t_{0} - d, T] \times J, R_{F}^{2})$ and $(u^{*}, v^{*})^{T} (., σ) \in C^{1} (J, ℝ_{F}^{2})$ for all t ∈ [t₀ – d, T] .

Proof. Assume that u, v are (i) – gH differentiable. The problem (18) can be transformed to the following functional-integral system: $\begin{matrix} (u, v)^{T} (t; σ) = & (η (t_{0}) \oplus \int_{t_{0}}^{t} v (s; σ) ds, D η (t_{0}) \\ \oplus \int_{t_{0}}^{t} F (s, σ, u, v) ds)^{T} \end{matrix}$ (19) for t ∈ [t₀, T] and $(u, v)^{T} (t; σ) = (η, D η)^{T} (t)$ , for t ∈ [t₀ – d, t₀], where $\begin{matrix} F (s, σ, u, v) \\ = f (s, u (s; σ), v (s; σ), u (s - d; σ), v (s - d; σ)) . \end{matrix}$

Now let us take the operator $A : C ([t_{0} - d, T] \times J, ℝ_{F}^{2}) ⟶ C ([t_{0} - d, T] \times J, ℝ_{F}^{2})$ , defined by $A (u, v)^{T} (t; σ) : = (\begin{matrix} A_{1} (u, v)^{T} \\ A_{2} (u, v)^{T} \end{matrix}),$ be the right hand side of (19) for t ∈ [t₀, T].

Let $X = C ([t_{0} - d, T] \times J, ℝ_{F}^{2})$ . It is similar to the proof of Theorem 4.1, from conditions (C₁) – (C₂), the operator A : (X, D) ⟶ (X, D) is a Picard operator.

For t ∈ [t₀ – d, t₀], we have $\begin{matrix} D (A (u_{1}, v_{1})^{T}, A (u_{2}, v_{2})^{T}) \\ = D ((η, D η)^{T}, (η, D η)^{T}) = 0 . \end{matrix}$

For t ∈ [t₀, T]

$\begin{matrix} D (A (u_{1}, v_{1})^{T}, A (u_{2}, v_{2})^{T}) \\ \leq (\begin{matrix} H (\int_{t_{0}}^{t} v_{1} (s; σ) ds, \int_{t_{0}}^{t} v_{2} (s; σ) ds) \\ H (\int_{t_{0}}^{t} F (s; σ, u_{1}, v_{1}) ds, \int_{t_{0}}^{t} F (s; σ, u_{2}, v_{2}) ds) \end{matrix}) \\ \leq & (\begin{matrix} (t - t_{0}) H (v_{1} (s; σ), v_{2} (s; σ)) \\ \int_{t_{0}}^{t} \frac{L_{1}}{2} [ρ (u_{1} (s; σ), {\tilde{u}}_{1} (s; σ)) + ρ (u_{2} (s; σ), {\tilde{u}}_{2} (s; σ))] \\ + \frac{L_{2}}{2} [ρ (v_{1} (s; σ), {\tilde{v}}_{1} (s; σ)) + ρ (v_{2} (s; σ), {\tilde{v}}_{2} (s; σ))] ds \end{matrix}) . \end{matrix}$

By putting s – d = v, s ∈ [a, t], we receive $\begin{matrix} D (A (u_{1}, v_{1})^{T}, A (u_{2}, v_{2})^{T}) \\ \leq Q . D ((u_{1}, v_{1})^{T}, (u_{2}, v_{2})^{T}) . \end{matrix}$

Because Q converges to zero matrix (the Hypotheses (C₂)), by Perov fixed theorem we implies that the operator A has a unique fixed point in $C^{1} (J, R_{F}^{2})$ . That is the unique mild solution of problem (18). Now we will prove the second task of the theorem.

In fact, let (u^*, v^*) ^T be the unique fixed point of A. Suppose that there exists (∂u^*/∂σ, ∂v^*/∂σ) ^T, from (18) – (19), we obtain $\begin{matrix} \frac{\partial u^{*}}{\partial σ} = & \int_{t_{0}}^{t} \frac{\partial v^{*} (s; σ)}{\partial σ} ds \\ \frac{\partial v^{*}}{\partial σ} = & \int_{t_{0}}^{t} \frac{\partial F (s, σ, u^{*}, v^{*})}{\partial u^{*} (s; σ)} . \frac{\partial u^{*} (s; σ)}{\partial σ} ds \\ + \int_{t_{0}}^{t} \frac{\partial F (s, σ, u^{*}, v^{*})}{\partial v^{*} (s; σ)} . \frac{\partial v^{*} (s; σ)}{\partial σ} ds \\ + \int_{t_{0}}^{t} \frac{\partial F (s, σ, u^{*}, v^{*})}{\partial u^{*} (s - d; σ)} . \frac{\partial u^{*} (s - d; σ)}{\partial σ} ds \\ + \int_{t_{0}}^{t} \frac{\partial F (s, σ, u^{*}, v^{*})}{\partial v^{*} (s - d; σ)} . \frac{\partial v^{*} (s - d; σ)}{\partial σ} ds \\ + \int_{t_{0}}^{t} \frac{\partial F (s, σ, u^{*}, v^{*})}{\partial σ} ds \end{matrix}$ for all t ∈ [t₀, T] , σ ∈ J. Then we consider following operator

$\begin{matrix} C : X \times X & ⟶ X \\ ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}) & \mapsto C ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}), \end{matrix}$

where C ((x₁₂, z₁₂) ^T, (u₁₃, u₂₄) ^T) (t, σ) =0 for t ∈ [t₀ – d, t₀] , σ ∈ J and $\begin{matrix} C ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}) (t, σ) \\ = (\begin{matrix} C_{1} ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}) \\ C_{2} ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}) \end{matrix}) \end{matrix}$ and $\begin{matrix} C_{1} ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}) (t, σ) \\ : = \int_{t_{0}}^{t} \frac{\partial v (s, σ)}{\partial σ} ds = \int_{t_{0}}^{t} u_{2} (s, σ) ds . \end{matrix}$ $\begin{matrix} C_{2} ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}) (t, σ) \\ : = \int_{t_{0}}^{t} \frac{\partial F (s, σ, u, v)}{\partial u (s, σ)} . u_{1} (s, σ) ds \\ + \int_{t_{0}}^{t} \frac{\partial F (s, σ, u, v)}{\partial v (s, σ)} . u_{2} (s, σ) ds \\ + \int_{t_{0}}^{t} \frac{\partial F (s, σ, u, v)}{\partial u (s - d, σ)} . u_{3} (s, σ) ds \\ + \int_{t_{0}}^{t} \frac{\partial F (s, σ, u, v)}{\partial v (s - d, σ)} . u_{4} (s, σ) ds + \int_{t_{0}}^{t} \frac{\partial F (s, σ, u, v)}{\partial σ} ds \end{matrix}$

for t ∈ [t₀, T] , σ ∈ J. Here we use the notations $u_{1} (s, σ) : = \frac{\partial u (s, σ)}{\partial σ}$ , $u_{2} (s, σ) : = \frac{\partial v (s, σ)}{\partial σ},$ $u_{3} (s, σ) : = \frac{\partial u (s - d, σ)}{\partial σ},$ $u_{4} (s, σ) : = \frac{\partial v (s - d, σ)}{\partial σ}$ .

We shall prove the statement that C ((x₁₂, z₁₂) ^T, (. , .) ^T) is a Q-contraction.

For t ∈ [t₀ – d, t₀]: $D (C ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}), C ((x_{12}, z_{12})^{T}, ({\bar{u}}_{13}, {\bar{u}}_{24})^{T})) = (\begin{matrix} 0 \\ 0 \end{matrix}) .$

For t ∈ [t₀, T], one has

$\begin{matrix} D (C_{1} ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}), C_{1} ((x_{12}, z_{12})^{T}, ({\bar{u}}_{13}, {\bar{u}}_{24})^{T})) \\ = H (\int_{t_{0}}^{t} u_{2} (s, σ) ds, \int_{t_{0}}^{t} {\bar{u}}_{2} (s, σ) ds) \\ \leq \int_{t_{0}}^{t} H (u_{2} (s, σ), {\bar{u}}_{2} (s, σ)) ds \leq (T - t_{0}) H (u_{2}, {\bar{u}}_{2}) \end{matrix}$

and

$\begin{matrix} D (C_{2} ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}), C_{2} ((x_{12}, z_{12})^{T}, ({\bar{u}}_{13}, {\bar{u}}_{24})^{T})) \\ \leq \int_{t_{0}}^{t} H (\frac{\partial f}{\partial u (s, σ)} . u_{1} (s, σ), \frac{\partial f}{\partial u (s, σ)} . {\bar{u}}_{1} (s, σ)) ds \\ + \int_{t_{0}}^{t} H (\frac{\partial f}{\partial u (s - d, σ)} . u_{3} (s, σ), \frac{\partial f}{\partial u (s - d, σ)} . {\bar{u}}_{3} (s, σ)) ds \\ + \int_{t_{0}}^{t} H (\frac{\partial f}{\partial v (s, σ)} . u_{2} (s, σ), \frac{\partial f}{\partial v (s, σ)} . {\bar{u}}_{2} (s, σ)) ds \\ + \int_{t_{0}}^{t} H (\frac{\partial f}{\partial v (s - d, σ)} . u_{4} (s, σ), \frac{\partial f}{\partial v (s - d, σ)} . {\bar{u}}_{4} (s, σ)) ds \\ \leq & \int_{t_{0}}^{t} \frac{L_{1}}{2} [H (u_{1}, {\bar{u}}_{1}) + H (u_{3}, {\bar{u}}_{3})] ds \\ + \int_{t_{0}}^{t} \frac{L_{2}}{2} [H (u_{2}, {\bar{u}}_{2}) + H (u_{4}, {\bar{u}}_{4})] ds . \end{matrix}$

By putting v = s – d, s ∈ [a, t] , then v ∈ [a – d, t – d] and $\begin{matrix} D (C_{2} ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}), C_{2} ((x_{12}, z_{12})^{T}, ({\bar{u}}_{13}, {\bar{u}}_{24})^{T})) \\ \leq (t - t_{0}) [L_{1} . H (u_{1}, {\bar{u}}_{1}) + L_{2} . H (u_{2}, {\bar{u}}_{2})] \\ \leq (T - t_{0}) [L_{1} . H (u_{1}, {\bar{u}}_{1}) + L_{2} . H (u_{2}, {\bar{u}}_{2})] . \end{matrix}$

Thus,

$\begin{matrix} D (C ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T}), C ((x_{12}, z_{12})^{T}, ({\bar{u}}_{13}, {\bar{u}}_{24})^{T})) \\ \leq (T - t_{0}) Q . (\begin{matrix} H (u_{1}, {\bar{u}}_{1}) \\ H (u_{2}, {\bar{u}}_{2}) \end{matrix}) \\ = Q . D ({((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T})}^{T}, {((x_{12}, z_{12})^{T}, ({\bar{u}}_{13}, {\bar{u}}_{24})^{T})}^{T}) . \end{matrix}$

From Theorem 4.1, P is a Picard operator. Thus the sequences $\begin{matrix} (x_{12}^{n + 1}, z_{12}^{n + 1})^{T} : = & A (x_{12}^{n}, z_{12}^{n})^{T} \\ (u_{13}^{n + 1}, u_{24}^{n + 1})^{T} : = & C ((x_{12}^{n}, z_{12}^{n})^{T}, (u_{13}^{n}, u_{24}^{n})^{T}), n \in ℕ . \end{matrix}$ Let ${((x_{12}^{*}, z_{12}^{*})^{T}, (u_{13}^{*}, u_{24}^{*})^{T})} = Fix (F)$ .

Then we have, $D (A (x_{12}, z_{12})^{T}, C ((x_{12}, z_{12})^{T}, (u_{13}, u_{24})^{T})) = D ((x_{12}^{n + 1}, z_{12}^{n + 1})^{T}, (u_{13}^{n + 1}, u_{24}^{n + 1})^{T})$ converges to $((x_{12}^{*}, z_{12}^{*})^{T}, (u_{13}^{*}, u_{24}^{*})^{T})$ with respect to t ∈ [t₀ – d, T] , σ ∈ J. In other words, $(x_{12}^{n + 1}, z_{12}^{n + 1})^{T}$ converges to $(x_{12}^{*}, z_{12}^{*})^{T}$ and $(u_{13}^{n + 1}, u_{24}^{n + 1})^{T}$ converges to $(u_{13}^{*}, u_{24}^{*})^{T} .$

If we take $(x_{12}^{0}, z_{12}^{0})^{T} = (0, 0)^{T}$ and $(u_{13}^{0}, u_{24}^{0})^{T} = (\partial {x_{12}}^{0} / \partial σ, \partial {z_{12}}^{0} / \partial σ)^{T} = (0, 0)^{T}$ . Then by induction, we have $(u_{13}^{n}, u_{24}^{n})^{T} = (\partial {x_{12}}^{n} / \partial σ, \partial {z_{12}}^{n} / \partial σ)^{T}$ converges to $(u_{13}^{*}, u_{24}^{*})^{T}, \forall n \in ℕ .$ Since $(x_{12}^{n}, z_{12}^{n})^{T}$ converges to $(x_{12}^{*}, z_{12}^{*})^{T}$ and (∂x₁₂ⁿ/∂σ, ∂z₁₂ⁿ/∂σ) ^T converges to $(u_{13}^{*}, u_{24}^{*})^{T}$ as n→ ∞. Then there exists $(\partial {x_{12}}^{*} / \partial σ, \partial {z_{12}}^{*} / \partial σ)^{T} = (u_{13}^{*}, u_{24}^{*})^{T} .$ Thus, (u^*, v^*) ^T (, , σ) is differentiable.

The case when u is (i)-gH differentiable and v is (ii)-gH differentiable, we can prove in the same arguments.□

7 Stability of the problem

For $f \in C ([t_{0}, T] \times ℝ_{F}^{4}, ℝ_{F}), ɛ > 0, d > 0,$ we consider the following inequation

${\begin{matrix} ρ (D y (t), w (t)) \leq ɛ \\ ρ (D w (t), f (t, y (t), w (t), y (t - d), w (t - d))) \leq ɛ \end{matrix}$ (20) with initial condition

${\begin{matrix} y (t) \leq η (t) \\ w (t) \leq D η (t) \end{matrix}, t \in [t_{0} - d, t_{0}] .$ (21)

Definition 7.1. The system (4) is Ulam-Hyers stable if there exists c > 0 such that for each ɛ > 0 and for each mild solution $(y, w)^{T} \in C^{2} ([t_{0} - d, T], ℝ_{F}^{2})$ of (20), there exists a mild solution $(u, v)^{T} \in C^{2} ([t_{0} - d, T], ℝ_{F}^{2})$ of (4) – (5) such that D ((y, w) ^T, (u, v) ^T) ≤ cɛ.

Theorem 7.1. Suppose that all hypotheses of Theorem 6.1 are satisfied. Then the system (4) is Ulam-Hyers stable.

Proof. We consider the operator E : W_d ([t₀, T]) → W_d ([t₀, T]), defined by the right hand side of (6), for t ∈ [t₀ – d, T]. From Corollary 4.1, E is a weakly Picard operator. Moreover, we have $\begin{matrix} D ((u, v)^{T}, E^{\infty} (u, v)) \leq & D ((u, v)^{T}, E (u, v)) \\ + D (E (u, v), E^{\infty} (u, v)) . \end{matrix}$ From $(x_{n + 1}, z_{n + 1})^{T} = E (x^{n}, z^{n}) = E^{n + 1} (u, v)$ we have $\begin{matrix} D (E (u, v), E^{\infty} (u, v)) \\ = lim_{n \to \infty} D (E (u, v), E (x^{n}, z^{n})) \\ \leq Q lim_{n \to \infty} D ((u, v)^{T}, (x^{n}, z^{n})^{T}) \\ = Q . D ((u, v)^{T}, E^{\infty} (u, v)) . \end{matrix}$ It implies $\begin{matrix} [1 - (T - t_{0}) Q] D ((u, v)^{T}, E^{\infty} (u, v)) \\ \leq D ((u, v)^{T}, E (u, v)) . \end{matrix}$ Therefore $\begin{matrix} D ((u, v)^{T}, E^{\infty} (u, v)) \\ \leq [1 - (T - t_{0}) Q]^{- 1} D ((u, v)^{T}, E (u, v)) . \end{matrix}$ Let (y^*, w^*) ^T be a solution of (20) – (21). Then there exists (u^*, v^*) ^T ∈ Fix (E) such that (u^*, v^*) ^T = E^∞ (y^*, w^*) and (u^*, v^*) ^T is a solution of (4) – (5). On the other hand, $\begin{matrix} D ((y^{*}, w^{*})^{T}, (u^{*}, v^{*})^{T}) \\ = D ((y^{*}, w^{*})^{T}, E^{\infty} (y^{*}, w^{*})) \\ \leq c . D ((y^{*}, w^{*})^{T}, E (y^{*}, w^{*})) \leq c ɛ . \end{matrix}$ Thus the system (6) is Ulam-Hyers stable.□

Example 7.1. We consider the following model $\begin{matrix} D^{2} u (t) = & \frac{π}{8} u (t - 1) + 2 K (1 - \frac{π}{16} t^{2}) \\ for t \in [0, \frac{4}{\sqrt{π}}] \end{matrix}$ (22) with initial condition

$u (t) = K cos t, t \in [- 1, 0],$ (23) where $D^{2} u$ is gH-derivative of fuzzy-valued function u in order 2, [K] ^κ = [κ, 1 – κ] , κ ∈ [0, 1] , is a triangular fuzzy number.

It is clear that the right hand side $f (t, u (t), D u (t), u (t - 1), D u (t - 1)) = \frac{π}{8} u (t - 1) + 2 K (1 - \frac{π}{16} t^{2})$ satisfies Lipschitz condition with Lipschitz constant is $L_{1} = \frac{π}{8}, L_{2} = \frac{π}{16},$ since $\begin{matrix} ρ (f (t, u (t), D u (t), u (t - 1), D u (t - 1)), f (t, v (t), \\ D v (t), v (t - 1), D v (t - 1))) \leq \frac{π}{8} ρ (u (t - 1), v (t - 1)) . \end{matrix}$ Recall that a square matrix $Q = (\begin{matrix} a b \\ c d \end{matrix})$ converges to zero if and only if a + d < min {2, 1 + ad – bc} (see [5]). Let $Q = T (\begin{matrix} 0 1 \\ \frac{π}{8} \frac{π}{16} \end{matrix})$ . It is easy to see that $\frac{π}{16} < min {2, 1 - \frac{π}{8}} = 1 - \frac{π}{8}$ then Qⁿ → 0 as n→ ∞. Therefore all conditions of Theorem 4.1 are satisfied. (22) – (23) has a unique mild solution u^* in type 1.

By calculating, we can point out that $u (t) = {\begin{matrix} K (t + 1)^{2} & for t \in [0, \frac{4}{\sqrt{π}}] \\ K cos t & for t \in [- 1, 0] \end{matrix}$ is a mild solution in type 1 of problem (22) – (23). Moreover it also satisfies Theorem 5.1, Theorem 5.2 and Theorem 5.3, then this unique solution is continous and monotony dependence with respect to datas.

Remark 7.1. Problem (4) can be generalized Hyers-Ulam-Rassias stable with respect to Φ provides that function Φ must satisfy some bounded conditions (see Theorem 4.2 in [14]).

8 Conclusions

This paper deals with the well-posedness of Cauchy problem for second-order fuzzy DEs with delay in the partially ordered fuzzy metric spaces. By using the theory of weakly Picard operator and Perov fixed point theorems, the existence of the uniqueness and properties of four kinds of mild solutions of fuzzy functional differential equantion in order 2 are proved. For the next step of future work, we can study some classes of fuzzy DEs with granular derivatives.

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