渐近半伪压缩映射的收敛性定理及其应用(英文).docx

渐近半伪压缩映射的收敛性定理及其应用(英文)
ConvergenceTheoremandItsApplicationsofAsymptoticallySemi-Pseudo-ContractiveMaps
Introduction
Inrecentyears,thestudyofasymptoticallysemi-pseudo-contractivemapshasgainedattentionasapowerfultooltoanalyzetheconvergenceofvariousiterativemethodssuchasfixedpointiterations,gradientbasedoptimizationmethods,anddynamicalsystemswithapplicationtomorecomplicatedproblemsinphysics,economics,andotherfields.Inthispaper,wepresentaconvergencetheoremforasymptoticallysemi-pseudo-contractivemapsandprovidesomeapplicationsofthistheorem.
ConvergenceTheorem
LetXbeanon-emptyboundedclosedconvexsubsetofarealHilbertspaceH,andletT:X→Xbeanasymptoticallysemi-pseudo-contractivemap,inthesensethatthereexistsasequence(αn)∈(0,1)suchthat
||Tn+1(x)-Tn(x)||≤αn||Tn(x)-x||,forallx∈Xandforalln∈N.
If(x_n)isanysequencegeneratedbytheiterativeprocessx_{n+1}=T_n(x_n),then(x_n)convergesweaklytoafixedpointofT.
Proof:
LetxbeaweaklyconvergentsequenceinXwithlimitpointa.Thenwehave
T_n(x)↔T(a)asn→∞,andT(a)=abythedefinitionofTasafixedpoint.Thus,
||T_n(x)-a||≤||T_n(x)-T(a)||+||T(a)-a||
≤α_n||x-a||+||T(a)-a||,(bythedefinitionofsemi-pseudo-contractivemap)
whichimpliesthat||T_n(x)-a||isboundedbyasummablesequence,asα_n∈(0,1).BytheBanach-Alaoqlutheorem,wecanobtainasubsequenceofT_n(x)thatconvergesweaklytosome
y∈H.ButT_n(x)waschosenarbitrarily,soanysubsequence(y_{n_k})of(T_n(x))isconvergent.
Since(y_{n_k})convergestoyweakly,andTisasymptoticallysemi-pseudo-contractive,thefollowinginequalityholds:
||y-a||^2=lim_{k→∞}||y_{n_k}-a||^2
≤lim_{k→∞}||T_{n_k}(y_{n_k})-a||^2
≤lim_{k→∞}α_{n_k}^2||y_{n_k}-a||^2,
byapplyingthePythagoreaninequalityforeachtermininequality.Itfollowsthatlim_{k→∞}||y_{n_k}-a||=0,so(T_{n_k}(x))convergestoaweaklyasitssubsequenceconvergestoaweakly.Therefore,wehaveprovedthatanysequencegeneratedbytheiterativeprocessx_{n+1}=T_n(x_n)convergesweaklytoafixedpointofT.
Applications
Asymptoticallysemi-pseudo-contractivemapshavenumerousapplications,especiallyinoptimizationproblems,inverseproblems,andchaoti