Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 27, 1-12; http://www.math.u-szeged.hu/ejqtde/

Existence of solutions to neutral differential equations with deviated argument

M. Muslim Department of Mathematics Indian Institute of Science, Bangalore - 560 012, India Email: malikiisc@gmail.com

and

D. Bahuguna Department of Mathematics and Statistics Indian Institute of Technology Kanpur, India Email: dhiren@iitk.ac.in

Abstract: In this paper we shall study a neutral differential equation with deviated argument in an arbitrary Banach space X. With the help of the analytic semigroups theory and fixed point method we establish the existence and uniqueness of solutions of the given problem. Finally, we give examples to illustrate the applications of the abstract results.

Keywords: Neutral differential equation with deviated argument, Banach fixed point theorem, Analytic semigroup.

AMS Subject Classification: 34K30, 34G20, 47H06.

1 Introduction

We consider the following neutral differential equation with deviated argument in a Banach space X:

\{  \begin{matrix}\frac{d}{dt}[u(t) + g(t, u(a(t)))] + A[u(t) + g(t, u(a(t)))] \\ = f(t, u(t), u[h(u(t), t)]), 0 < t ≤ T < ∞, \\ u(0) = u_{0}, \end{matrix}  (1.1)

where −A is the infinitesimal generators of an analytic semigroup. f, g, h and a are suitably defined functions satisfying certain conditions to be specified later.

Initial results related to the differential equations with deviated arguments can be found in some research papers of the last decade but still a complete theory seems to be missing . For the initial works on existence, uniqueness and stability of various types of solutions of different kind of differential equations, we refer to [1]-[10] and the references cited in these papers.

Adimy et al [1] have studies the existence and stability of solutions of the following general class of nonlinear partial neutral functional differential equations:

\frac{d}{dt}(u(t) − g(t, u_{t})) = A(u(t) − g(t, u_{t})) + f(t, u_{t}), t ≥ 0,
u_{0} = φ ∈ C_{0},  (1.2)

where the operator A is the Hille-Yosida operator not necessarily densely defined on the Banach space B. The functions g and f are continuous from [0, ∞) × C_{0} into B.

In this paper, we use the Banach fixed point theorem and analytic semigroup theory to prove existence and uniqueness of different kind of solutions to the given problem (1.1). The plan of the paper is as follows. In Section 3, we prove the existence and uniqueness of local solutions and in Section 4, the existence of global solution for the problem (1.1) is given. In the last section, we have given an example.

The results presented in this paper easily can be apply to the same problem (1.1) with nonlocal condition under some modified assumptions on the function f and operator A.

2 Preliminaries and Assumptions

We note that if −A is the infinitesimal generator of an analytic semigroup then for c > 0 large enough, −(A + cI) is invertible and generates a bounded analytic semigroup. This allows us to reduce the general case in which −A is the infinitesimal generator of an analytic semigroup to the case in which the semigroup is bounded and the generator is invertible. Hence without loss of generality we suppose that

∥S(t)∥ ≤ M for t ≥ 0

and

0 ∈ ρ(−A),

where ρ(−A) is the resolvent set of −A. It follows that for 0 ≤ α ≤ 1, A^{α} can be defined as a closed linear invertible operator with domain D(A^{α}) being dense in X. We have X_{κ} ↪ X_{α} for 0 < α < κ and the embedding is continuous. For more details on the fractional powers of closed linear operators we refer to Pazy [11]. It can be proved easily that X_{α} := D(A^{α}) is a Banach space with norm ∥x∥_{α} = ∥A^{α}x∥ and it is equivalent to the graph norm of A^{α}. Also, for each α > 0, we define X_{−α} = (X_{α})^{∗}, the dual space of X_{α} is a Banach space endowed with the norm ∥x∥_{−α} = ∥A^{−α}x∥.

It can be seen easily that C_{t}^{α} = C([0, t]; X_{α}), for all t ∈ [0, T], is a Banach space endowed with the supremum norm,

∥ψ∥_{t,α} := sup_{0≤η≤t} ∥ψ(η)∥_{α}, ψ ∈ C_{t}^{α}.

We set,

C_{T}^{α−1} = C([0, T]; X_{α−1}) = {y ∈ C_{T}^{α} : ∥y(t) − y(s)∥_{α−1} ≤ L|t − s|, ∀ t, s ∈ [0, T]},

where L is a suitable positive constant to be specified later and 0 ≤ α < 1. We assume the following conditions:

(A1): 0 ∈ ρ(−A) and −A is the infinitesimal generator of an analytic semigroup {S(t) : t ≥ 0}.

(A2): Let U_{1} ⊂ Dom(f) be an open subset of ℝ_{+} ×X_{α} ×X_{α−1} and for each (t, u, v) ∈ U_{1} there is a neighborhood V_{1} ⊂ U_{1} of (t, u, v). The nonlinear map f : ℝ_{+}×X_{α}×X_{α−1} → X satisfies the following condition,

∥f(t, x, ψ) − f(s, y, ψ̃)∥ ≤ L_{f}[|t − s|^{θ_{1}} + ∥x − y∥_{α} + ∥ψ − ψ̃∥_{α−1}],

where 0 < θ_{1} ≤ 1, 0 ≤ α < 1, L_{f} > 0 is a positive constant, (t, x, ψ) ∈ V_{1}, and (s, y, ψ̃) ∈ V_{1}.

(A3): Let U_{2} ⊂ Dom(h) be an open subset of X_{α} × ℝ_{+} and for each (x, t) ∈ U_{2} there is a neighborhood V_{2} ⊂ U_{2} of (x, t). The map h : X_{α} × ℝ_{+} → ℝ_{+} satisfies the following condition,

|h(x, t) − h(y, s)| ≤ L_{h}[∥x − y∥_{α} + |t − s|^{θ_{2}}],

where 0 < θ_{2} ≤ 1, 0 ≤ α < 1, L_{h} > 0 is a positive constant, (x, t), (y, s) ∈ V_{2} and h(., 0) = 0.

(A4): Let U_{3} ⊂ Dom(g) be an open subset of [0, T] × X_{α−1} and for each (t, x) ∈ U_{3} there is a neighborhood V_{3} ⊂ U_{3} of (x, t). The function g : [0, T] × X_{α−1} → X_{α} is continuous for (t, u) ∈ [0, T] × X_{α−1} such that

∥A^{α}g(t, x) − A^{α}g(s, y)∥ ≤ L_{g}{|t − s| + ∥x − y∥_{α−1}},

where 0 ≤ α < 1, L_{g} > 0 is a positive constant and (x, t), (y, s) ∈ V_{3}.

(A5): The function a : [0, T] → [0, T] satisfies the following two conditions:

(i) a satisfies the delay property a(t) ≤ t for all t ∈ [0, T];

(ii) The function a is Lipschitz continuous; that is, there exist a positive constant L_{a} such that

|a(t) − a(s)| ≤ L_{a}|t − s|, for all t, s ∈ [0, T] and 1 > ∥A^{−1}∥L_{a}.

Definition 2.1 A continuous function u ∈ C_{T}^{α−1} ∩ C_{T}^{α} is said to be a mild solution of equation (1.1) if u is the solution of the following integral equation

u(t) = S(t)[u(0) + g(0, u_{0})] − g(t, u(a(t)))
+ ∫_{0}^{t} S(t − s)f(s, u(s), u[h(u(s), s)])ds, t ∈ [0, T]  (2.1)

and satisfies the initial condition u(0) = u_{0}.

Definition 2.2 By a solution of the problem (1.1), we mean a function u : [0, T] → X_{α} satisfying the following four conditions:

(i) u(.) + g(., u(a(.))) ∈ C_{T}^{α−1} ∩ C^{1}((0, T), X) ∩ C([0, T], X),

(ii) u(t) ∈ D(A), and (t, u(t), u[h(u(t), t)]) ∈ U_{1},

(iii) \frac{d}{dt}[u(t) + g(t, u(a(t)))] + A[u(t) + g(t, u(a(t)))] = f(t, u(t), u[h(u(t), t)]) for all t ∈ (0, T],

(iv) u(0) = u_{0}.

3 Existence of Local Solutions

We can prove that assumptions (A2)–(A3), for 0 ≤ α < 1, 0 < T_{0} ≤ T, and u ∈ C_{T_{0}}^{α} imply that f(s, u(s), u[h(u(s), s)]) is continuous on [0, T_{0}]. Therefore, we can show that there exists a positive constant N such that

∥f(s, u(s), u[h(u(s), s)])∥ ≤ N= L_{f}[T_{0}^{θ_{1}} + δ(1 + LL_{h}) + LL_{h}T_{0}^{θ_{2}}] + N_{0},

where N_{0} = ∥f(0, u_{0}, u_{0})∥. Similarly with the help of the assumptions (A4)–(A5), we can show easily that ∥A^{α}g(t, u(a(t)))∥ ≤ L_{g}[T_{0} + δ] + ∥g(0, u_{0})∥_{α} = N_{1}. Also, we denote ∥A^{−1}∥ = M_{2} and ∥A^{−α}∥ = M_{3}.

Theorem 3.1 Let us assume that the assumptions (A1)-(A5) are hold and u_{0} ∈ D(A^{α}) for 0 ≤ α < 1. Then, the differential equation (1.1) has a unique local mild solution if

(L_{g} + C_{α}L_{f}[2 + LL_{h}]\frac{T_{0}^{1−α}}{1 − α}) < 1.  (3.1)

Proof. Now for a fixed δ > 0, we choose 0 < T_{0} ≤ T such that

∥(S(t) − I)A^{α}[u_{0} + g(0, u_{0})]∥ + L_{g}[T_{0} + δ] ≤ \frac{δ}{2}, for all t ∈ [0, T_{0}]  (3.2)

and

C_{α}N \frac{T_{0}^{1−α}}{1 − α} ≤ \frac{δ}{2}.  (3.3)

We set

W = {u ∈ C_{T_{0}}^{α} ∩ C_{T_{0}}^{α−1} : u(0) = u_{0}, ∥u − u_{0}∥_{T_{0},α} ≤ δ}.

Clearly, W is a closed and bounded subset of C_{T}^{α−1}.

We define a map F : W → W given by

(Fu)(t) = S(t)[u_{0} + g(0, u_{0})] − g(t, u(a(t)))
+ ∫_{0}^{t} S(t − s)f(s, u(s), u[h(u(s), s)])ds, t ∈ [0, T].  (3.4)

In order to proved this theorem first we need to show that Fu ∈ C_{T_{0}}^{α−1} for any u ∈ C_{T_{0}}^{α−1}. Clearly, F : C_{T}^{α} → C_{T}^{α}.

If u ∈ C_{T_{0}}^{α−1}, T > t_{2} > t_{1} > 0, and 0 ≤ α < 1, then we get

∥(Fu)(t_{2}) − (Fu)(t_{1})∥_{α−1}
≤ ∥(S(t_{2}) − S(t_{1}))(u_{0} + g(0, u_{0}))∥_{α−1}
+∥A^{−1}∥∥A^{α}g(t_{2}, u(a(t_{2}))) − A^{α}g(t_{1}, u(a(t_{1})))∥
+ ∫_{0}^{t_{1}} ∥(S(t_{2} − s) − S(t_{1} − s))A^{α−1}∥∥f(s, u(s), u[h(u(s), s)])∥ds
+ ∫_{t_{1}}^{t_{2}} ∥S(t_{2} − s)A^{α−1}∥∥f(s, u(s), u[h(u(s), s)])∥ds.  (3.5)

We have,

∥(S(t_{2}) − S(t_{1}))(u_{0} + g(0, u_{0}))∥_{α−1} ≤ ∫_{t_{1}}^{t_{2}} ∥A^{α−1}S′(s)(u_{0} + g(0, u_{0}))∥ds
= ∫_{t_{1}}^{t_{2}} ∥A^{α}S(s)(u_{0} + g(0, u_{0}))∥ds
≤ ∫_{t_{1}}^{t_{2}} ∥S(s)∥[∥u_{0}∥_{α} + ∥g(0, u_{0})∥_{α}]ds
≤ C_{1}(t_{2} − t_{1}),  (3.6)

where C_{1} = [∥u_{0}∥_{α} + ∥g(0, u_{0})∥_{α}]M.

Also, we can see that

∥A^{α−1}g(t_{2}, u(a(t_{2}))) − A^{α−1}g(t_{1}, u(a(t_{1})))∥
≤ ∥A^{−1}∥∥A^{α}g(t_{2}, u(a(t_{2}))) − A^{α}g(t_{1}, u(a(t_{1})))∥
≤ ∥A^{−1}∥L_{g}[(t_{2} − t_{1}) + ∥u(a(t_{2})) − u(a(t_{1}))∥_{α−1}]
≤ ∥A^{−1}∥[L_{g} + LL_{a}](t_{2} − t_{1}).  (3.7)

We observe that,

∥(S(t_{2} − s) − S(t_{1} − s))∥_{α−1} ≤ ∫_{0}^{t_{2}−t_{1}} ∥A^{α−1}S′(l)S(t_{1} − s)∥dl

≤ ∫_{0}^{t_{2}−t_{1}} ∥S(l)A^{α}S(t_{1} − s)∥dl
≤ MC_{α}(t_{2} − t_{1})(t_{1} − s)^{−α}.  (3.8)

Now we use the inequality (3.8) to get the inequality given below,

∫_{0}^{t_{1}} ∥(S(t_{2} − s) − S(t_{1} − s))A^{α−1}∥∥f(s, u(s), u[h(u(s), s)])∥ds ≤ C_{2}(t_{2} − t_{1}),  (3.9)

where C_{2} = NMC_{α} \frac{T_{0}^{1−α}}{1−α} .

Similarly,

∫_{t_{1}}^{t_{2}} ∥S(t_{2} − s)A^{α−1}∥∥f(s, u(s), u[h(u(s), s)])∥ds ≤ C_{3}(t_{2} − t_{1}),  (3.10)

where C_{3} = ∥A^{α−1}∥MN.

We use the inequalities (3.6) (3.7) (3.9) and (3.10) in inequality (3.5) and get the following inequality,

∥(Fu)(t_{2}) − (Fu)(t_{1})∥_{α−1} ≤ L|t_{2} − t_{1}|,  (3.11)

where, L = \frac{C_{1}+C_{2}+C_{3}+∥A^{−1}∥L_{g}}{1−∥A^{−1}∥L_{a}} . Hence, F : C_{T_{0}}^{α−1} → C_{T_{0}}^{α−1}. Our next task is to show that F : W → W. Now, for t ∈ (0, T_{0}] and u ∈ W, we have

∥(Fu)(t) − u_{0}∥_{α}

≤ ∥(S(t) − I)A^{α}[u_{0} + g(0, u_{0})]∥ + ∥A^{α}g(s, u(a(s))) − A^{α}g(0, u(a(0)))∥

+ ∫_{0}^{t} ∥S(t − s)A^{α}∥∥f(s, u(s), u[h(u(s), s)])∥ds

≤ ∥(S(t) − I)A^{α}[u_{0} + g(0, u_{0})]∥ + L_{g}[T_{0} + δ] + C_{α}N \frac{T_{0}^{1−α}}{1 − α}.

Hence, from inequalities (3.2) and (3.3), we get

∥Fu − u_{0}∥_{T_{0},α} ≤ δ.

Therefore, F : W → W.

Now, if t ∈ (0, T_{0}] and u, v ∈ W, then

∥(Fu)(t) − (Fv)(t)∥_{α}
≤ ∥A^{α}g(t, u(a(t))) − A^{α}g(t, v(a(t)))∥
+ ∫_{0}^{t} ∥S(t − s)A^{α}∥∥f(s, u(s), u[h(u(s), s)]) − f(s, v(s), v[h(u(s), s)])∥ds. (3.12)

We have the following inequalities,

∥A^{α}g(t, u(a(t))) − A^{α}g(t, v(a(t)))∥ ≤ L_{g}∥A^{−1}∥∥u − v∥_{T_{0},α},  (3.13)

∥f(s, u(s), u[h(u(s), s)]) − f(s, v(s), v[h(v(s), s)])∥ ≤ L_{f}[2 + LL_{h}]∥u − v∥_{T_{0},α}. (3.14)

We use the inequalities (3.13) and (3.14) in the inequality (3.12) and get

∥(Fu)(t) − (Fv)(t)∥_{α} ≤ (L_{g}∥A^{−1}∥ + C_{α}L_{f}[2 + LL_{h}]\frac{T_{0}^{1−α}}{1 − α}) ∥u − v∥_{T_{0},α}.  (3.15)

Hence from inequality (3.1), we get the following inequality given below

∥Fu − Fv∥_{T_{0},α} < ∥u − v∥_{T_{0},α}.

Therefore, the map F has a unique fixed point u ∈ W which is given by,

u(t) = S(t)[u_{0} + g(0, u_{0})] − g(t, u(a(t)))
+ ∫_{0}^{t} S(t − s)f(s, u(s), u[h(u(s), s)])ds, t ∈ [0, T_{0}].  (3.16)

Hence, the mild solution u of equation (1.1) is given by the equation (3.16) and belong to C_{T_{0}}^{α} ∩ C_{T_{0}}^{α−1}.

Theorem 3.2 Let us assume that the assumptions (A1)-(A5) are hold and u_{0} ∈ D(A^{α}) for 0 ≤ α < 1. Then, the differential equation (1.1) has a unique local solution in the sense of the Definition 2.2.

Proof. In order to prove this theorem, we first need to prove that the mild solution u is Hölder continuous on (0, T_{0}]. From Theorem 2.6.13 in Pazy [11], it follows that for every 0 < β < 1 − α, t > s > 0 and every 0 < h < 1, we have

∥(S(h) − I)A^{α}S(t − s)∥ ≤ C_{β}h^{β}∥A^{α+β}S(t − s)∥
≤ Ch^{β}(t − s)^{−(α+β)},  (3.17)

where C = C_{β}C_{α+β}.

For 0 < t < t + h ≤ T_{0}, we have

∥u(t + h) − u(t)∥_{α}
≤ ∥((S(h) − I)S(t)(u_{0} + g(0, u_{0}))∥_{α}
+∥A^{α}g(t_{2}, u(a(t_{2}))) − A^{α}g(t_{1}, u(a(t_{1})))∥
+ ∫_{0}^{t} ∥(S(h) − I)A^{α}S(t − s)∥∥f(s, u(s), u[h(u(s), s)])∥ds
+ ∫_{t}^{t+h} ∥S(t + h − s)A^{α}∥∥f(s, u(s), u[h(u(s), s)])∥ds.  (3.18)

We calculate the first term of the above inequality (3.18) as follows;

∥(S(h) − I)S(t)A^{α}(u_{0} + g(0, u_{0}))∥ ≤ Ct^{−(α+β)}{∥u_{0}∥ + ∥g(0, u_{0})∥}h^{β}
≤ M_{1}h^{β},  (3.19)

where M_{1} = Ct^{−(α+β)}{∥u_{0}∥ + ∥g(0, u_{0})∥} depends on t and blows up as t decreases to zero. Second term of the above inequality (3.18) we calculate as follows,

∥A^{α}g(t + h, u(a(t + h))) − A^{α}g(t, u(a(t)))∥
≤ ∥A^{α}g(t + h, u(a(t + h))) − A^{α}g(t, u(a(t)))∥
≤ L_{g}[h + ∥A∥∥u(a(t + h)) − u(a(t))∥_{α−1}]
≤ L_{g}[h + ∥A∥LL_{a}h]
≤ M_{2}h,  (3.20)

where M_{2} = L_{g}[1 + ∥A∥LL_{a}] is a constant independent of t.

Third and the fourth term of the inequality (3.18) can be calculated as follows:

∫_{0}^{t} ∥(S(h) − I)A^{α}S(t − s)∥∥f(s, u(s), u[h(u(s), s)])∥ds
≤ Ch^{β}N ∫_{0}^{t}(t − s)^{−(α+β)}ds
≤ M_{3}h^{β},  (3.21)
∫_{t}^{t+h} ∥A^{α}S(t + h − s)∥∥f(s, u(s), u[h(u(s), s)])∥ds, ≤ C_{α}N ∫_{t}^{t+h}(t + h − s)^{−α}ds
≤ M_{4}h^{β},  (3.22)

where M_{3} and M_{4} can be chosen to be independent of t.

Therefore,

∥u(t + h) − u(t)∥_{α} ≤ C′h^{β},

where C′ is a positive constant. Thus, u is locally Hölder continuous on (0, T_{0}].

Hence,

∥f(t, u(t), u[h(u(t), t)]) − f(s, u(s), u[h(u(s), s)])∥
≤ L_{f}{|t − s|^{θ_{1}} + ∥u(t) − u(s)∥_{α} + L|h(u(t), t) − h(u(s), s)|}
≤ L_{f}{|t − s|^{θ_{1}} + ∥u(t) − u(s)∥_{α} + LL_{h}[|t − s|^{θ_{2}} + ∥u(t) − u(s)∥_{α}]}
≤ L_{f}{|t − s|^{θ_{1}} + C′|t − s|^{β} + LL_{h}[|t − s|^{θ_{2}} + C′|t − s|^{β}]}.  (3.23)

Hence, the map t ↦ f(t, u(t), u[h(u(t), t)]) is locally Hölder continuous. Therefore,

f(t, u(t), u[h(u(t), t)]) ∈ C([0, T], X) ∩ C^{β′} ((0, T], X),

where 0 < β′ < min{θ_{1}, β, θ_{2}}. Similarly, we can prove that u(.) + g(., u(a(.))) is also Hölder continuous on (0, T_{0}]. Therefore from Theorem 3.1 pp. 110 and Corollary 3.3, pp. 113, Pazy [11], the function u(.) + g(., u(a(.))) ∈ C_{T_{0}}^{α−1} ∩ C^{1}((0, T_{0}), X) ∩ C([0, T_{0}], X) and u(.) is the unique solution of the problem (1.1) in the sense of Definition 2.2. This completes the proof of the Theorem. □

4 Existence of Global Solutions

Theorem 4.1 Suppose that 0 ∈ ρ(−A) and the operator −A generates the analytic semigroup S(t) with ∥S(t)∥ ≤ M, for t ≥ 0, the conditions (A1)–(A4) are satisfied and u_{0} ∈ D(A^{α}). If there are continuous nondecreasing real valued function k_{1}(t), k_{2}(t) and k_{3}(t) such that

∥f(t, x, y)∥ ≤ k_{1}(t)(1 + ∥x∥_{α} + ∥y∥_{α−1}), for all t ≥ 0, x ∈ X_{α}, y ∈ X_{α−1}, (4.1)
|h(z, t)| ≤ k_{2}(t)(1 + ∥z∥_{α}), for all t ≥ 0, z ∈ X_{α},  (4.2)
∥g(t, v)∥_{α} ≤ k_{3}(t)(1 + ∥v∥_{α−1}), for all t ≥ 0, v ∈ X_{α−1},  (4.3)

then the initial value problem (1.1) has a unique solution which exists for all t ∈ [0, T].

Proof: By theorem (3.1) we can continue the solution of equation (1.1) as long as ∥u(t)∥_{α} stays bounded. It is therefore sufficient to show that if u exists on [0, T [ then ∥u(t)∥_{α} is bounded as t ↑ T.

We have the following inequality,

∥u[h(u(s), s)]∥_{α−1} ≤ ∥u[h(u(s), s)] − u(0)∥_{α−1} + ∥u_{0}∥_{α−1}
≤ L|h(u(s), s)| + ∥u_{0}∥_{α−1}
≤ Lk_{2}(T) + Lk_{2}(T)∥u∥_{s,α}] + ∥u_{0}∥_{α−1}.  (4.4)

For t ∈ [0, T [, we have

∥u(t)∥_{α} ≤ ∥S(t)A^{α}[u_{0} + g(0, u_{0})]∥ + ∥g(t, u(a(t)))∥_{α}
+ ∫_{0}^{t} ∥A^{α}S(t − s)∥∥f(s, u(s), u[h(u(s), s)])∥ds
≤ M[∥u_{0}∥_{α} + k_{3}(T){1 + ∥u_{0}∥_{α}}] + k_{3}(T)[1 + ∥A^{−1}∥∥u∥_{t,α}]
+ C_{α} ∫_{0}^{t}(t − s)^{−α}k_{1}(T)[1 + ∥u∥_{s,α} + ∥u[h(u(s), s)]∥_{α−1}]ds,
≤ M[∥u_{0}∥_{α} + k_{3}(T){1 + ∥u_{0}∥_{α}}] + k_{3}(T) + k_{3}(T)∥A^{−1}∥∥u∥_{t,α}
+ k_{1}(T)C_{α} ∫_{0}^{t}(t − s)^{−α}ds + k_{1}(T)C_{α} ∫_{0}^{t}(t − s)^{−α}∥u∥_{s,α}ds
+ Lk_{2}(T)k_{1}(T)C_{α} ∫_{0}^{t}(t − s)^{−α}ds + ∥u_{0}∥_{α−1}Lk_{2}(T)k_{1}(T)C_{α} ∫_{0}^{t}(t − s)^{−α}ds
+ Lk_{2}(T)k_{1}(T)C_{α} ∫_{0}^{t}(t − s)^{−α}∥u∥_{s,α}ds.

Hence,

∥u∥_{t,α} ≤ C_{1} + C_{2} ∫_{0}^{t}(t − s)^{−α}∥u∥_{s,α}ds,  (4.5)

where C_{1} = \frac{M}{1−k_{3}(T)}[∥u_{0}∥_{α} + k_{3}(T)\{1 + ∥u_{0}∥_{α}\}] + \frac{k_{3}(T)}{1−k_{3}(T)} + \frac{k_{1}(T)C_{α}T^{1−α}}{(1−k_{3}(T))(1−α)} + \frac{Lk_{2}(T)k_{1}(T)C_{α}T^{1−α}}{(1−k_{3}(T))(1−α)}

+ ∥u_{0}∥_{α−1} \frac{Lk_{2}(T)k_{1}(T)C_{α}T^{1−α}}{(1−k_{3}(T))(1−α)} and C_{2} = \frac{k_{1}(T)C_{α}[1+Lk_{2}(T)]}{1−k_{3}(T)}. Hence by applying the Gronwall’s lemma to the above inequality (4.5), we get the required results. This completes the proof of the theorem. □

5 Examples

Let X = L^{2}(0, 1). We consider the following partial differential equations with deviated argument,

\{ \begin{matrix}∂_{t}[w(t, x) + ∂_{x}f_{1}(t, w(a(t), x))] − ∂_{x}^{2}[w(t, x) + f_{1}(t, w(a(t), x))] \\ = f_{2}(x, w(t, x)) + f_{3}(t, x, w(t, x)), x ∈ (0, 1), t > 0, \\ w(t, 0) = w(t, 1) = 0, t ∈ [0, T], 0 < T < ∞, \\ w(0, x) = u_{0}, x ∈ (0, 1), \end{matrix}  (5.1)

where

f_{2}(x, w(t, x)) = ∫_{0}^{x} K(x, s)w(s, h(t)(a_{1}|w(s, t)| + b_{1}|w_{s}(s, t)|))ds.

The function f_{3} : ℝ_{+} × [0, 1] × ℝ → ℝ is measurable in x, locally Hölder continuous in t, locally Lipschitz continuous in w and uniformly in x. Further we assume that a_{1}, b_{1} ≥ 0, (a_{1}, b_{1}) ≠ (0, 0), h : ℝ_{+} → ℝ_{+} is locally Hölder continuous in t with h(0) = 0 and K : [0, 1] × [0, 1] → ℝ. The function f_{1} : ℝ_{+} × ℝ → ℝ is locally Hölder continuous in t, locally Lipschitz continuous in w.

We define an operator A, as follows,

Au = −u′′ with u ∈ D(A) = {u ∈ H_{0}^{1}(0, 1) ∩ H^{2}(0, 1) : u′′ ∈ X}.  (5.2)

Here clearly the operator A is self-adjoint with compact resolvent and is the infinitesimal generator of an analytic semigroup S(t). Now we take α = 1/2, D(A^{1/2}) = H_{0}^{1}(0, 1) is the Banach space endowed with the norm,

∥x∥_{1/2} := ∥A^{1/2}x∥, x ∈ D(A^{1/2})

and we denote this space by X_{1/2}. Also, for t ∈ [0, T], we denote

C_{t}^{1/2} = C([0, t]; D(A^{1/2})),

endowed with the sup norm

∥ψ∥_{t,1/2} := sup_{0≤η≤t} ∥ψ(η)∥_{α}, ψ ∈ C_{t}^{1/2}.

We observe some properties of the operators A and A^{1/2} defined by (5.2). For u ∈ D(A) and λ ∈ ℝ, with Au = −u′′ = λu, we have ⟨Au, u⟩ = ⟨λu, u⟩; that is,

⟨−u′′, u⟩ = |u′|_{L^{2}}^{2} = λ|u|_{L^{2}}^{2}

so λ > 0. A solution u of Au = λu is of the form

u(x) = C cos(\sqrt{λ}x) + D sin(\sqrt{λ}x)

and the conditions u(0) = u(1) = 0 imply that C = 0 and λ = λ_{n} = n^{2}π^{2}, n ∈ ℕ. Thus, for each n ∈ ℕ, the corresponding solution is given by

u_{n}(x) = D sin(\sqrt{λ_{n}}x).

We have ⟨u_{n}, u_{m}⟩ = 0 for n ≠ m and ⟨u_{n}, u_{n}⟩ = 1 and hence D = \sqrt{2}. For u ∈ D(A), there exists a sequence of real numbers {α_{n}} such that

u(x) = ∑_{n∈ℕ}α_{n}u_{n}(x), ∑_{n∈ℕ}(α_{n})^{2} < +∞ and ∑_{n∈ℕ}(λ_{n})^{2}(α_{n})^{2} < +∞.

We have

A^{1/2}u(x) = ∑_{n∈ℕ} \sqrt{λ_{n}} α_{n} u_{n}(x)

with u ∈ D(A^{1/2}); that is, ∑_{n∈ℕ} λ_{n}(α_{n})^{2} < +∞. X_{− \frac{1}{2}} = H^{1}(0, 1) is a Sobolev space of negative index with the equivalent norm ∥.∥_{− \frac{1}{2}} = ∑_{n=1}^{∞} |⟨., u_{n}⟩|^{2}. For more details on the Sobolev space of negative index, we refer to Gal [6].

The equation (5.1) can be reformulated as the following abstract equation in X = L^{2}(0, 1):

\frac{d}{dt}[u(t) + g(t, u(a(t)))] + A[u(t) + g(t, u(a(t)))] = f(t, u(t), u[h(u(t), t)]) t > 0,
u(0) = u_{0},  (5.3)

where u(t) = w(t, .) that is u(t)(x) = w(t, x), x ∈ (0, 1). The function g : ℝ_{+} × X_{1/2} → X, such that g(t, u(a(t)))(x) = ∂_{x}f_{1}(t, w(a(t), x)) and the operator A is same as in equation (5.2).

The function f : ℝ_{+} × X_{1/2} × X_{−1/2} → X, is given by

f(t, ψ, ξ)(x) = f_{2}(x, ξ) + f_{3}(t, x, ψ),  (5.4)

where f_{2} : [0, 1] × X → H_{0}^{1}(0, 1) is given by

f_{2}(t, ξ) = ∫_{0}^{x} K(x, y)ξ(y)dy,  (5.5)

and

∥f_{3}(t, x, ψ)∥ ≤ Q(x, t)(1 + ∥ψ∥_{H^{2}(0,1)})  (5.6)

with Q(., t) ∈ X and Q is continuous in its second argument. We can easily verify that the function f satisfies the assumptions (H1)-(H4). For more details see [6].

For the function a we can take

(i) a(t) = kt, where t ∈ [0, T] and 0 < k ≤ 1.

(ii) a(t) = kt^{n} for t ∈ I = [0, 1] k ∈ (0, 1] and n ∈ ℕ;

(iii) a(t) = k sin t for t ∈ I = [0, \frac{π}{2} ], and k ∈ (0, 1].

Acknowledgements: The first author would like to thank the National Board for Higher Mathematics for providing the financial support to carry out this work under its research project No. NBHM/2001/R&D-II. The same author also would like to thank the UGC for their support to the department of mathematics.

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(Received June 18, 2008)