The Solutions For Initial Value Problems Of Second Order Integro-Differential Equations Of Mixed Type In Banach Spaces

In this paper, the following initial value problems(IVP)for second orderintegro-differential equations of mixed type in Banach space are considered:u ′′ (t )=f(t,u(t),u′(t),(Tu)(t),(Su)(t)), (2.1)u ( 0)= u0 ,u′(0)=u1。 (2.2)wheret ∈ J=[0,1], f ∈ C[ J×E×E×E×E,E],wheret ∈ J=[0,1], f ∈ C[ J×E×E×E×E,E],k ∈ C[ D,R⁺],h ∈ C[ J×J,R⁺],D = {( t,s)∈J×J:t≥s},The cone P induces a partial ordering on E.Suppose that the following assumptions are satisfied:( H₁)there exist v₀, w₀ ∈ C¹[ J,E]∩C₂[J,E],with v₀ (t )≤ w₀(t),andv₀′ (t )≤w₀′(t), ?t ∈J。Where v₀ (t), w₀ (t) are the lower and upper solutions ofthe IVP(2.1)—(2.2) and satisfyv₀′′ ≤f( t,v₀,v₀′,Tv₀,Sv₀), ?t ∈J,v 0 ( 0)≤ u0, v 0′ ( 0)≤u1。w 0′′ ≥f(t ,w0,w0′,Tw0,Sw0), ?t ∈J,w 0 ( 0)≥ u0, w 0′ ( 0)≥u1。( H 2)there exist M ≥0, M 1 >0, N >0 such thatf (t ,u,v,w,z)? f(t,u,v,w,z)≥?M(u?u)?M1 (v?v)?N(w?w), ?t ∈J。wherev 0 (t )≤ u≤u≤w0(t), v 0′ (t )≤v≤v≤w0′(t),(T v 0 )(t)≤ w≤w≤(Tw0)(t), ( Sv 0 )(t)≤ z≤z≤(Sw0)(t),( H 3)there exist constants ci ≥ 0, (i=1,2,3,4),such thatα ( f (J,V1 ,V2,V3,V4))≤ c1α(V1)+c2α(V2)+c3α(V3)+c4α(V4),where Vi ? E is bounded, i =1,2,3,4, α is the Kuratowski's measure ofnoncompactness of a bounded subset in E。We consider the situation that the lower solution is less than the uppersolution。First we prove a comparison result。 Then by using the theory of coneand the monotone iterative technique,we have obtained the existence ofminimal and maximal solutions of IVP(2.1)—(2.2)with the noncompactcondition。The main conclusions are as follows:Theorem 3.1 Assume that E is a real Banach space,P is a normal cone,and( H 1, H 2 , H 3) are satisfied。Further4( c1 + 2c2+c3k0+c4h0+2Nk0+2M+M1)<1,KtkteMt1( )≤ 0 ()?,N M′≤1。wherek 0 (t)∈ C[J,R+],and ()1N ∫0 t k0 sds≤MM1<;∫M ′ =mt∈ [ 0a,1x] 0t k*(t,s)ds,and * ( ,)1()((,))ktrdrNktseMtsMts= ? +∫。Then IVP(2.1)-(2.2) exists the minimal solution u (t) and maximal solutionu * (t) in [ v 0 ,w0],which is included in C 2 [J,E]。Furthermore ,letv tFtt KitsFnsdsiin ( )n1 ()1(1)0(,)?1()∞?== +∑ ?∫, ?t ∈J;w tGtt KitsGnsdsiin ( )n1 ()1(1)0(,)?1()∞?== +∑ ?∫, ?t ∈J。whereK 1 (t ,s)= M(t?s)+M1+N∫s t (t?r)k(r,s)dr,K i (t ,s)= ∫s t K1 (t,r)Ki?1(r,s)dr,?(s,t)∈D,(i=2,3),∫????????+?′???=++tnnnnnnnntsfsvsvsTvsSvsMvsNTvsMvsdsFtuMuut0 1111111110101[()((,(),(),()(),()())()()())()]()()∫????????+?′???=++tnnnnnnnntsfswswsTwsSwsMwsNTwsMwsdsGtuMuut0 1111111110101[()((,(),(),()(),()())()()())()]()()And the monotone sequences {v n (t)}, {w n (t)} such that nl ?im >∞ vn(t)=u(t),nl i?m >∞ wn(t)=u* (t) uniformly on J , where u (t), u * (t) are the minimal andmaximal solutions of IVP(2.1)-(2.2)in [ v 0 ,w0]。Moreover u (t) and u * (t)satisfyv 0 (t )≤ v1(t)≤≤vn (t)≤≤u(t)≤≤u*(t)≤wn(t)≤≤w1(t)≤w0(t), ?t ∈J。Theorem 3.2 Assume that E is a real Banach space ,P is a regularcone ,and ( H 1, H 2) are satisfied 。FurtherKtkteMt1( )≤ 0 ()?,N M′≤1,wherek 0 (t)∈ C[J,R+], and ()1N ∫0 t k0 sds≤MM1<;∫M ′ =mt∈ [ 0a,1x] 0t k*(t,s)ds,and * ( ,)1()((,))ktrdrNktseMtsMts= ? +∫。and for any r >0, the set f ( J,Br ,Br,Br,Br)is bounded , whereBr = { x∈E:x≤r},then the conclusion of the theorem 3.1 is also correct。...