Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 50, 1-8; http://www.math.u-szeged.hu/ejqtde/

GLOBAL SOLUTIONS FOR ABSTRACT FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

EDUARDO HERNÁNDEZ M^{∗} AND SUELI M. TANAKA AKI

Abstract. In this paper we study the existence of global solutions for a class of abstract functional differential equation with nonlocal conditions. An application is considered.

1. Introduction

In this paper, we discuss the existence of global solutions for an abstract functional differential equation with nonlocal conditions in the form

\{ \begin{matrix}u′(t) & = Au(t) + f(t, u_{t}, u(ρ(t))), t ∈ I = [0, ∞), \\ u_{0} & = g(u, φ) ∈ B, \end{matrix} (1.1)

where A is the infinitesimal generator of a C_{0}-semigroup of bounded linear operators (T (t))_{t≥0} defined on a Banach space (X, ∥ · ∥); the history u_{t} : (−∞, 0] → X, defined by u_{t}(θ) := u(t + θ), belongs to some abstract phase space B defined axiomatically, φ ∈ B and ρ, f, g are appropriated functions.

The study of initial value problems with nonlocal conditions arises to deal specially with some situations in physics. For the importance of nonlocal conditions in different fields, we refer the reader to [1, 2, 3, 4, 5] and the references contained therein.

Differential systems with nonlocal conditions are considered in different works. In Byszewski [1] is studied the existence of mild, strong and classical solutions for the abstract Cauchy problem

x′(t) = Ax(t) + f(t, x(t)), t ∈ [0, a],
x(0) = x_{0} + q(t_{1}, t_{2}, t_{3}, ..., t_{n}, x(·)) ∈ X,

where A is the generator of a C_{0}-semigroup of linear operators on a Banach space X; f, q are appropriated functions; 0 < t_{1} < .... < t_{n} ≤ a are prefixed numbers and the symbol q(t_{1}, t_{2}, t_{3}, ..., t_{n}, x(·)) is used in the sense that the function x(·) can be evaluated only in the points t_{i}, for instance q(t_{1}, t_{2}, t_{3}, ..., t_{n}, x(·)) = ∑_{i=1}^{n} α_{i}x(t_{i}).

The literature concerning differential equations with nonlocal conditions also include ordinary differential equations, partial differential equations, abstract partial functional differential, integro-differential equations. Related to this matter, we cite

EJQTDE, 2009 No. 50, p. 1 among others works, [1, 2, 3, 4, 5, 6, 7, 8, 9]. To the best of our knowledge, the study of the existence of “ global ” solutions for equations described in the abstract form (1.1), is a untreated topic, and this fact, is the main motivation of this work.

* Corresponding author.

2000 Mathematics Subject Classification. 35R10, 34K30, 49K25, 47D06.

Key words and phrases. Abstract Cauchy problem, semigroup of linear operators.

In this paper, A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (T (t))_{t≥0} defined on a Banach space (X, ∥ · ∥) and M, δ are positive constants such that ∥ T (t) ∥≤ Me^{−δt}, for every t ≥ 0. In this work, the phase space B is a linear space of functions mapping (−∞, 0] into X endowed with a seminorm ∥ · ∥_{B} and verifying the following axioms.

(A) If x : (−∞, a) → X, a > 0, is continuous on [0, a) and x_{0} ∈ B, then for every t ∈ [0, a) the following conditions hold:

(i) x_{t} is in B.

(ii) ∥ x(t) ∥≤ H ∥ x_{t} ∥_{B}.

(iii) ∥ x_{t} ∥_{B}≤ K(t) sup{∥ x(s) ∥: 0 ≤ s ≤ t} + M(t) ∥ x_{0} ∥_{B},

where H > 0 is a constant; K, M : [0, ∞) → [1, ∞), K is continuous, M is locally bounded and H, K, M are independent of x(·).

(A1) For the function x(·) in (A), the function t → x_{t} is continuous from [0, a)

into B. (B) The space B is complete.

Remark 1. In this paper, 𝔎 is a positive constant such that sup_{t≥0}{K(t), M(t)} ≤ 𝔎. We know from that the functions M(·), K(·) are bounded if, for instance, B is a fading memory space, see [10, p. 190] for additional details.

Example 1.1. The phase space C_{r} × L^{p}(ϱ, X)

Let r ≥ 0, 1 ≤ p < ∞, and let ϱ : (−∞, −r] ↦ ℝ be a nonnegative measurable function which satisfies the conditions (g-5)-(g-6) of [10]. Briefly, this means that ϱ is locally integrable and there exists a non-negative locally bounded function γ on (−∞, 0] such that ϱ(ξ + θ) ≤ γ(ξ)ϱ(θ), for all ξ ≤ 0 and θ ∈ (−∞, −r) \ N_{ξ}, where N_{ξ} ⊆ (−∞, −r) is a set whose Lebesgue measure zero.

The space B = C_{r} × L^{p}(ϱ, X) consists of all classes of functions ψ : (−∞, 0] ↦ X such that ψ is continuous on [−r, 0], Lebesgue-measurable, and ϱ∥ψ∥^{p} is Lebesgue integrable on (−∞, −r). The seminorm in C_{r} × L^{p}(ϱ, X) is defined as follows:

∥ ψ ∥_{B}:= sup{∥ψ(θ)∥ : −r ≤ θ ≤ 0} + (∫_{−∞}^{−r} ϱ(θ)∥ψ(θ)∥^{p}dθ)^{1/p} .

The space B = C_{r} × L^{p}(ϱ, X) satisfies axioms (A), (A1), and (B). Moreover, when r = 0 and p = 2, one can then take H = 1, M(t) = γ(−t)^{1/2} and K(t) =

1 + (∫_{−t}^{0} ϱ(θ) dθ)^{1/2} for t ≥ 0 (see [10, Theorem 1.3.8]). We also note that if the conditions (g-5)-(g-7) of [10] hold, then B is a uniform fading memory.

For additional details concerning phase space, we cite [10].

In this paper, h : [0, ∞) → ℝ is a positive, continuous and non-decreasing function such that h(0) = 1 and lim_{t→∞} h(t) = ∞. In the sequel, C_{0}(X) and C_{h}^{0}(X) are the

EJQTDE, 2009 No. 50, p. 2 spaces defined by

C_{0}(X) = {x ∈ C([0, ∞) : X) : lim_{t→∞} ∥ x(t) ∥= 0},
C_{h}^{0}(X) = \{x ∈ C([0, ∞) : X) : lim_{t→∞} \frac{∥ x(t) ∥}{h(t)} = 0\},

endowed with the norms ∥ x ∥_{0}= sup_{t≥0} ∥ x(t) ∥ and ∥ x ∥_{h}= sup_{t≥0} \frac{∥x(t)∥}{h(t)} respectively. Additionally, we define the space

BC_{h}^{0}(X) = {u ∈ C(ℝ, X) ; u_{0} ∈ B, u|_{[0,∞)} ∈ C_{h}^{0}(X)}

endowed with the norm ∥ u ∥_{BC_{h}^{0}}=∥ u_{0} ∥_{B} + ∥ u_{|_{[0,∞)}} ∥_{h}.

We recall here the following compactness criterion.

Lemma 1.1. A set B ⊂ C_{h}^{0}(X) is relatively compact in C_{h}^{0}(X) if, and only if, B is equicontinuous, B(t) = {x(t) : x ∈ B} is relatively compact in X for all t ≥ 0 and lim_{t→∞} \frac{∥x(t)∥}{h(t)} = 0 uniformly for x ∈ B.

2. Existence of mild solutions

In this section, we study the existence of mild solutions for the nonlocal abstract Cauchy problem (1.1). To treat this system, we introduce the following conditions.

H_{1} The function ρ : [0, ∞) → [0, ∞) is continuous and ρ(t) ≤ t for every t ≥ 0. H_{2} The function f : I × B × X → X is continuous and there exist an integrable

function m : [0, ∞) → [0, ∞) and a nondecreasing continuous function W : [0, ∞) → (0, ∞) such that ∥ f(t, ψ, x) ∥≤ m(t)W (∥ ψ ∥_{B} + ∥ x ∥), for every (t, ψ, x) ∈ [0, ∞) × B × X.

H_{3} The function g : BC_{h}^{0}(X) × B → B is continuous and there exists L_{g} ≥ 0 such that

∥ g(u, φ) − g(v, φ) ∥_{B}≤ L_{g} ∥ u − v ∥_{BC_{h}^{0}}, u, v ∈ BC_{h}^{0}(X).

H_{4} The function g(·, φ) : BC_{h}^{0}(X) → B is completely continuous and uniformly bounded. In the sequel, N is a positive constant such that ∥ g(ψ, φ) ∥_{B}≤ N for every ψ ∈ BC_{h}^{0}(X).

In this work, we adopt the following concept of mild solutions for (1.1).

Definition 2.1. A function u ∈ C(ℝ, X) is called a mild solution of system (1.1) if u_{0} = g(u, φ) and

u(t) = T (t)g(u, φ)(0) + ∫_{0}^{t} T (t − s)f(s, u_{s}, u(ρ(s)))ds, t ∈ I = [0, ∞).

Now, we can establish the principal results of this section.

Theorem 2.1. Assume H_{1}, H_{2} and H_{4} be hold and that the followings properties are verified.

EJQTDE, 2009 No. 50, p. 3 (a) The set {T (t)f(s, ψ, x) : (s, ψ, x) ∈ [0, t] × B_{r}(0, B) × B_{r}(0, X)} is relatively

compact in X for every t ≥ 0 and all r > 0.

(b) For every K > 0, lim_{t→∞} \frac{1}{h(t)} ∫_{0}^{t} m(s)W (Kh(s))ds = 0.

If M𝔎(1+H) ∫_{0}^{∞} m(s)ds < ∫_{c}^{∞} \frac{ds}{W (s)}, where c = 𝔎N(1 + H)(1 + MH), then there exists a mild solution for system (1.1).

Proof. Let Γ : BC_{h}^{0}(I : X) → BC_{h}^{0}(X) be the operator defined by (Γu)_{0} = g(u, φ) and

Γu(t) = T (t)g(u, φ)(0) + ∫_{0}^{t} T (t − s)f(s, u_{s}, u(ρ(s)))ds,  t ≥ 0.

Next we prove that Γ verifies the conditions in [11, Theorem 6.5.4]. To begin, we note that for u ∈ BC_{h}^{0}(X) and t ≥ 0, ∥ u(t) ∥≤ ∥ u_{|_{I}} ∥_{h} h(t). If 𝔎 is the constant in Remark 1, then

\frac{∥ Γu(t) ∥}{h(t)} ≤ \frac{MNH}{h(t)} + \frac{M}{h(t)} ∫_{0}^{t} m(s)W (2𝔎 ∥ u ∥_{BC_{h}^{0}} h(s))ds,

which from (b) permits us to conclude that Γ is a function from BC_{h}^{0}(X) into BC_{h}^{0}(X).

Let (x^{n})_{n∈ℕ} be a sequence in BC_{h}^{0}(X) and x ∈ BC_{h}^{0}(X) such that x^{n} → x in BC_{h}^{0}(X). Let ε > 0 and 𝔑 = sup_{n∈ℕ} ∥ x^{n} ∥_{BC_{h}^{0}}. From condition (b), we can select L > 0 such that

\frac{2MNH}{h(t)} + \frac{M}{h(t)} ∫_{0}^{t} m(s)W (2𝔎𝔑h(s))ds ≤ \frac{ε}{2}, t ≥ L. (2.1)

From the properties of the functions f, g, we can choose N_{ε} ∈ ℕ such that

(2.2)  ∥ g(x^{n}, φ) − g(x, φ) ∥_{B} ≤ \frac{ε}{4(MH + 1)},
(2.3) ∫_{0}^{L} ∥ f(s, x_{s}^{n}, x^{n}(ρ(s))) − f(s, x_{s}, x(ρ(s))) ∥ ds ≤ \frac{ε}{4M} ,

for every n ≥ N_{ε}. Under these conditions, we see that

sup\{\frac{∥ Γx^{n}(t) − Γx(t) ∥}{h(t)} : t ∈ [0, L], n ≥ N_{ε}\} ≤ \frac{ε}{2}. (2.4)

Moreover, for t ≥ L and n ≥ N_{ε}, we find that

\frac{∥ Γx^{n}(t) − Γx(t) ∥}{h(t)} ≤ M \frac{∥ g(x^{n}, φ)(0) − g(x, φ)(0) ∥}{h(t)}

+\frac{M}{h(t)} ∫_{0}^{t} ∥ f(s, x_{s}^{n}, x^{n}(ρ(s))) − f(s, x_{s}, x(ρ(s))) ∥ ds

≤ \frac{2MNH}{h(t)} + \frac{M}{h(t)} ∫_{0}^{t} m(s)W (2𝔎𝔑h(s))ds,

EJQTDE, 2009 No. 50, p. 4 and hence,

sup\{\frac{∥ Γx^{n}(t) − Γx(t) ∥}{h(t)} : t ≥ L, n ≥ N_{ε}\} ≤ \frac{ε}{2}. (2.5)

From the inequalities (2.2), (2.4) and (2.5), it follows that ∥ Γx^{n} − Γx ∥_{BC_{h}^{0}}≤ ε, for every n ≥ N_{ε}, which proves that Γ is continuous.

To prove that Γ is completely continuous, we introduce the decomposition Γ = Γ_{1} + Γ_{2}, where (Γ_{1}u)_{0} = g(u, φ), (Γ_{2}u)_{0} = 0 and

(2.6)  Γ_{1}u(t) = T (t)g(u, φ)(0),  t ≥ 0,
(2.7)  Γ_{2}u(t) = ∫_{0}^{t} T (t − s)f(s, u_{s}, u(ρ(s)))ds,  t ≥ 0.

From Lemma 1.1 and the properties of the semigroup (T (t))_{t≥0}, it is easy to see that Γ_{1} is completely continuous. Next, we prove that Γ_{2} is also completely continuous. From [12, Lemma 3.1], we infer that Γ_{2}B_{r}(0, BC_{h}^{0}) |_{[0,a]}= {Γ_{2}x_{|_{[0,a]}}; x ∈ B_{r}(0, BC_{h}^{0})} is relatively compact in C([0, a], X) for every a > 0. Considering this property, Lemma 1.1 and the fact that

\frac{∥ Γ_{2}x(t) ∥}{h(t)} ≤ \frac{M}{h(t)} ∫_{0}^{t} m(s)W (2𝔎rh(s))ds → 0, as t → ∞,

uniformly for x ∈ B_{r}(0, BC_{h}^{0}), we can conclude that the set Γ_{2}(B_{r}(0, BC_{h}^{0})) is relatively compact in BC_{h}^{0}(X) for every r > 0. Thus, Γ is completely continuous.

To finish the proof, we obtain a priori estimates for the solutions of the integral equation u = λΓu, λ ∈ (0, 1)}. Let λ ∈ (0, 1) and u^{λ} ∈ BC_{h}^{0}(X) be a solution of λΓz = z. By using the notation α^{λ}(t) = 𝔎(sup_{s∈[0,t]} ∥ u^{λ}(s) ∥ + ∥ u_{0}^{λ} ∥_{B}), for t ∈ I we see that

∥ u^{λ}(t) ∥ ≤ MNH + M ∫_{0}^{t} m(s)W (∥ u_{s}^{λ} ∥_{B} + ∥ u^{λ}(ρ(s)) ∥)ds,
≤ MNH + M ∫_{0}^{t} m(s)W ((1 + H)α^{λ}(s))ds,

and hence,

α^{λ}(t) ≤ 𝔎 ∥ g(u, φ) ∥_{B} +𝔎MNH + 𝔎M ∫_{0}^{t} m(s)W (1 + H)α^{λ}(s))ds,
≤ 𝔎N(1 + MH) + 𝔎M ∫_{0}^{t} m(s)W ((1 + H)α^{λ}(s))ds.

Defining by β_{λ} the right hand side of the last inequality, we see that β′_{λ}(t) ≤ 𝔎Mm(t)W ((1 + H)β_{λ}(t)), which implies that

∫_{(1+H)β_{λ}(0)=c}^{(1+H)β_{λ}(t)} \frac{ds}{W (s)} ≤ M𝔎(1 + H) ∫_{0}^{∞} m(s)ds < ∫_{c}^{∞} \frac{ds}{W (s)}.

EJQTDE, 2009 No. 50, p. 5

This inequality permits us to conclude that the set of functions {β_{λ} : λ ∈ (0, 1)} is bounded in C(ℝ) and as consequence that the set {u^{λ} : λ ∈ (0, 1)} is bounded in BC_{h}^{0}(X).

Now the assertion is a consequence of [11, Theorem 6.5.4]. The proof is complete.

Theorem 2.2. Assume assumptions H_{1}, H_{2} and H_{3} be hold. If the conditions (a) and (b) of Theorem 2.1 are valid and

(2.8)  L_{g}(1 + MH) + M lim inf_{r→∞} ∫_{0}^{+∞} \frac{m(s)W ((1 + 2𝔎)rh(s))}{rh(s)}ds < 1,

then there exists a mild solution of (1.1).

Proof. Let Γ, Γ_{1}, Γ_{2} be the operators introduced in the proof of Theorem 2.1. We claim that there exists r > 0 such that Γ(B_{r}) ⊂ B_{r}, where B_{r} = B_{r}(0, BC_{h}^{0}(X)). In fact, if this property is false, then for every r > 0, there exist x^{r} ∈ B_{r} and t^{r} ≥ 0 such that r <∥ (Γx^{r})_{0} ∥_{B} + ∥ \frac{Γx^{r}(t^{r})}{h(t^{r})} ∥. Consequently,

r ≤ ∥ g(x^{r}, φ) ∥_{B} +M \frac{∥ g(x^{r}, φ)(0) ∥}{h(t^{r})}

+\frac{M}{h(t^{r})} ∫_{0}^{t} m(s)W (𝔎 ∥ x_{0}^{r} ∥_{B} +𝔎 sup_{θ∈[0,s]} ∥ x^{r}(θ) ∥ + ∥ x^{r}(ρ(s)) ∥)ds

≤ L_{g} ∥ x^{r} ∥_{BC_{h}^{0}} + ∥ g(0, φ) ∥_{B} +\frac{MH}{h(t^{r})}L_{g} ∥ x^{r} ∥_{BC_{h}^{0}}

+\frac{M}{h(t^{r})} ∥ g(0, φ) ∥_{B} +\frac{M}{h(t^{r})} ∫_{0}^{t^{r}} m(s)W (𝔎r + 𝔎rh(s) + rh(s))ds

≤ L_{g}(1 + MH)r + (1 + M) ∥ g(0, φ) ∥_{B} +M ∫_{0}^{t^{r}} \frac{m(s)((2𝔎 + 1)rh(s))}{h(s)}ds

and then,

1 ≤ L_{g}(1 + MH) + M lim inf_{r→∞} ∫_{0}^{+∞} \frac{m(s)W ((2𝔎 + 1)rh(s))}{rh(s)}ds,

which is contrary to (2.8).

Let r > 0 such that Γ(B_{r}) ⊂ B_{r}. From the proof of Theorem 2.1, we know that Γ_{2} is a completely continuous on B_{r}. Moreover, from the estimate,

∥ Γ_{1}u − Γ_{1}v ∥_{BC_{h}^{0}}
≤ L_{g} ∥ u − v ∥_{BC_{h}^{0}} +H ∥ g(u, φ) − g(v, φ) ∥_{B}≤ L_{g}(1 + H) ∥ u − v ∥_{BC_{h}^{0}},

we infer that Γ_{1} is a contraction on B_{r} from which we conclude that Γ is a condensing operator on B_{r}.

The assertion is now a consequence of [13, Theorem 4.3.2].

EJQTDE, 2009 No. 50, p. 6

3. An application

To complete this work, we study the existence of global solutions for a concrete partial differential equation with nonlocal conditions. In the sequel, X = L^{2}[0, π] and A : D(A) ⊂ X → X is the operator Ax = x′′ with domain D(A) := {x ∈ X : x′′ ∈ X, x(0) = x(π) = 0}. It is well known that A is the infinitesimal generator of an analytic semigroup (T (t))_{t≥0} on X. Furthermore, A has discrete spectrum with eigenvalues of the form −n^{2}, n ∈ ℕ, and corresponding normalized eigenfunctions given by z_{n}(x) = (\frac{2}{π})^{\frac{1}{2}} sin(nx); the set of functions \{z_{n} : n ∈ ℕ\} is an orthonormal basis for X and T (t)x = ∑_{n=1}^{∞} e^{−n^{2}t}⟨x, z_{n}⟩z_{n} for every x ∈ X and all t ≥ 0. It follows from this last expression that (T (t))_{t≥0} is a compact semigroup on X and that ∥ T (t) ∥≤ e^{−t} for every t ≥ 0.

As phase space, we choose the space B = C_{0} × L^{2}(ϱ, X), see Example 1.1, and we assume that the conditions (g5) and (g7) in [10] are verified. Under these conditions, B is a fading memory space, and as consequence, there exists 𝔎 > 0 such that sup_{t≥0}{M(t), K(t)} ≤ 𝔎.

Consider the delayed partial differential equation with nonlocal conditions

(3.1) \frac{∂}{∂t}w(t, ξ) = \frac{∂^{2}}{∂ξ^{2}} w(t, ξ) + ∫_{−∞}^{t} a(t, s − t) w(s, ξ) ds + b(t, w(ρ(t), ξ)),
(3.2) w(t, 0) = w(t, π) = 0,  t ≥ 0,
(3.3) w(s, ξ) = ∑_{i=1}^{∞} L_{i}w(t_{i} + s, ξ) + φ(s, ξ),  s ≤ 0, ξ ∈ [0, π],

for t ≥ 0 and ξ ∈ [0, π], and where (L_{i})_{i∈ℕ}, (t_{i})_{i∈ℕ} are sequences of real numbers and φ ∈ B.

To treat this system in the abstract form (1.1), we assume ∑_{i=1}^{∞} | L_{i} | h(t_{i}) < ∞, the functions ρ : [0, ∞) → [0, ∞), a : ℝ^{2} → ℝ are continuous, ρ(t) ≤ t and L_{1}(t) = (∫_{−∞}^{0} \frac{a^{2}(t,s)}{ϱ(s)}ds)^{1/2} < ∞, for every t ≥ 0. We also assume that b : [0, ∞) × ℝ → ℝ is continuous and that there exists a continuous function L_{2} : [0, ∞) → [0, ∞) such that |b(t, x)| ≤ L_{2}(t)|x|, for every t ≥ 0 and x ∈ ℝ.

By defining the functions g : BC_{h}^{0}(X) → B, f : [0, ∞) × B × X → X by g(u, φ) = φ + ∑_{i=1}^{∞} L_{i}u_{t_{i}} and

f(t, ψ, x)(ξ) = ∫_{−∞}^{0} a(t, s) ψ(s, ξ)ds + b(t, x(ρ(t), ξ)),

we can transform system (3.1)-(3.3) into the abstract system (1.1). Moreover, it is easy to see that f, g are continuous, ∥ f(t, ψ, x) ∥≤ L_{1}(t) ∥ ψ ∥_{B} +L_{2}(t) ∥ x ∥, for all (t, ψ, x) ∈ [0, ∞) and g verifies conditions H_{3} with L_{g} = 𝔎 ∑_{i=1}^{∞} L_{i}h(t_{i}).

The next result is a direct consequence of Theorem 2.2. We omit the proof.

Proposition 3.1. If 𝔎 ∑_{i=1}^{∞} L_{i}h(t_{i}) + (1 + 2𝔎) ∫_{0}^{∞}(L_{1}(s) + L_{2}(s))ds < 1, then there exists a mild solution of (3.1)-(3.3).

EJQTDE, 2009 No. 50, p. 7

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(Received May 5, 2009)

Eduardo Hernández M & Sueli M. Tanaka Aki Departamento de Matemática, Instituto de Ciências Matemáticas de São Carlos, Universidade de São Paulo, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil

E-mail address: lalohm@icmc.usp.br
E-mail address: smtanaka@icmc.usp.br

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