NONLINEAR RANDOM OPERATOR EQUATIONS AND INEQUALITIES IN BANACH SPACES

In this paper we give some new existence theorems for nonlinear random equations and inequalities involving operators of monotone type in Banach spaces. A random Hammerstein integral equation is also studied. In order to obtain random solutions we use some results from the existing deterministic theory as well as from the theory of measurable multifunctions and, in particular, the measurable selection theorems of Kuratowski/Ryll-Nardzewski and of Saint-Beuve.

Tz, (d) qu-bounded if to each M >0 there x n D, x n z D and Tx n y imply y= corresponds a constant K(M) such that whenever x D, (Tx, x) M d g it follows that Tz K(M), (e) on0toe if (Tx-Tg, x-) 0 for MI x, D, (f) mimM monotone if it is monotone d there is no proof extension of T that is Mso a monotone operator, (g) pseudomonotone if (D is a closed, convex subset of X) the following conditions e satisfied: (P1) For y finite dimensional subspace F of X, T is continuous from F D into X*, endowed with the weak topology; (P2) If x n D, z n z D, Txnf d limsup(Tzn, xn x) 0 then f=7 =d lim(7.,.)=(y,), (h) o if .D,.D,T.y=d limsup(Tzn, zn-x) 0 imply that f Tx.
For operator T:flxDX* we will write T(w)x for the vMue of T at (w,z) fi flxD.Then T is called rdom, if, for y x D, T(. )x is meurable.A rdom operator T is cMled crcive if here exists a function c'R + R, with c(r) + r + , such that (T(), ) c( II)" for n d D.
A rdom operator T is sd to be monotone, demicontinuous, etc., if, for every w fl, T(w)(is monotone, demicontinuous, etc.We symbolize by B(,X) the set of meurable functions .x such that aup { ()II' a} < .
We ben with a random fixed point threm which generMizes Threm 6 of cceri [9].In this d in the following sections we fix (fl,,) to denote a complete, a-finite, meure spe.THEOREM 3.1.t H be a sepable Hilbert space d D a convex, closed d unded subset of H with 0 intD.Le A" x D H be a random and wetly continuous operator such tht (a(), ) 2 fo n D d a.The a admits a rdom fixed point, i.e., there exists a meurable " D such that A(w) (w)= (w) for 1 w E a. PROOF.For every fl, we define a mapping R" 2 D by R(w) {x D'x A(w)x}.By Threm 6 of cceri [9] we have R(w) for M1 w ft.Let {gn}n N be a dense sequence into H.We note that Gn= {(, ) z axD--()=0} {(, ): (-(), .)0} n=l Now, for each n fi N, the mapping fn" x D R defined by fn (, z) (x -'A (w) x, gn) is meurable in w d continuous in z.By [10, Theorem 6.1] fn is jointly meurable, so Gr R B(D).Applng Threm 3 of Sainte-Beuve [11], there exists a meurable selection of R, i.e., a meurable : D such that ()6() () fo n Z a.
We study now perturbations of rdom mimM monotone operators by rdom operators of ty (M).The deterministic case of the following theorem has been obtained in [12].THEOREM 3.2.Let X be a separable, reflexive Banach space and D1, D 2 subsets of X.Let L: fl D X* be a jointly measurable, maximal monotone and weakly closed operator with L(w)O 0 for all w (5 12. Also let T: 12D 2 X* be a random, quasi-bounded, coercive and of type (M) operator.Suppose that there exists a dense linear subspace X 0 of X, which is contained in D2, such that for each finite dimensional subspace F of X 0, the operator T: fl F---} X* is demicontinuous.Then, for each y (5 B(12,X*) there exists z (5 B(12,X) such that: L()x() + T()x() y() for all w (5 . PROOF.We may assume, without loss of generality, that y(w)= 0 for all w (5 12. Also, by a result of Trojanski, we may suppose that the spaces X and X* are locally uniformly convex.Thus, the mapping j-1.X* X is continuous from the strong topology of X* to the strong topology of X (cf.[3]).For > 0 and ,o f, let Le(w) be the Yosida approximant of L(w) defined by: L()x (L(w) -1 + J-1)-lx.
Let {Xn}nq be an increing sequence of finite dimensionM subspaces of X0, such that U X n is dense in X.For each n , let Jn be the injection mapping of Xn into X d j its n=l du.Clely, the operator K,'xX,X defined by K,:()z=j(L()+T())j,z is rdom d continuous.Since Le()0 0 and T is crcive, Kne is Mso crcive.
Let Fne(w)=weakcl{xi(w)'i>n}, for every n(bN. Under the weak topology M(0)--{x X. x < M) is metrizable space.Thus, by [10, Theorem 5.6] the multifunctions Fne are weakly measurable.Then the multifunction F e" 12 --} 2 BM(O) defined by r() [q Fn (w), for each w (5 f/ n=l is also weakly measurable [10, Theorem 4.1].By the well-known theorem of Kuratowski and Ryll- Nardzewski there is a weakly measurable selector xe" 12 BM(O of Fe.Because of the separability A. KARAMOLEGOS AND D. KRAVVARTIS of X, ze is also measurable when BM(O has the norm topology.For a fixed w ( , there is a subsequence of {xne(w)} (denoted again by {xne(w)} such that .(,)(,), ,oo Besides, we may assume that une(w w ue(w) and vne(w v(w) as n oo.
It is clear that u(w) + re(w) 0 for all w n Let {eS}sCN be a sequence of positive numbers such that es-"0, as --,oo.We set xs(w) z%(w) and () f3 w {(): _> } As before, we deduce that there exists a measurable mapping x: --X such that x(w) 6 R(w) for M1 w ft.For a fixed w fl, there exists a subsequence of {x(w)} (which, we agMn denote by {xs(w)} such that () (), Clearly, us(w) u%(w) u(w) and u() + () 0 for e n As in the deterministic case [12], one can prove that u(w) L(w)x(w) and v(w)= T(w)x(w) So, L(w)x(w) + T(w)x(w) 0 for all co e n i.e., x(. is the desired solution. REMARK.The sumptions on L e stisfied when L is rdom, wetly continuous, mimM monotone d L()0 0 for M1 ft.(In pticul, when L is rdom, line, mimM monotone, with D X).Then, clely, L is wetly closed.In addition, L is jointly meurble.Indd, if V is y element of X, the operator (, x) (L()x, V) is Cthdory function, hence meurble.It follows that L is wetly jointly meurable d, since X* is sepble, L is M jointly meurble.

NONLINEAR RANDOM INEQUALITIES.
The threm which follows gives a rdom version of Threm 4 i [9].THEOREM 4.1.t X be a sepable, reflexive Bch spce, D a convex, closed subset of X with intAff(D)D # O (i.e. the interior of D, relative to the ne spce generated by D, is non- empty).If O: flx D X* is multifunction such that: 1) 0() is non-empty, convex d wetly compact subset of X* for M1 fl d x D.
2) The functionM inf (x*, V), where x D, is lower semicontinuous for MI V D-D d 3) There exists a compact set K C_ D and a point Y0 K with the propert inf (z*, x V0) > 0 for all x D\K and w 4) The graph of the multifunction [ftxK (restriction of to txK) belongs to (R) B(K) (R) B(X*).
Then there exist measurable mappings : fl K and r/: ft X* such that, for all rt(w _ '(w)(w) and (r/(w), (w)-y) < 0 for M1 y fi; D. PROOF.We consider the multifunction F:fK x X* defined by F() {(x, z) 6 K x X*" z 6 q)()z and (z, a"-U) < 0 for all y D} By Theorem 4 of Ricceri [91, we have R() # 0 for all ft.Let {n}n be a sequence of points of D, dense into D. Note that: F() {(, ) x X*.
So GrFneE@B(gxX*) which in turn implies GrF EB(K x X*).By the selection threm of Sainte-Beuve, there exists a meurable mapping h" + K x X* such that h(w) F(w) for M1 w e .If we put h ({, y) it follows that {" K and y" X* e meurable mappings, that y(w) e (w){(w) for all w e d that ((w), {(w)-y)0 forMlyeDdw.
We shM1 nd the concept of sepability for a rdom function " x X R, where X is a separable, metrizable d complete space (cf.Shucha-aeid [7]).
Sepable random functions are chacterized in the following way (for the prf, s Papageorgiou [14]): A random function " x X R is separable, if, d only if, there exists a countable, dense set D g X and a N , (N) 0, such that for w N and for x X, there exists a sequence x n D, such that x n x and (w, xn) (w, x).THEOREM 4.2.Let X be a sepable, reflexive Bach space d K a closed, convex d bounded subset of X.If T" x K + X* is a random, monotone d demicontinuous operator d : x K R is a rdom, convex, lower semicontinuous d sepable function, then for each meurable y" X* there exists a meurable x" K such that (T(w)x(w)-y(w), x(w)-z) (w, z)-(w, x(w)) for M1 z g d for Mmost M1 w e .
By Theorem 3.1 of Papageorgiou [14] the mapping (w, x) (T(w)x, x x,) (w, x,) + p(w, x) from (f\N)K to I is jointly measurable.So, GrRn f\N(R)B(K) and, consequently, Gr R f'l Gr Rn EFt\g (R) B(K).Since the measure space (f\N, Et2\g, #) is complete, applying n--1 Sainte-Beuve's selection theorem, we get measurable z: fl\N K such that z(w) R(w) for all w I\N, i.e., (T()(), ()-Now we prove an existence theorem for random inequalities involving a pseudomonotone operator T and a random, continuous, convex function o.It would be interesting to prove this theorem with the same assumptions on as in the previous theorem.
THEOREM 4.3.If T: Q K X* is a random, pseudomonotone and bounded operator and 0: flx K R is a random, convex and continuous function, then for each measurable y: Q X* th it mm : a X h that (T()()-(), ()-) _< (,, )-(, ()) o all z K and Ft.PROOF.We may assume that y(w)=0 for all t.Let {Xn}nN be an increasing sequence of finite dimensional subspaces of X, whose union is dense in X.We denote in: Xn X, the injection mapping of Xn into X, and "* X* 3n: Xn, its adjoint.We then define Tn" flx Kn X, where Kn K Xn, by Tn(w)x j T(w)jnx.We consider the multifunction Rn(w) {x Kn: (Tn()x, x z) < o(w, z) o(w, x) for all z Kn}.
From Browder [15], we know that Rn(w)# O for all w ft and n N. Let {Zm} m N be a dense sequence into Kn.We have Rn(w) o {x K,: (Tn(w)x, x-zm) <_ p(w, zm)-p(w, x)}.rn=l As in the proof of the previous theorem, we deduce that GrRn E(R) B(Kn) and, by Sainte- Beuve's selection theorem, there exists xn: f-K, measurable, such that xn(w Rn(w), for all w fl, i.e.

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 5. A RANDOM HAMMERSTEIN INTEGRAL EQUATION.

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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