4 Differentiability wrt parameters

In this section we study differentiability of solutions of the IVP (3.1)–(3.2) wrt the initial func-tion, ϕ, and the parametersξ,λandθof the functions τ,ρand f, respectively.

Fix ¯γ := (ϕ, ¯¯ ξ, ¯λ, ¯θ) ∈ M. Let the positive constants α and δ, the parameter set P := B_Γ(γ;¯ δ), and the compact and convex sets M₁, M₂, M₃ and M₄ be defined by Theorem 3.3, and let

M₅ := B_Θ(_θ;^¯ δ)_, M₆ := B_Ξ(_ξ;^¯ δ)_{, and} M₇ :=B_Λ(_λ;^¯ δ)_{, (4.1)} as in the proof of Theorem3.3.

First we define a few notations will be used throughout this section. We introduce the space

X:=_R×C×_Rⁿ×_Rⁿ×_Rⁿ×_Θ, and for a vector x:= (t,ψ,u,v,w,θ)∈Xits norm is defined by

|_x|_{X :}=|t|+|ψ|_C+|u|+|v|+|w|+|θ|_Θ.

Fix a vector ¯x := (t, ¯^¯ ψ, ¯u, ¯v, ¯w, ¯θ) ∈ [0,α]×M₁×M₂× M₃×M₄×M₅ ⊂ X. We define the notation

ω_f(x,¯ x):= f(x)− f(x¯)−D₂f(x¯)(ψ−ψ^¯)−D₃f(x¯)(u−u¯)−D₄f(x¯)(v−v¯)

−D₅f(x¯)(w−w¯)−D₆f(x¯)(θ−θ^¯)

for x := (t,^¯ ψ,u,v,w,θ) ∈ [0,α]×_Ω1×_Ω2×_Ω3×_Ω4×_Ω5 ⊂ X. Note that the first com-ponents of ¯x and x are identical. Assumption (A1) (ii) yields the function (_ψ,u,v,w,θ) 7→

f(t,^¯ ψ,u,v,w,θ)is continuously differentiable onΩ1×_Ω2×_Ω3×_Ω4×_Ω5, and in particular, it is differentiable at(ψ, ¯^¯ u, ¯v, ¯w, ¯θ). Hence

|ω_f(x,¯ x)|

|x−x¯|_X →0, as|x−x¯|_X →0. (4.2) LetL₁= L₁(α,M₁,M2,M3,M₄,M5)be defined by (A1) (i). Then it follows from (A1) (i) that

|ω_f(x,¯ x)| ≤ |f(x)− f(x¯)|+|D₂f(x¯)(ψ−ψ^¯)|+|D₃f(x¯)(u−u¯)|+|D₄f(x¯)(v−v¯)|

+|D₅f(x¯)(w−w¯)|+|D₆f(x¯)(θ−θ^¯)|

≤

L₁+ max

j=2,...,5|D_jf(x¯)|_Xj

|x−x¯|_X (4.3)

for x := (t,^¯ ψ,u,v,w,θ) ∈ [0,α]×M₁×M₂×M₃×M₄×M₅ ⊂ X. Note here X₂, . . . ,X₅ are defined in assumption (A1) (ii).

Similarly, we define the normed linear space

A:=_R×C×_Ξ, |a|_A :=|t|+|ψ|_C+|ξ|_Ξ for a:= (t,ψ,ξ)∈A.

We fix ¯a := (t, ¯^¯ ψ, ¯ξ)∈[0,α]×M₁×M₆⊂A. Introduce the notation

ωτ(a,¯ a):=τ(a)−τ(a¯)−D2τ(a¯)(ψ−ψ^¯)−D3τ(a¯)(ξ−ξ^¯)

fora:= (t,^¯ ψ,ξ)∈[0,α]×_Ω1×_Ω6 ⊂ A. LetL₂ = L₂(α,M₁,M₆)be defined by (A2) (i). Then, similarly to (4.2) and (4.3), we get

|ω_τ(a,¯ a)|

|a−a¯|_A →_{0, as} |_a−a¯|_A →_{0, (4.4)} and

|ωτ(a,¯ a)|

|a−a¯|_A ≤L₂+max

j=2,3|D_jτ(a¯)|_Yj, a6=a,¯ a:= (t,^¯ ψ,ξ)∈[0,α]×M₁×M₆⊂A, (4.5) whereY2 andY3are defined in (A2) (ii).

We also define

B:=_R×C×_Λ, |_b|_B:=|t|+|ψ|_C+|λ|_Λ for b:= (t,ψ,λ)∈ B.

Fix a vector ¯b:= (^¯t, ¯ψ, ¯λ)∈[0,α]×M₁×M7⊂ B. We define

ω_ρ(b,^¯ b):=ρ(b)−ρ(b^¯)−D₂ρ(b^¯)(ψ−ψ^¯)−D₃ρ(b^¯)(λ−λ^¯)

forb:= (t,¯ ψ,λ)∈ [_0,α]×_Ω1×_Ω7 ⊂B. Let L4 = _L4(_α,M1,M7)be defined by (A4) (i). Then we can obtain

|ω_ρ(_b,^¯ _b)|

|b−b^¯|_B →0, as|b−b^¯|_B →0, (4.6)

and

|ω_ρ(b,^¯ b)|

|_b−_b^¯|_B ≤ L₄+_max

j=2,3|D_jρ(_b^¯)|_{Zj, b}6=_b,^¯ _b:= (t,¯ ψ,λ)∈[_0,α]×M₁×M₇ ⊂_{B, (4.7)} where Z₂ andZ₃are defined in (A4) (ii).

Letx: [−r,α]→_Rⁿbe continuously differentiable. We have

|x(t)−x(t^¯)−x˙(t^¯)(t−t^¯)|=

Z _t

t¯

(x˙(u)−x˙(^¯t))du

≤_Ωx˙(|t−t^¯|)|t−t^¯|, t, ¯t∈[−r,α], (4.8) where

Ωx˙(ε):=supn

|x˙(t)−x˙(t^¯)|: |t−t^¯| ≤ε, t, ¯t∈[−r,α]^o. Note that the uniform continuity of ˙xon[−r,α]impliesΩx˙(ε)→0 asε→0.

It is easy to show (see [14] or [23]) that the partial derivatives of a function of the form F: [0,α]×W^1,∞×_Ξ→_Rⁿ, F(t,ψ,ξ):=ψ(−τ(t,ψ,ξ))are given by

D₂F(t,ψ,ξ)u=−ψ^˙(−τ(t,ψ,ξ))D₂τ(t,ψ,ξ)u+u(−τ(t,ψ,ξ)), u∈W^1,∞, D3F(t,ψ,ξ)v=−ψ^˙(−τ(t,ψ,ξ))D3τ(t,ψ,ξ)v, v∈ _Ξ

fort ∈[0,α],ψ∈ C¹andξ ∈ Ξ. Using these relations, we can formulate the linear variational equation corresponding to Equation (3.1) in the following way. (See also [14,15,18] for the form of the variational equations associated to other classes of SD-DDEs.)

Letγ= (ϕ,ξ,λ,θ)∈ M ∩Pbe fixed, andx(t):= x(t,γ)be the corresponding solution of the IVP (3.1)–(3.2) on[−r,α]. Note that Theorem3.3yields thatxis continuously differentiable on [−r,α]. We will use the short vector notation

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

for the argument of f in Equation (3.1), and the vectors a(t)_:=t,x_t,ξ

and b(t)_:= t,x_t,λ

for the arguments ofτandρ, respectively.

Fixh= (h^ϕ,h^ξ,h^λ,h^θ)∈_Γ, and consider the variational equation

˙

z(t) =D₂f(_x(t))z_t

+D₃f(x(t))^h−x˙(t−τ(a(t)))ⁿD₂τ(a(t))z_t+D₃τ(a(t))h^ξo

+z(t−τ(a(t)))ⁱ +D₄f(_x(t))z˙(t−µ(t))

+D₅f(_x(t))^h−x¨(t−ρ(_b(t)))ⁿD₂ρ(_b(t))z_t+D₃ρ(_b(t))h^λo

+z˙(t−ρ(_b(t)))ⁱ

+D₆f(x(t))h^θ, a.e.t ∈[0,α], (4.9)

z(t) =h^ϕ(t), t ∈[−r, 0]. (4.10)

This is an inhomogeneous linear time-dependent but state-independent NFDE forz. Note that forγ∈ M ∩Pthe term ¨x(t−ρ(b(t)))is defined only for a.e.t∈ [0,α]. Since the neutral terms on the right-hand side of (4.9) do not depend on values of ˙zon the interval(t−r₀,t], it is easy

to prove using the method of steps with the intervals [ir₀,(i+1)r₀], that the IVP (4.9)–(4.10) has a unique solution,z(t) =z(t,γ,h), which depends linearly onh. The boundedness of the mapΓ→_Rⁿ,h7→ z(t,γ,h)for each t∈[0,α]follows from Lemma4.1below.

We introduce the following notation L(t,γ)(ψ,h^ξ,h^λ,h^θ)

:=D₂f(x(t))ψ

+D3f(x(t))^h−x˙(t−τ(a(t)))ⁿD2τ(a(t))ψ+D3τ(a(t))h^ξo

+ψ(−τ(a(t)))ⁱ +D₄f(x(t))ψ^˙(−µ(t))

+D₅f(x(t))^h−x¨(t−ρ(b(t)))ⁿD₂ρ(b(t))ψ+D₃ρ(b(t))h^λo

+ψ^˙(−ρ(b(t)))ⁱ

+D₆f(x(t))h^θ (4.11)

for a.e.t ∈[0,α],γ∈ M ∩P andψ∈W^1,∞, h^ξ ∈ _Ξ, h^λ ∈ _Λ, h^θ ∈ Θ. With this notation (4.9) can be rewritten as

˙

z(t) = L(t,γ)(zt,h^ξ,h^λ,h^θ), a.e.t∈ [0,α]. (4.12) We introduce the Lipschitz constants L₁ = L₁(α,M₁,M₂,M₃,M₄,M₅), L₂ = L₂(α,M₁,M₆), L3:= L3(α)and L₄ = L₄(α,M₁,M7)from (A1) (i), (A2) (i), (A3) and (A4) (i), respectively. Let N and L^∗ = L^∗(γ) be defined by (3.31) and (3.33), respectively. Then (A1), (A2), (A4) and Remarks3.1and3.4yield

|D_jf(x(t))|_Xj ≤ L₁, |D₃τ(a(t))|_Y3 ≤ L₂, |D₃ρ(b(t))|_Z3 ≤L₃, t ∈[0,α], j=3, 4, 5, 6, (4.13) whereX₃,X₄,X₅,X₆,Y₃andZ₃are defined in (A1) (ii), (A2) (ii) and (A4) (ii), respectively. We claim

|D₂f(x(t))ψ| ≤L₁|ψ|_C, |D₂τ(a(t))ψ| ≤ L₂|ψ|_C, |D₂ρ(b(t))ψ| ≤L₃|ψ|_C, ψ∈W^1,∞, t∈ [0,α] (4.14) fort∈ [0,α]. We show only the first estimate, the proofs of the second and third relations are similar. Letψ∈ W^1,∞,|ψ|_C 6= 0 and t ∈ [0,α]be fixed. Let ∆ := (0,ψ, 0, 0, 0, 0)∈ X(here the 0-s are the zero vectors of the respective spaces). It follows from Remark3.4thatx_t ∈ M₁, and for large enoughk∈_Nwe havext+ ¹_kϕ∈ M₁. Then assumption (A1) (1) implies for suchk

|D₂f(_x(t))ψ|

|ψ|_C = |D₂f(x(t))¹_kψ|

|¹_kψ|_C

≤ |f(x(t) +¹_k∆)− f(x(t))|

|¹_kψ|_C +|f(x(t) + ¹_k∆)− f(x(t))−D₂f(x(t))¹_kψ|

|¹_kψ|_C

≤L₁+|f(x(t) + ¹_k∆)− f(x(t))−D2f(x(t))¹_kψ|

|¹_kψ|_C ^, which yields the first estimate of (4.14) ask→_∞.

Then, combining (3.31), (3.33), (4.13) and (4.14) with (4.11), we get

|L(t,γ)(ψ,h^ξ,h^λ,h^θ)| ≤L₁|ψ|_C+L₁

NL₂(|ψ|_C+|h^ξ|_Ξ) +|ψ|_C+L₁|P(ψ^˙)|_L∞ r0

+L₁

L^∗L₄(|ψ|_C+|h^λ|_Λ) +|P(ψ^˙)|_L∞r0

+L₁|h^θ|_Θ

≤ N₀

|ψ|_C+|P(ψ^˙)|_L∞r0 +|h^ξ|_Ξ+|h^λ|_Λ+|h^θ|_Θ (4.15)

for a.e.t∈ [0,α], ψ∈W^1,∞, h^ξ ∈_Ξ, h^λ∈ _Λ, h^θ ∈ _{Θ, where}N₀:= L₁(NL₂+L^∗L₄+2). The next lemma shows that the linear maps Γ 3 h 7→ z(t,γ,h) ∈ _Rⁿ and Γ 3 h 7→

˙

z(t,γ,h) ∈ _Rⁿ are bounded for t ∈ [−r,α] and for a.e. t ∈ [−r,α], respectively, and for γ∈ M ∩P. We also prove that ˙zsatisfies a certain special inequality, which will be important in the proof of Lemma4.2below.

Lemma 4.1. Assume (A1)–(A4), let α > 0 and P ⊂ _Π be defined by Theorem3.3. Then for every γ∈ M ∩P there exist constants N₁ ≥ 0 and N2 ≥ 0such that the solution of the IVP (4.9)–(4.10) satisfies

|z(t,γ,h)| ≤N₁|h|_Γ, t ∈[−r,α], h∈_Γ, (4.16)

|z˙(t,γ,h)| ≤N₂|h|_Γ, a.e. t∈[−r,α], h∈ _Γ. (4.17) Moreover, for every γ ∈ M ∩P there exist constants N₃ ≥ 0, N₄ ≥ 0, N₅ ≥ 0, N₆ ≥ ₀ and a monotone increasing functionΩ: [0,α]→[0,∞)with the propertyΩ(ε)→0asε→0+such that

|z˙(t,γ,h)−z˙(t,^¯ γ,h)| ≤N₃|t−t^¯||h|_Γ+N₄Ω(|t−t^¯|)|h|_Γ

+N₅|x¨(t−ρ(b(t)))−x¨(t^¯−ρ(b(t^¯)))||h|_Γ +N₆

|z˙(t−µ(t),γ,h)−z˙(t^¯−µ(t^¯),γ,h)|

+|z˙(t−ρ(b(t)),γ,h)−z˙(t^¯−ρ(b(t^¯)),γ,h)| (4.18) for a.e. t, ¯t ∈[0,α], h∈_Γ.

Proof. Let γ ∈ M ∩P be fixed. For simplicity we use the notations h = (h^ϕ,h^ξ,h^θ,h^λ) ∈ _Γ, x(t) := x(t,γ) and z(t) := z(t,γ,h). Let δ,M₁,M₂,M₃ and M₄ be defined by Theorem 3.3, M₅, M₆ andM₇ be defined by (4.1), L₁, . . . ,L₄ be the corresponding Lipschitz constants from (A1)–(A4), and let L^∗ = L^∗(γ)andN₀= N₀(γ)be defined by (3.33) and (4.15), respectively.

Let m := [α/r₀] (here [·] denotes the greatest integer part), t_j := jr₀ for j = 0, 1, . . . ,m, t_m+1:=_{α, and let}t∈ [t₀,t₁]. Integrating (4.12) from 0 to t, and using estimate (4.15) we get

|z(t)| ≤ |h^ϕ(0)|+

Z _t

0

|L(s,γ)(z_s,h^ξ,h^λ,h^θ)|ds

≤ |h^ϕ|_C+N₀r₀(|h^ξ|_Ξ+|h^λ|_Λ+|h^θ|_Θ) +N₀ Z _t

0

(|zs|_C+|P(z˙s)|_L∞ r0)ds

=|h^ϕ|_C+N₀r₀(|h^ξ|_Ξ+|h^λ|_Λ+|h^θ|_Θ) +N₀ Z _t

0

(|z_s|_C+ _{ess sup}

−r≤ζ≤−r₀

|z˙(s+ζ)|)ds

=|h^ϕ|_C+N₀r₀(|h^ξ|_Ξ+|h^λ|_Λ+|h^θ|_Θ) +N₀ Z _t

0

(|z_s|_C+ ess sup

−r≤ζ≤−r0

|h^˙ϕ(s+ζ)|)ds

≤(1+N₀r₀)|h|_Γ+N₀ Z _t

0

|z_s|_Cds, t∈ [t₀,t₁].

Note that|z₀|_C≤ (1+N₀r₀)|h|_Γ, hence Gronwall’s inequality yields

|z(t)| ≤c0|h|_Γ, t ∈[t0,t₁], wherec₀ := (1+N₀r₀)e^N0r0. Sincec₀≥1, it follows that

|z(t)| ≤c₀|h|_Γ, t ∈[−r,t₁]_.

Using Equation (4.12) and relation (4.15) we get

≤ |(D₂f(x(t))−D₂f(x(t^¯)))z_t|+|D₂f(x(t^¯))(z_t−zt¯)|

with and appropriate constantK₅. Similarly,

|a(t)−a(t^¯)|_A =|b(t)−b(t^¯)|_B ≤ |t−t^¯|+|x_t−xt¯|_C ≤K₆|t−t^¯|, t, ¯t ∈[0,α], (4.21) whereK6:=1+N. Then (4.13), (4.14), (4.16), (4.17), (4.19), (4.20), (4.21) and the definitions of ΩD f,ΩDτ andΩDρ yield

|z˙(t)−z˙(t^¯)| ≤_{ΩD f}(|t−^¯t|)N₁|h|_Γ+L₁N2|t−t^¯||h|_Γ+_{ΩD f}(|t−^¯t|)NL2(N₁+1) +N₁

|h|_Γ +L₁L^∗(1+L₂K₆)|t−t^¯|L₂(N₁+1)|h|_Γ

+L₁N

ΩDτ(|t−t^¯|)N₁|h|_Γ+L2N2|t−^¯t||h|_Γ+_ΩDτ(|t−t^¯|)|h|_Γ +L₁N₂(1+L₂K₆)|t−t^¯||h|_Γ

+_{ΩD f}(|t−t^¯|)N2|h|_Γ+L₁|z˙(t−µ(t))−z˙(t^¯−µ(t^¯))|

+_{ΩD f}(|t−t¯|)L^∗L₄(N₁+₁)|h|_Γ+N₂|h|_Γ

+L₁|x¨(t−ρ(b(t)))−x¨(t^¯−ρ(b(t^¯)))|L₄(N₁+1)|h|_Γ +L₁L^∗

ΩDρ(|t−t¯|)N₁|h|_Γ+L₄N₂|t−t¯||h|_Γ+_ΩDρ(|t−t¯|)|h|_Γ +L₁L^∗

z˙(t−ρ(b(t)))−z˙(t^¯−ρ(b(t^¯)))

+_{ΩD f}(|t−t¯|)|h|_Γ, a.e.t, ¯t ∈[0,α]. (4.22) We defineΩ(ε):=max{_{ΩD f}(ε),ΩDτ(ε),ΩDρ(ε)}. Clearly,Ωis monotone increasing on[0,α]. The functionsD_jf(_x(s))for j = 2, . . . , 6 are continuous, and therefore uniformly continuous on[0,α]. Hence (A1) (ii) and the definition of ΩD f(ε)yieldΩD f(ε)→0 asε →0+. Similarly, ΩDτ(ε) → 0 and ΩDρ(ε) → 0, hence Ω(ε) → 0 as ε → 0+. Therefore, it follows from (4.22) that there exist nonnegative constants N₃, N₄, N₅ and N₆ such that (4.18) holds for a.e.

t, ¯t ∈[0,α].

The next estimate is the key step of the proof of our main result, Theorem4.3below. In its proof we need a weak version of Lipschitz continuity of ˙z(t,γ,h), wrtt. In order to obtain such a result we apply estimate (4.18), but we also need more smoothness for the first component ofh= (h^ϕ,h^ξ,h^λ,h^θ). We introduce the parameter spaceΓ2:=W^2,∞×_Ξ×_Λ×Θ, and a norm by|h|_Γ2 :=|h^ϕ|_W2,∞+|h^ξ|_Ξ+|h^λ|_Λ+|h^θ|_Θ. We note thatM ⊂_Γ2, and theΓ2-norm is stronger than theΓ-norm onΓ2.

Lemma 4.2. Suppose (A1)–(A4), let α > 0 and P ⊂ _Π be defined by Theorem 3.3, and let γ = (_ϕ,ξ,λ,θ)∈ M ∩P, h_k = (h^ϕ_k,h^ξ_k,h^λ_k,h^θ_k)∈ _Γ2be such that γ+h_k ∈ P for k∈ _Nand|h_k|_Γ2 → 0 as k→_{∞. Let x}(t):=x(t,γ), x^k(t):= x(t,γ+h_k), z^k(t):= z(t,γ,h_k)and

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

,

x^k(t):= t,x_t^k,x^k(t−τ(t,x^k_t,ξ+h^ξ_k)), ˙x^k(t−µ(t)), ˙x^k(t−ρ(t,x^k_t,λ+h^λ_k)),θ+h^θ_k . Then there exist nonnegative constants N₇, N₈, and a sequence of nonnegative functions g_k: [0,α]→ [0,∞)satisfying

klim→_∞

1

|h_k|_Γ2

Z _α

0 g_k(s)ds=0 (4.23)

such that

f(_x^k(s))− f(_x(s))−L(s,γ)(z^k_s,h^ξ_k,h^λ_k,h^θ_k)

≤g_k(s) +N₇

x_s^k−x_s−z^k_s

C+N₈

P(x^˙k_s−x˙_s−z^˙k_s)

L^∞_r0, a.e. s∈ [0,α], k ∈_N. (4.24) Proof. Let α,M₁,M₂,M₃ and M₄ be defined by Theorem 3.3, M₅, M₆ and M₇ be defined by (4.1), and L₁, . . . ,L₄ be the corresponding Lipschitz constants from (A1)–(A4), and let L^∗ = L^∗(γ) and N₀ = N₀(γ)be defined by (3.33) and (4.15), respectively. We use the short vector notations

a^k(s):=s,x^k_s,ξ+h^ξ_k

and b^k(s):=s,x^k_s,λ+h^λ_k

, k ∈_N.

The definitions of L(s,γ),ω_f,ω_τ andω_ρyield for a.e.s ∈[0,α] f(x^k(s))− f(x(s))−L(s,γ)(z^k_s,h^ξ_k,h^λ_k,h^θ_k)

=ω_f(x(s),x^k(s)) +D₂f(x(s))^hx^k_s−xs−z^k_si

+D₃f(_x(s))ⁿx^k(s−τ(_a^k(s)))−x(s−τ(_a^k(s)))−z^k(s−τ(_a^k(s)))

+x(s−τ(a^k(s)))−x(s−τ(a(s))) +x˙(s−τ(a(s)))τ(a^k(s))−τ(a(s))

−x˙(s−τ(a(s)))ωτ(a(s),a^k(s))−x˙(s−τ(a(s)))D₂τ(s,xs,ξ)^hx_s^k−xs−z^k_si +z^k(s−τ(a^k(s)))−z^k(s−τ(a(s)))^o

+D₄f(x(s))ⁿx^˙k(s−µ(s))−x˙(s−µ(s))−z^˙k(s−µ(s))^o

+D₅f(x(s))ⁿx^˙k(s−ρ(b^k(s)))−x˙(s−ρ(b^k(s)))−z^˙k(s−ρ(b^k(s)))

+x˙(s−ρ(b^k(s)))−x˙(s−ρ(b(s))) +x¨(s−ρ(b(s)))ρ(b^k(s))−ρ(b(s))

−x¨(s−ρ(_b(s)))ω_ρ(_b(s),b^k(s))−x¨(s−ρ(_b(s)))D₂ρ(s,x_s,λ)^hx_s^k−x_s−z^k_si +z^˙k(s−ρ(b^k(s)))−z^˙k(s−ρ(b(s)))^o. (4.25) Using (3.32) and estimates similar to (3.44) and (3.45), we have that

|x^k(s)−x(s)|_X =|x^k_s−xs|_C+|x^k(s−τ(a^k(s)))−x(s−τ(a(s)))|

+|x^˙k(s−µ(s))−x˙(s−µ(s))|+|x^˙k(s−ρ(_b^k(s)))−x˙(s−ρ(_b(s)))|+|h^θ_k|_Θ

≤2|x^k_s−xs|_C+L2N(|x^k_s−xs|_C+|h^ξ_k|_Ξ)

+₂|x^k_s−x_s|_W1,∞+L₄L^∗(|x^k_s−x_s|_C+|h^λ_k|_Λ) +|h^θ_k|_Θ

≤K₇|h_k|_Γ, a.e.s ∈[0,α], k ∈_N, (4.26) with some constantK7. Similarly, (3.32) yields

|a^k(s)−a(s)|_A =|x^k_s−x_s|_C+|h^ξ_k|_Ξ≤(L+₁)|h_k|_Γ, s ∈[_0,α]_, k ∈_N, (4.27) and

|b^k(s)−b(s)|_B =|x^k_s−x_s|_C+|h^λ_k|_Λ ≤(L+1)|h_k|_Γ, s∈[0,α], k∈_N. (4.28)

Using (A2) (ii) and (3.32), we get (4.4), (4.6), (4.13), (4.14), (4.26)–(4.31), we get from (4.25)

|h_k|_Γ≤ |h_k|_Γ2, (4.26), (4.27) and (4.28), we have

In the remaining part of the proof we show

klim→_∞

Note thatu, vand v_k are strictly monotone increasing functions on [0,α]. Assumptions (A3) and (A4) (i) imply

−r ≤u(t)≤t−r0, −r≤ v(t)≤t−r0 and −r≤ v_k(t)≤t−r0, t∈[0,α], k∈ _N.

(4.37) Define the constantK₉ :=max{1+L₃, 1+L₄K₆}. Then it is easy to check that

|u(t)−u(t¯)| ≤K₉|t−t¯| _and |v(t)−v(t¯)| ≤K₉|t−t¯|_, t, ¯t ∈[−r,α]_{. (4.38)}

Also, (A3) and (3.29) yield

Then the Mean Value Theorem, (4.17), (4.34) and (4.41) imply Z _α This proves the limit relation (4.36) in case (1).

(2) Next supposev(α)>0.

On the interval [η⁰⁰_k,0,α] both v and v_k take positive values. Therefore, we can use (4.18) to Next the function w can be substituted with the functions u or v. Note that both u and v are strictly monotone increasing on [0,α], both are Lipschitz continuous with the Lipschitz constantK₉, and the essential infimum of ˙uand ˙vare bounded below by the positive constant m^∗. So the estimates we present below work for both functions. Next we show relation (4.45) for w = u and w = v. We consider two subcases (similarly to cases (1) and (2)): either w(v(α))≤0 orw(v(α))>0.

Then the Mean Value Theorem, (4.17) and (4.46) yield w, but this dependence is omitted in the notation. Then it is easy to check that η⁰⁰_k,0 < η_k,1⁰ <

v⁻¹(w⁻¹(0))<η⁰⁰_k,1 <α, andw(v(s)),w(v_k(s))≤0 fors∈ [η_k,0⁰⁰ ,η_k,1⁰ ]andw(v(s)),w(v_k(s))≥0

+N₆ Z _α

η⁰⁰_k,1

|z^˙k(u(w(v_k(s))))−z^˙k(u(w(v(s))))|ds +N6

Z _α

η⁰⁰_k,1

|z^˙k(v(w(v_k(s))))−z^˙k(v(w(v(s))))|ds. (4.49) We have |v(w(v_k(s)))−v(w(v(s)))| ≤ K₉²K8|h_k|_Γ2, s ∈ [0,α]. Hence Lemma 2.4 with y = x,¨ p(s) =v(w(v(s))), p^k(s) = v(w(v_k(s)))andω_k =K₉²K₈|h_k|_Γ2 yields that the second term is of order o(|h_k|_Γ2). Therefore, we need to show that

Z _α

η_k,1⁰⁰

|z^˙k(w₂(w₁(v_k(s))))−z^˙k(w₂(w₁(v(s))))|ds=o(|h_k|_Γ2), forw₁,w₂ ∈ {u,v}. (4.50) As before, we consider two cases.

(2.2.1)v(α)>0,w₁(v(α))>0 andw2(w₁(v(α)))≤0

In this case we havew₂(w₁(v(s)))≤ 0 fors∈ [η⁰⁰_k,1,α], and, similarly to the estimates used in cases (1) and (2.1), it is easy to see that (4.50) holds.

(2.2.2)v(α)>0,w₁(v(α))>0 andw₂(w₁(v(α)))>0.

Recall that η_k,0⁰⁰ = v⁻¹(_∆k) _and η⁰⁰_k,1 = v⁻¹(w⁻₁¹(₀) +_∆k)_{, and} η⁰⁰_k,0 < η_k,1⁰⁰ < α. From the assumption it follows w⁻₁¹(0)< w⁻₁¹(w⁻₂¹(0)) <v(α). We assume thatk is large enough that v(₀)< −_∆k <_∆k <w⁻₁¹(₀)−_∆k < w⁻₁¹(₀) +_∆k <w⁻₁¹(w⁻₂¹(₀))−_∆k <w⁻₁¹(w⁻₂¹(₀)) +_∆k <

v(α). Then we defineη_k,2⁰ :=v⁻¹(w⁻₁¹(w⁻₂¹(0))−_∆k)andη_k,2⁰⁰ := v⁻¹(w⁻₁¹(w⁻₂¹(0)) +_∆k). The above inequalities yield η_k,0⁰⁰ < η_k,1⁰⁰ < η⁰_k,2 < η_k,2⁰⁰ < α. Then, similarly to (4.49), we can obtain an estimate of (4.50) where on the righ-hand side we have integrals of the form

Z _α

η_k,2⁰⁰

|x¨(v(w₂(w₁(v_k(s)))))−x¨(v(w₂(w₁(v(s)))))||h_k|_Γ2ds (4.51)

and Z _α

η_k,2⁰⁰

|z^˙k(w₃(w₂(w₁(v_k(s)))))−z^˙k(w₃(w₂(w₁(v(s)))))|ds (4.52) where w₃,w₂,w₁ ∈ {u,v}_{. Lemma2.4}yields that the integral in (4.51) is of order o(|h_k|_Γ2)_{. If} w₃(w₂(w₁(v(α)))) ≤ 0, we get an explicit estimate of the integral in (4.52), and it is of order o(|h_k|_Γ2). But if w₃(w₂(w₁(v(α)))) > 0, we can continue the recursive estimating described above. Clearly, this recursion will end, since

wm(· · ·(w₃(w₂(w₁(v(α)))))· · ·)≤α−(m+1)r₀ <0, w₁,w₂, . . . ,wm∈ {u,v}. Hence, after no more than m number of iterations, the above described iterative procedure ends. Note, that in each step of the second case, we may double the number of integrals similar to (4.52) used in the estimate. But, since we may have only finitely many terms, and each terms have order o(|h_k|_Γ2), this completes the proof of (4.36). Hence C_k → 0 as k → _∞, and relations (4.32), (4.33), (4.35) and (4.36) prove the statement of the lemma with g_k(s):=C_k|h_k|_Γ+g_k,1(s) +g_k,2(s) +g_k,3(s).

Next we prove differentiability of the functionsx(t,γ)andxt(·,γ)wrtγusing theΓ2-norm on the parameter set. We denote this differentiation by D₂x. We note that the differentiability is proved at a parameter value γ which belongs to M ∩P, i.e., where the compatibility con-dition is satisfied. But in computing the partial derivative we use solutions corresponding to parameter valuesγ+h,h ∈_Γ2 with small norm, hence these solutionsx(·,γ+h), in general, do not satisfy the compatibility condition.

Theorem 4.3. Assume (A1)–(A4). Letγ¯ ∈ M, and letδ >0, P := B_Γ(γ;¯ δ), andα> 0be defined by Theorem3.3, and x(t;γ)be the solution of the IVP(3.1)–(3.2)on[−r,α]forγ∈P. Then the map

Γ2⊃ P∩_Γ2→_Rⁿ, γ7→x(t,γ) is differentiable at everyγ∈ M ∩P and t∈[0,α], and

D₂x(t,γ)h= z(t,γ,h)_, h∈_Γ2, t ∈[_0,α]_, γ∈ M ∩P, where z is the solution of the IVP(4.9)–(4.10).

Moreover, the map

Γ2⊃ P∩_Γ2 →C, γ7→x(·,γ)_t

is also differentiable atγ∈ M ∩P and t∈[0,α], and its derivative is given by D₂x_t(·,γ)h= z_t(·,γ,h), h∈_Γ2, t∈[0,α], γ∈ M ∩P.

Proof. Let γ ∈ M ∩P be fixed, and let h_k = (h_k^ϕ,h^ξ_k,h^λ_k,h^θ_k) ∈ _Γ2 be a sequence such that

|h_k|_Γ2 →0 ask →_∞andγ+h_k ∈Pfork∈N. For brevity, we use the notationsx(t):=x(t,γ), x^k(t):= x(t,γ+h_k), z^k(t):=z(t,γ,h_k), and

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

x^k(t) =t,x^k_t,x^k(t−τ(t,x^k_t,ξ+h^ξ_k))_{, ˙}x^k(t−µ(t))_{, ˙}x^k(t−ρ(t,x^k_t,λ+h^λ_k))_,θ+h^θ_k . Define the functionw^k(t):= x^k(t)−x(t)−z^k(t). Equations (3.1) and (4.12) imply

w˙^k(t) = f(_x^k(t))− f(_x(t))−L(t,γ)(z^k_t,h^ξ_k,h^λ_k,h^θ_k)_{, a.e.}t ∈[_0,α]_{. (4.53)} Then Lemma4.2yields

|w^˙k(t)| ≤g_k(t) +N₇|w^k_t|_C+N₈|P(w^˙k_t)|_L∞

r0, a.e.t∈[0,α], (4.54) where N₇,N₈ ≥ 0, and the nonnegative function g_k satisfies (4.23). Letm := [α/r₀](here [·] denotes the greatest integer part),t_i :=ir₀ fori=_{0, 1, ..,}m, t_m+1 := α. Integrating (4.53) from t_i totand using inequality (4.54) we have

|w^k(t)| ≤ |w^k(t_i)|+

Z _ti+1

ti

g_k(s)ds+

Z _t

ti

(N₇|w^k_s|_C+N₈|P(w^˙k_s)|_L∞

r0)ds, t ∈[t_i,t_i+1]. (4.55) Then, using thatw^k(s) =_{0 for}s ∈[−r, 0]_{, we get}

|w^k(t)| ≤

Z _t1

t₀ g_k(s)ds+

Z _t

t₀ N₇|w^k_s|_Cds, t ∈[t₀,t₁]_, hence Gronwall’s lemma implies

|w^k(t)| ≤a^k₀, t ∈[t0,t₁], k∈_N, wherea^k₀:= Rt₁

t₀ g_k(s)ds

e^N7r0. Note that (4.23) yields

klim→_∞

a^k_i

|h_k|_Γ2 =0 (4.56)

fori=0. Sincew^k(t) =0 fort∈[−r, 0], we get from (4.54) that

|w^˙k(t)| ≤g_k(t) +b^k₀, a.e.t∈[t₀,t₁], with b^k₀ := N₇a^k₀. We note that (4.56) with i=0 yields

klim→_∞

b^k_i

|h_k|_Γ2 =0 (4.57)

fori=0. Suppose

|w^k(t)| ≤a^k_i, t∈[t_i,t_i+1], and |w^˙k(t)| ≤g_k(t) +b^k_i, a.e.t ∈[t_i,t_i+1]

for i= 0, 1, . . . ,jwith some j < m, moreover, a^k₀ ≤ a^k₁ ≤ · · · ≤ a^k_j, b₀^k ≤ b₁^k ≤ · · · ≤b^k_j, and a^k_i andb^k_i satisfy (4.56) and (4.57), respectively, fori=0, 1, . . . ,j. Then (4.55) implies

|w^k(t)| ≤ |w^k(t_j+1)|+

Z _tj+2

t_j+1

g_k(s)ds+

Z _t

t_j+1

(N₇|w^k_s|_C+N₈|P(w^˙k_s)|_L∞r0)ds

≤a^k_j +

Z _tj+2

t_j+1

g_k(s)ds+N₈b^k_jr₀+

Z _t

t_j+1

N₇|w^k_s|_Cds, t∈ [t_j+1,t_j+2], hence an application of Gronwall’s lemma gives

|w^k(t)| ≤a^k_j+1, t∈ [t_j+1,t_j+2], where a^k_j+1 := a^k_j +Rtj+2

tj+₁ g_k(s)ds+N₈b^k_jr₀

e^N7r0. We observe that a^k_j+1 ≥ a^k_j, and a^k_j+1 also satisfies (4.56). Then (4.54) implies

|w^k(t)| ≤g_k(t) +b^k_j+1, a.e.t∈ [t_j+1,t_j+2]

with b^k_j+1 := N₇a^k_j+1+ N₈b^k_j. Therefore, the constants a^k_i and b^k_i can be defined for i = 0, 1, . . . ,m, so thata₀^k ≤ · · · ≤a^k_m hold, and (4.56) is satisfied fori=mtoo. Then we get

|x^k(t)−x(t)−z^k(t)| ≤ |x^k_t −x_t−z^k_t|_C ≤a^k_m, t∈[0,α], k∈_N, (4.58) and both claims of the theorem follow from (4.56) withi=m.

We note that the results of this manuscript can be extended to the case of more explicit state-dependent delays in the equation.

Acknowledgements

This research was supported by the Hungarian National Foundation for Scientific Research Grant No. K120186 and the TKP2020-IKA-07 project financed under the 2020-4.1.1-TKP2020 Thematic Excellence Programme by the National Research, Development and Innovation Fund of Hungary.

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In document parameters in a class of neutral differential equations with state-dependent delays (Pldal 21-41)