Generalized Vector Equilibrium-Like Problems without Pseudomonotonicity in Banach Spaces

• Main Authors: Guu Sy-Ming, Yao Jen-Chih, Ceng Lu-Chuan
• Format: Article
• Language: English
• Published: SpringerOpen 2007-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2007/061794

<p/> <p>Let <inline-formula><graphic file="1029-242X-2007-061794-i1.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2007-061794-i2.gif"/></inline-formula> be real Banach spaces, <inline-formula><graphic file="1029-242X-2007-061794-i3.gif"/></inline-formula> a nonempty closed convex subset of <inline-formula><graphic file="1029-242X-2007-061794-i4.gif"/></inline-formula>, and <inline-formula><graphic file="1029-242X-2007-061794-i5.gif"/></inline-formula> a multifunction such that for each <inline-formula><graphic file="1029-242X-2007-061794-i6.gif"/></inline-formula> is a proper, closed and convex cone with <inline-formula><graphic file="1029-242X-2007-061794-i7.gif"/></inline-formula>, where <inline-formula><graphic file="1029-242X-2007-061794-i8.gif"/></inline-formula> denotes the interior of <inline-formula><graphic file="1029-242X-2007-061794-i9.gif"/></inline-formula>. Given the mappings <inline-formula><graphic file="1029-242X-2007-061794-i10.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2007-061794-i11.gif"/></inline-formula>, and <inline-formula><graphic file="1029-242X-2007-061794-i12.gif"/></inline-formula>, we study the generalized vector equilibrium-like problem: find <inline-formula><graphic file="1029-242X-2007-061794-i13.gif"/></inline-formula> such that <inline-formula><graphic file="1029-242X-2007-061794-i14.gif"/></inline-formula> for all <inline-formula><graphic file="1029-242X-2007-061794-i15.gif"/></inline-formula> for some <inline-formula><graphic file="1029-242X-2007-061794-i16.gif"/></inline-formula>. By using the KKM technique and the well-known Nadler result, we prove some existence theorems of solutions for this class of generalized vector equilibrium-like problems. Furthermore, these existence theorems can be applied to derive some existence results of solutions for the generalized vector variational-like inequalities. It is worth pointing out that there are no assumptions of pseudomonotonicity in our existence results.</p>