The Study On Solvability Of Two Classes Of Second Order Differential Inclusion Problems

In this dissertation,we study the existence of positive solutions,global structure and nodal solutions of second order differential inclusion problems with periodic boundary conditions and Dirichlet boundary conditions respectively by using the fixed point theory on cones of set-valued maps and bifurcation theory.The main results are described as follows:1.We use the fixed point theory on cones of set-valued maps to study the exis-tence of positive solutions for second order differential inclusion periodic boundary value problem Where q ? C([0,2?],[0,?))is a periodic function of 2? and q(?)0,t ?[0,27?],F:[0,2?]× R?2R\(?)is a multi-valued mapping.When the nonlinearity F is single,the problem degenerated into the problem studied by Graef et al.in[Appl.Math.Lett.,2008],therefore the results obtained supplement their work.2.We consider the global structure of the positive solutions set for differential inclusion problem Where ?>0 is a constant,F:[0,2?]×R?2R\(?)is a multi-valued mapping.Firstly,we obtain the branch of the positive solution from a simple eigenvalue through Rabinowitz global bifurcation theory.Then by virtue of the idea of homotopy and bifurcation theory of differential inclusion,thus we determine the direction of the branch of positive solutions.Finally,it is proved that the component is unbounded.3.We use the bifurcation theory to study the nodal solutions for second order differential inclusion problem with Dirichlet boundary conditions Where the nonlinearity F is discontinuous at u=0,k?C1([0,1],(0,?)),g ?C([0,1]× R,R).The main difficulties to be overcome in this section are,the nonlinearity F is discontinuous at u=0,resulting in the differential operator cannot be transformed into the corresponding integral operator,and the bifurcation theory cannot be di-rectly applied.Therefore,we construct an auxiliary problem,and use a family of functions {fl}l?N to approximate F.Then,the Krein-Rutman theory is applied to the corresponding auxiliary problems,and the unbounded connectivity of the solutions of the two columns Cn,l± is obtained for each fixed l.Finally,the unbounded connect-ed branch Cn± of the problems is obtained by using the method of taking limit of connected branches in Ma et al.[Nonlinear.Anal.,2009].The issues considered in this part are compared to the work studied by Ma et al.in[Nonlinear.Anal.,2004],the nonlinearity is allowed to have discontinuities,so the problem is considered more widely.