On a general class of multi-valued weakly Picard mappings

The concept of weak contraction from the case of single-valued mappings is extended to multi-valued mappings and then corresponding convergence theorems for the Picard iteration associated to a multi-valued weak contraction are obtained. The main results in this paper extend, improve and unify a multitude of classical results in the fixed point theory of single and multi-valued contractive mappings and also improve recent results from the paper [P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 192 (1995), 655-666].     

Fixed point theorems for multi-valued contractions in complete metric spaces

In this paper the concept of a contraction for multi-valued mappings in a metric space is introduced and the existence theorems for fixed points of such contractions in a complete metric space are proved. Presented results generalize and improve the recent results of Y. Feng, S. Liu [Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103-112], D. Klim, D. Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132-139] and several others. The method used in the proofs of our results is new and is simpler than methods used in the corresponding papers. Two examples are given to show that our results are genuine generalization of the results of Feng and Liu and Klim and Wardowski.     

Approximation of common fixed points of weakly compatible pairs using the Jungck iteration

We show that the Jungck iteration scheme can be used to approximate the common fixed points of some weakly compatible pairs of generalized quasicontractive operators defined on metric spaces. The existence of coincidence points are also discussed for those pair of maps. The results are generalizations of well known results of the convergence of Picard iterations for single self maps of Banach spaces. In particular, the results improve, generalize and extend the recent results of Berinde [V. Berinde, A common fixed point theorem for compatible quasi contractive self mappings in metric spaces, Applied Mathematics and Computation 213 (2009) 348-354] and answers the open question posed in the paper.     

Some fixed point results for a generalized ψ-weak contraction mappings in orbitally metric spaces

Samet and Vetro [Samet B, Vetro C. Berinde mappings in orbitally complete metric spaces. Chaos Solitons Fract 2011;44:1075–9.] studied a fixed point theorem for a self-mapping satisfying a general contractive condition of integral type in orbitally complete metric spaces. In this paper, we introduce the notion of a generalized ψ-weak contraction mapping and establish some results in orbitally complete metric spaces. Our results generalize several well-known comparable results in the literature. As an application of our results we deduce the result of Samet and Vetro. Some examples are given to illustrate the useability of our results.     

Mann and Ishikawa-Type Iterative Schemes for Approximating Fixed Points of Multi-valued Non-Self Mappings

A Mann-type iterative scheme which converges strongly to a fixed point of a multi-valued nonexpansive non-self mapping T is constructed in a real Hilbert space H. We also constructed a Mann-type sequence which converges to a fixed point of a multi-valued quasi-nonexpansive non-self mapping under appropriate conditions. In addition, an Ishikawa-type iterative scheme which approximates the fixed points of multi-valued Lipschitz pseudocontractive non-self mappings is constructed in Banach spaces. The results obtained in this paper improve and extend the known results in the literature.

     

Some new results and generalizations in metric fixed point theory

The main purpose of this paper is the study of the generalization of some results given in [M. Berinde, V. Berinde, On a general class of multi-valued weakly Picard mappings, J. Math. Anal. Appl. 326 (2007) 772-782] and references therein. Some generalizations of the Mizoguchi-Takahashi fixed point theorem, Kannan’s fixed point theorems and Chatterjea’s fixed point theorems are established by using our new fixed point theorems.     

Nadler’s type principle with high order of convergence

Recently Proinov [P.D. Proinov, A generalization of the Banach contraction principle with high order of convergence of successive approximations, Nonlinear Analysis (2006), doi:10.1016/j.na.2006.09.008] generalized Banach contraction principle with high order of convergence. We extend some results of Proinov to the case of multi-valued maps from a complete metric space X$X$ into the space of all nonempty proximinal closed subsets of X$X$. Our results not only generalize Nadler’s fixed-point theorem (in the case when T$T$ is a mapping from a complete metric space X$X$ into the space of all nonempty proximinal closed subsets of X$X$) but also gives high order of convergence. As an application, we obtain an existence theorem for first-order initial value problem.     

Fixed point theorems for weak contractions in the sense of Berinde on partial metric spaces

In this paper, we introduce the notion of $(\delta ,L)$ weak contraction and $(\phi ,L)$ weak contraction in the sense of Berinde in partial metric space. Then we give some fixed point results in partial metric space using these new concepts.     

Fixed Points for Directional Multi-Valued <Emphasis Type="Italic">k</Emphasis>(·)-Contractions

A fixed point theorem for directional multi-valued k(·)-contractions acting m a complete metric space is established which extends similar results both for k(·)-contractions and directional contractions. Such theorem enables to obtain fixed points theorems for the former class of set-valued maps from those valid for the latter one without metrical convexity or proximinality assumptions, thereby contributing to unify the current setting of the theory. Connections with several recent advances on this subject are also examinated.Mathematics Subject Classifications (2000): 47H10, 54H25     

序Banach空间中的随机不动点定理

该文获得了序Banach空间中随机序压缩映射存在不动点的充要条件.利用随机Mann迭代序列,给出了几个随机不动点收敛定理,改进了最近文献的相应结果.     

Stability of tripled fixed point iteration procedures for monotone mappings

The new concept of tripled fixed point introduced recently by Berinde and Borcut (Nonlinear Anal. 74:4889–4897, 2011) directed to several researches on this subject, in partially metric spaces and in cone metric spaces. In this paper, we introduce the notion of stability definition of tripled fixed point iteration procedures and establish stability results for monotone mappings which satisfy various contractive conditions. Our results extend and complete some existing results in the literature.     

Iteration processes for approximating fixed points of operators of monotone type

In this paper, the unique fixed points of multi-valued and single-valued operators of monotone type are approximated by Ishikawa and Mann iteration processes with errors in real Banach spaces. The operators may not satisfy the Lipschitzian conditions. The results presented improve and extend some recent results.

     

Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings

In this paper, a necessary and sufficient conditions for the strong convergence to a common fixed point of a finite family of continuous pseudocontractive mappings are proved in an arbitrary real Banach space using an implicit iteration scheme recently introduced by Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Fuct. Anal. Optim. 22 (2001) 767-773] in condition αn∈(0,1], and also strong and weak convergence theorem of a finite family of strictly pseudocontractive mappings of Browder-Petryshyn type is obtained. The results presented extend and improve the corresponding results of M.O. Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73-81].     

Moser iteration for (quasi)minimizers on metric spaces

We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisub- and quasisuperminimizers. This is done in metric spaces equipped with a doubling measure and supporting a weak (1, p)-Poincaré inequality. The metric space is not required to be complete. We also provide an example which shows that the dilation constant from the weak Poincaré inequality is essential in the condition on the balls in the Harnack inequality. This fact seems to have been overlooked in the earlier literature on nonlinear potential theory on metric spaces.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates and asymptotic behavior of extremal function to Hardy-Sobolev type inequality on the H-type group*

This paper is devoted to discuss the regularity of the weak solution to a class of non-linear equations corresponding to Hardy-Sobolev type inequality on the H-type group. Combining the Serrin's idea and the Moser's iteration, L^p estimates of the weak solution are obtained, which generalize the results of Garofalo and Vassilev in [6, 14]. As an application, asymptotic behavior of the weak solution has been discussed. Finally, doubling property and unique continuation of the weak solution are given. *This material is based upon work funded by Zhejiang Provincial Natural Science Foundation of China under Grant No. Y606144.     

Multi-dimensional scalar balance laws with discontinuous flux

We consider scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect to both the space variable x and the unknown quantity u. We formulate the definition of entropy weak solutions and provide existence and uniqueness to the considered problem. The problem is formulated in the framework of multi-valued mappings. The notion of entropy measure-valued solutions is used to prove the so-called contraction principle and comparison principle.     

Nonempty Intersection Theorems and Generalized Multi-objective Games in Product <Emphasis Type="Italic">FC</Emphasis>-Spaces

A new class of generalized multi-objective games is introduced and studied in FC-spaces where the number of players may be finite or infinite, and all payoff are all set-valued mappings and get their values in a topological space. By using an existence theorems of maximal elements for a family of set-valued mappings in product FC-spaces due to author, some new nonempty intersection theorems for a family of set-valued mappings are first proved in FC-spaces. As applications, some existence theorems of weak Pareto equilibria for the generalized multi-objective games are established in noncompact FC-spaces. These theorems improve, unify and generalize the corresponding results in recent literatures.     

Remarks on regularity and uniqueness of the Dirac–Klein–Gordon equations in one space dimension

In recent work, the authors extended the local and global well-posedness theory for the 1D Dirac–Klein–Gordon equations, but the uniqueness of the solutions was only known in the contraction spaces (of Bourgain–Klainerman–Machedon type). Here we prove some unconditional uniqueness results [that is, uniqueness in the larger space C([0,T];X ₀), where X ₀ denotes the data space]. We also prove a result about persistence of higher regularity, which is stronger than the standard version obtained from the contraction argument, since our result allows to independently increase the regularity of the spinor and scalar fields, whereas in the standard result they must be increased by the same amount.     

Weak stability of the Ishikawa iteration procedures for -hemicontractions and accretive operators

Let X be an arbitrary Banach space, K be a nonempty closed convex subset of X, and T : K → K be a Lipschitzian and hemicontractive mapping with the property lim inf_t→∞((t)/t) > 0. It is shown that the Ishikawa iteration procedures are weakly T-stable. As consequences, several related results deal with the weak stability of these procedures for the iteration proximation of solutions of nonlinear equations involving accretive operators. Our results improve and extend those corresponding results announced by Osilike.