The Solvability of a New System of Nonlinear Variational-Like Inclusions

• Main Authors: Ume JeongSheok, Kang ShinMin, Liu Zeqing, Liu Min
• Format: Article
• Language: English
• Published: SpringerOpen 2009-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2009/609353

<p/> <p>We introduce and study a new system of nonlinear variational-like inclusions involving <inline-formula> <graphic file="1687-1812-2009-609353-i1.gif"/></inline-formula>-<inline-formula> <graphic file="1687-1812-2009-609353-i2.gif"/></inline-formula>-maximal monotone operators, strongly monotone operators, <inline-formula> <graphic file="1687-1812-2009-609353-i3.gif"/></inline-formula>-strongly monotone operators, relaxed monotone operators, cocoercive operators, <inline-formula> <graphic file="1687-1812-2009-609353-i4.gif"/></inline-formula>-relaxed cocoercive operators, <inline-formula> <graphic file="1687-1812-2009-609353-i5.gif"/></inline-formula>-<inline-formula> <graphic file="1687-1812-2009-609353-i6.gif"/></inline-formula>-relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique associated with <inline-formula> <graphic file="1687-1812-2009-609353-i7.gif"/></inline-formula>-<inline-formula> <graphic file="1687-1812-2009-609353-i8.gif"/></inline-formula>-maximal monotone operators and Banach contraction principle, we demonstrate the existence and uniqueness of solution for the system of nonlinear variational-like inclusions. The results presented in the paper improve and extend some known results in the literature.</p>