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The strict opositivity of 4th order symmetric tensor may apply to detect vacuum stability of general scalar potential. For finding analytical expressions of (strict) opositivity of 4th order symmetric tensor, we may reduce its order to 3rd order to better deal with it. So, it is provided that several analytically sufficient conditions for the copositivity of 3th order 2 dimensional (3 dimensional) symmetric tensors. Subsequently, applying these conclusions to 4th order tensors, the analytically sufficient conditions of copositivity are proved for 4th order 2 dimensional and 3 dimensional symmetric tensors. Finally, we apply these results to present analytical vacuum stability conditions for vacuum stability for $\mathbb{Z}_3$ scalar dark matter., Comment: 21 pages

In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking copositivity and positive definiteness of tensors given by such scalar potentials. Because the tensors given by general scalar potential are fourth-order and symmetric, our work mainly focuses on finding precise expressions to test copositivity and positive definiteness of fourth-order tensors in this paper. First of all, an analytically sufficient and necessary condition of positive definiteness is provided for fourth-order 2-dimensional symmetric tensors. For fourth-order 3-dimensional symmetric tensors, we give two analytically sufficient conditions of (strictly) copositivity by using proof technique of reducing orders or dimensions of such a tensor. Furthermore, an analytically sufficient and necessary condition of copositivity is showed for fourth-order 2-dimensional symmetric tensors. We also give several distinctly analytically sufficient conditions of (strict) copositivity for fourth-order 2-dimensional symmetric tensors. Finally, these results may be applied to check lower bound of scalar potentials, and to present analytical vacuum stability conditions for potentials of two real scalar fields and the Higgs boson.

The Z \begin{document}$ _1 $\end{document} -eigenvalue of tensors (hypermatrices) was widely used to discuss the properties of higher order Markov chains and transition probability tensors. In this paper, we extend the concept of Z \begin{document}$ _1 $\end{document} -eigenvalue from finite-dimensional tensors to infinite-dimensional tensors, and discuss the upper bound of such eigenvalues for infinite-dimensional generalized Hilbert tensors. Furthermore, an upper bound of Z \begin{document}$ _1 $\end{document} -eigenvalue for finite-dimensional generalized Hilbert tensor is obtained also.

In this paper, we intend to solve the multi-linear system by preconditioned iterative method based on tensor splitting. We propose a new preconditioner I + S m a x . In order to compare the convergence rates of preconditioned iterative methods with preconditioner I + S m a x and preconditioner I + S α presented by Li et al. (2018), we propose a new splitting method for tensors. The convergence and comparison theorems of newly proposed splitting method with two preconditioner are given.

The uniqueness of positive Z-eigenvector of nonnegative tensors has many important applications in both theory and practical application. In this paper, we give a sufficient condition that a nonnegative tensor has unique positive Z-eigenvector by using its entries. For the third order transition probabilities tensor, we show that its positive Z-eigenvector corresponding to eigenvalue 1 is unique.

Cryopreservation of tissue cells is an important method to maintain cell viability and cellular function. However, cell viability and function are less than ideal by conventional cell cryopreservation methods, which may result in apoptosis and necrosis of cells in cryopreservation. Trehalose plays a role in maintaining cell structure and protecting cells from stress. However, owing to the difficulty in transport of trehalose across the cell membrane, its antifreeze effect is limited. A large amount of trehalose (up to 237 ± 8.5 mM) can be delivered to smooth muscle cells incubated in a medium containing trehalose and apatite nanomaterials at 37 °C for 6 h. Our data showed that trehalose was efficiently delivered intracellularly with the aid of nanoparticles (NP), with a loading efficiency up to 137.3 ± 34.5%, thus allowing for cryopreservation of LMC with nontoxic sugar as the sole cryoprotectant. Colloidal bioelastic apatite NP were used as bioactive promoters for the cryopreservation of tissue cells with trehalose. The addition of apatite NP in the medium substantially increased aortic smooth muscle cell cryosurvival, up to 83.6% (30% improvement over control without NP), a level comparable to that associated with the traditional Me2SO cryoprotective regimen. Furthermore, the cytotoxicity of nanocapsules in the intracellular delivery of trehalose was negligible. This method provides a new option to enhance the activity of valvular cells for cryopreservation.

In particle physics, the vacuum stability of scalar potentials is to check positive definiteness (or copositivity) of its coupling tensors, and such a coupling tensor is a 4th order and symmetric tensor. In this paper, we mainly discuss precise expressions of positive definiteness of 4th order tensors. More specifically, two analytically sufficient conditions of positive definiteness for 4th order 2 dimensional symmetric tensors are given by reducing orders of tensors, and applying these conclusions, some sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors are derived. We also present several other sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors. Finally, we test and verify the vacuum stability of general scalar potentials of two real singlet scalar fields and the Higgs boson by using these results.

This paper looks at the tensor eigenvalue complementarity problem (TEiCP) which arises from the stability analysis of finite dimensional mechanical systems and is closely related to the optimality conditions for polynomial optimization. We investigate two monotone ascent spectral projected gradient (SPG) methods for TEiCP. We also present a shifted scaling-and-projection algorithm (SPA), which is a great improvement of the original SPA method proposed by Ling et al. (Comput. Optim. Appl. 63, 143–168 2016). Numerical comparisons with some existing gradient methods in the literature are reported to illustrate the efficiency of the proposed methods.

Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors that include positive semi-definite tensors as special cases. The inheritance property and invariant property of column sufficient tensors are presented. Then, various spectral properties of symmetric column sufficient tensors are given. It is proved that all H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient tensors and the handicap of tensors are defined. We prove that a tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of tensors are presented. Finally, as an application of column sufficient tensors, several results on tensor complementarity problems are established.

In this paper, one of our main purposes is to prove the boundedness of the solution set of tensor complementarity problems such that the specific bounds depend only on the structural properties of such a tensor. To achieve this purpose, firstly, we prove that this class of structured tensors is strictly semi-positive. Subsequently, the strictly lower and upper bounds of operator norms are given for two positively homogeneous operators. Finally, with the help of the above upper bounds, we show that the solution set of tensor complementarity problems has the strictly lower bound. Furthermore, the upper bounds of spectral radius are obtained, which depends only on the principal diagonal entries of tensors.

In this paper, we introduce the concept of an $m$-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_{n}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}-m+a},\ a\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=1,2,\cdots,n, $$ and show that its $H$-spectral radius and its $Z$-spectral radius are smaller than or equal to $M(a)n^{m-1}$ and $M(a)n^{\frac{m}{2}}$, respectively, here $M(a)$ is a constant only dependent on $a$. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for $a\geq1$. For an $m$-order infinite dimensional generalized Hilbert tensor $\mathcal{H}_{\infty}$ with $a>0$, we prove that $\mathcal{H}_{\infty}$ defines a bounded and positively $(m-1)$-homogeneous operator from $l^{1}$ into $l^{p}\ (1, Comment: 15 pages

Fermented papaya extracts (FPEs) are obtained by fermentation of papaya by Aspergillus oryzae and yeasts. In this study, we investigated the protective effects of FPEs on mammary gland hyperplasia induced by estrogen and progestogen. Rats were randomly divided into 6 groups, including a control group, an FPE-alone group, a model group, and three FPE treatment groups (each receiving 30, 15, or 5 ml/kg FPEs). Severe mammary gland hyperplasia was induced upon estradiol benzoate and progestin administration. FPEs could improve the pathological features of the animal model and reduce estrogen levels in the serum. Analysis of oxidant indices revealed that FPEs could increase superoxide dismutase (SOD) and glutathione peroxidase (GSH-Px) activities, decrease malondialdehyde (MDA) level in the mammary glands and serum of the animal models, and decrease the proportion of cells positive for the oxidative DNA damage marker 8-oxo-dG in the mammary glands. Additionally, estradiol benzoate and progestin altered the levels of serum biochemical compounds such as aspartate transaminase (AST), total bilirubin (TBIL), and alanine transaminase (ALT), as well as hepatic oxidant indices such as SOD, GSH-Px, MDA, and 8-oxo-2′-deoxyguanosine (8-oxo-dG). These indices reverted to normal levels upon oral administration of a high dose of FPEs. Taken together, our results indicate that FPEs can protect the mammary glands and other visceral organs from oxidative damage.

Tensor eigenvalue problem is one of important research topics in tensor theory. In this manuscript, we consider the properties of Z-eigenpair of irreducible nonnegative tensors. By estimating the ratio of the smallest and largest components of a positive Z-eigenvector for a nonnegative tensor, we present some bounds for the eigenvector and Z-spectral radius of an irreducible and weakly symmetric nonnegative tensor. The proposed bounds complement and extend some existing results. Finally, several examples are given to show that such a bound is different from one given in the literature.

Trehalose is a disaccharide molecule consisting of two molecules of glucose. Industrially, trehalose is derived from corn starch and utilized as a drug. This study aims to examine whether the integration of nanoparticle-encapsulated trehalose to the Ice-Free Cryopreservation (IFC) method for preserving heart valves has better cell viability, benefits to protect the extracellular matrix (ECM), and reduce immune response after storage. For the experiment to be carried out, we obtained materials, and the procedures were carried out in the following manner. The initial step was the preparation of hydroxyapatite nanoparticles, followed by precipitation to acquire Apatite colloidal suspensions. Animals were obtained, and their tissue isolation and grouping were done ethically. All samples were then divided into four groups, Control group, Conventional Frozen Cryopreservation (CFC) group, IFC group, and IFC + T (IFC with the addition of 0.2 M nanoparticle-encapsulated Trehalose) group. Histological analysis was carried out via H&E staining, ECM components were stained with Modified Weigert staining, and the Gomori Ammonia method was used to stain reticular fibers. Alamar Blue assay was utilized to assess cell viability. Hemocompatibility was evaluated, and samples were processed for immunohistochemistry (TNFα and IL-10). Hemocompatibility was quantified using Terminal Complement Complex (TCC) and Neutrophil elastase (NE) as an indicator. The results of the H&E staining revealed less formation of extracellular ice crystals and intracellular vacuoles in the IFC + T group compared with all other groups. The CFC group’s cell viability showed better viability than the IFC group, but the highest viability was exhibited in the IFC + T group (70.96 ± 2.53, P < 0.0001, n = 6). In immunohistochemistry, TNFα levels were lowest in both IFC and IFC + T group, and IL-10 expression had significantly reduced in IFC and IFC + T group. The results suggested that the nanoparticle encapsulated trehalose did not show significant hemocompatibility issues on the cryopreserved heart valves.

Earthquake ground‐motion prediction models usually define site conditions based on the time‐averaged shear‐wave velocity in the upper 30 m ( V S 30). Proxy‐based estimations of V S 30 are commonly used, if velocity measurements are not available. We compile a soil‐profile database for the Beijing plain area (China), using data from research documents and technical reports. The database contains 479 soil profiles, 463 of which have depths greater than 30 m. We develop regional relationships for the Beijing plain area for extrapolating the time‐averaged shear‐wave velocity to a given depth less than 30 m to V S 30, and then compare the performance of available models. We find that the second‐order polynomial model (Boore et al. , 2011), based on data from Japan, provides an overprediction, whereas the linear model (Boore, 2004) calibrated on data from California underestimates V S 30. We develop relationships for estimating V S 30 based on proxies such as ground slope gradients from radar‐derived digital elevation models (DEMs) and surface geology at different scales. We find that local V S 30 data in the Beijing plain are generally lower than existing 30 arcsec gradient‐based global models. Regression results show a modest correlation between V S 30 and topographic ground slope for several DEM resolutions (3, 15, 30, and 60 arcsec). Geology‐based proxies are more effective than ground slope for V S 30 estimation in the analyzed area. We propose a bilinear model based on geologic ages and depositional environments for estimating V S 30, which shows a statistically significant trend for application in the Beijing plain area. Online Material: Figures showing topographic ground slopes and correlations of V S 30 with topographic slope from digital elevation model (DEM) data and a table summarizing data from the 463 boreholes.

In this paper, we introduce the concept of monotone α-nonexpansive mappings in an ordered Banach space E with the partial order ź, which contains monotone nonexpansive mappings as special case. With the help of the Mann iteration, we show some existence theorems of fixed points of monotone α-nonexpansive mappings in uniformly convex ordered Banach space. Also, we prove some weak and strong convergence theorems of the Mann iteration for finding an order fixed point of monotone α-nonexpansive mappings under the condition lim sup n ź ∞ β n ( 1 - β n ) 0 or lim inf n ź ∞ β n ( 1 - β n ) 0 .

High order tensor arises more and more often in signal processing, data analysis, higher-order statistics, as well as imaging sciences. In this paper, an adaptive gradient (AG) method is presented for generalized tensor eigenpairs. Global convergence and linear convergence rate are established under some suitable conditions. Numerical results are reported to illustrate the efficiency of the proposed method. Comparing with the GEAP method, an adaptive shifted power method proposed by Kolda and Mayo (SIAM J Matrix Anal Appl 35:1563---1581, 2014) the AG method is much faster and could reach the largest eigenpair with a higher probability.

The tensor complementarity problem is a specially structured nonlinear complementarity problem, then it has its particular and nice properties other than ones of the classical nonlinear complementarity problem. In this paper, it is proved that a tensor is an S-tensor if and only if the tensor complementarity problem is feasible, and each Q-tensor is an S-tensor. Furthermore, the boundedness of solution set of the tensor complementarity problem is equivalent to the uniqueness of solution for such a problem with zero vector. For the tensor complementarity problem with a strictly semi-positive tensor, we proved the global upper bounds for solution of such a problem. In particular, the upper bounds keep in close contact with the smallest Pareto $H-$($Z-$)eigenvalue., Comment: Journal of Optimization Theory and Applications, 2016

Objectives Grb2-associated binder 1 (Gab1), a scaffolding adaptor protein, plays an important role in transmitting key signals that control cell growth, migration, and function from multiple tyrosine kinase receptors. This study was designed to investigate the influence of upregulation of Gab1 in endothelial progenitor cells (EPCs) stimulated with hepatocyte growth factor (HGF), and the underlying molecular mechanisms. Materials and Methods Endothelial progenitor cells isolated from human umbilical cord blood were identified and divided into four groups. EPCs in the Control group were cultured normally; those in the Control+HGF group were treated with HGF stimulation; those in the AD-Gab1 group were transfected with adenovirus containing the Gab1 gene but not treated with HGF stimulation; and, those in the AD-Gab1+HGF group were treated with both HGF stimulation and transfection with adenovirus containing the Gab1 gene. Subsequently, Gab1 expression and proliferation and migration ability were compared for EPCs grown under different conditions. Furthermore, we measured phosphorylation levels of three key proteins Gab1, SHP2, and ERK1/2. Results The AD-Gab1+HGF group had the highest expression of Gab1 and higher proliferation and migration than the other three groups. Conclusions Upregulation of Gab1 promoted HGF-induced EPC proliferation and migration. Mechanistically, HGF stimulated Gab1 tyrosine phosphorylation in EPCs, thus leading to activation of extracellular regulated MAP kinase 1/2, which is involved in proliferation and migration signaling.

At present, the biggest obstacle to the protection of endangered species in China is the lack of basic information. In order to help the relevant departments to collect species information better, a deep convolutional neural network method based on migration learning and data enhancement technology is proposed to realize real-time identification of species. First extract the bottleneck descriptor of the model on the ImageNet dataset, and then use the TensorFlow official script to train the new dataset. In the experiment, using data enhancement technology and real-time image distortion, the Inception3 and MobileNet models were trained respectively, and their own optimal parameters selection scheme was proposed. During the experiment, the best training model MobileNet_1.0_224 was obtained by weighing the time complexity and space complexity of the model. The accuracy of the test set was 89%. Finally, the trained model is transplanted into the Android device to realize the real-time classification of rare animals. The experiment proves that the proposed method has high accuracy and stable running performance.

In this paper, we consider the Halpern iteration for $$\alpha $$ -nonexpansive mapping semigroups; such semigroups contain classical nonexpansive mapping semigroups as special cases. The strong convergence theorems of Halpern iteration sequence are shown for finding a common fixed point of an $$\alpha $$ -nonexpansive semigroup, which exactly is a metric projection point of the anchor point u to the common fixed point set of such semigroups. Furthermore, the strong convergence of the Halpern iteration for an $$\alpha $$ -nonexpansive mapping sequence is given also.

In this paper, the concepts of Pareto $H$-eigenvalue and Pareto $Z$-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto $H$-eigenvalue (Pareto $Z$-eigenvalue). Furthermore, the minimum Pareto $H$-eigenvalue (or Pareto $Z$-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor $\mathcal{A}$ is copositive if and only if every Pareto $H$-eigenvalue ($Z-$eigenvalue) of $\mathcal{A}$ is non-negative., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1302.6084

In this paper, for an $\lambda$-strict pseudocontraction $T$, we prove strong convergence of the modified Mann's iteration defined by $$x_{n+1}=\beta_{n}u+\gamma_nx_n+(1-\beta_{n}-\gamma_n)[\alpha_{n}Tx_n+(1-\alpha_{n})x_n],$$ where $\{\alpha_{n}\}$, $ \{\beta_{n}\}$ and $\{\gamma_n\}$ in $(0,1)$ satisfy: (i) $0 \leq \alpha_{n}\leq \frac{\lambda}{K^2}$ with $\liminf\limits_{n\to\infty}\alpha_n(\lambda-K^2\alpha_n)> 0$; (ii) $\lim\limits_{n\to\infty}\beta_n= 0$ and $\sum\limits_{n=1}^\infty\beta_n=\infty$; (iii) $\limsup\limits_{n\to\infty}\gamma_n, Comment: 8 pages, 2014

In this paper, for Lipschitz accretive operator A, an iteration scheme is defined as follows: x n + 1 = ( 1 - α n ) x n + α n ( u - β n Ax n ) . Its strong convergence is established for finding some zero of A whenever α n , β n ∈ ( 0 , 1 ) satisfying conditions: lim n → ∞ α n = 0 , ∑ n = 1 + ∞ α n = + ∞ , lim n → ∞ β n = 0 . Furthermore, some applications for equilibrium problems are given also. In particular, the iteration coefficient is simpler and more general.

We show that an (eventually) strongly increasing and positively homogeneous mapping T defined on a Banach space can be turned into an Edelstein contraction with respect to Hilbert’s projective metric. By applying the Edelstein contraction theorem, a nonlinear version of the famous Krein-Rutman theorem is presented, and a simple iteration process {Tkx/‖Tkx‖} (∀ x ∈ P+) is given for finding a positive eigenvector with positive eigenvalue of T. In particular, the eigenvalue problem of a nonnegative tensor A can be viewed as the fixed point problem of the Edelstein contraction with respect to Hilbert’s projective metric. As a result, the nonlinear Perron-Frobenius property of a nonnegative tensor A is reached easily.

Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, for a tensor with nonnegative entries (which we refer it as a nonnegative tensor), efficient numerical schemes have been proposed to calculate its maximum eigenvalue based on a Perron–Frobenius-type theorem. In this paper, we consider a new class of tensors called essentially nonnegative tensors, which extends the concept of nonnegative tensors, and examine the maximum eigenvalue of an essentially nonnegative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially nonnegative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in its turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper and lower estimates for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor. Numerical examples are also presented to illustrate the significance of the results.

The Fredholm alternative type results are proved for eigenvalues ($E$-eigenvalues, $H$-eigenvalues, $Z$-eigenvalues) of a higher order tensor $\mathcal{A}$. For the positively homogeneous operators...

In this paper, the $m-$order infinite dimensional Hilbert tensor (hypermatrix) is intrduced to define an $(m-1)$-homogeneous operator on the spaces of analytic functions, which is called Hilbert tensor operator. The boundedness of Hilbert tensor operator is presented on Bergman spaces $A^p$ ($p>2(m-1)$). On the base of the boundedness, two positively homogeneous operators are introduced to the spaces of analytic functions, and hence the upper bounds of norm of such two operators are found on Bergman spaces $A^p$ ($p>2(m-1)$). In particular, the norms of such two operators on Bergman spaces $A^{4(m-1)}$ are smaller than or equal to $��$ and $��^\frac1{m-1}$, respectively., 17 pages

In this paper, the concept of eigenvalue is introduced to the monotone homogeneous mapping pairs ( f , g ) , and its existence and uniqueness are established successfully under the boundedness of some orbits of f , g in the Hilbert semi-norm. Also, the Collatz–Wielandt min–max type property is obtained for such a class of mapping pairs. In particular, the nonlinear Perron–Frobenius property for nonnegative tensor pairs ( A , B ) is obtained without involving the calculation of the tensor inversion. By particularizing the mapping g or f , several results can emerge as corollaries. They lead to further existence results and open problems.

The main objectives of this paper are to employ a new proof technique to prove the strong convergence of { x n } and { y n } , defined respectively by x n + 1 = α n u + β n x n + ( 1 − α n − β n ) J r n A x n and y n + 1 = β n y n + ( 1 − β n ) J r n A ( α n u + ( 1 − α n ) y n ) , to some zero of accretive operator A in a uniformly convex Banach space E with a uniformly Gâteaux differentiable norm (or with a weakly continuous duality mapping J φ ) whenever { α n } and { β n } are sequences in ( 0 , 1 ) and { r n } ⊂ ( 0 , + ∞ ) satisfying the conditions: lim n → ∞ α n = 0 , ∑ n = 1 + ∞ α n = + ∞ , lim sup n → ∞ β n 1 , lim inf n → ∞ r n > 0 .

In this paper, for a monotone operator T, we shall show strong convergence of the regularization method for Rockafellar's proximal point algorithm under more relaxed conditions on the sequences {r k } and {t k }, $$\lim\limits_{k\to\infty}t_k = 0;\quad \sum\limits_{k=0}^{+\infty}t_k = \infty;\quad\ \liminf\limits_{k\to\infty}r_k > 0.$$ Our results unify and improve some existing results.

An iterative algorithm is provided to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of some variational inequality in a Hilbert space. Using this result, we consider a strong convergence result for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Our results include the previous results as special cases and can be viewed as an improvement and refinement of the previously known results.

In Shahzad and Zegeye (Nonlinear Anal. 71 (2009), no. 3-4, 838{844), the authors introduced several Ishikawa iterative schemes for xed points of multi-valued mappings in Banach spaces, and proved some strong convergence theorems by using their iterations. In their proofs of the main results, it seems reasonable and simpler to prove for the iteration fxng to be a Cauchy sequence. In this paper, we modify and improve the proofs of the main results given by Shahzad and Zegeye. Two concrete examples also are given.

Strong convergence of two iterative schemes is proved to approach some zero of multivalued accretive operators in a Banach space. The first one is a regularization method for Rockafellar's proximal point algorithm of the resolvent and the second one is a kind of Halpern type iteration process of the resolvent. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In this paper, three iterations are designed to approach zeros of set-valued accretive operators in Banach spaces. The first one is the continuous Picard type iteration involving the resolvent, the second one is the approximate Picard type iteration involving the resolvent and the third one is the Halpern type iteration involving the resolvent. Some strong convergence theorems for three iterations are proved.

Under the framework of a real Banach space with an uniformly Gâteaux differentiable norm, two new iteration algorithms are introduced to obtain strong convergence to a fixed point of an asymptotically nonexpansive mapping. Furthermore, the proof technique is independent of the implicit iteration-path.

The main aim of this paper is to study the strong convergence of Halpern iteration for firmly type nonexpansive mappings defined on a Banach space with a uniformly Gâteaux differentiable norm, and the proof techniques are simpler and different from much existing literature also, which gives the affirmative answer of the Halpern open problem for this class of mapping.