Applications of Coupled xed points for multivalued mappings in cone metric spaces

In this article we present a variant of the basic result of "T. Gnana. Bhaskar and V. Lakshmikanthan, Fixed point theorem in partially ordered metric spaces and applications, Nonlinear Anal. TMA, 65 (2006), 1379–1393." to …nd the coupled …xed points of multivalued mappings without mixed monotone property in cone metric space involving non-normal cones. As applications we prove the existence of certain type of non-linear Fredholm integral equations in two variables and present a non-trivial example. In this way we provide a gateway to work in this applicable theory of coupled …xed and coincidence points. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2010 Mathematics Subject Classi…cation: 47H10; 54H25, Keywords and Phrases: ordered cone metric space; multi-valued mappings; coupled …xed points; partial di¤erential equations 1 Introduction In [1] Guo and Lakshamikantham proved the …rst important result for coupled …xed points of nonlinear operators and presented some applications in di¤erential equations. After that in [2] Bhaskar and Lakshamikantham prove the following; Thoerem[2] Let F : X X ! X be a continuous mapping having the mixed monotone property on X: Assume that there exists a k 2 [0; 1) with d (F (x; y) ; F (u; v)) k 2 [d(x; u) + d(y; v)]; for all x u; y v: If there exists x0; y0 2 X such that x0 F (x0; y0) and y F (y0; x0): Then there exist x; y 2 X such that x = F (x; y) and y = F (y; x): The coupled …xed point theory has many applications in the existence theory of many types of operators. Many authors divert their e¤orts in this direction and generalize and present many variants of the above result in many directions. A mutivalued result is presented in [3] by Samet and Vetro. Some researchers proved coupled …xed point results without monotone property (see [4-9]). In [10] Samet et al. show that