Research Article | Open | Published:
Regularity for Solutions of Second-Order Nonlinear Integrodifferential Functional Equations
Journal of Inequalities and Applicationsvolume 2010, Article number: 280463 (2010)
We deal with the well-posedness for solutions of nonlinear integrodifferential equations of second-order in Hilbert spaces by converting the problem into the contraction mapping principle with more general conditions on the principal operators and the nonlinear terms and obtain a variation of constant formula of solutions of the given nonlinear 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:
where the nonlinear term is given by
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  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 . 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]).
has a unique solution which satisfies what follows: if , , and, then
By different methods, results similar to those in result mentioned above were established by Browder , Barbu , Lions and Strauss , and many others. Tanabe  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 .  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.
2. Linear Hyperbolic Equations
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
Therefore, we assume that has a stronger topology than that of and, for the brevity, we may regard that
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,
(ii) is symmetric, that is,
(iii) satisfies the Gårding's inequality, that is,
Let be the operator such that . Then, as seen in [13, Theorem 2.2.3], the operator is positive, definite, and self-adjoint, , and
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:
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:
By virtue of Theorem 8.2 of , we have the following result on the corresponding linear equation of (2.8) in case .
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
where is a constant depending on . Moreover, the mapping
is a linear continuous map of .
3. Hyperbolic Equations with Nonlinear Perturbations
Let be a nonlinear mapping such that is measurable and
for a positive constant . We assume that for the sake of simplicity.
For we set
where belongs to .
Let , . Then ,
Moreover if , then
By Hölder inequality we obtain
From (F) and using the Hölder inequality, it is easily seen that
The proof of the second paragraph is similar.
The following lemma is from Brézis (, Lemma A.5)
Let satisfying for all and let be a constant. Let be a continuous function on satisfying the following inequalities:
Let . Then invoking Proposition 2.2, we obtain that the problem
has a unique solution .
The following Lemma is one of the useful integral inequalities.
Let and consider the following inequality:
Let be the solutions of (3.8) with replaced of by , respectively. Then the following inequality holds:
For we consider the following equation:
Then, we have that
for . Acting on both sides of (3.15) by , we have
By integration of parts, it holds that
Consequently, since is self-adjoint, we have
and by Lemma 3.1,
By integrating over , (3.16) implies that
which yields that
From (3.22) and Schwarz's inequality it follows that
Integrating (3.23) over we have
By Lemma 3.2, we get
where , that is,
Hence, inequality (3.11) is obtained from (3.22) and (3.26), which implies that
So, by using Lemma 3.1, we obtain (3.12).
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 .
Let us fix such that
We are going to show that is strictly contractive from to itself if condition (3.28) is satisfied. The norm in is given by
From (3.11) and (3.12) it follows that
Starting from initial value , consider a sequence satisfying
Then from (3.30) it follows that
Hence, we obtain that
So by virtue of condition (3.28) the contraction principle gives that there exists such that
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.
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
We establish the estimates of solution. Let be the solution of
Then if is a solution of (1.1), since
by multiplying and using the monotonicity of , then we obtain
By the procedure similar to (3.33) we have
Then from Proposition 2.2, we have that
for some positive constant . Noting that by Lemma 3.1
and by Proposition 2.2
it is easy to obtain the norm estimate of in satisfying (3.35).
Now from (3.41) it follows that
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).
Let be another Hilbert space and let be a bounded linear operator from to . We consider the following semilinear control equation of second-order:
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.
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
for a positive constant ; therefore, our results can be obtained directly.
Barbu V: Analysis and Control of Nonlinear Infinite-Dimensional Systems, Mathematics in Science and Engineering. Volume 190. Academic Press, Boston, Mass, USA; 1993:x+476.
Duzaar F, Mingione G: Second order parabolic systems, optimal regularity, and singular sets of solutions. Annales de l'Institut Henri Poincare (C) Non Linear Analysis 2005, 22(6):705–751. 10.1016/j.anihpc.2004.10.011
Strauss WA: On continuity of functions with values in various Banach spaces. Pacific Journal of Mathematics 1966, 19: 543–551.
Baiocchi C: Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert. Annali di Matematica Pura ed Applicata, Series 4 1967, 76(1):233–304. 10.1007/BF02412236
Garrido-Atienza MJ, Real J: Existence and uniqueness of solutions for delay evolution equations of second order in time. Journal of Mathematical Analysis and Applications 2003, 283(2):582–609. 10.1016/S0022-247X(03)00297-X
Da Prato G, Iannelli M: Existence and regularity for a class of integro-differential equations of parabolic type. Journal of Mathematical Analysis and Applications 1985, 112(1):36–55. 10.1016/0022-247X(85)90275-6
Jeong J-M, Kim J-R, Kim H-G: Regularity for solutions of nonlinear second order evolution equations. Journal of Mathematical Analysis and Applications 2008, 338(1):209–222. 10.1016/j.jmaa.2007.05.010
Brézis H: Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. In Contributions to Nonlinear Functional Analysis. Academic Press, New York, NY, USA; 1971:101–156.
Brézis H: Opérateurs Maximaux Monotones et Semigroupes de Contractions dans un Espace de Hilbert. North Holland, Amsterdam, The Netherlands; 1973.
Browder FE: On non-linear wave equations. Mathematische Zeitschrift 1962, 80: 249–264. 10.1007/BF01162382
Barbu V: Nonlinear Semigroups and Differential Equations in Banach Spaces. Nordhoff Leiden, Bucharest, Romania; 1976.
Lions J-L, Strauss WA: Some non-linear evolution equations. Bulletin de la Société Mathématique de France 1965, 93: 43–96.
Tanabe H: Equations of Evolution, Monographs and Studies in Mathematics. Volume 6. Pitman, London, UK; 1979:xii+260.
Strauss WA: On weak solutions of semi-linear hyperbolic equations. Anais da Academia Brasileira de Ciências 1970, 42: 645–651.
Kartsatos AG, Markov LP: An -approach to second-order nonlinear functional evolutions involving -accretive operators in Banach spaces. Differential and Integral Equations 2001, 14(7):833–866.
Ladyzenskaya QA: Linear Differential Equations in Banach Spaces. Nauka, Moscow, Russia; 1967.
Lions J-L: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York, NY, USA; 1971:xi+396.
Lions JL, Magenes E: Non-Homogeneous Boundary Value Problèms and Applications. Springer, Berlin, Germany; 1972.
This work was supported by the Research Foundation funded by Dong-A University.