ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

Using the properties of almost nonexpansive curves introduced by B. Djafari Rouhani, we study the asymptotic behavior of solutions of nonlinear functional differential equation du(t)/dt + Au(t)+ G(u)(t) f(t), where A is a maximal monotone operator in a nilbert space H,f E LI(0,:H) and G:C([O,c):D(A))LI(O,c:H)is a given mapping.


du(t)
dt + Au(t) + G(u)(t) 9 f(t), 0 < < c (1.1) where A is a maximal monotone (possibly multivalued) operator defined on a subset D(A) contained in H, x E D(A),f E Loc ([0,c):H) and G is a given mapping G:C([O,T]:D(A))LI(O,T:H), for all T > 0. (1.2) Problems of the type (1.1) have been considered by many authors (see [1]- [7]).Crandall and  Nohel  [4] treated the problem in connection with the study of a related nonlinear Volterra equation, and obtained the existence result of generalized solution of (1.1), provided that G satisfies a Lipschitz type condition.In particular, under some suitable hypotheses on A and G, Aizicovici [1] obtained nice asymptotic results of generalized solutions of (1.1), which are the natural analogs of the evolution case (i.e., G 0).Using the convergence condition of Pazy [8], Mitidieri  [5] studied the strong convergence of solutions of (1.1).
The purpose of this paper is to continue the study initiated by Aizicovici, using the properties of almost nonexpansive curve, which was introduced by Djafari Rouhani [9].In section 2, we describe the notations and contain some definitions and known results.Section 3 contains the several results [Theorem 1, 2, Corollary 1] concerning the asymptotic behavior of almost nonexpansive curve.Main results are given in Section 4. First we establish criterions for the weak convergence in H, as to of generalized solutions of (1.1) [Theorem 3, Corollary 3].Next, we study the weak convergence of the Ceshro mean of the generalized solutions [Theorem 4]. 9.. PRELIMINARIIS.
Let H be a real Hilbert space with inner product (,) and norm II.Let m be maximal monotone (possibly multivalued) operator defined on subset D(A) C H.
As usual, we will put [z,V] 5 A//E Az.We denote by F the (possibly empty) set F A-'0 {: D(A), a% 0} where Ag denotes the element of minimum norm in the closed convex set Ay. Clearly, F is a closed convex subset of H.For background material concerning maximal monotone operators, see [10], [11].
We will use "w-lira" or "-" to indicate weak convergence in H.The symbol D denotes the closure of the set D. For a function u: [0, c)H, we denote by Ww(U(t)) the weak -limit set of u, i.e., w(u(t)) {y H: y w-lira u(t,) for some sequence and by -e-dw.,(u(t)) the closed convex hull of w(u(t)), respectively.Let u:[0, cx)H be a bounded function.With the function u(t), we associate the functional (y) lim sup u(t)-y =t--*OO Then is a continuous, strictly convex function on H, satisfying (y)x as u II--', d therefore has a unique minimum in H.The unique point c E H satisfying (c) min (y)   is called the asymptotic center of u(t) and it is denoted by c AC(u(t)).
In our next results, we will use the following notation: E(u(t)) {q e H:lim u(t)-q exists}.
Then, /a) For each z 6 D/A) and f BVocl[O, oo):H), problem 1.1) has a unique strong solution defined on [0, o).
(b) For each z E D(A)and f Loc([O, oo):H), problem (1.1) has a unique generalized solution defined on [0,).
3. ASYMPTOTIC BEHAVIOR OF CURVES IN H.
In this section, we study asymptotic behavior of almost nonexpansive curve, which was introduced by Djafari Rouhani [9].The following results are essentially in spirit of Djafari Rouhani.For the study and completeness, we give several results similar to those in [9] with detail and slightly different proofs.
As a direct consequence of Lemma 3 and Theorem 1, we have the following: COROLLARY 1.Let {u(t)} be an ANEC.Then the following condition which implies that limt_oo =(t)-p exists and hence p e E(u(t)).THEOREM 1.Let {u(t)} be an ANEC in H. Then the following are equivalent: (i) w limt.ou(texists. (ii) E(u(t)) and wo(u(t)) C E(u(t)).Moreover, if w-limt.u(texists, then it is the asymptotic center of {u(t)}.
We turn now to the weak convergence of Cesro mean a(t) of curve {u(t)}.The following lemma is the principal ingredient.
Let p e(a(t)).
for any r, s _> 0. Let e > 0 be given and choose > 0 so that e(h,t) < for all h, > I. Then taking r > l, s > 0 fixed and letting t,oo, we obtain u( / ,)-p _< u()p /, Therefore by the same argument as in Lemma 3, we conclude that limt_.ou(t)p exists and hence p E(u(t)).
Then {u(t)} is bounded, and hence {a(t)} is also bounded.Then since a Hilbert space is reflexive, {a(t)} has a subnet {a(tn) which converges wetly to some pH.By Lemma 4, w(a(t)) C E(u(t)), and so p S(u(t)).If there w another subnet {a(h)} which converges weakly to some q e H, then we Mso have q e E(u(t)).Hence the net 2(u(t),q p) + P q =(t)p =(t)q has a limit as t--}o, i.e., limt,oo(u(t), q-p) exists.Therefore (p, q p) (q, q-p), which implies p-q 0, d hence p q. Hence every weakly convergent subnet of a(t) converges weakly to p, and hence w limt_ooa(t p.
PROPOSITION 2. Let A be a maximal monotone operator on H. Assume that (C1), (C2), (C3), (C4) d (C5) hold.If u is a generalized solution of (1.1) d if {u(t)} is bounded on [0,), then the curve {u(t)} is an ANEC in H. PROOF.By (C4), we have r>s Thus the result follows from Corollary 2 and Remark 1 (b) by taking [ f(r + (r-s))-f(r)II d f _> (,) .o] f(r + (r-s))-f(r)11 dr if s > r.Now, we apply Theorem 1 and 2 to study the asymptotic behavior of the generalized solution u of (1.1).We need the next known result.
As a direct consequence, we have the following: COROLLARY 4. Let A be a maximal monotone operator on H. Assume that (C1), (C2), (C3), (C4) and (C5) hold.Let u be a generalized solution of (1.1).If F and w-limt_.oo(u(t+h)-u(t))=Ofor any h >0, then u(t)converges weakly as t--.oo to the asymptotic center of the curve {u(t)}.PROOF.By Lemma 8, we have F C E(u(t)).Thus the result follows from Theorem 1 and Corollary 3.
Thus the result follows from Theorem 2.
As a consequence, we also have the following: COROLLARY 5. Let A be a maximal monotone operator on H. Assume that (C1), (C2), and (C5) hold.Let u be a generalized solution of (1.1) and.a(t)= fotU(r)dr.If (C3), (C4) F , then a(t) converges weakly as too to the asymptotic center of the curve {u(t)}.
PROOF.By Lemma S, we have F C E(u(t)).Thus the result follows from Theorem 4. lMARK 2. (a) Under the hypotheses of Theorem 3, the fact that the following condition (iv) F -} and ww(u(t)) C F is is equivalent to (i) in Theorem 3 was proved by Aizicovici [1].Consequently, all the conditions (i) and (ii)in Theorem 3, (iii)in Corollary 3 and this (iv) are equivalent.
(b) The case in which G _=0 was previously considered by Moros.anu [14]and Djafari Rouhani [9].Theorem 3 is a new result even in the case in which G 0. (c) Properties of the metric projection were not used in Theorem 4 in contrast to ([1, Theorem 2.3]).

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning (t)-(t)II 5 (,)-()II + f' Y()-()II d (.x) for 0GsGt<.

of Applied Mathematics and Decision Sciences Special Issue on Intelligent Computational Methods for Financial Engineering
u(t) + /tb(ts)Au(s)ds g(t), o (o) (o).t>OJournal

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation