12

positive

n

MIECZYSLAW ALTMAN

{r.} and q1

(1~i~n)

with

1

(a)

l::

a.krk~qri(i=1,

... ,n)

k=1

1

conditions for a matrix to satisfy (a) are given by Matkowski [1,2].

THEOREM. If P is closed on D and -PXEfx(P) for all XED with

such that

llh-11· B.(

II

p

II·'·

1 1 - 1 X 1

h={h.(x)}

1

for BiElB;, (i=1, ... ,n), (see (10.3), then Px=O has a solution XED.

15. GENERALIZED DIRECTIONAL CONTRACTIONS. Let X=X

1

x ••• x

Xn and F;:X +X;

be nonlinear. Then F={F;} is a generalized directional contraction if for

arbitrary x,yEX there exist Oe1 such that

n

I!F.(x+ey)-

F.xl!.~e

l::

a.ki!Ykllk'

1 1 1

k= 1

1

for i = 1, ... ,n, where {aik} is a given matrix with property (a) given in

14. If y is replaced by y={Fix-x;}, then F is called a generalized

directional contraction in the narrow sense.

A FIXED POINT THEOREM. A closed generalized directional contraction in the

narrow sensed has a fixed point, i.e., F;x=xi for some x={xi}. The above

fixed point theorem generalizes the fixed point theorem of Matkowski [1,2],

Pavaloiu [1], and Rus [1].

16. THE METHODOLOGY OF PROOFS IN THE THEORY OF CONTRACTOR DIRECTIONS. Four

different methods of proofs are now available in the theory of contractor

directions:

1. The transfinite induction argument.

2. An argument based on the Brezis-Browder principle on ordered sets and its

generalization.

3. The method of interations.

4. The Caristi-Kirk fixed point argument.

REFERENCES

[1] M. Altman, Contractors and Contractor Directions. Theory and

Applications. M. Dekker 1977.

[2] Recent development of the theory of contractors and contractor

directions and its applications, International Conference on Nonlinear Analysis

and Applications, Memorial University of Newfoundland, St. John's, Newfound-

land, Canada, 1981. Ed. S.P. Singh.

positive

n

MIECZYSLAW ALTMAN

{r.} and q1

(1~i~n)

with

1

(a)

l::

a.krk~qri(i=1,

... ,n)

k=1

1

conditions for a matrix to satisfy (a) are given by Matkowski [1,2].

THEOREM. If P is closed on D and -PXEfx(P) for all XED with

such that

llh-11· B.(

II

p

II·'·

1 1 - 1 X 1

h={h.(x)}

1

for BiElB;, (i=1, ... ,n), (see (10.3), then Px=O has a solution XED.

15. GENERALIZED DIRECTIONAL CONTRACTIONS. Let X=X

1

x ••• x

Xn and F;:X +X;

be nonlinear. Then F={F;} is a generalized directional contraction if for

arbitrary x,yEX there exist Oe1 such that

n

I!F.(x+ey)-

F.xl!.~e

l::

a.ki!Ykllk'

1 1 1

k= 1

1

for i = 1, ... ,n, where {aik} is a given matrix with property (a) given in

14. If y is replaced by y={Fix-x;}, then F is called a generalized

directional contraction in the narrow sense.

A FIXED POINT THEOREM. A closed generalized directional contraction in the

narrow sensed has a fixed point, i.e., F;x=xi for some x={xi}. The above

fixed point theorem generalizes the fixed point theorem of Matkowski [1,2],

Pavaloiu [1], and Rus [1].

16. THE METHODOLOGY OF PROOFS IN THE THEORY OF CONTRACTOR DIRECTIONS. Four

different methods of proofs are now available in the theory of contractor

directions:

1. The transfinite induction argument.

2. An argument based on the Brezis-Browder principle on ordered sets and its

generalization.

3. The method of interations.

4. The Caristi-Kirk fixed point argument.

REFERENCES

[1] M. Altman, Contractors and Contractor Directions. Theory and

Applications. M. Dekker 1977.

[2] Recent development of the theory of contractors and contractor

directions and its applications, International Conference on Nonlinear Analysis

and Applications, Memorial University of Newfoundland, St. John's, Newfound-

land, Canada, 1981. Ed. S.P. Singh.