Solutions to Second Order Differential Inclusions in Banach Spaces

The purpose of this paper is to study the existence of solutions to second order differential inclusions in Banach spaces, the problems discussed are as follows (1) if multifunction F satisfies some condition as (H1) ∃ M>0,∀ x1,x2,y∈v0,w0,x1≤x2,∀ u1∈F(t,x1,y),∀ u2∈F(t,x2,y), we have u2-u1≥-M(x2-x1) (H2) there exists const L,M>L≥0, for any x,∀ y1,y2∈v0,w0,y1≤y2,∀ v1∈F(t,x,y1),∀ v2∈F(t,x,y2), we have v1-v2≥-L(y2-y1) (H3) there exists const K,M> K> L,for any x,y∈v0,w0,x≤y,∀ u∈F(t,x,y),∀ v∈F(t,y,x), we have u-v≥(K+L)(y-x) (H4) for ∀ uF∈F(t,v0,w0),∀ vF∈F(t,w0,v0), we have -vő+L(w0-v0)≤uF, v0(0)≥v0(1)=θ -wő-L(w0-v0)≥vF, w0(0)≤wo(1)=θ we get the result that there is a solution in v0,w0 to inclusion (1). The resluts presented in this paper generalize results in and simplify the proof to.