A mapping is called - -convex (or - -concave) with respect to in second argument if, for any nonempty finite subset,,,, with and, there exists such that.
(1 For any, is open; (2) is closed; (3) is lsc; (4 for all and, ; (5)for all, is -concave with respect to in second argument; (6 for all, is lsc; (7) is usc, where for all.
If is a - -convex mapping (or a - -concave mapping) with respect to in second argument, then for any, is a -convex mapping (or a -concave mapping) with respect to.
A trimapping is generalized pseudo-contractive with respect to map in first argument of with constant and generalized -pseudo-contractive wito respect to in the second argument of with constant, Lipschitz continuous with respect to first, second, and third arguments with positive constants, respectively.
Firstly, following Singh and Jain [14], let be the set of all real continuous functions, non decreasing in first argument, and satisfying the following conditions: (i for,, or implies that, (ii) implies that.
Let be -strongly monotone and -Lipschitz continuous, and let be a -Lipschitz continuous random operator, and let be -strongly accretive with respect to and -Lipschitz continuous in the first argument, and -Lipschitz continuous in the second argument, -Lipschitz continuous in the third argument, respectively.
Let be a -Lipschitz continuous random operator and be -Lipschitz continuous in the first argument, -Lipschitz continuous in the second argument and -Lipschitz continuous in the third argument, respectively.
Theorem 2.1 Let h be positively homogeneous in the second argument, η be linear in the first argument and η ( x, x ) = 0, ∀ x ∈ K.
For ∖ let be all the same as in Theorem 3.2, be -expanding, be Lipschitz continuous in the first, second and third arguments with constants respectively, and be -relaxed cocoercive with respect to in the first argument, be -cocoercive with respect to in the second argument, be -relaxed Lipschtz with respect to in the third argument.
For ∖ let be all the same as in Theorem 3.2, be Lipschitz continuous in the first, second and third arguments with constants respectively, and be -relaxed cocoercive with respect to in the first argument, be -relaxed Lipschitz with respect to in the second argument, be -relaxed monotone with respect to in the third argument.
Clearly, F is g.h.c. in the first argument and l.s.c. in the second argument.
