An iteration method for integral equations arising from axisymmetric loading problems

Let the concentrated forces and the centers of pressure with unknown density functions x(ξ) and y(ξ) respectively be distributed along the axis z outside the solid, then one can reduce an axismmetric loading problem of solids of revolution to two simultaneous Fredholm integral equations. An iteration method for solving such equations is duscussed. A lemma equivalent to E. Rakotch’s contractive mapping theorem and a theorem concerning the convergent proof of the iteration method are presented.     

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…     

ON REICH''''S OPEN QUESTION

1　IntroductionandPreliminariesThroughoutthispaperweassumethatEisarealBanachspace ,E isthedualspaceofE ,DisanonemptysubsetofEandJ:E → 2 E isthenormalizeddualitymappingdefinedbyJ(x) =f∈E ,　〈x ,f〉 =‖x‖‖f‖ ,　‖x‖ =‖f‖ 　 (x∈E) . (1 )　　Definition 1　LetT :D →Dbeamapping .1 .Tissaidtobeasymptoticallynonexpansive[1],ifthereexistsasequence kn [1 ,∞ )withlimn→∞kn =1suchthat‖Tnx-Tny‖ ≤kn‖x-y‖　forall　x ,y∈D ,n≥ 1 ;　　 2 .Tissaidtobenonexpansive,ifthesequence kn app…     

Some results of variational inclusion problems and fixed point problems with applications

This paper introduces a general iterative algorithm to approximate a common element in the solution set of quasi-variational inclusion problems and the common fixed point set of an infinite family of nonexpansive mappings. It is proven that the iterative sequences generated in the proposed iterative algorithm converge strongly to some common element in the framework of the real Hilbert spaces.     

Solving the velocity parameter of the 2-D wave inverse problems with the integration-characteristic method

For the 2 -D wave inverse problems introduced from geophysical exploration, in this paper, the author presents integration-characteristic method to solve the velocity parameter, and then applies it to common shotpoint model data, in noise-free case. The accuracy is quite good.     

Existence of solutions for nonlinear impulsive Hammerstein integral equations in Banach spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｔｈｅｂａｓｉｃｃｈａｒａｃｔｅｒｉｓｔｉｃｓｏｆｎｏｎｌｉｎｅａｒｉｎｔｅｇｒａｌｅｑｕａｔｉｏｎｏｆＶｏｌｔｅｒｒａｔｙｐｅａｒｅｆｕｎｄａｍｅｎｔａｌｌｙｄｉｆｆｅｒｅｎｔｆｒｏｍｔｈａｔｏｆＨａｍｍｅｒｓｔｅｉｎｔｙｐｅ:Ｈａｍｍｅｒｓｔｅｉｎｉｎｔｅｇｒａｌｅｑｕａｔｉｏｎｈａｓａｆｉｘｅｄｕｐｐｅｒｌｉｍｉｔｏｆｉｎｔｅｇｒａｔｉｏｎ.Ｔｈｉｓｍｅａｎｓｔｈａｔａｓｏｌｕｔｉｏｎｍｕｓｔａｌｗａｙｓｂｅｄｅｆｉｎｅｄｏｖｅｒｅ…     

On uniform convergence of linear operators on the probabilistic normed space

In this paper, we introduced the notion of uniform convergence of the linear operators on the probabilistic normed space, and the notion of probabilistic distance between the operators, which describes the above convergence completely. In terms of these notions, we obtained the essential features of the continuity of operators, and of the uniform convergence of operator sequences, and we also obtained the closure of continuity and complete continuity under the operation of the limit of uniform convergence.     

On the Plane Orthotropic Stress Problem of Quasi-Static Thermoelasticity

The plane stress boundary value problem of quasi-static linear orthotropic thermoelasticity is discussed. The thermoelastic system on a bounded simply-connected domain is decoupled. The decoupled temperature equation is investigated by using accurate estimation and the contraction mapping principle. Representations of solutions of the field equation are obtained, and some solvability results are proved. The results are of both theoretical and numerical interest.     

ON PROBLEM OF NEAREST COMMON FIXED POINT OF NONEXPANSIVE MAPPINGS

The purpose is by using the viscosity approximation method to study the convergence problem of the iterative scheme for an infinite family of nonexpansive mappings and a given contractive mapping in a reflexive Banach space. Under suitable conditions, it was proved that the iterative sequence converges strongly to a common fixed point which was also the unique solution of some variational inequality in a reflexive Banach space. The results presented extend and improve some recent results.     

Fixed point index of uniform limit of set-Valued strict set-contractive mappings and its applications

In this paper,the class of uniform limit mappings of set-valued,strick set-contractivemappings is discussed.Furthermore,the fixed point index theory for the uniform limitmappings is established.Using the fixed point index theory,some positive fixed pointtheorems are proved.Our theorems generalize some results in[1,4,5,7].     

Intuitionistic Menger inner product spaces and applications to integral equations

In this paper, first introduce and define an intuitionistic Menger inner product space, and then, obtain a new fixed point theorem in a complete intuitionistic Menger inner product space. As an application, the results are used to study the existence and uniqueness of the solution to a linear Volterra integral equation.     

ON A CLASS OF NEW KKM THEOREM WITH APPLICATIONS

ＯＮＡＣＬＡＳＳＯＦＮＥＷＫＫＭＴＨＥＯＲＥＭＷＩＴＨＡＰＰＬＩＣＡＴＩＯＮＳＺｈａｎｇＳｈｉｓｈｅｎｇ（张石生），ＺｈａｎｇＸｉａｎ（张宪）（ＲｅｃｅｉｖｅｄＭａｙ１５，１９９５）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ａｃｌａｓｓｏｆｎｅｗＫＫ...     

Algorithms of common solutions to quasi variational inclusion and fixed point problems

The purpose of this paper is to present an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with multivalued maximal monotone mapping and inverse-strongly monotone mappings and the set of fixed points of nonexpansive mappings in Hilbert space.Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper not only improve and extend the main results in Korpelevich（Ekonomika i Matematicheskie Metody,1976,12（4）：747-756）,but also extend and replenish the corresponding results obtained by Iiduka and Takahashi（Nonlinear Anal TMA,2005,61（3）：341-350）,Takahashi and Toyoda（J Optim Theory Appl,2003, 118（2）：417-428）,Nadezhkina and Takahashi（J Optim Theory Appl,2006,128（1）：191- 201）,and Zeng and Yao（Taiwanese Journal of Mathematics,2006,10（5）：1293-1303）.     

One approximate solution of the Nekrasov problem

An approximate solution ω = A[ω, μ] of the nonlinear integral Nekrasov equation is obtained by successive replacement of the kernel of the integral operator by a close one. The solution is sought not directly at the bifurcation point μ₁ = 3 of the linearized equation ω = μL[ω] but at the point μ = 1 at which operator A[ω, μ], remaining nonlinear in ω, is linear in μ. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 50–56, November–December, 2007.     

Fredholm and Compact Mappings Sharing a Value

1　ＰｒｅｌｉｍｉｎａｒｉｅｓＩｎｔｈｉｓｐａｐｅｒ,ｗｅａｓｓｕｍｅｔｈａｔＸ ,ＹａｎｄＺａｒｅＢａｎａｃｈｓｐａｃｅｓｏｖｅｒＫ =ＲｏｒＫ =Ｃ .ＯｎｅｍｅｔｈｏｄｔｏｓｏｌｖｅｔｈｅｅｑｕａｔｉｏｎＦ(ｘ) =0 ( 1 )ｉｓｔｈｅｃｏｎｔｉｎｕａｔｉｏｎｍｅｔｈｏｄ[1- 10 ].Ｔｈｉｓｍｅｔｈｏｄｃｏｎｓｉｓｔｓｏｆｅｍｂｅｄｄｉｎｇ ( 1 )ｉｎａｃｏｎｔｉｎｕｕｍｏｆｐｒｏｂｌｅｍｓＨ(ｔ,ｘ) =0 ,　　 0 ≤ｔ≤ 1 ,( 2 )ｗｈｅｒｅ ( 2 )ｉｓｅａｓｉｌｙｓｏｌｖｅｄｗｈｅｎｔ=0 .Ｉｆｉｔｉｓｐｏｓｓｉｂｌｅ…     

Problem of periodic solutions for neutral functional differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＣ(ｋ- 1)2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓ (ｋ -1 )＿ｔｈｏｒｄｅｒｃｏｎｔｉｎｕｏｕｓｄｉｆｆｅｒｅｎｔｉａｂｌｅａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　Ｃ2π =ｈ(ｔ) ｜ｈ :Ｒ →Ｒｉｓｃｏｎｔｉｎｕｏｕｓａｎｄｈ(ｔ+ 2π) ≡ｈ(ｔ) ,　　‖ｈ(ｔ)‖ =ｓｕｐｔ∈ [0 ,2π] ｜ｈ(ｔ) ｜ ,　　‖ｈ(ｔ)‖Ｐｋ- 1 =ｍａｘ‖ｈ(ｔ)‖ ,‖ｈ′(ｔ)‖ ,… ,‖ｈ(ｋ- 1) (ｔ)‖ ,　　ｘ(ｍ) (ｔ+ ·) (θ) =ｘ(ｍ) (ｔ+θ)　　θ∈Ｒ (ｍ =0 ,1 ,2 ,… ,ｋ-1 ) .Ｃｌｅａｒｌｙ ,ｘ(ｍ) (ｔ + ·) ∈Ｃ2π, …     

Existence and Approximation of Solutions to Variational Inclusions with Accretive Mappings in Banach Spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒｗｅｓｕｐｐｏｓｅｔｈａｔＸｉｓａｒｅａｌＢａｎａｃｈｓｐａｃｅ,Ｘ ｉｓｉｔｓｄｕａｌｓｐａｃｅ ,〈· ,·〉ｉｓｔｈｅｐａｉｒｉｎｇｏｆＸａｎｄＸ .Ｄ(Ｔ)ａｎｄＲ(Ｔ)ｄｅｎｏｔｅｔｈｅｄｏｍａｉｎａｎｄｔｈｅｒａｎｇｅｏｆＴｒｅｓｐｅｃｔｉｖｅｌｙ .ＬｅｔＴ ,Ａ :Ｘ →Ｘ ,ｇ :Ｘ→Ｘ ｂｅｔｈｒｅｅｍａｐｐｉｎｇｓａｎｄφ :Ｘ →Ｒ∪ ∞ ａｐｒｏｐｅｒｃｏｎｖｅｘｌｏｗｅｒｓｅｍｉ＿ｃｏｎｔｉｎｕｏｕｓｆｕｎｃｔｉｏｎ .Ｉｎ 1 999,ｔｈ…     

Elxed Point Theorem of Composition g-Contraction Mapping and its Applications

ＩｎｔｒｏｄｕｃｔｉｏｎＭａｎｙｓｔｕｄｉｅｓｈａｖｅｂｅｅｎｄｏｎｅｏｎｃｏｎｔｒａｃｔｉｖｅｍａｐｐｉｎｇｓ,ｅ .ｇ .,Ｒｈｏａｄｅｓ[1]ｍａｄｅａｃｏｍｐａｒｉｓｏｎｏｆｖａｒｉｏｕｓｄｅｆｉｎｉｔｉｏｎｓ (ｍｏｒｅｔｈａｎ 1 0 0ｔｙｐｅｓｖａｒｉｅｄｆｒｏｍ 2 5ｂａｓｉｃｔｙｐｅｓ)ｏｆｃｏｎｔｒａｃｔｉｖｅｍａｐｐｉｎｇｏｎｃｏｍｐｌｅｔｅｍｅｔｒｉｃｓｐａｃｅｉｎ 1 977.Ａｎｄｕｐｔｏｎｏｗ ,ｓｕｃｈｓｔｕｄｙｉｓｓｔｉｌｌｇｏｉｎｇｏｎ ,ｅ .ｇ .,[2 ] .Ｉｎｔｈｉｓｐａｐｅｒ,ｔｈｅｓｅ…     

The uniqueness and existence of solution of the characteristic problem on the generalized KdV equation

The generalized KdV equationu ₁+auu_a+μu_a3+eu_a5=0^[1] is a typical integrable equation. It is derived studying the dissemination of magnet sound wave in cold plasma^[2], the isolated wave in transmission line^[3], and the isolated wave in the boundary surface of the divided layer fluid^[4]. For the characteristic problem of the generalized KdV equation, this paper, based on the Riemann function, designs a suitable structure, then changes the characteristic problem to an equivalent integral and differential equation whose corresponding fixed point, the above integral differential equation has a unique regular solution, so the characteristic problem of the generalized KdV equation has a unique solution. The iteration solution derived from the integral differential equation sequence is uniformly convegent in .     

UNIFORM NORMAL STRUCTURE AND SOLUTIONS OF REICH’S OPEN QUESTION

The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich＇ s open question is given. The results presented extend and improve Zhang Shisheng＇ s recent ones in the following aspects ： （i） Zhang＇ s stronger condition that the sequence of iterative parameters converges to zero is removed; （ii） Zhang＇ s stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; （iii） Zhang＇ s stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi,Ueda and Wittmann.