Regularity for Solutions of Second-Order Nonlinear Integrodifferential Functional Equations

Let and be two complex Hilbert spaces. Assume that is a dense subspace in and the injection of into is continuous. If is identified with its dual space then we may write densely and the corresponding injections are continuous. The norm on , , and will be denoted by , and , respectively.

The subset of this paper is to consider the initial value problem of the following second-order nonlinear integrodifferential control system on Hilbert spaces:

(1.1)

where the nonlinear term is given by

(1.2)

Let be a continuous linear operator from into which is assumed to satisfy Gårding's inequality. Here stands for the dual space of . The nonlinear mapping is Lipschitz continuous from into .

The regularity results of the evolution equations of second-order in case are given by [1–3]. We also call attention to the method of Baoicchi [4] who has obtained the results directly, without duality, under slightly different hypotheses, which is valid for first- and second-order equations. The well-posedness of solutions for delay evolution equations of second-order in time was referred to in [5]. As for the equation of first order, it is well known as the quasi-autonomous differential equation (see Theorem 2.6 in [1, 11] and [6, 7]).

The paper is a generalization of a particular case of the equation by Brézis [8, 9] based on the Lipschitz continuity of nonlinear term where the nonlinear differential equation of hyperbolic type

(1.3)

has a unique solution which satisfies what follows: if , , and , then

(1.4)

By different methods, results similar to those in result mentioned above were established by Browder [10], Barbu [11], Lions and Strauss [12], and many others. Tanabe [13] proved the existence of local solution of (1.3) when the nonlinear mapping is locally Lipschitz continuous. Even under the weakest assumption on nonlinear term, the existence of weak solution can be shown though the uniqueness is not vertain. For example, it is referred to in a work by Strauss [14]. [15] dealt with an -approach to second-order nonlinear functional evolutions involving -accretive operators in Banach spaces.

In this paper, under the assumption that and the local Lipschitz continuity of the nonlinear mapping from into (not from into itself), we deal with regularity for the solution of the given equation (1.1) which will enable us to obtain a global existence theorem for the strict solution belonging to ; namely, assuming the Lipschitz continuity of nonlinear terms, we show that the well-posedness and stability properties for a class with nonlinear perturbation of second-order are similar to those of its corresponding linear system.

we begin to study well-posedness and stability properties for a class with nonlinear perturbation of second-order.

We will develop and apply the existence theory for first-order differential equations (see [16, 17]) to study certain second-order differential equations associated with nonlinear maximal monotone operators in Hilbert spaces.

The duality pairing between the element of and the element of is denoted by , which is the ordinary inner product in if . By considering , we may write where and denote the dual spaces of and , respectively. For we denoted by the value of at . The norm of as element of is given by

(2.1)

Therefore, we assume that has a stronger topology than that of and, for the brevity, we may regard that

(2.2)

Definition.

Let and be complex Banach spaces. An operator from to is called antilinear if and for and for .

Let be a quadratic form defined on which is linear in and antilinear in .

We make the following assumptions:

(i) is bounded, that is,

(2.3)

(ii) is symmetric, that is,

(2.4)

(iii) satisfies the Gårding's inequality, that is,

(2.5)

Let be the operator such that . Then, as seen in [13, Theorem 2.2.3], the operator is positive, definite, and self-adjoint, , and

(2.6)

It is also known that the operator is a bounded linear from to . The realization of in which is the restriction of to is also denoted by , which is structured as a Hilbert space with the norm . Then the operator generates an analytic semigroup in both of and . Thus we have the following sequence:

(2.7)

where each space is dense in the next one which is continuous injection.

If is a Banach space and , then is the collection of all strongly measurable functions from into the th powers whose norms are integrable and is the set of all functions whose derivatives up to degree in the distribution sense belong to , and is the set of all -times continuously differentiable functions from into . Let and be complex Banach spaces. Denote by (resp., the set of all bounded linear (resp., antilinear) operators from and . Let .

First, consider the following linear hyperbolic equation:

(2.8)

By virtue of Theorem 8.2 of [18], we have the following result on the corresponding linear equation of (2.8) in case .

Proposition.

Suppose that the assumptions for the principal operator stated above are satisfied. Then the following properties hold: For and , , there exists a unique solution of (2.8) belonging to

(2.9)

and satisfying

(2.10)

where is a constant depending on . Moreover, the mapping

(2.11)

is a linear continuous map of .

As for the solution which belongs to spaces, we refer to Chapter IV of [17]. The applications of these problems are mixed problems in the sense of Ladyzenskaya [16].

Let be a nonlinear mapping such that is measurable and

(F)

for a positive constant . We assume that for the sake of simplicity.

For we set

(3.1)

where belongs to .

Lemma.

Let , . Then ,

(3.2)

Moreover if , then

(3.3)

Proof.

By Hölder inequality we obtain

(3.4)

From (F) and using the Hölder inequality, it is easily seen that

(3.5)

The proof of the second paragraph is similar.

The following lemma is from Brézis ([9], Lemma A.5)

Lemma 3.2.

Let satisfying for all and let be a constant. Let be a continuous function on satisfying the following inequalities:

(3.6)

Then,

(3.7)

Let . Then invoking Proposition 2.2, we obtain that the problem

(3.8)

has a unique solution .

The following Lemma is one of the useful integral inequalities.

Lemma.

Let and consider the following inequality:

(3.9)

Then,

(3.10)

Lemma.

Let be the solutions of (3.8) with replaced of by , respectively. Then the following inequality holds:

(3.11)

(3.12)

where and

(3.13)

Proof.

For we consider the following equation:

(3.14)

Then, we have that

(3.15)

for . Acting on both sides of (3.15) by , we have

(3.16)

By integration of parts, it holds that

(3.17)

Consequently, since is self-adjoint, we have

(3.18)

and by Lemma 3.1,

(3.19)

where

(3.20)

By integrating over , (3.16) implies that

(3.21)

which yields that

(3.22)

From (3.22) and Schwarz's inequality it follows that

(3.23)

Integrating (3.23) over we have

(3.24)

By Lemma 3.2, we get

(3.25)

where , that is,

(3.26)

Hence, inequality (3.11) is obtained from (3.22) and (3.26), which implies that

(3.27)

So, by using Lemma 3.1, we obtain (3.12).

Theorem.

Let the assumption (F) be satisfied. Assume that and . Then there exists a time such that the functional differential equation (1.1) admits a unique solution in .

Proof.

Let us fix such that

(3.28)

We are going to show that is strictly contractive from to itself if condition (3.28) is satisfied. The norm in is given by

(3.29)

From (3.11) and (3.12) it follows that

(3.30)

Starting from initial value , consider a sequence satisfying

(3.31)

Then from (3.30) it follows that

(3.32)

Hence, we obtain that

(3.33)

So by virtue of condition (3.28) the contraction principle gives that there exists such that

(3.34)

Since and the operator is bounded linear from to , it follows from (1.1) that the solution belongs to . Thus, it completes the proof of theorem.

From now on, we give a norm estimation of the solution of (1.1) and establish the global existence of solutions with the aid of norm estimations.

Theorem.

Let the assumption (F) be satisfied. Assume that and . Then the solution of (1.1) exists and is unique in , and there exists a constant depending on such that

(3.35)

Proof.

We establish the estimates of solution. Let be the solution of

(3.36)

Then if is a solution of (1.1), since

(3.37)

by multiplying and using the monotonicity of , then we obtain

(3.38)

By the procedure similar to (3.33) we have

(3.39)

Put

(3.40)

Then from Proposition 2.2, we have that

(3.41)

for some positive constant . Noting that by Lemma 3.1

(3.42)

and by Proposition 2.2

(3.43)

it is easy to obtain the norm estimate of in satisfying (3.35).

Now from (3.41) it follows that

(3.44)

So, we can solve the equation in and obtain an analogous estimate to (3.41). Since condition (3.28) is independent of initial values, the solution of (1.1) can be extended to the internal for natural number , that is, for the initial in the interval , as analogous estimate (3.41) holds for the solution in . Furthermore, the estimate (3.35) is easily obtained from (3.41) and (3.43).

Example.

Let

(3.45)

Let be another Hilbert space and let be a bounded linear operator from to . We consider the following semilinear control equation of second-order:

(3.46)

where belongs to . Let the assumption (F) be satisfied. Assume that . Then the solution of (3.46) exists and is unique in , and the variation of constant formula (3.35) is established.

Remark.

Since the operator is an unbounded operator, we will make use of the hypothesis (F). If is a bounded operator from into itself, then we may assume that is a nonlinear mapping satisfying

(3.47)

for a positive constant ; therefore, our results can be obtained directly.