Appl. Math. Left. Vol. 4, No. 6, pp. 73-75, 1991 0893-9659191 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright@ 1991 Pergamon Press plc

Phi-Pseudo-Monotonicity and Approximation-Solvability of Nonlinear Equations

RAM VERMA AND LOKENATH DEBNATH

Department of Mathematics, University of Central Florida

(Received June 1991)

Abstract. We generalize the results on the approximation-solvability of nonlinear functional equations to the caSe of O-pseudo-monotone mappings-a new and rather general class of mappings. An application is considered.

1. INTRODUCTION

Petryshyn [l] considered an upgraded approximation scheme, and applied it to the general approximation-solvability of nonlinear equations in general normed spaces. This scheme is especially more application-oriented toward the numerical methods in the sense that we can approximate solutions of an equation in some function space X by solutions of equations in some discrete space, say X, = R”. For example, one can consider an integral equation

I l x(t) = qt, s) x(s) ds + j(t)

on C([O, 11) with k E C([O, l] x [0, l]),’ and apply the convergent mechanical quadrature to

approximate scr y(s) ds, that is, we write

I 1

0 2 anj Y(%j) i- m(Y), j=l

with coefficients a,j 1 0, interpolation points 0 5 s,r < . . . < s,,,, < 1 and the remainder r,, such that r,,(y) 4 0 on C([O,l]). Th us, the integral equation is approximated by the following discrete system:

C*i = e a,j k(S,;, Snj) Cnj + f(S,i)

j=l

for i = 1,. . . ,7I,

obtained by neglecting T, and taking t = s,i for i = 1,. . . , n. Our motivation of this paper is two-fold. First, we generalize the results on approximation-

solvability, by applying the theorem of Petryshyn [l] to the case of @-pseudomonotone mappings-a new class of mappings. Second, we give an application to the numerical range- a generalization of the Zarantonello numerical range [2].

DEFINITION. Let X and Y be complex Banach spaces, and 0 : X -+ Y be such that

(i) Q(X) is dense in Y; and (ii) for each x E X and each t 1 0, @(ix) = t O(x) and 1(9x1( _< [[xl/.

A mapping A : X + Y’ from X to Y*, the dual of Y, is said to be Q-pseudo-monotone if there exists a constant d > 0 such that

I[Ax - Au, @(x - u)]I 2 d 1(x - I# for all x, 21 E X,

where [., .] is the pairing between Y’ and Y.

When Y = X (reflexive) and Q = I, A is pseudo-monotone [3].

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73

74 R. VERMA, L. DEBNATH

2. APPROXIMATION-SOLVABILITY

Let X be a separable reflexive complex Banach space with dimX = 00, and A : X -+ X’ from X to its dual X’ be @-pseudo-monotone and continuous. Let (X,) be a Galerkin scheme in X such that

X, = span(el,, . . . , eb), n= 1,2,... .

Let E,, : X,, -+ X be the embedding mapping corresponding to X,, c X. We construct R,, : X + X, in such a manner that, for each x E X, there exists at least an element R,x such that

[Ix - R,,xll = dist(x,X,,).

We consider the equation

Ax = b, x E x, bEX*, (1)

along with approximate equations

E;AE,,x, = E;;b, % E X,, n = 1,2,. . . , (2)

with respect to an approximation scheme T = {X,, O,, E,,, R,,XA, Ei}, represented by a diagram

x2x A

- x*

En T R, E. IT 1 E:, (3)

AlI x, 5 x, - x;

where A ,, = Ei A E,, and a,, = R,@ E,, are continuous. For more details on approximation- solvability, see [ 1,4-71.

Now we discuss our main result on the approximation-solvability.

THEOREM 2.1. Let X be a separable reffetive complex Banach space with dim X = CO. Let T = {X,, O,, E,,, R,, , Xl:, E;} be an approximation scheme for the pair (X, X’) represented by the diagram (3), and let @ : X 2 X be continuous such that Q(u + V) = Ou + Cpv for all u, v E X, and

R,@E,,x=@x for all I E X, and each n.

If A : X -* X’ is cP-pseudc+monotone and continuous, then, for each b E X’, the equation

Ax = b, 2 E x,

is uniquely approximation-solvable. For 0 = I, we arrive at the following corollary:

COROLLARY 2.2. If X is a separable reflexive complex Banach space with dim X = 00, TO = {X,, E,,, R,, XA, Ez} is an approximation scheme for the pair (X,X*), and A : X + X’ is pseudc+monotone and continuous, then, for each b E X”, the equation

Ax=b,xEX,

is uniquely approximation-solvable.

Monotonicity and solvability of nonlinear equations 75

3. APPLICATION TO NUMERICAL RANGE

In this section, we consider the approximation-solvability of the equations involving nu- merical ranges.

DEFINITION (NUMERICAL RANGE). Let A : X - X* be a mapping from a separable reflexive complex Banach space X to its dual X’. The set

is called the numerical range of A, where @ : X -+ X is continuous such that

(i) Q(X) is dense in X; and (ii) for each x E X and each t 1 O,@(tx) = t@(x) and II@zll 5 ((x((.

THEOREM 3.1. Let X be a separable reflexive real Banach space and X’ its dual with dimX = 00. If A : X + X’ is continuous, B : X -+ X* is continuous such that

[Bu, @u] = 11~11~ and [Bu - Bv, @(u - v)] = llzl - ~11~ for all u v E X,

and if the number A E K (field) is at a positive distance from the numerical range ofA, no[A], i.e., d = dist(X, no[A]) > 0, then, for each b E X’, the equation

Ax-XBx=b,xEX,

is uniquely approximation-solvable.

COROLLARY 3.2. When X is a Hilbert space, and Q = I, Theorem 3.1 reduces to a gener- alization ([6], Theorem 2.2) of the theorem ([7], Theorem 34.C).

The proofs will appear in the expanded version of the paper.

REFERENCES

1. W. Petryshyn, Nonlinear equations involving noncompact operators, Nonlinear Fvnclional Analysis, Volume 18 (Presented at the Proc. Sympos. Pure Math., Chicago, 1968), pp. 206-233, Amer. Math. Sot., Providence, (1970).

2. E. ZarantoneIlo, The closure of the numerical range contains the spectrum, Bull. Amer. Math. Sec. 70, 781-787 (1964).

3. M.M. Vainberg, Variational Method and Method for Monotone Operators in ihe Theory of Nonlinear

Equa2iona, John Wiley and Sons, New York, (1973). 4. K. DeimIing, Nonlinear Functional Analysis, Springer-Verlag, New York, (1985). 5. I.G. Rosen, Convergence of Gale&n approximation for operators Riccati equations: A nonlinear ev*

Iution equation approach, J. Math. Anal. Appl. 155, 226-248 (1991). 6. R. Verma and R. Mohapatra, Application of numerical range to approximation-solvability of nonlinear

functional equations, Pan American Math. J. 1, 46-54 (1991). 7. E. Zeidler, Nonlinear Functional Analysis and Ils Applications II/B, Springer-Verlag, New York,

(1990). 8. R. Verma, Joint spatial numerical range of tensor products, Proc. Royal Irish Acad. 90A, 201-204

(1990). 9. R. Verma and L. Debnath, On joint numerical status and tensor products, Internat. J. Math. E4 Math.

Sci. 13, 807-811. (1990).

Department of Mathematics, University of Central Florida, Orlando, FL 32816