| In this paper, we study the existence and ergodicity on invariant measures of set-valued mappings. Our study bases on following environment and conditions:H-1: The measurable set-valued mapping F : X - CC(X) is e-bounded almost lower semicontinuos (a.l.s.c.) where X is a closed, bounded and convex subset of a n-dimension Banach space (n > 1).H-2: The measurable set-valued mapping F : X - CC(X) is -bounded (n+1)-lower semicontinuos where X is a closed, bounded and convex subset of a n-dimension Banach space (n > 1).H- 3: X is a closed, bounded and convex subset of a 1-dimension normed linear space, and the measurable set-valued mapping F : X - CC(X) is 2-lower semicontinuous and has closed, bounded and convex images.H- 4: X is a compact metric space, and the measurable set-valued mapping F : X - 2X is weakly lower semicontinuous and has closed and convex images.H- 5: X is a compact metric space, and the set-valued mapping F : X -2X is lower semicontinuous and has closed and convex images.H-6: X is a 1-dimension normed linear space, and the measurable set-valued mapping F : X - Cbc(X) is 2-lower semicontinuous and has closed, bounded and convex images. Further, there exists a sequence of {an} C .M1 and a measure M1 with n = 1/2 Pin - P- for some P PF.H- 7: X is a Polish space, and the measurable set-valued mapping F : X - 2X is weakly lower semicontinuous and has closed and convex images. Further, there exists a sequence of {an} M1 and a measure 6 M1 with n = 1/2 for some P PF.H-8: A" is a Polish space, and the measurable set-valued mapping F : X - 2X is lower semicontinuous and has closed and convex images. Further, there exists a sequence of {an} c M\ and a measure p, G M.\ with //?= 1/2 Pvn for some F PF.Firstly, we apply the definition of Markov invariant measure on set-valued mappings and the equivalent property between it and Aubin's definition on set-valued mappings (Theorem A in [15]), and prove that the set-valued mappings have invariant measures . We sum up the results as follows:1) If one of the assumptions H-1-H- 3 holds, then(1) The mapping -F has invariant measures in M2(A", F);(2) There are minimal invariant measures of M(X,F) in M2(X,F);(3) TF 0 has both minimal elements and the greatest one, and for each P e PF, MP 0.2) If the assumption H- 4 holds, then(1) The mapping F has invariant measures in M2(X, F);(2) TF 0 has the greatest element, and for each P e PF, Mp 0. When the assumption H- 5 is satisfied, the following (3) also holds besides(1) and (2),(3) The case M(A,F) has the greatest element in M2(A, F) . 3) If one of the assumptions H- 6 - H- 8 holds, then(1) TF 0 has both minimal elements and the greatest one, and for each P PF, MP 0.(2) The mapping F has invariant measures HQ 6 M2(A", F), further fj,Q 6In literature [1], the existence theorem of invariant measures on set- valued mappings was established under the assumption that X is compact metric space and F has a closed graph. Our conditions on the existence on invariant measures of set-valued mappings are different from them. The paper [16] studied the existence on invariant measures of set- valued mappings which are lower sernicontinuous with closed, bounded and convex images on a compact Banach space. H- 1 and H- 2 are different type of conditions and H- 3 - H- 8 are weaker than them.After the discussion of the existence on invariant measures of set- valued mappings we discuss the ergodicity on invariant measures of set-valued mappings. Our work fill up that vacancy in this respect, as follows:1. We originally give the definition of ergodic measures on set-valued mappings, that is: let (X,B(X),n) is a probability space, \i is an invariant measure of the set- valued mapping F : X - Y 1X with closed images, if the only members B ?B(X) with F(B) C B satisfy //(B) = 0 or /j(B) = 1, then F is called ergodic for //.2. We build other ways of stating the ergodicity condition as following : (1) and (2) are both equivalent to the case that F is ergodic. (3) is...
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