# Global Existence for Functional Differential Equations with State-Dependent Delay

Authors: Mouffak Benchohra; Imene Medjadj; Juan J. Nieto; P. Prakash
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Our aim in this work is to study the existence of solutions of a functional differential equation with state-dependent delay. We use Schauder's fixed point theorem to show the existence of solutions.


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## Body

## 1. Introduction

The theory of functional differential equations has emerged as an important branch of nonlinear analysis. Differential delay equations, and functional differential equations, have been used in modeling scientific phenomena for many years. Often, it has been assumed that the delay is either a fixed constant or is given as an integral in which case it is called a distributed delay [1–5]. In 1806, Poisson [6] published one of the first papers on functional differential equations and studied a geometric problem leading to an example with a state-dependent delay (see also [7]). An extensive theory is developed for evolution equations [8, 9]. Uniqueness and existence results have been established recently for different evolution problems in the papers by Baghli and Benchohra for finite and infinite delay in [10–12]. However, complicated situations in which the delay depends on the unknown functions have been considered in recent years. These equations are frequently called equations with state-dependent delay: see, for instance, [3, 13–15]. Existence results were derived recently for functional differential equations when the solution is depending on the delay for impulsive problems. We refer the reader to the papers by Abada et al. [16], Ait Dads and Ezzinbi [17], Anguraj et al. [18], Hartung et al. [19, 20], Hernández et al. [21], and Li et al. [22]. Over the past several years it has become apparent that equations with state-dependent delay arise also in several areas such as in classical electrodynamics [23], in population models [24–27], in models of commodity price fluctuations [28, 29], in models of blood cell productions [30–33], and in drilling [34].In this work, we prove the existence of solutions of a class of functional differential equations. Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the real axisℝ. We will use Schauder's fixed point theorem combined with the semigroup theory to have the existence of solutions of the following functional differential equation with state-dependent delay:
(1)y′(t)=Ay(t)+f(t,yρ(t,yt)),a.e.t∈J:=[0,+∞)y(t)=ϕ(t),t∈(-∞,0],
where f:J×ℬ→E is a given function, A:D(A)⊂E→E is the infinitesimal generator of a strongly continuous semigroup T(t),t∈J,ℬ is the phase space to be specified later, ϕ∈ℬ, ρ:J×ℬ→(-∞,+∞), and (E,|·|) is a real Banach space. For any function y defined on (-∞,+∞) and any t∈J we denote by yt the element of ℬ defined by yt(θ)=y(t+θ),θ∈(-∞,0]. Here yt(·) represents the history of the state from time -∞ up to the present time t. We assume that the histories yt belong to some abstract phase space ℬ, to be specified later. To our knowledge, the literature on the global existence of evolution equations with delay is very limited, so the present paper can be considered as a contribution to this question.


## 2. Preliminaries

In this section, we present briefly some notations, a definition and a theorem which are used throughout this work.In this paper, we will employ an axiomatic definition of the phase spaceℬ introduced by Hale and Kato in [1] and follow the terminology used in [3]. Thus, (ℬ,∥·∥ℬ) will be a seminormed linear space of functions mapping (-∞,0] into E and satisfying the following axioms.(A1)
Ify:(-∞,b)→E,b>0, is continuous on J and y0∈ℬ, then for every t∈J the following conditions hold:
(i)
yt∈ℬ;(ii)
there exists a positive constantH such that |y(t)|≤H∥yt∥ℬ;(iii)
there exist two functionsL(·),M(·):ℝ+→ℝ+ independent of y with L continuous and bounded and M locally bounded such that
(2)∥yt∥ℬ≤L(t)sup{|y(s)|:0≤s≤t}+M(t)∥y0∥ℬ.(A2)
For the functiony in (A1), yt is a ℬ-valued continuous function on J.(A3)
The spaceℬ is complete. Denote
(3)l=sup{L(t):t∈J},m=sup{M(t):t∈J}.Remark 1.
(
1
)
(
A
1
) (ii) is equivalent to |ϕ(0)|≤H∥ϕ∥ℬ for every ϕ∈ℬ.
(
2
) Since ∥·∥ℬ is a seminorm, two elements ϕ,ψ∈ℬ can verify ∥ϕ-ψ∥ℬ=0 without necessarily ϕ(θ)=ψ(θ) for all θ≤0.
(
3
) From the equivalence in the first remark, we can see that, for all ϕ,ψ∈ℬ such that ∥ϕ-ψ∥ℬ=0. We necessarily have that ϕ(0)=ψ(0).
By BUC we denote the space of bounded uniformly continuous functions defined from(-∞,0] to E.
ByBC:=BC(-∞,+∞) we denote the Banach space of all bounded and continuous functions from (-∞,+∞) into E equipped with the standard norm
(4)∥y∥BC=supt∈(-∞,+∞)|y(t)|.
Finally, by BC′:=BC′([0,+∞)) we denote the Banach space of all bounded and continuous functions from [0,+∞) into E equipped with the standard norm
(5)∥y∥BC′=supt∈[0,+∞)|y(t)|.Definition 2.
A mapf:J×ℬ→E is said to be Carathéodory if(i)
t→f(t,y) is measurable for all y∈ℬ;(ii)
y→f(t,y) is continuous for almost each t∈J.Theorem 3 (see Schauder fixed point [35]).
LetB be a closed, convex, and nonempty subset of a Banach space E. Let N:B→B be a continuous mapping such that N(B) is a relatively compact subset of E. Then N has at least one fixed point in B. That is, there exists an y∈B such that Ny=y.Lemma 4 (see Corduneanu [36]).
LetD⊂BC([0,+∞),E). Then D is relatively compact if the following conditions hold.(a)
D is bounded in BC.(b)
The function belonging toD is almost equicontinuous on [0,+∞), that is, equicontinuous on every compact of [0,+∞).(c)
The setD(t):={y(t):y∈D} is relatively compact on every compact of [0,+∞).(d)
The function fromD is equiconvergent that is, given ϵ>0,it corresponds a T(ϵ)>0 such that |u(t)-limt→+∞u(t)|<ϵ, for any t≥T(ϵ) and u∈D.


## 3. Existence of Mild Solutions

Now we give our main existence result for problem (1). Before starting and proving this result, we give the definition of the mild solution.Definition 5.
We say that a continuous functiony:(-∞,+∞)→E is a mild solution of problem (1) if y(t)=ϕ(t),t∈(-∞,0], and the restriction of y(·) to the interval [0,+∞) is continuous and satisfies the following integral equation:
(6)y(t)=T(t)ϕ(0)+∫0t‍T(t-s)f(s,yρ(s,ys))ds,t∈J.
Set
(7)ℛ(ρ-)={ρ(s,ϕ):(s,ϕ)∈J×ℬ,ρ(s,ϕ)≤0}.We always assume thatρ:J×ℬ→ℝ is continuous. Additionally, we introduce the following hypothesis:(Hϕ)
the functiont→ϕt is continuous from ℛ(ρ-) into ℬ, and there exists a continuous and bounded function ℒϕ:ℛ(ρ-)→(0,∞) such that
(8)∥ϕt∥≤ℒϕ(t)∥ϕ∥foreveryt∈ℛ(ρ-).Remark 6.
The condition(Hϕ) is frequently verified by functions continuous and bounded. For more details, see, for instance, [3].Lemma 7 ([21, Lemma 2.4]).
Ify:(-∞,+∞)→E is a function such that y0=ϕ, then
(9)∥ys∥ℬ≤(M+ℒϕ)∥ϕ∥ℬ+lsup{|y(θ)|;θ∈[0,max{0,s}]},11111111111111111s∈ℛ(ρ-)∪J,
where ℒϕ=supt∈ℛ(ρ-)ℒϕ(t).Let us introduce the following hypotheses.(H1)A:D(A)⊂E→E is the infinitesimal generator of a strongly continuous semigroup T(t),t∈J, which is compact for t>0 in the Banach space E. Let M′=sup{∥T∥B(E):t≥0}.(H2) The function f:J×ℬ→E is Carathéodory.(H3) There exists a continuous function k:J→[0,+∞) such that
(10)|f(t,u)-f(t,v)|≤k(t)∥u-v∥ℬ,t∈J,u,v∈ℬ,k*:=supt∈J∫0t‍k(s)ds<∞.(H4) The function t→f(t,0)=f0∈L1(J,[0,+∞)) with F*=∥f0∥L1.Theorem 8.
Assume that(H1)−(H4),(Hϕ) hold. If k*M′l<1, then the problem (1) has at least one mild solution on BC.Proof.
Transform problem (1) into a fixed point problem. Consider the operator N:BC→BC defined by
(11)N(y)(t)={ϕ(t),ift∈(-∞,0],T(t)ϕ(0)+∫0tT(t-s)×f(s,yρ(s,ys))ds,ift∈J.
Letx(·):(-∞,+∞)→E be the function defined by
(12)x(t)={ϕ(t),ift∈(-∞,0];T(t)ϕ(0),ift∈J.
Then x0=ϕ. For each z∈BC with z(0)=0, we denote by z¯ the function
(13)z¯(t)={0,ift∈(-∞,0];z(t),ift∈J.
If y satisfies y(t)=N(y)(t), we can decompose it as y(t)=z(t)+x(t),t∈J, which implies yt=zt+xt for every t∈J, and the function z(·) satisfies
(14)z(t)=∫0t‍T(t-s)f(s,zρ(s,zs+xs)+xρ(s,zs+xs))ds,t∈J.
Set
(15)BC0′={z∈BC′:z(0)=0},
and let
(16)∥z∥BC0′=sup{|z(t)|:t∈J},z∈BC0′.BC0′ is a Banach space with the norm ∥·∥BC0′. We define the operator 𝒜:BC0′→BC0′ by
(17)𝒜(z)(t)=∫0t‍T(t-s)f(s,zρ(s,zs+xs)+xρ(s,zs+xs))ds,t∈J.
We will show that the operator 𝒜 satisfies all conditions of Schauder's fixed point theorem. The operator A maps BC0′ into BC0′; indeed, the map 𝒜(z) is continuous on [0,+∞) for any z∈BC0′, and for each t∈J we have
(18)|𝒜(z)(t)|≤M′∫0t‍|f(s,zρ(s,zs+xs)+xρ(s,zs+xs))11111111111-f(s,0)+f(s,0)f(s,zρ(s,zs+xs)+xρ(s,zs+xs))|ds≤M′∫0t‍|f(s,0)|ds+M′∫0tk(s)∥zρ(s,zs+xs)+xρ(s,zs+xs)∥ℬds≤M′F*+M′∫0t‍k(s)×(l|z(s)|+(m+ℒϕ+lM′H)∥ϕ∥ℬ)ds.
Set
(19)C:=(m+ℒϕ+lM′H)∥ϕ∥ℬ.
Then, we have
(20)|𝒜(z)(t)|≤M′F*+M′C∫0t‍k(s)ds+M′∫0tl|z(s)|k(s)ds≤M′F*+M′Ck*+M′l∥z∥BC0′k*.
Hence, 𝒜(z)∈BC0′.
Moreover, letr>0 be such that
(21)r≥M′F*+M′Ck*1-M′k*l,
and let Br be the closed ball in BC0′ centered at the origin and of radius r. Let z∈Br, and let t∈[0,+∞). Then,
(22)|𝒜(z)(t)|≤M′F*+M′Ck*+M′k*lr.
Thus,
(23)∥𝒜(z)∥BC0′≤r,
which means that the operator 𝒜 transforms the ball Br into itself.
Now we prove that𝒜:Br→Br satisfies the assumptions of Schauder's fixed theorem. The proof will be given in several steps.
Step
1. (𝒜 is continuous in Br). Let {zn} be a sequence such that zn→z in Br. At first, we study the convergence of the sequences (zρ(s,zsn)n)n∈ℕ,s∈J.
Ifs∈J is such that ρ(s,zs)>0, then we have
(24)∥zρ(s,zsn)n-zρ(s,zs)∥ℬ≤∥zρ(s,zsn)n-zρ(s,zsn)∥ℬ+∥zρ(s,zsn)-zρ(s,zs)∥ℬ≤L∥zn-z∥ℬ+∥zρ(s,zsn)-zρ(s,zs)∥ℬ,
which proves that zρ(s,zsn)n→zρ(s,zs) in ℬ as n→∞ for every s∈J such that ρ(s,zs)>0. Similarly, if ρ(s,zs)<0, we get
(25)∥zρ(s,zsn)n-zρ(s,zs)∥ℬ=∥ϕρ(s,zsn)n-ϕρ(s,zs)∥ℬ=0,
which also shows that zρ(s,zsn)n→zρ(s,zs) in ℬ as n→∞ for every s∈J such that ρ(s,zs)<0. Combining the pervious arguments, we can prove that zρ(s,zs)n→ϕ for every s∈J such that ρ(s,zs)=0. Finally,
(26)|𝒜(zn)(t)-𝒜(z)(t)|≤M′∫0t‍|f(s,zρ(s,zsn+xs)n+xρ(s,zsn+xs))11111111-f(s,zρ(s,zs+xs)+xρ(s,zs+xs))(s,zρ(s,zsn+xs)n+xρ(s,zsn+xs))|ds≤M′∫0t‍|f(s,zρ(s,zsn+xs)+xρ(s,zsn+xs))111111111-f(s,zρ(s,zs+xs)+xρ(s,zs+xs))|ds.
Then by (H2), we have
(27)f(s,zρ(s,zsn+xs)n+xρ(s,zsn+xs))⟶f(s,zρ(s,zs+xs)hhhhhhhhhhhhhhhhhhhhhhh+xρ(s,zs+xs)),1111111111111111111hhhhhhhhhiasn⟶∞,
and by the Lebesgue dominated convergence theorem we get
(28)∥𝒜(zn)-𝒜(z)∥BC0′⟶0,asn⟶∞.
Thus, 𝒜 is continuous.
Step
2. 𝒜(Br)⊂Br which is clear.
Step
3. 𝒜(Br) is equicontinuous on every compact interval [0,b] of [0,+∞) for b>0. Let τ1,τ2∈[0,b] with τ2>τ1; we have
(29)|𝒜(z)(τ2)-𝒜(z)(τ1)|≤∫0τ1‍∥T(τ2-s)-T(τ1-s)∥B(E)×|f(s,zρ(s,zs+xs)+xρ(s,zs+xs))|ds+∫τ1τ2‍∥T(τ2-s)∥B(E)×|f(s,zρ(s,zs+xs)+xρ(s,zs+xs))|ds≤∫0τ1‍∥T(τ2-s)-T(τ1-s)∥B(E)×|f(s,zρ(s,zs+xs)+xρ(s,zs+xs))-f(s,0)|ds+∫0τ1‍∥T(τ2-s)-T(τ1-s)∥B(E)|f(s,0)|ds+∫τ1τ2‍∥T(τ2-s)∥B(E)×|f(s,zρ(s,zs+xs)+xρ(s,zs+xs))-f(s,0)|ds+∫τ1τ2‍∥T(τ2-s)∥B(E)|f(s,0)|ds≤C∫0τ1‍∥T(τ2-s)-T(τ1-s)∥B(E)k(s)ds+rL∫0τ1‍∥T(τ2-s)-T(τ1-s)∥B(E)k(s)ds+∫0τ1‍∥T(τ2-s)-T(τ1-s)∥B(E)|f(s,0)|ds+C∫τ1τ2‍∥T(τ2-s)∥B(E)k(s)ds+rL∫τ1τ2‍∥T(τ2-s)∥B(E)k(s)ds+∫τ1τ2‍∥T(τ2-s)∥B(E)|f(s,0)|ds.
When τ2→τ1, the right-hand side of the above inequality tends to zero; since T(t) is a strongly continuous operator, and the compactness of T(t) for t>0 implies the continuity in the uniform operator topology (see [37]), this proves the equicontinuity.
Step
4.  𝒜(Br)(t) is relatively compact on every compact interval of t∈[0,∞). Let t∈[0,b] for b>0, and let ε be a real number satisfying 0<ε<t. For z∈Br, we define
(30)𝒜ε(z)(t)=T(ε)∫0t-ε‍T(t-s-ε)111111×f(s,zρ(s,zs+xs)+xρ(s,zs+xs))ds.
Note that the set
(31){∫0t-ε‍T(t-s-ε)f(s,zρ(s,zs+xs)+xρ(s,zs+xs))ds:z∈Br}
is bounded. Since T(t) is a compact operator for t>0, the set
(32){𝒜ε(z)(t):z∈Br}
is precompact in E for every ε, 0<ε<t. Moreover, for every z∈Br we have
(33)|𝒜(z)(t)-𝒜ε(z)(t)|≤∫t-εt‍T(t-s)f(s,zρ(s,zs+xs)+xρ(s,zs+xs))ds≤M′F*ε+M′C∫t-εt‍k(s)ds+rM′∫t-εtlk(s)ds,⟶0asε⟶0.
Therefore, the set{𝒜(z)(t):z∈Br} is precompact, that is, relatively compact.
Step
5  (𝒜(Br) is equiconvergent). Let t∈[0,+∞) and z∈Br; we have
(34)|𝒜(z)(t)|≤M′∫0t|f(s,zρ(s,zs+xs)+xρ(s,zs+xs))|ds≤M′F*+M′C∫0t‍k(s)ds+M′r∫0t‍Lk(s)ds≤M′F*+M′C∫0t‍k(s)ds+M′rl∫0t‍k(s)ds.
Then by (37), we have
(35)|𝒜(z)(t)|⟶M′F*,ast⟶+∞.
Hence,
(36)|𝒜(z)(t)-𝒜(z)(+∞)|⟶0,ast⟶+∞.
As a consequence of Steps 1–4, with Lemma4, we can conclude that 𝒜:Br→Br is continuous and compact. From Schauder's theorem, we deduce that 𝒜 has a fixed point z*. Then y*=z*+x is a fixed point of the operators N, which is a mild solution of problem (1).


## 4. An Example

Consider the following functional partial differential equation:(37)∂∂tz(t,x)-∂2∂x2z(t,x)=e-t∫-∞0‍z(s-σ1(t)σ2(∫0π‍a(θ)|z(t,θ)|2dθ),x)ds,11111111111111111111111111x∈[0,π],t∈[0,+∞)z(t,0)=z(t,π),t∈[0,+∞),z(θ,x)=z0(θ,x),t∈(-∞,0],x∈[0,π],
where z0≇0. Set
(38)f(t,ψ)(x)=∫-∞0‍e-tψ(s,x)ds,ρ(t,ψ)=t-σ1(t)σ2(∫0π‍a2(θ)|ψ(t,θ)|2dθ),
where σi:ℝ+→ℝ+,i=1,2, and a:ℝ→ℝ are continuous functions.TakeE=L2[0,π] and define A:E→E by Aω=ω′′ with domain
(39)D(A)={ω∈E,ω,ω′areabsolutelycontinuous,ω′′∈E,ω(0)=ω(π)=0}.Then(40)Aω=∑n=1∞‍n2(ω,ωn)ωn,ω∈D(A),
where ωn(s)=2/πsinns,n=1,2,…, is the orthogonal set of eigenvectors in A. It is well known (see [37]) that A is the infinitesimal generator of an analytic semigroup T(t),t≥0, in E and is given by
(41)T(t)ω=∑n=1∞‍exp(-n2t)(ω,ωn)ωn,ω∈E.
Since the analytic semigroup T(t) is compact, there exists a positive constant M such that
(42)∥T(t)∥B(E)≤M.
Let ℬ=BCU(ℝ-;E), and let ϕ∈ℬ then (Hϕ).The functionf(t,ψ)(x) is Carathéodory, and
(43)|f(t,ψ1)(x)-f(t,ψ2)(x)|≤e-t|ψ1(t,x)-ψ2(t,x)|.
Thus, k(t)=e-t; moreover, we have
(44)k*=sup{∫0t‍e-sds,t∈[0,+∞)}=1,f0≡0.
Then problem (1) in an abstract formulation of the problem (37) and conditions (H1)–(H4),(Hϕ) are satisfied. Theorem 8 implies that the problem (37) has at least one mild solution on BC.



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