Sentence examples for applied propositions from inspiring English sources

In the first inequality, we used Hölder's inequality, in the second inequality, we applied Propositions 3.1 and 3.2 in [1], and in the last one, we used condition (1.5).

(II) In 2004, Guo [33] applied Proposition 6.2 to prove existence theorems of solutions of the complementarity problems for multivalued monotone operator in Banach spaces.

(II) In 1998, Park and Kum [17] applied Proposition 5.1 with A = B to a general existence theorem on a generalized variational inequalities and some of its consequences.

(III) In 2000, Lee and Kum [18] applied Proposition 5.1 to several existence theorems of solutions of implicit vector variational inequalities for multimaps with or without generalized pseudomonotonicity.

(V) In 2002, Kum and Lee [20] applied Proposition 5.1 to some existence of solutions of the implicit vector variational inequality problem (IVVI) for noncompact-valued multimaps.

In 2004, Guo [33] applied Proposition 6.2 to prove existence theorems of solutions of the complementarity problems for multivalued monotone operator in Banach spaces.

In order to prove some existence results for noncompact settings, many authors applied Proposition 5.1 as follows: (I) In 1998, Lee and Kum [16] applied Proposition 5.1 to obtain two theorems on the existence of solutions of VVI problems with a noncompact setting in a Hausdorff topological vector space.

In 1998, Lee and Kum [16] applied Proposition 5.1 to obtain two theorems on the existence of solutions of VVI problems with a noncompact setting in a Hausdorff topological vector space.

In the first inequality we used Hölder's inequality, and in the second inequality we applied Proposition 3.2 in [33], and in the last one we used condition (1.6).

(VII) In 2005, Kum and Kim [22] applied Proposition 5.1 to develop the scheme of vector variational inequalities with operator solutions due to Domokos and Kolumbán from the single-valued case into the multi-valued one.

(VI) In 2002, Lin and Cheng [21] applied Proposition 5.1 to the existence results of two types of equilibrium problems--the constrained or the competitive Nash type equilibrium problems with multivalued payoff functions.