ON THE ITERATIVE APPROXIMATION PROBLEM OF FIXED POINTS FOR ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒｗｅａｓｓｕｍｅｔｈａｔＥｉｓａｒｅａｌＢａｎａｃｈｓｐａｃｅ ,Ｅ ｉｓｔｈｅｄｕａｌｓｐａｃｅｏｆＥ ,ＤｉｓａｎｏｎｅｍｐｔｙｓｕｂｓｅｔｏｆＥａｎｄＪ:Ｅ → 2 Ｅ ｉｓｔｈｅｎｏｒｍａｌｉｚｅｄｄｕａｌｉｔｙｍａｐｐｉｎｇｄｅｆｉｎｅｄｂｙＪ(ｘ) =ｆ∈Ｅ :〈ｘ ,ｆ〉 =‖ｘ‖·‖ｆ‖ ,　‖ｆ‖ =‖ｘ‖ (ｘ∈Ｅ) .　　Ｄｄｆｉｎｉｔｉｏｎ 1 1　ＬｅｔＴ :Ｄ →Ｄｂｅａｍａｐｐｉｎｇ1 )Ｔｉｓｓａｉｄｔｏｂｅａｓｙｍｐｔｏｔｉｃａｌｌｙｎｏ…     

Ishikawa Iterative Process in Uniformly Smooth Banach Spaces

ＬｅｔＥｂｅａｕｎｉｆｏｒｍｌｙｓｍｏｏｔｈＢａｎａｃｈｓｐａｃｅ ,ＫｂｅａｎｏｎｅｍｐｔｙｃｌｏｓｅｄｃｏｎｖｅｘｓｕｂｓｅｔｏｆＥ ,ａｎｄｓｕｐｐｏｓｅＴ :Ｋ→ＫｉｓａｃｏｎｔｉｎｕｏｕｓΦ＿ｓｔｒｏｎｇｌｙｐｓｅｕｄｏｃｏｎｔｒａｃｔｉｖｅｏｐｅｒａｔｏｒ.ＤｅｎｏｔｅｔｈｅｄｕａｌｓｐａｃｅｏｆＥｂｙＥ .ＷｅｄｅｎｏｔｅｂｙＪｔｈｅｄｕａｌｉｔｙｍａｐｆｒｏｍＥｔｏ 2 Ｅ ｄｅｆｉｎｅｄｂｙＪ(ｘ) =ｆ∈Ｅ :〈ｘ ,ｆ〉=‖ｘ‖2 =‖ｆ‖2 . ( 1 )Ｉｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｉｆＥｉｓｕ…     

Iterative Process for Certain Nonlinear Mappings with Lipschitz Condition

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｅｌｉｍｉｎａｒｉｅｓＴｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒ,ｗｅａｓｓｕｍｅｔｈａｔＸｉｓａｒｅａｌＢａｎａｃｈｓｐａｃｅａｎｄＸ ｉｓｔｈｅｄｕａｌｓｐａｃｅｏｆＸ ,〈· ,·〉ｄｅｎｏｔｅｓｔｈｅｐａｉｒｉｎｇｏｆＸａｎｄＸ .ＴｈｅｍａｐｐｉｎｇＪ:Ｘ → 2 Ｘ ｄｅｆｉｎｅｄｂｙＪ(ｘ) =ｊ∈Ｘ :〈ｘ ,ｊ〉=‖ｘ‖·‖ｊ‖ ,‖ｊ‖ =‖ｘ‖ ,　　ｘ∈Ｘｉｓｃａｌｌｅｄｔｈｅｎｏｒｍａｌｉｚｅｄｄｕａｌｉｔｙｍａｐｐｉｎｇ .Ｄｅｆｉｎｉｔｉｏｎ 1 1　ＬｅｔＸｂｅａ…     

ISHIKAWA ITERATIVE PROCESS FOR CONSTRUCTING SOLUTIONS OF m-ACCRETIVE OPERATOR EQUATIONS

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｅｌｉｍｉｎａｒｉｅｓＬｅｔＸｂｅａｒｅａｌＢａｎａｃｈｓｐａｃｅｗｉｔｈｔｈｅｎｏｒｍ‖·‖ ,Ｘ ｂｅｔｈｅｄｕａｌｓｐａｃｅｏｆＸ ,ａｎｄ〈· ,·〉ｂｅｔｈｅｇｅｎｅｒａｌｉｚｅｄｄｕａｌｉｔｙｐａｉｒｉｎｇｂｅｔｗｅｅｎＸａｎｄＸ .ＴｈｅｎｏｒｍａｌｉｚｅｄｄｕａｌｉｔｙｍａｐｐｉｎｇＪ:Ｘ →2 Ｘ ｉｓｄｅｆｉｎｅｄｂｙＪ(ｘ) =ｆ∈Ｘ :〈ｘ ,ｆ〉=‖ｘ‖2 =‖ｆ‖2 ,　　ｘ∈Ｘ .　　ＲｅｃａｌｌｔｈａｔｔｈｅｍｏｄｕｌｕｓρＸ( ·)ｏｆｓｍｏｏｔｈｎｅｓｓｏ…     

Existence and iterative algorithms of solutions for generalized mixed quasi-variational inequalities

１　ＰｒｏｂｌｅｍｓＬｅｔＨｂｅａｒｅａｌＨｉｌｂｅｒｔｓｐａｃｅｗｉｔｈｎｏｒｍａｎｄｉｎｎｅｒｐｒｏｄｕｃｔｓｄｅｎｏｔｅｄｂｙ‖·‖ａｎｄ〈·,·〉ｒｅｓｐｅｃｔｉｖｅｌｙ．ＬｅｔＫｂｅａｃｌｏｓｅｄｃｏｎｖｅｘｓｕｂｓｅｔｉｎＨａｎｄｆ：Ｈ→Ｈｂｅａｓｉｎｇｌｅｖａｌｕｅｄｍａｐｐｉｎｇ．Ｔｈｅｃｌａｓｓｉｃａｌｖａｒｉａｔｉｏｎａｌｉｎｅｑｕａｌｉｔｙｐｒｏｂｌｅｍ（ＶＩＰ（ｆ,Ｋ））ｉｓｔｏｆｉｎｄ＾ｘ∈Ｋ（＾ｘ）,ｓｕｃｈｔｈａｔ〈ｆ（＾ｘ）,ｙ－＾ｘ〉≥０　　（ｙ∈Ｋ）．（１）…     

REMARK ON STABILITY OF ISHIKAWA ITERATIVE PROCEDURES

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｅｌｉｍｉｎａｒｉｅｓＳｕｐｐｏｓｅＥｉｓａｒｅａｌＢａｎａｃｈｓｐａｃｅａｎｄＴｉｓａｓｅｌｆｍａｐｏｆＥ .Ｓｕｐｐｏｓｅｘ0 ∈Ｅａｎｄｘｎ+1=ｆ(Ｔ ,ｘｎ)ｄｅｆｉｎｅｓａｎｉｔｅｒａｔｉｏｎｐｒｏｃｅｄｕｒｅｗｈｉｃｈｙｉｅｌｄｓａｓｅｑｕｅｎｃｅｏｆｐｏｉｎｔｓ ｘｎ ∞ｎ=0 ｉｎＥ .Ｆｏｒａｎｅｘａｍｐｌｅ ,ｔｈｅｆｕｎｃｔｉｏｎｉｔｅｒａｔｉｏｎｘｎ+1=ｆ(Ｔ ,ｘｎ) =Ｔｘ0 .ＳｕｐｐｏｓｅＦ(Ｔ) =ｘ∈Ｅ :Ｔｘ=ｘ ≠ ａｎｄｔｈａｔ ｘｎ ｃｏｎｖｅｒｇｅｓｓ…     

SOME BOUNDED RESULTS OF θ(t)-TYPE SINGULAR INTEGRAL OPERATORS

1　ＴｈｅＣｏｎｃｅｐｔｉｏｎｓａｎｄＲｅｓｕｌｔｓＬｅｔＢｂｅａＢａｎａｃｈｓｐａｃｅｉｎＵＭＤ(ｕｎｃｏｎｄｉｔｉｏｎａｌｉｔｙｏｆｍａｒｔｉｎｇａｌｅｄｉｆｆｅｒｅｎｃｅｓ)ｗｉｔｈａｎｕｎｃｏｎｄｉｔｉｏｎａｌｂａｓｉｓ.Ｂ＿ｖａｌｕｅｄｆｕｎｃｔｉｏｎｆ(ｘ)ｏｎＲｎｉｓｓａｉｄｔｏｂｅｉｎＬｐＢ(Ｒｎ),(1≤ｐ< ∞)ｉｆ‖ｆ‖ＬｐＢ=∫Ｒｎ‖ｆ(ｘ)‖ｐｄｘ1/ｐ< ∞　(1≤ｐ< ∞),ａｎｄｔｈｅｎｏｒｍａｓ　‖ｆ‖ＬｐＢ=∫Ｒｎ‖ｆ(ｘ)‖ｐｄｘ1/ｐ　(1≤ｐ< ∞).　　Ｆｏｒｔｈｅｓｉｎｇｕｌａｒｉｎｔｅ…     

On the existence of common fixed points for a pair of lipschitzian mappings in banach spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＫｂｅａｎｏｎｅｍｐｔｙｓｕｂｓｅｔｏｆａＢａｎａｃｈｓｐａｃｅＸ .ＴｈｅｎａｍａｐｐｉｎｇＴ :Ｋ→ＫｉｓｓａｉｄｔｏｂｅａＬｉｐｓｃｈｉｔｚｉａｎｍａｐｐｉｎｇｉｆ,ｆｏｒｅａｃｈｉｎｔｅｇｅｒｎ≥ 1 ,ｔｈｅｒｅｅｘｉｓｔｓａｃｏｎｓｔａｎｔｋｎ >0ｓｕｃｈｔｈａｔ‖Ｔｎｘ-Ｔｎｙ‖ ≤ｋｎ‖ｘ-ｙ‖　ｆｏｒａｌｌｘ ,ｙ∈Ｋ .ＡＬｉｐｓｃｈｉｔｚｉａｎｍａｐｐｉｎｇＴｉｓｓａｉｄｔｏｂｅｕｎｉｆｏｒｍｌｙｋ＿Ｌｉｐｓｃｈｉｔｚｉａｎｉｆｋｎ =ｋｆｏｒａｌｌｎ ≥ 1 ;ｎｏ…     

On the Problem of Dissipative Perturbations of Nonexpansive Mappings

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｅｌｉｍｉｎａｒｉｅｓＴｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒ,ｗｅａｓｓｕｍｅｔｈａｔＨｉｓａｒｅａｌＨｉｌｂｅｒｔｓｐａｃｅ ,〈· ,·〉ｉｓｔｈｅｉｎｎｅｒｐｒｏｄｕｃｔｏｎＨ ,ＰｉｓａｃｏｎｅｉｎＨ .ＢｙｖｉｒｔｕｅｏｆｔｈｅｃｏｍｅＰ ,ａｎｏｒｄｅｒ“≤”ｉｓｉｎｄｕｃｅｄｉｎＨ ,ｉ.ｅ .,ｆｏｒａｎｙｇｉｖｅｎｘ,ｙ∈Ｈ ,ｘ≤ｙｉｆａｎｄｏｎｌｙｉｆｙ -ｘ∈Ｐ .Ａｍｕｌｔｉ＿ｖａｌｕｅｄｍａｐｐｉｎｇＡ :Ｄ(Ａ) Ｈ → 2 Ｈｉｓｓａｉｄｔｏｂｅａｃｃｒｅｔｉｖ…     

Quasi-equilibrium problems in noncompact generalized convex spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＸａｎｄＹｂｅｎｏｎｅｍｐｔｙｓｅｔｓａｎｄ2ＸｂｅｔｈｅｆａｍｉｌｙｏｆａｌｌｓｕｂｓｅｔｓｏｆＸ.ＬｅｔＴ:Ｘ→Ｙｂｅａｓｉｎｇｌｅ＿ｖａｌｕｅｄｍａｐｐｉｎｇ,Ａ:Ｘ→2Ｘｂｅａｓｅｔ＿ｖａｌｕｅｄｍａｐｐｉｎｇａｎｄｆ:Ｘ×Ｙ→Ｒ∪±∞ｂｅａｆｕｎｃｔｉｏｎ.Ｔｈｅｑｕａｓｉ＿ｅｑｕｉｌｉｂｒｉｕｍｐｒｏｂｌｅｍＱＥＰ(Ｔ,Ａ,ｆ)ｉｓｔｏｆｉｎｄ^ｘ∈Ｘｓｕｃｈｔｈａｔ^ｘ∈Ａ(^ｘ),ｆ(^ｘ,Ｔ^ｘ)≤ｆ(ｙ,Ｔ^ｘ),　　ｙ∈Ａ(^ｘ).(1)ＴｈｅＱＥＰ(Ｔ,Ａ,ｆ)ｗａｓｉｎｔｒｏｄ…     

Iterative approximation with errors of fixed point for a class of nonlinear operator with a bounder range

Let X be a uniformly smooth real Banach space. Let T:X → X be continuos and strongly accretive operator. For a given f ε X, define S: X → X by Sx =f−Tx+x, for all x ε X. Let {a_n} _n=0 ^∞ , {β_n} _n=0 ^∞ be two real sequences in (0, 1) satisfying:

Iterative solution of nonlinear equations with strongly accretive operators in Banach spaces

1IntroductionandPreliminariesLetXbearealBanachspacewithnormIJ'11andadualX'.ThenormalizeddualitymappingJ:X~ZxisdefinedbyJx={x'eX*I(x,x')=11x112=11x if'},where',')denotesthegeneralizeddualitypairing.Itiswell-knownthatifX isstrictlyconvex,Jissingle-valuedandJ(tx)=tjxforallt201xeX.IfX*isuniformlyconvex,thenJisuniformlycontinuousonanyboundedsubsetSofX(of.Browde,fljandBarbuL2]).AnoperatorTwithdomainD(T)andrangeR(T)inXissaidtobeaccretiveifforeveryx,y6D(T),thereexistsajeJ(x--y)suchthat(T…     

Initial value problems for a class of nonlinear integro-partial differential equations

１ＰｒｏｂｌｅｍｓａｎｄＭａｉｎＲｅｓｕｌｔｓＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｔｕｄｙｔｈｅｎｏｎｌｉｎｅａｒｖｉｂｒａｔｉｏｎｓｏｆｉｎｆｉｎｉｔｅｒｏｄｓｗｉｔｈｖｉｓｃｏｅｌａｓｔｉｃｉｔｙ．Ｔｈｅｃｏｎｓｔｉｔｕｔｉｏｎｌａｗｏｆｔｈｅｒｏｄｓ...     

Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｈａｌｌｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓ (ＢＶＰ)ｕ″ ｇ(ｔ)ｆ(ｕ) =0 ,　　 0 <ｔ<1 ,αｕ(0 ) -βｕ′(0 ) =0 ,　　γｕ(1 ) δｕ′(1 ) =0 ,(1 )ｗｈｅｒｅα ,β,γ ,δ≥ 0 ,ρ:=βγ αγ αδ>0 ,ｆ∈Ｃ([0 ,∞ ) ,[0 ,∞ ) ) ,ｇｍａｙｂｅｓｉｎｇｕｌａｒａｔｔ=0ａｎｄ/ｏｒｔ=1 .Ｔｈｉｓｐｒｏｂｌｅｍａｒｉｓｅｓｎａｔｕｒａｌｌｙｉｎｔｈｅｓｔｕｄｙｏｆｒａｄｉａｌｌｙｓｙｍｍｅｔ…     

On the operator of symmetric differentiation on a compact Riemannian manifold with boundary

We state a particular case of one of the theorems which we shall prove. Let Ω be a bounded open set in ℝ ⁿ with smooth boundary and let σ=(σ _ij )be a symmetric second-order tensor with components σ _ij εH ^k(Ω) for some (positive or negative) integer k; H ^k are Sobolev spaces on Ω. Then we have for some u _i εH ^k +1(Ω),i=1,...,n, if and only if (if k<0, the integral is in fact a duality) for any symmetric tensor (ω with components and such that ). Some applications in the theory of elasticity are also given.     

Periodicity and strict oscillation for generalized lyness equations

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｄｅｌａｙｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎｘｎ 1=ｘｎ(ａ ｂｘｎ)ｘｎ- 1,　　ｎ=0 ,1 ,2 ,… ,(1 )ｗｈｅｒｅａ ,ｂ∈ [0 ,∞ )ｗｉｔｈａ ｂ>0 (2 )ａｎｄｗｈｅｒｅｔｈｅｉｎｉｔｉａｌｖａｌｕｅｓｘ- 1ａｎｄｘ0 ａｒｅａｒｂｉｔｒａｒｙｐｏｓｉｔｉｖｅｎｕｍｂｅｒｓ.Ｅｑ .(1 )ｉｓｒｅｇａｒｄｅｄａｓａｇｅｎｅｒａｌｉｚｅｄＬｙｎｅｓｓｅｑｕａｔｉｏｎｂｙＧ .Ｌａｄａｓｉｎ [1 ] .Ｏｂｖｉｏｕｓｌｙ ,ｕｎｄｅｒｃｏｎｄｉｔｉｏｎ (2 ) ,…     

Ellipsoidal Domains: Piecewise Nonuniform and Impotent Eigenstrain Fields

In association with multi-inhomogeneity problems, a special class of eigenstrains is discovered to give rise to disturbance stresses of interesting nature. Some previously unnoticed properties of Eshelby’s tensors prove useful in this accomplishment. Consider the set of nested similar ellipsoidal domains {Ω₁, Ω₂,⋯,Ω _N+1}, which are embedded in an infinite isotropic medium. Suppose that

Numerical study of a nonlinear integrodifferential equation

I.IntroductionItiswell-knobal.nthatKorteweg-deVriesequationisacanonicalmodeltodescribethebalanceofthenonlineareffectandthedispersiveeffectofaphysicalsystem.Thisequationpossessestheso-called'soliton"solution,whichhasbeenfoundnumericallybyZabuskyandKruskall'].Ho-c'Jlever,sometimesthebalanceofnonlinearityanddispersionofasystemmayleadtoa,integroditTerentialequationinsteadofadifferentialequation.Forinstance,inthestudyofvortexbreakdownofanunboundedrotatingfluidLeibovich12]derivedfollowingnonline…     

Generalized diagonalization of matrices over quaternion field

Ｉｎｒｅｃｅｎｔｙｅａｒｓ,ａｐｐｌｉｃａｔｉｏｎｓｏｆｑｕａｔｅｒｎｉｏｎｍａｔｒｉｃｅｓａｒｅｂｅｃｏｍｉｎｇｍｏｒｅａｎｄｍｏｒｅｉｍｐｏｒｔａｎｔａｎｄｅｘｔｅｎｓｉｖｅｉｎｒｉｇｉｄｍｅｃｈａｎｉｃｓ,ｑｕａｎｔｕｍｍｅｃｈａｎｉｃｓ,ｃｏｎｔｒｏｌｔｈｅｏｒｙａｎｄｈｅｌｉｃａｌｔｅｃｈｎｏｌｏｇｙ［１～３］．Ｗｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｔｈｅａｂｏｖｅｄｉｓｃｉｐｌｉｎｅｓ,ｉｔｉｓｇｅｔｔｉｎｇｍｏｒｅａｎｄｍｏｒｅｎｅｃｅｓｓａｒｙｆｏｒｕｓｔｏｆｕｒｔｈ…     

Positive periodic solution of a neutral predator-prey system

ＩｎｔｒｏｄｕｃｔｉｏｎＡｔｐｒｅｓｅｎｔ,ｔｈｅｒｅａｒｅｏｎｌｙａｆｅｗｐａｐｅｒｓ［１～３］ｈａｖｉｎｇｂｅｎｐｕｂｌｉｓｈｅｄｏｎｔｈｅｇｌｏｂａｌｅｘｉｓｔｅｎｃｅｏｆｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｆｏｒｎｅｕｔｒａｌｄｅｌａｙｐｏｐｕ...