parameters in a class of neutral differential equations with state-dependent delays

Differentiability of solutions with respect to

parameters in a class of neutral differential equations with state-dependent delays

Ferenc Hartung

University of Pannonia, H-8201 Veszprém, P.O. Box 158, Hungary Received 11 November 2019, appeared 28 July 2021

Communicated by Hans-Otto Walther

Abstract. In this paper we consider a class of nonlinear neutral functional differential equations with state-dependent delays. We study well-posedness and differentiability of the solution with respect to the parameters in a pointwise sense and also in the supremum norm.

Keywords: neutral differential equation, state-dependent delay, differentiability with respect to parameters.

2010 Mathematics Subject Classification: 34K05, 34K40.

1 Introduction

In this paper we consider a neutral functional differential equation with state-dependent de- lays (SD-NFDEs) of the form

˙

x(t) = f

t,xt,x(t−τ(t,xt,ξ)), ˙x(t−µ(t)), ˙x(t−ρ(t,xt,λ)),θ

, a.e.t ≥0. (1.1) Here x_t denotes the solution segment function defined by x_t(ζ) = x(t+ζ) for ζ ∈ [−r, 0], where r > 0 is a fixed finite constant. ξ ∈ _Ξ, λ ∈ _Λ and θ ∈ _Θ represent parameters in the formula of τ, ρ and f, respectively. The parameter spaces Ξ, Λ and Θ are finite or infinite dimensional normed linear spaces. The dependence of f on the second argument represent delay terms which are not state-dependent (since we will assume differentiability of the function f with respect to its second argument). Also, the fourth argument of f contains a neutral term with a time-dependent delay. The terms in the third and fifth arguments are delayed terms of x and ˙x with explicitly given state-dependent delays. For the simplicity of the presentation only single explicit state-dependent delays in the retarded and neutral terms, and a single time-dependent delay in the neutral term assumed to be present in the equation.

Differential equations with state-dependent delays (SD-DDEs) are studied intensively in the last decades (see, e.g., [3,9,11,23,27–29,33,35,36,42] for some recent work and a survey

BEmail: hartung.ferenc@uni-pannon.hu

for SD-DDEs). SD-NFDEs received much less attention in the literature (see, e.g., [1,2,7,8,12, 13,16,21,22]). Walther in [39–41] studied a class of neutral equations of the form

˙

x(t) =g(∂x_t,x_t)

under conditions which allow state-dependent delays both in the retarded and in the neutral terms. Here g : C×C¹ ⊃ W → _Rⁿ, W is open, and ∂ : C¹ → C denotes the continuous linear operator of differentiation (see Section 2 for definition of the function spaces). Using the state-space of continuously differentiable functions, he proved the existence of continuous semiflows corresponding to certain subsets of initial functions, and a principle of linear sta- bility. For well-posedness results corresponding to a class of NFDEs (without state-dependent delays) of the form ˙x(t) = f(t,x_t, ˙x_t)we refer to [32]. Here the existence of solutions is proved using a variant of the Krasnoselskii fixed point theorem, andW^1,pwith finite pis used as the state-space of the solutions. Note that for the case of state-dependent delays in the neutral term the conditions of [32] are not applicable to prove the existence of solutions since ˙x_t is assumed to be only an L^p-function.

In this paper we discuss differentiability of solutions with respect to (wrt) parameters, including the initial function. Differentiability of solutions wrt parameters in equations of the form

˙

x(t) = f

t,x(t),x(t−τ(t,x_t,ξ)),θ

was first proved in [14,24], and differentiability wrt parameters including the initial time for a slightly more general equations of the form

˙

x(t) = f

t,x_t,x(t−τ(t,x_t,ξ)),θ

was proved later in [17,20]. In [5] and [19], beside of the second order differentiability, the first order differentiability was also proved. Note that in [19] the conditions assumed for the first order differentiability are weaker than the conditions assumed in earlier papers.

In [14] the differentiability of the map(ϕ,ξ,θ)7→ x_t(·,ϕ,ξ,θ)is proved for a fixedt, where the C-norm is used on the state-space of the solutions, and here ϕ is the initial function associated to the equation. The key assumption was that the parameters and the continuously differentiable initial function ϕsatisfy the compatibility condition

˙

ϕ(0−) = f

0,ϕ(0),ϕ(−τ(0,ϕ,ξ)),θ

.

This condition together with the continuity of f andτimply that the solutionxcorresponding to the parameters(ϕ,ξ,θ)is differentiable wrt the parameters at a fixed parameter value where the compatibility condition is satisfied. Walther in [37,38] proved the existence ofC¹-smooth solution semiflow for retarded functional differential equations containing large classes of SD- DDEs restricting the set of initial functions to those which satisfy the compatibility condition.

A different assumption and a different technique was used in [24] to prove differentiabiliy of the map(ϕ,ξ,θ)7→ xt(·,ϕ,ξ,θ). It was assumed that(ϕ,ξ,θ)are parameters such that the corresponding solutionxgenerates a strictly monotone time lag function, more precisely,

ess inf

t∈[0,α]

d dt

t−τ(t,x_t,ξ))>0

for some α > 0. Here W^1,p (with finite p) was used as the state-space of the solutions, but the initial functions are assumed to belong toW^1,∞. Such monotonicity assumption was also

used in [5,17]. In [19,20] the strict monotonicity was relaxed to the assumption that the time lag function is piecewise monotone. Note that in [19] an example is given to demonstrate that in the case when the time lag function is constant on a time interval, the solution may not be differentiable wrt parameters.

Differentiability wrt parameters for “implicit” SD-NFDEs of the form d

dt

x(t)−g(t,x(t−η(t)))= f

t,x_t,x(t−τ(t,x_t,ξ)),θ

was discussed in [15], and of the form d

dt

x(t)−g(t,xt,x(t−ρ(t,xt,χ)),λ)= f

t,xt,x(t−τ(t,xt,ξ)),θ

in [18]. In both manuscripts differentiability results was proved at parameter values where a compatibility condition is satisfied, theC-norm is used on the state-space of the solutions and the W^1,∞-norm is used for the initial functions. Note that a similar compatibility condition was used in [30] for NFDEs in order to guarantee the existence of a continuous semiflow on a subset ofC¹. Another important assumption used in both papers is that the delay functions ηand ρ in the neutral term is bounded below by a positive constant. Similar condition was used in [26,34,39–41].

The structure of the paper is the following: In Section 2 we introduce some notations and preliminary results. In Section 3 we give conditions which imply the well-posedness of the initial value problem (IVP) associated to the “explicit” SD-NFDE (1.1). By a solution of Equation (1.1) we mean an absolutely continuous function x which satisfies (1.1) for a.e.

t∈ [_0,α]_{for some} α>_{0, and} x(t) = ϕ(t)_fort ∈[−r, 0]with some associated initial function ϕ. We assume ϕ ∈ W^1,∞ throughout this paper. Then ˙x in (1.1) is defined only for a.e.t, so a condition is needed for the measurability of the composite function ˙x(t−µ(t)). A simple way to guarantee it is to assume that the functiont−µ(_t)is strictly monotone increasing. For the same reason, we will pose conditions which imply that the solution generates a strictly monotone time lag function in the neutral term with state-dependent delay. We show the existence and uniqueness of the solutions in a small neighbourhood P of a fixed parameter (ϕ, ¯¯ ξ, ¯λ, ¯θ) ∈ M, where M is a special parameter set. In the definition ofM (see details in Section3 below) we assume that ϕ∈ W^2,∞, the parameters satisfy a compatibility condition, and a condition which implies that the time lag function t 7→ t−ρ(t,xt,λ) of the second neutral term is strictly monotone increasing, more precisely,

ess inf

t∈[0,α]

d dt

t−ρ(t,x_t,ξ))>0

for some α > 0. We show thatW^1,∞ andW^2,∞ initial functions inP generateW^1,∞ andW^2,∞

solution segment functions, respectively, and the segment functions are uniformly bounded in the respective norms on [0,α] wrt the parameters from P. The solutions are Lipschitz continuous wrt parameters in theW^1,∞-norm (in a restricted sense) with a Lipschitz constant independent of the selection of the parameters from P. Note that the proof uses standard techniques, but it is presented for completeness, and since the uniform estimates mentioned above are important for the proofs of Section4.

In Section4 we prove the differentiability of the solutions wrt parameters in a pointwise sense, i.e., differentiability of the function (ϕ,ξ,λ,θ)7→ x(t,ϕ,ξ,λ,θ)for any fixedt ∈ [0,α], and the differentiability of the function(_ϕ,ξ,λ,θ)7→x_t(·_,ϕ,ξ,λ,θ), where we use theC-norm

on the solution segments. In both cases the differentiability result is proved at parameter values which belong toM ∩P, the initial functions are restricted toW^2,∞-functions, and we use theW^2,∞-norm for the initial functions. See Theorem4.3below for the precise formulation of the statement.

2 Notations and preliminaries

A fixed norm onRⁿ and the corresponding matrix norm on R^n×n are both denoted by | · |. The open ball in a normed linear space(X,| · |_X)around a point x₀ with radiusRis denoted byB_X(x₀;R)_{, i.e.,}B_X(x₀;R) _:= {x ∈ X : |x−x₀|_X < R}, and the corresponding closed ball byB_X(x₀;R). The space of bounded linear operators between normed linear spacesX andY is denoted byL(X,Y), and the norm on it is by| · |_L(X,Y).

The derivative of a single variable functionv(t)_wrttis denoted by ˙v. Note that all deriva- tives we use in this paper are Fréchet derivatives. Suppose the function F(x₁, . . . ,xm) takes values inRⁿ. The partial derivatives of Fwrt its first, second, etc. arguments are denoted by D1F,D2F, etc. In the case when the argument x1of Fis real, we simply writeD1F(_{x1, . . . ,xm}) instead of the more precise notationD₁F(x₁, . . . ,xm)1, i.e., here D₁Fdenotes the vector inRⁿ instead of the linear operator L(_R,Rⁿ). In the case when, let say, x₂ ∈ _Rⁿ, then we identify the linear operatorD2F(x₁, . . . ,xm)∈ L(_Rⁿ,Rⁿ)by ann×n matrix.

The spaces of continuous and continuously differentiable functions from[−r, 0]to Rⁿ are denoted by C and C¹, respectively, where the norms are defined by |ψ|_C := max{|ψ(ζ)| : ζ ∈ [−r, 0]}and|ψ|_C1 :=max{|ψ|_C,|ψ^˙|_C}. The L^∞-norm of an essentially bounded Lebesgue measurable function ψ: [−r, 0] → _Rⁿ is defined by |ψ|_L∞ := ess sup{|ψ(ζ)| : ζ ∈ [−r, 0]}. W^1,p (1 ≤ p < _{∞) and} W^1,∞ denote the spaces of absolutely continuous functions ψ : [−r, 0]→_Rⁿof finite norm|ψ|_W1,p := R0

−r|ψ(s)|^p+|ψ^˙(s)|^pds1/p

, 1≤ p< _{∞, and}|ψ|_W1,∞ := max{|ψ|_C,|ψ^˙|_L∞}, respectively. We note thatψ∈W^1,∞, if and only ifψis Lipschitz continuous.

W^2,∞ is the space of continuously differentiable functions from [−r, 0] _{to R}ⁿ with Lipschitz continuous first derivative. The norm onW^2,∞is defined by|ψ|_W2,∞ :=max{|ψ|_C,|ψ^˙|_C,|ψ^¨|_L∞}. For a given 0< r₀ < r, the space of continuous functions χ: [−r,−r₀] → _Rⁿ is denoted by C_r0, and the norm is |χ|_Cr

0 := _max{|χ(ζ)|: ζ ∈ [−r,−r₀]}. Similarly, we use the notation L^∞_r0 := L^∞([−r,−r₀],Rⁿ), and|χ|_L∞

r0 :=ess sup{|χ(ζ)| : ζ ∈[−r,−r₀]}.

The following version of the well-known Gronwall’s lemma is used in the manuscript.

Lemma 2.1. Suppose u: [a−r,b]→[0,∞)is continuous, u(t)≤ A+B

Z _t

a

|u_s|_Cds, t∈ [a,b], (2.1) and

|u_a|_C≤ A. (2.2)

Then

u(t)≤ |u_t|_C ≤ Ae^B(t−a), t∈[a,b]. (2.3) Proof. It is easy to see that (2.1) and (2.2) imply that the functionv(t):=|ut|_C satisfies

v(t)≤ A+B Z _t

a v(s)ds, t∈ [a,b], which yields (2.3).

The Mean Value Theorem will be used in the following form througout the manuscript.

Lemma 2.2. Supposeψ∈W^1,∞([a,b],Rⁿ). Then

|ψ(t₁)−ψ(t₂)| ≤ |ψ^˙|_L∞([a,b]_,Rⁿ)|t₁−t₂|, t₁,t₂∈ [a,b]. The following lemma from [4] is a key result to prove Lemma2.4.

Lemma 2.3([4]). Let p∈[1,∞), g∈ L^p([−r,α],Rⁿ),ε>0, and u∈ A(ε), where

A(ε):={v∈W^1,∞([0,α],[−r,α]) : ˙v(s)≥εfor a.e. s∈ [0,α]}. (2.4)

Then Z _α

0

|g(u(s))|^pds≤ ¹ ε

Z _α

−r

|g(s)|^pds.

Moreover, if the sequence u^k ∈ A(ε)is such that|u^k−u|_C([0,α],R)→₀as k→_∞, then

klim→_∞ Z _α

0

g(u^k(s))−g(u(s))

p

ds=0.

Next we recall the following estimate from [17].

Lemma 2.4. Let y ∈ W^1,∞([−r,α],Rⁿ), and let ω_k ∈ (0,∞) (k ∈ _N) be a sequence satisfying ω_k →0as k→_{∞. Let}ε>0,A(ε)be defined by(2.4), and p,p^k ∈ A(ε)be such that

|p^k−p|_C([0,α],R) ≤ω_k, k ∈_N. (2.5) Then

klim→_∞

1 ω_k

Z _α

0

y(p^k(s))−y(p(s))−y˙(p(s))(p^k(s)−p(s))

ds=0. (2.6)

The following result is a simplified version of Lemma 2.5 from [19].

Lemma 2.5. Suppose g ∈ L^∞([c,d],R), and u: [a,b]→ [c,d]is an absolutely continuous function, and

ess inf

s∈[a,b] u˙(s)>0. (2.7)

Then the composite function g◦u∈ L^∞([a,b],R), and|g◦u|_L∞([a,b],R) ≤ |g|_L∞([c,d],R).

3 Well-posedness and continuous dependence on parameters

Consider the SD-NFDE

˙

x(t) = f

t,x_t,x(t−τ(t,x_t,ξ))_{, ˙}x(t−µ(t))_{, ˙}x(t−ρ(t,x_t,λ))_,θ

, a.e.t ∈[_0,T]_{, (3.1)} where T>0 is finite orT =∞, in which case[0,T]denotes the interval[0,∞).

We associate the initial condition

x(t) = ϕ(t), t ∈[−r, 0]. (3.2) Next we list our assumptions on the SD-NFDE (3.1) which are used throughout this paper.

LetΘ,ΞandΛbe normed linear spaces with norms | · |_Θ,| · |_Ξ and| · |_Λ, respectively, and let Ω1 ⊂ C,Ω2 ⊂ _Rⁿ,Ω3 ⊂ _Rⁿ, Ω4 ⊂_Rⁿ, Ω5 ⊂ _{Θ, Ω6} ⊂_Ξ andΩ7 ⊂ _Λbe open subsets of the respective spaces. Let 0 <r₀ <rbe fixed constants. We assume:

(A1) (i) f: R×C×_Rⁿ×_Rⁿ×_Rⁿ×_Θ⊃[0,T]×_Ω1×_Ω2×_Ω3×_Ω4×_Ω5→_Rⁿ is locally Lipschitz continuous in the following sense: for every finite α ∈ (0,T], for every closed subset M₁ ⊂ _Ω1 of C which is also a bounded subset of W^1,∞, compact subset M_j ⊂ _Ωj (j =2, 3, 4) of Rⁿ, and closed and bounded subset M₅ ⊂ _Ω5 of Θ, there exists a constantL₁= L₁(α,M₁,M2,M3,M₄,M5)such that

|f(t,ψ,u,v,w,θ)− f(t, ¯^¯ ψ, ¯u, ¯v, ¯w, ¯θ)|

≤L₁

|t−¯t|+|ψ−ψ¯|_C+|u−u¯|+|v−v¯|+|w−w¯|+|θ−θ^¯|_Θ, fort, ¯t∈ [0,α],ψ, ¯ψ∈ M₁,u, ¯u∈ M2,v, ¯v∈ M3,w, ¯w∈ M₄ andθ, ¯θ ∈ M5;

(ii) f is differentiable wrt its second, third, fourth, fifth and sixth variables, and the functions

R×C×_Rⁿ×_Rⁿ×_Rⁿ×_Θ⊃[0,T]×_Ω1×_Ω2×_Ω3×_Ω4×_Ω5 →X_j, (t,ψ,u,v,w,θ)7→ D_jf(t,ψ,u,v,w,θ)

are continuous for j= 2, 3, 4, 5, 6, whereX2 := L(C,Rⁿ), X3 := X₄ := X5 := _R^n×n, andX₆ :=L(_Θ,Rⁿ);

(A2) (i) τ: R×C×_Ξ⊃[0,T]×_Ω1×_Ω6 →_Rsatisfies

0≤τ(t,ψ,ξ)≤r, fort∈ [0,T], ψ∈_Ω1 andξ ∈ _Ω6,

and it is locally Lipschitz continuous in the following sense: for every finiteα ∈ (0,T], closed subset M₁ ⊂ _Ω1 of C which is also a bounded subset of W^1,∞, and closed and bounded subsetM₆ ⊂_Ω6ofΞthere exists a constantL₂= L₂(α,M₁,M₆) such that

|τ(t,ψ,ξ)−τ(t, ¯^¯ ψ, ¯ξ)| ≤L₂

|t−t^¯|+|ψ−ψ^¯|_C+|ξ−ξ^¯|_Ξ fort, ¯t∈ [0,α],ψ, ¯ψ∈ M₁andξ, ¯ξ ∈ M₆;

(ii) τis differentiable wrt its second and third variables, and the maps R×C×_Ξ⊃ [0,T]×_Ω1×_Ω6 →Y_j, (t,ψ,ξ)7→ D_jτ(t,ψ,ξ) are continuous forj=2, 3, whereY₂:=L(C,R)andY₃ :=L(_Ξ,R);

(A3) µ: [_0,T]→[r₀,r]is a contraction on any finite time interval, i.e., for every finiteα∈(_0,T] there existsL₃ =L₃(α)<1 such that

|µ(t)−µ(t^¯)| ≤L₃|t−^¯t|, t, ¯t∈[0,α]; (A4) (i) ρ: R×C×_Λ⊃ [0,T]×_Ω1×_Ω7 →_Rsatisfies

0<r₀≤ ρ(t,ψ,λ)≤r, t ∈[0,T], ψ∈_Ω1, λ∈_Ω7,

and it is locally Lipschitz continuous in the following sense: for every finiteα ∈ (0,T], closed subset M₁ ⊂ _Ω1 of C which is also a bounded subset of W^1,∞, and bounded and closed subsetM7 ⊂_Ω7ofΛthere existsL₄ = L₄(α,M₁,M7)such that

|ρ(_t,ψ,λ)−ρ(¯t, ¯ψ, ¯λ)| ≤ L4

|t−t^¯|+ _max

ζ∈[−r,−r0]|ψ(ζ)−ψ^¯(ζ)|+|λ−λ^¯|_Λ fort, ¯t∈ [_0,α]_{,ψ, ¯}ψ∈ M₁, andλ, ¯λ∈ M₇;

(ii) ρ is continuously differentiable, i.e.,ρis differentiable wrt all its variables, and the maps

R×C×_Λ⊃[0,T]×_Ω1×_Ω7→ Z_j, (t,ψ,λ)7→ D_jρ(t,ψ,λ)

are continuous forj=1, 2, 3, whereZ₁ :=_R, Z₂ := L(C,R)andZ₃ := L(_Λ,R). (iii) The partial derivatives D₁ρ and D₂ρ are uniformly continuous on the sets [0,α]×

M₁×M₇ ⊂ [0,T]×_Ω1×_Ω7 such that α > 0 is finite, M₁ is a closed subset ofC which is also a bounded subset of W^1,∞, and M₇ is a bounded and closed subset of Λ.

Remark 3.1. It follows from (A4) (i), (ii) that ρ(t,ψ,λ)depends only on the restriction ofψto the interval [−r,−r₀], since ifψ(ζ) =ψ^¯(ζ)forζ ∈ [−r,−r₀], thenρ(t,ψ,λ) =ρ(t, ¯ψ,λ)and

D_jρ(t,ψ,λ) =D_jρ(t, ¯ψ,λ), j=1, 2, 3. (3.3) Moreover,

|D₂ρ(t,ψ,λ)h| ≤ |D₂ρ(t,ψ,λ)|_L(C,R) max

ζ∈[−r,−r0]

|h(ζ)| (3.4)

fort ∈[0,T],ψ∈_Ω1,λ∈ _Ω7 andh∈C.

We prove (3.3) forj=2. The proofs forj=1 andj=3 are similar. Leth∈Cbe a non-zero function, and define the sequenceη_k := ¹_khfork∈_N. We have|η_k|_C→0 ask→_∞, and

|[D₂ρ(t,ψ,λ)−D₂ρ(t, ¯ψ,λ)]h|

|h|_C = |[D₂ρ(t,ψ,λ)−D₂ρ(t, ¯ψ,λ)]η_k|

|η_k|_C

≤ |ρ(t,ψ+η_k,λ)−ρ(t,ψ,λ)−D₂ρ(t,ψ,λ)η_k|

|η_k|_C

+|ρ(t, ¯ψ+η_k,λ)−ρ(t, ¯ψ,λ)−D2ρ(t, ¯ψ,λ)η_k|

|η_k|_C ^.

Since the right-hand side goes to 0 ask→∞, we get (3.3) with j=2.

To prove (3.4), fixh∈ Csuch thathis nonconstant on[−r₀, 0]. Define h¯(ζ):=

(h(ζ), −r≤ζ ≤ −r₀, h(−r₀), −r₀ <ζ ≤0,

and the sequence of functions χ_k := ¹_k(h−h^¯). Then ¯h,χ_k ∈ C, |χ_k|_C 6= 0 for all k ∈ _{N, and}

|χ_k|_C →0 ask→∞. Therefore,

klim→_∞

|ρ(_t,ψ+χ_k,λ)−ρ(_t,ψ,λ)−D2ρ(_t,ψ,λ)χ_k|

|χ_k|_C =0.

On the other hand, sinceχ_k(ζ) =0 for−r ≤ζ ≤ −r₀, we have fork ∈_N

|ρ(t,ψ+χ_k,λ)−ρ(t,ψ,λ)−D₂ρ(t,ψ,λ)χ_k|

|χ_k|_C = |D₂ρ(t,ψ,λ)χ_k|

|χ_k|_C = |D₂ρ(t,ψ,λ)(h−h^¯)|

|h−h^¯|_C =0.

Therefore, D2ρ(t,ψ,λ)h= D2ρ(t,ψ,λ)h, so^¯

|D₂ρ(t,ψ,λ)h|=|D₂ρ(t,ψ,λ)h^¯| ≤ |D₂ρ(t,ψ,λ)|_L(C,R)|h^¯|_C =|D₂ρ(t,ψ,λ)|_L(C,R) max

ζ∈[−r,−r0]

|h(ζ)|, which proves (3.4).

Note that assumptions (A1) and (A2) are very similar to those used in [24] for SD-DDEs.

The key assumptions in this paper are thatρandµare bounded below byr0 >0 (see (A3) and (A4) (i)), andρ(t,ψ,λ) depends only on the restriction ofψto the interval [−r,−r₀]. Similar assumption is used for SD-NFDEs in [15], see condition (g1) in [39], [41], and for PDEs with state-dependent delays in [34].

Assumptions (A1), (A2) and (A4) are naturally satisfied for the case when the parame- ter spaces Ξ, Λ and Θ are finite dimensional. Another typical situation when the assumed Lipschitz continuity conditions can be satisfied is the case when the parameters are functions.

For example, we can consider the example whenΛ=W^1,∞([0,T],R), andρhas the form ρ(t,ψ,λ) =ρ¯

t,ψ(−ν¹(t)), . . . ,ψ(−ν^`(t)), Z _−r0

−r B(t,ζ)ψ(ζ)dζ,λ(t)

,

where t ∈ [0,T], ψ ∈ C, λ ∈ Λ. It is easy to formulate natural assumptions on ¯ρ, ν¹, . . . ,ν^` and B which guarantee (A4). Similar specific examples can be given for the parameters θ andξ, and for the particular form of f andρ when conditions (A1) and (A2) hold naturally.

See Lemma 3.4 in [18] for such related results. We comment that the Arzelà–Ascoli Theorem yields that closed subsets ofCwhich are bounded inW^1,∞ are compact in C.

For the rest of the manuscript we will denote the restriction of a functionψ: [−r, 0]→_Rⁿ to the interval[−r,−r₀]by

P(ψ):=ψ|_[−r,−r

0].

For a function ψ ∈ C its continuous extension to the interval (0,∞)by a constant value will be denoted by

ψe(t):= (

ψ(t), t∈[−r, 0],

ψ(0), t>0. (3.5)

Assumeα₁> 0 is such thatα₁ ≤min{r₀,T}, and consider the IVP (3.1)–(3.2) on the small time interval[_0,α₁]_{. Since}α₁ ≤r₀, it follows

x_t(ζ) =x(t+ζ) = ϕ(t+ζ) = ϕ_e(t+ζ) = ϕ_et(ζ), t∈ [0,α₁], ζ ∈ [−r,−r₀], therefore (A3) and (A4) (i) yield

t−µ(t)≤0 and t−ρ(t,x_t,λ) =t−ρ(t,ϕe_t,λ)≤0, t ∈[0,α₁]. (3.6) Hence, on[0,α₁], Equation (3.1) is equivalent to the SD-DDE

˙

x(t) = f(t,x_t,x(t−τ(t,x_t,ξ)), ˙ϕ(t−µ(t)), ˙ϕ(t−ρ(t,ϕe_t,λ)),θ), a.e.t ∈[0,α₁], (3.7) where ϕis the initial function from (3.2). It is known (see, e.g., [6,14]) that the initial function must be Lipschitz continuous in order to guarantee the uniqueness of the solutions of (3.7).

But then ˙ϕis only almost everywhere differentiable, and we need to ensure that ˙ϕ(t−µ(t)) and ˙ϕ(t−ρ(t,ϕet,λ)) are both defined for a.e. t ∈ [0,α₁]. An easy way to guarantee it is to use the condition formulated in Lemma2.5, where it is assumed that the essential infimum of the time derivative of the inner function is positive, therefore the inner function is strictly monotone increasing. Note that an other (more technical) assumption could be to assume piecewise monotonicity of the inner function (see [19] for precise definition and for more details). In this manuscript we will assume conditions which imply that the inner functions

in the neutral terms in (3.1) (and the arguments of ˙ϕin (3.7)) are strictly monotone increasing in the sense of (2.7).

Note that (A3) implies ˙µ(t)≤ L₃(α), and so _dt^d(t−µ(t))≥ 1−L₃(α)>0 for a.e. t ∈[0,α] on any finite interval [0,α]. For the reason described above, we will restrict the parameter λ and the initial function ϕso that

ess inf

t∈[0,α₁]

d dt

t−ρ(t,ϕet,λ)>0 (3.8) with some 0<α₁ ≤r0.

The parameter space is defined as Γ := W^1,∞×_Ξ×_Λ×Θ, and we use the short no- tation γ = (ϕ,ξ,λ,θ) or γ = (γ^ϕ,γ^ξ,γ^λ,γ^θ) as a parameter vector, and the product norm

|γ|_Γ := |ϕ|_W1,∞+|ξ|_Ξ+|λ|_Λ+|θ|_Θ is used as the norm onΓ. We introduce the set of feasible parameters

Π:= ⁿ(ϕ,ξ,λ,θ)∈_Γ: ϕ∈_Ω1, ϕ(−τ(0,ϕ,ξ))∈_Ω2, θ ∈_Ω5, ξ ∈ _Ω6, λ∈_Ω7^o. Next define the special parameter set

M :=ⁿ(ϕ,ξ,λ,θ)∈ _Π : ϕ∈W^2,∞, D₁ρ(0,ϕ,λ) +D₂ρ(0,ϕ,λ)ϕ˙ <1,

˙

ϕ(−µ(₀))∈_Ω3, ϕ˙(−ρ(_0,ϕ,λ))∈_Ω4,

˙

ϕ(0−) = f

0,ϕ,ϕ(−τ(0,ϕ,ξ)), ˙ϕ(−µ(0)), ˙ϕ(−ρ(0,ϕ,λ)),θ o

. As an example, letΛ= C¹([_0,T]_,R), and supposeρhas the formρ(t,ψ,λ) =ρ¯(ψ(₀)_,λ(t)) for some function ¯ρ ∈C¹(_Rⁿ×_R,R). Then forϕ∈C¹it follows

D₁ρ(0,ϕ,λ) +D₂ρ(0,ϕ,λ)ϕ˙ =D₂ρ¯(ϕ(0),λ(0))λ^˙(0) +D₁ρ¯(ϕ(0),λ(0))ϕ˙(0).

Clearly, if ˙λ(0) = 0 = ϕ˙(0), then condition D₁ρ(0,ϕ,λ) +D₂ρ(0,ϕ,λ)ϕ˙ < 1 holds for any

¯

ρ. It also holds for the special case when ˙λ(0) = 0 and D₁ρ¯(ϕ(0),λ(0))ϕ˙(0) < 1. It is easy to satisfy the compatibility condition in the case when the parameter θ, or part of it appears in an additive way in the formula of f. Suppose, e.g., Θ = C¹([0,T],R^d)×C¹([0,T],Rⁿ), θ = (θ₁,θ₂) ∈ _Θ, and f has the form f(t,ψ,u,v,w,θ) = f^¯(t,ψ,u,v,w,θ₁(t)) +θ₂(t). If θ2(0) = ϕ˙(0−)− f^¯(0,ϕ,ϕ(−τ(0,ϕ,ξ)), ˙ϕ(−µ(0)), ˙ϕ(−ρ(0,ϕ,λ)),θ₁(0)), then the compatibil- ity condition inM holds. The above simple examples demonstrate that the conditions inM can be satisfied for certain classes of f,τandρ. We assume in this manuscript that f,τandρ are such that the setM is non-empty.

In the proof of Theorem 3.3 we will need the following result. We show that if a fixed parameter ¯γ belongs to M, in particular, if ¯ϕ ∈ W^2,∞ and D₁ρ(0, ¯ϕ, ¯λ) +D₂ρ(0, ¯ϕ, ¯λ)ϕ˙¯ < 1, then there exists α₁ > 0 such that (3.8) holds for ϕ and λ close to ¯ϕ and ¯λ. Note that the method of the proof is similar to that of Lemma 5.2 from [24].

Lemma 3.2. Supposeρsatisfies (A4), and letϕ¯ ∈W^2,∞∩_Ω1andλ¯ ∈_Ω7be such that D₁ρ(0, ¯ϕ, ¯λ) +D2ρ(0, ¯ϕ, ¯λ)ϕ˙¯ <1.

Then there exist finite constants0< α₁^∗≤ min{r₀,T},0<ε^∗ <1andδ^∗ >0such that ϕe_t ∈_Ω1for t∈[_0,α^∗₁], and

d dt

t−ρ(t,ϕe_t,λ)≥ε^∗, a.e. t∈[0,α^∗₁], ϕ∈ B_W1,∞(ϕ;¯ δ^∗), λ∈ B_Λ(λ;^¯ δ^∗). (3.9)

Proof. We note that ¯ϕ∈C¹ by the assumption ¯ϕ∈W^2,∞. Letε^∗₀be such that 0<ε^∗₀<1/2 and D₁ρ(0, ¯ϕ, ¯λ) +D₂ρ(0, ¯ϕ, ¯λ)ϕ˙¯ <1−2ε^∗₀. (3.10) Letm₁^∗>0 be fixed. The openness of Ω1 andΩ7, and the assumed continuity ofD₁ρandD₂ρ and Remark3.1 yield that there exist finite constants 0 < T₁^∗ ≤ T, κ₁^∗ > 0, ε^∗₁ > 0 andδ₁^∗ > 0 such thatB_C(ϕ;¯ κ₁^∗)⊂_Ω1,B_Λ(λ;^¯ δ₁^∗)⊂_Ω7, and

D2ρ(t,ψ,λ)−D2ρ(0, ¯ϕ, ¯λ)

L(_C,R) <m^∗₁ (3.11)

and

D₁ρ(t,ψ,λ) +D₂ρ(t,ψ,λ)χ˙−D₁ρ(0, ¯ϕ, ¯λ)−D₂ρ(0, ¯ϕ, ¯λ)ϕ˙¯

<ε^∗₀ (3.12) for t ∈ [_0,T₁^∗]_, ψ ∈ B_C(ϕ;¯ κ^∗₁)_, λ ∈ B_Λ(_λ;^¯ δ₁^∗) _and χ ∈ C¹ satisfying |P(χ˙)− P(ϕ˙¯)|_Cr

0 ≤ ε^∗₁. Define the constant m^∗ := D₂ρ(0, ¯ϕ, ¯λ)

L(_C,R)+m^∗₁. Combining the definition of m^∗ with (3.11), and (3.10) with (3.12) we get

D2ρ(_t,ψ,λ)

L(_C,R)

<_m^∗ _(3.13)

and

D₁ρ(t,ψ,λ) +D₂ρ(t,ψ,λ)χ˙ <1−ε^∗₀ (3.14) fort ∈[0,T₁^∗], ψ∈ B_C(ϕ;¯ κ₁^∗),λ∈ B_Λ(λ;^¯ δ₁^∗)andχ∈C¹satisfying|P(χ˙)− P(ϕ˙¯)|_Cr

0 ≤ε^∗₁. Define the constant

α^∗₁ :=









 minn

κ^∗₁

|ϕ¯|_C1,_|ϕ¨¯^ε_|^∗1

L∞,r₀,T₁^∗o

, |ϕ¨¯|_L∞ 6=0, |ϕ¯|_C1 6=0, minn

κ^∗₁

|ϕ¯|_C1,r0,T₁^∗o

, |ϕ¨¯|_L∞ =0, |ϕ¯|_C1 6=0, min{r₀,T₁^∗}, |ϕ¯|_C1 =0,

and introduce the extension of ¯ϕ∈C¹ to[−r,∞)by Φ(t):=

(ϕ¯(t), t∈ [−r, 0],

˙¯

ϕ(0−)t+ϕ¯(0), t>0. (3.15) ThenΦis continuously differentiable on[−r,∞). Sinceα^∗₁ ≤r₀, we have

e¯

ϕ_t(ζ) =eϕ¯(t+ζ) =ϕ¯(t+ζ) =_Φ(t+ζ) =_Φt(ζ), ζ ∈ [−r,−r₀], t∈[0,α₁^∗], (3.16) so Remark3.1yields

ρ(t,eϕ¯_t, ¯λ) =ρ(t,Φt, ¯λ), t ∈[0,α^∗₁]. The definitions ofΦ, eϕ,¯ α^∗₁ and the Mean Value Theorem imply

|_Φt−ϕ¯|_C = _max

−r≤ζ≤0|_Φ(t+ζ)−_Φ(ζ)| ≤ |ϕ¯|_C1α^∗₁ ≤κ₁^∗, t∈[_0,α^∗₁]_,

|ϕe¯_t−ϕ¯|_C = max

−r≤ζ≤0|eϕ¯(t+ζ)−eϕ¯(ζ)| ≤ |ϕ¯|_C1α^∗₁ ≤κ₁^∗, t∈[0,α₁^∗]. HenceΦt∈ _Ω1and ϕe¯_t∈ _Ω1 fort∈[0,α^∗₁], and

|P(_Φ^˙_t)− P(ϕ˙¯)|_Cr

0 = sup

−r≤ζ≤−r₀

|ϕ˙¯(t+ζ)−ϕ˙¯(ζ)| ≤ |ϕ¨¯|_L∞α^∗₁ ≤ε^∗₁, t∈[0,α₁^∗].

Therefore, it follows from (3.14) that

D₁ρ(t,Φt,λ) +D2ρ(t,Φt,λ)_Φ^˙_t <1−ε^∗₀, t∈ [0,α₁^∗], λ∈ B_Λ(λ;^¯ δ^∗₁), (3.17) so

d dt

t−ρ(t,eϕ¯_t,λ)= ^d

dt(t−ρ(t,Φt,λ)) =1−D₁ρ(t,Φt,λ)−D₂ρ(t,Φt,λ)_Φ^˙_t >ε^∗₀,

fort ∈[0,α₁^∗]andλ∈ B_Λ(λ;^¯ δ₁^∗). Next we show that a similar lower estimate can be obtained in a small neighbourhood of ¯ϕ.

The set{eϕ¯_t: t ∈[0,α^∗₁]} ⊂_Ω1is a compact subset of C, since the map[0,α₁^∗]3t7→ eϕ¯_t ∈C is continuous. Then there exists 0 < δ^∗₂ < δ₁^∗ such that its closed neighbourhood with radius δ^∗₂ belongs toΩ1. Define the set M₁^∗ := {ϕ_et: t ∈ [_0,α₁^∗]_,|ϕ−ϕ¯|_W1,∞ ≤ δ₂^∗} ∪ {_Φt: t ∈ [_0,α^∗₁]}. Then M^∗₁ ⊂ _Ω1, since |ϕ_et−eϕ¯_t|_C ≤ |ϕ−ϕ¯|_C ≤ δ₂^∗ for t ∈ [0,α^∗₁] and ϕ ∈ B_W1,∞(ϕ;¯ δ₂^∗). It is easy to check that M₁^∗ is closed inCand it is bounded inW^1,∞, so it is a compact subset ofC.

Define the closed ball M^∗₇ := B_Λ(λ;^¯ δ₂^∗). Sinceδ₂^∗ <δ₁^∗, it follows M^∗₇ ⊂ _Ω7. We introduce the notation

ω^∗_ρ(ˆt, ˆψ,t,ψ,λ):=ρ(t,ψ,λ)−ρ(t, ˆˆ ψ,λ)−D₁ρ(t, ˆˆ ψ,λ)(t−ˆt)−D₂ρ(t, ˆˆ ψ,λ)(ψ−ψˆ) fort, ˆt ∈[0,α^∗₁],ψ, ˆψ∈ M^∗₁ andλ∈ M₇^∗. Sinceρis continuously Fréchet differentiable, we have

ω_ρ^∗(t, ˆ^ˆ ψ,t,ψ,λ):=

Z ₁

0

nh D₁ρ

tˆ+ν(t−t^ˆ), ˆψ+ν(ψ−ψ^ˆ),λ)−D₁ρ(t, ˆ^ˆ ψ,λ)ⁱ(t−t^ˆ) +^hD₂ρ

ˆt+ν(t−tˆ), ˆψ+ν(ψ−ψˆ),λ)−D₂ρ(t, ˆˆ ψ,λ)ⁱ(ψ−ψˆ)^odν. (3.18) Define the function

Ωρ(δ):=supn max

|D₁ρ(t,ψ,λ)−D₁ρ(t, ˆ^ˆ ψ,λ)|,|D₂ρ(t,ψ,λ)−D₂ρ(t, ˆ^ˆ ψ,λ)|_L(C,R): t, ˆt ∈[0,α^∗₁], ψ, ˆψ∈ M₁^∗, λ∈ M^∗₇, |t−t^ˆ|+|ψ−ψ^ˆ|_C≤ δ

o

for δ > 0. Since M^∗₁ is closed in C and it is bounded in W^1,∞, and M^∗₇ is a closed and bounded subset ofΛ, assumption (A4) (iii) yields thatD₁ρandD₂ρare uniformly continuous on [0,α^∗₁]×M₁^∗×M₇^∗, therefore

Ωρ(δ)→0, asδ→0+. (3.19)

Relation (3.18) implies

|ω^∗_ρ(t, ˆˆ ψ,t,ψ,λ)| ≤_Ωρ|t−tˆ|+|ψ−ψˆ|_C|t−tˆ|+|ψ−ψˆ|_{C (3.20)} fort, ˆt∈ [0,α₁^∗],ψ, ˆψ∈ M^∗₁, λ∈ M^∗₇.

Fix 0<ε⁰ < ε^∗₀. Relation

Φt+h−_Φt h −_Φ^˙_t

C

→0, ash→0 uniformly on[0,α^∗₁]implies that there existsν₁>0 such that

m^∗

Φt+h−_Φt h −_Φ^˙_t

C

≤ε^∗₀−ε⁰, 0< |h|<ν₁, t,t+h∈[0,α^∗₁].

Then (3.13) and (3.17) yield

D₁ρ(t,Φt,λ) +D2ρ(t,Φt,λ)^Φt+h−_Φt h

≤ D₁ρ(t,Φt,λ) +D2ρ(t,Φt,λ)_Φ^˙_t+m^∗

Φt+h−_Φt h −_Φ^˙_t

C

≤1−ε⁰, 0<|h|<ν₁, t,t+h ∈[0,α^∗₁], λ∈ M^∗₇. (3.21) It is easy to check thatt 7→t−ρ(t,ϕe_t,λ)is Lipschitz continuous on[0,α^∗₁]for ϕ∈ B_W1,∞(ϕ;¯ δ₂^∗) andλ∈ M^∗₇, hence the map is a.e. differentiable on[0,α^∗₁].

Fix 0<ε^∗ <ε⁰. To show (3.9), it is enough to find 0< δ^∗≤ δ^∗₂ andν>0 such that 1

h

ρ(t+h,ϕe_t+h,λ)−ρ(t,ϕet,λ)≤1−ε^∗, t,t+h∈[0,α^∗₁], 0<|h|<ν, (3.22) for all ϕ∈ B_W1,∞(ϕ;¯ δ^∗),λ∈ M₇^∗.

Simple manipulations, (3.3) and (3.16) yield ρ(t+h,ϕe_t+h,λ)−ρ(t,ϕe_t,λ)

= D₁ρ(t,ϕet,λ)h+D₂ρ(t,ϕet,λ)(ϕ_et+h−ϕ_et) +ω^∗_ρ(t,ϕet,t+h,ϕe_t+h,λ)

= D₁ρ(t,Φt,λ)h+D₂ρ(t,Φt,λ)(_Φt+h−_Φt) +D₁ρ(t,ϕe_t,λ)−D₁ρ(t,Φt,λ)h +D₂ρ(t,ϕe_t,λ)−D₂ρ(t,Φt,λ)ϕ_et+h−ϕ_et

+D₂ρ(t,Φt,λ)ϕ_et+h−ϕ_et−_Φt+h+_Φt+ω_ρ^∗(t,ϕe_t,t+h,ϕe_t+h,λ)

= D₁ρ(t,Φt,λ)h+D2ρ(t,Φt,λ)(_Φt+h−_Φt) +D₁ρ(t,ϕet,λ)−D₁ρ(t,eϕ¯_t,λ)h +D₂ρ(t,ϕe_t,λ)−D₂ρ(t,eϕ¯_t,λ)ϕ_et+h−ϕ_et

+D₂ρ(t,Φt,λ)ϕ_et+h−ϕ_et−_Φt+h+_Φt+ω_ρ^∗(t,ϕe_t,t+h,ϕe_t+h,λ). (3.23) The Mean Value Theorem yields

|ϕ_et+h−ϕ_et|_C ≤ |ϕ˙|_L∞|h| ≤(|ϕ˙¯|_C+δ^∗₂)|h|, t,t+h∈[0,α₁^∗], ϕ∈ B_W1,∞(ϕ;¯ δ^∗₂). (3.24) Then, using (3.20) and (3.24), we get

|ω_ρ^∗(t,ϕet,t+h,ϕe_t+h,λ)|

|h| ≤N^∗Ωρ

N^∗|h|, t,t+h ∈[0,α^∗₁], ϕ∈ B_W1,∞(ϕ;¯ δ₂^∗), λ∈ M₇^∗ (3.25) with N^∗ := 1+|ϕ˙¯|_C+δ₂^∗. Let ν₁ be the corresponding constant from (3.21). Then it follows from (3.4), (3.13), (3.21), (3.23), (3.24), (3.25) and the definition of Ωρ for t,t+h ∈ [0,α₁^∗], ϕ∈ B_W1,∞(ϕ;¯ δ₂^∗),λ∈ M^∗₇ and 0< |h|<ν₁

ρ(t+h,ϕe_t+h,λ)−ρ(t,ϕe_t,λ) h

≤1−ε⁰+_Ωρ|ϕ_et−eϕ¯_t|_C+_Ωρ|ϕ_et−eϕ¯_t|_C(|ϕ˙¯|_C+δ^∗₂) + ^m

∗

|h||P(ϕ_et+h−_Φt+h−(ϕ_et−_Φt))|_Cr

0 +N^∗Ωρ

N^∗|h|

≤1−ε⁰+N^∗Ωρ

|ϕ−ϕ¯|_C+m^∗|ϕ−ϕ¯|_W1,∞+N^∗Ωρ

N^∗|h|. (3.26)