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Let X and Y be Banach spaces and Ω ⊆ X. Let f : Ω ⟶ Y be a single valued function which is nonsmooth. Suppose that F : X⇉ 2 Y is a set-valued mapping which has closed graph. In the present paper, we study the extended Newton-type method for solving the nonsmooth generalized equation 0 ∈ f x + F x and analyze its semilocal and local convergence under the conditions that f + F − 1 is Lipschitz-like and f admits a certain type of approximation which generalizes the concept of point-based approximation so-called n , α -point-based approximation. Applications of n , α -point-based approximation are provided for smooth functions in the cases n = 1 and n = 2 as well as for normal maps. In particular, when 0 < α < 1 and the derivative of f , denoted ∇ f , is ℓ , α -Hölder continuous, we have shown that f admits 1 , α -point-based approximation for n = 1 while f admits 2 , α -point-based approximation for n = 2 , when 0 < α < 1 and the second derivative of f , denoted ∇ 2 f , is K , α -Hölder. Moreover, we have constructed an n , α -point-based approximation for the normal maps f C + F when f has an n , α -point-based approximation. Finally, a numerical experiment is provided to validate the theoretical result of this study. [ABSTRACT FROM AUTHOR]

We extend the well-known characterizations of convergence in the spaces l p (1 ≤ p < ∞) of p -summable sequences and c 0 of vanishing sequences to a general characterization of convergence in a Banach space with a Schauder basis and obtain as instant corollaries characterizations of convergence in an infinite-dimensional separable Hilbert space and the space c of convergent sequences. "The method in the present paper is abstract and is phrased in terms of Banach spaces, linear operators, and so on. This has the advantage of greater simplicity in proof and greater generality in applications." Jacob T. Schwartz [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce a new hybrid iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, set of common fixed points of a family of quasi-φ-asymptotically nonexpansive mappings, and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. The results presented in the paper improve and extend the corresponding results announced by some authors. [ABSTRACT FROM AUTHOR]

In this paper, we propose the generalized M-iteration process for approximating the fixed points from Banach spaces to hyperbolic spaces. Using our new iteration process, we prove Δ -convergence and strong convergence theorems for the class of mappings satisfying the condition C λ and the condition E which is the generalization of Suzuki generalized nonexpansive mappings in the setting of hyperbolic spaces. Moreover, a numerical example is given to present the capability of our iteration process and the solution of the integral equation is also illustrated using our main result. [ABSTRACT FROM AUTHOR]

The results of three papers, in which the author inadvertently overlooks certain deficiencies in the descriptions of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a complex Banach space established in “On the Carleman Classes of Vectors of a Scalar Type Spectral Operator,” Int. J. Math. Math. Sci. 2004 (2004), no. 60, 3219–3235, are observed to remain true due to more recent findings. [ABSTRACT FROM AUTHOR]

In this paper, a new type of non-self-mapping, called Berinde MT-cyclic contractions, is introduced and studied. Best proximity point theorems for this type of mappings in a metric space are presented. Some examples illustrating our main results are also given. Our results generalize and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a common random fixed-point and some random fixed-point theorems for random weakly contractive operators in separable Banach spaces. A random Mann iterative process is introduced to approximate the fixed point. Finally, the main result is supported by an example and used to prove the existence and the uniqueness of a solution of a nonlinear stochastic integral equation system. [ABSTRACT FROM AUTHOR]

The purpose of this paper is, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of quasi-φ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the strong and weak convergence theorems of the implicit iteration processes for an infinite family of Lipschitzian pseudocontractive mappings in Banach spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the stability in the sense of Hyers-Ulam of differential equation of second order y″ + p(x)y′ + q(x)y + r(x) = 0. That is, if f is an approximate solution of the equation y″ + p(x)y′ + q(x)y + r(x) = 0, then there exists an exact solution of the equation near to f. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of a wide class of weighted best simultaneous approximation in normed linear spaces, and it establishes a new characterization result for the class of approximation by virtue of the notion of simultaneous regular point. [ABSTRACT FROM AUTHOR]

In this paper, we derive a fixed-point theorem for self-mappings. That is, it is shown that every isometric self-mapping on a weakly compact convex subset of a strictly convex Banach space has a fixed point. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to obtain common fixed point theorems by employing the recently introduced notion of weak reciprocal continuity. The new notion is a proper generalization of reciprocal continuity and is applicable to compatible mappings as well as noncompatible mappings. We demonstrate that weak reciprocal continuity ensures the existence of common fixed points under contractive conditions, which otherwise do not ensure the existence of fixed points. Our results generalize and extend Banach contraction principle and Meir-Keeler-type fixed point theorem. [ABSTRACT FROM AUTHOR]

Let X be a Banach space and let G be a closed bounded subset of X. For (x1,x2,…,xm)∈m, we set ρ(1,x2,…,m,G)=sup {max 1≤i≤m ‖xi−y‖:y∈G}. The set G is called simultaneously remotal if, for any (1,x2,…,m)∈m, there exists g∈G such that ρ(1,x2,…,m,G)=max 1≤i≤m ‖xi−g‖. In this paper, we show that if G is separable simultaneously remotal in X, then the set of ∞-Bochner integrable functions, L∞(I,G), is simultaneously remotal in L∞(I,X). Some other results are presented. [ABSTRACT FROM AUTHOR]

Hypersubstitutions are mappings which map operation symbols to terms of the corresponding arities. They were introduced as a way of making precise the concept of a hyperidentity and generalizations to M-hyperidentities. A variety in which every identity is satisfied as a hyperidentity is called solid. If every identity is anM-hyperidentity for a subsetMof the set of all hypersubstitutions, the variety is called M-solid. There is a Galois connection between monoids of hypersubstitutions and sublattices of the lattice of all varieties of algebras of a given type. Therefore, it is interesting and useful to know how semigroup or monoid properties of monoids of hypersubstitutions transfer under this Galois connection to properties of the corresponding lattices of M-solid varieties. In this paper, we study the order of each hypersubstitution of type (2,1), that is, the order of the cyclic subsemigroup of the monoid of all hypersubstitutions of type (2,1) generated by that hypersubstitution. [ABSTRACT FROM AUTHOR]

We introduce a new modified Ishikawa iterative process and a new W-mapping for computing fixed points of an infinite family of strict pseudo contractions mapping in the framework of q-uniformly smooth Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions. The results obtained in this paper extend and improve the recent results of Cai and Hu 2010, Dong et al. 2010, Katchang and Kumam 2011 and many others in the literature. [ABSTRACT FROM AUTHOR]

Let L be the space of complex-valued functions f on the set of vertices T of an infinite tree rooted at o such that the difference of the values of f at neighboring vertices remains bounded throughout the tree, and let Lw be the set of functions f ϵ L such that ∣f(v) - f(v-)∣ = O(∣v∣-1), where ∣v∣ is the distance between o and v and v- is the neighbor of v closest to o. In this paper, we characterize the bounded and the compact multiplication operators between L and Lw and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between Lw and the space L∞ of bounded functions on T and determine their operator norm and their essential norm. We establish that there are no isometries among the multiplication operators between these spaces. [ABSTRACT FROM AUTHOR]

It is our purpose in this paper to prove two convergents of viscosity approximation scheme to a common fixed point of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, it is the unique solution in F to a certain variational inequality, where F := ∩N=0∞n)F(Tn) stands for the common fixed-point set of the family of multivalued nonexpansive mapping {Tn}. [ABSTRACT FROM AUTHOR]

We introduce a concept of weak Bregman relatively nonexpansive mapping which is distinct from Bregman relatively nonexpansive mapping. By using projection techniques, we construct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. The main results in this paper develop, extend, and improve the corresponding results of Matsushita and Takahashi (2005) and Qin and Su (2007). [ABSTRACT FROM AUTHOR]

Let X be a real uniformly convex Banach space and C a closed convex nonempty subset of X. Let {Ti}i=1r be a finite family of nonexpansive self-mappings of C. For a given x1 ∈ C, let {xn} and {xn(i)}, i = 1, 2, …, r, be sequences defined xn(0) = xn, xn(1) = an1(1)T1xn(0) + (1 - an1(1)xn(0), xn(2) = an2(2)T2xn(1) + an1(2)T1xn + (1 - an2(2) - an1(2)xn, …, xn+1 + xn(r) = anr(r)Trxn(r-1) + an(r-1)(r)Tr-1xn(r-2) + ⋯ + an1(r)T1xn + (1 - an(r)(r) - an(r-1)(r) - ⋯ - an1(r))xn, n ≥ 1, where ani(j) ∈ [0, 1] for all j ∈ {1, 2, …, r], n ∈ ℕ and i = 1, 2, …, j. In this paper, weak and strong convergence theorems of the sequence {xn} to a common fixed point of a finite family of nonexpansive mappings Ti (i = 1, 2, …, r) are established under some certain control conditions. [ABSTRACT FROM AUTHOR]

This paper deals with some existence results of equilibrium problems (EPΨ) on convex and closed sets (either bounded or unbounded) in Banach spaces. Moreover, an application to the existence of solution for a differential inclusion is given. [ABSTRACT FROM AUTHOR]

The paper deals with the existence, uniqueness, and iterative approximations of solutions for the functional equations arising in dynamic programming ofmultistage decisionmaking processes in Banach spaces BC(S) and B(S) and completemetric space BB(S), respectively. Our main results extend, improve, and generalize the results due to several authors. Some examples are also given to demonstrate the advantage of our results over existing one in the literature. [ABSTRACT FROM AUTHOR]

As a counterpart to best approximation in Banach spaces, the best coapproximation was introduced by Franchetti and Furi (1972). In this paper, we shall consider the relation between coproximinality of a nonempty subset M of a Banach space X and of C(S, M) in C(S, X). [ABSTRACT FROM AUTHOR]

For a locally compact Hausdorff space K and a Banach space X, let C 0 K , X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ , we denote C 0 K , X as C 0 K . If A K be an extremely regular subspace of C 0 K and T : A K ⟶ C 0 S , X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c 0 , then the cardinality of K is less than that of S. Moreover, if T T − 1 < 3 and A K is also a subalgebra of C 0 K , the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λ X > 1 and T T − 1 < λ X , then K is a continuous image of a subspace of S. Here, λ X is the geometrical parameter introduced by Jarosz in 1989: λ X = inf max x + λ y : λ = 1 : x = y = 1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz. [ABSTRACT FROM AUTHOR]

Given the abstract evolution equation y′(t)=Ay(t),  t∈R, with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on R. The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R. [ABSTRACT FROM AUTHOR]

Let E be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let J = {T(t) : t ≥ 0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of E, with functions u, v : [0,∞) → [0,∞). Let F := F(J) = ∩t≥0F(T(t)) ≠ ∭ and f : K → K be a weakly contractive map. For some positive real numbers Λ and δ satisfying δ + Λ > 1, let G : E → E be a δ-strongly accretive and Λ-strictly pseudocontractive map. Let {tn} be an increasing sequence in [0,∞) with limn→∞tn = ∞, and let {αn} and {βn} be sequences in (0, 1] satisfying some conditions. Strong convergence of a viscosity iterative sequence to common fixed points of the family J of uniformly asymptotically regular asymptotically nonexpansive semigroup, which also solves the variational inequality ⟨(G - βf)p, j(p - x)⟩ ≤ 0, for all x ∈ F, is proved in a framework of a real Banach space. [ABSTRACT FROM AUTHOR]

By means of the fixed point index theory of strict-set contraction operator,we study the existence of positive solutions for the multipoint singular boundary value problem (-1)n-ku(n)(t) = f(t, u(t)), 0 < t < 1, n ≥ 2, 1 ≤ k ≤ n-1, u(0) = Σ i=1m-2 aiu(ξi), u(i)(0) = u(j)(1) = θ, 1 ≤ i ≤ k-1, 0 ≤ j ≤ n-k-1 in a real Banach space E, where θ is the zero element of E, 0 < ξ1 < ξ2 < ... < ξm-2 < 1, ai ∈ [0, +∞), i = 1, 2, ... , m - 2. As an application, we give two examples to demonstrate our results. [ABSTRACT FROM AUTHOR]

The concept of tangential for single-valued mappings is extended to multivalued mappings and used to prove the existence of a common fixed point theorem of Gregus type for four mappings satisfying a strict general contractive condition of integral type. Consequently, several known fixed point results generalized and improved the corresponding recent result of Pathak and Shahzad (2009) and many authors. [ABSTRACT FROM AUTHOR]

We establish the general solution for a mixed type functional equation of aquartic and a quadratic mapping in linear spaces. In addition, we investigate the generalized Hyers-Ulam stability in p- Banach spaces. [ABSTRACT FROM AUTHOR]

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E*. Let S={T(s):0≤s<∞} be a nonexpansive semigroup on E such that Fix(S):=⋂t≥0Fix(T(t))≠∅, and f is a contraction on E with coefficient 0<α<1. Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ+λ>1 and γ a positive real number such that γ<1/α(1-1-δ/λ). When the sequences of real numbers {αn} and {tn} satisfy some appropriate conditions, the three iterative processes given as follows: xn+1=αnγf(xn)+(I-αnF)T(tn)xn, n≥0, yn+1=αnγf(T(tn)yn)+(I-αnF)T(tn)yn, n≥0, and zn+1=T(tn)(αnγf(zn)+(I-αnF)zn), n≥0 converge strongly to x̃, where x̃ is the unique solution in Fix(S) of the variational inequality (F-γf)x̃,j(x-x̃)≥0, x∈Fix(S). Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others. [ABSTRACT FROM AUTHOR]

We prove that an operator is weak Dunford-Pettis if its adjoint is one but the converse is false in general, and we give some necessary and sufficient conditions under which each positive weak Dunford-Pettis operator has an adjoint which is weak Dunford-Pettis. [ABSTRACT FROM AUTHOR]

We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others. [ABSTRACT FROM AUTHOR]

We prove strong and weak convergence theorems of modified hybrid proximal-point algorithms for finding a common element of the zero point of a maximal monotone operator, the set of solutions of equilibrium problems, and the set of solution of the variational inequality operators of an inverse strongly monotone in a Banach space under different conditions. Moreover, applications to complementarity problems are given. Our results modify and improve the recently announced ones by Li and Song (2008) and many authors. [ABSTRACT FROM AUTHOR]

We construct a new iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of closed relatively quasinonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the properties of generalized f-projection operator. Using this result, we discuss strong convergence theorem concerning variational inequality and convex minimization problems in Banach spaces. Our results extend many known recent results in the literature. [ABSTRACT FROM AUTHOR]

We treat the classical concept of domain of holomorphy in ℂn when the holomorphic functions considered are restricted to lie in some Banach space. Positive and negative results are presented. A new view of the case n = 1 is considered. [ABSTRACT FROM AUTHOR]

The equation Lu = f, where L = A + B, with A being a K-positive definite operator and B being a linear operator, is solved in a Banach space. Our scheme provides a generalization to the so-called method of moments studied in a Hilbert space by Petryshyn (1962), as well as Lax and Milgram (1954). Furthermore, an application of the inverse function theorem provides simultaneously a general solution to this equation in some neighborhood of a point xo, where L is Fréchet differentiable and an iterative scheme which converges strongly to the unique solution of this equation. [ABSTRACT FROM AUTHOR]

We have shown that if the Toeplitz operator TΦ on the Bergman space La2 (D) belongs to the Schatten class Sp, 1 ≤ p < ∞, then Φ∼ ∈ Lp(D, dλ), where Φ∼ is the Berezin transform of Φ, dλ(z) = dA(z)/(1 - ∣z∣2)2, and dA(z) is the normalized area measure on the open unit disk D. Further, if Φ ∈ Lp(D, dλ) then Φ∼ ∈ Lp(D, dλ) and TΦ ∈ Sp. For certain subclasses of L∞(D), necessary and sufficient conditions characterizing Schatten class Toeplitz operators are also obtained. [ABSTRACT FROM AUTHOR]

We prove path convergence theorems and introduce a new iterative sequence for a countably infinite family of m-accretive mappings and prove strong convergence of the sequence to a common zero of these operators in uniformly convex real Banach space. Consequently, we obtain strong convergence theorems for a countably infinite family of pseudocontractive mappings. Our theorems extend and improve some important results which are announced recently by various authors. [ABSTRACT FROM AUTHOR]

Using the generalized Caristi's fixed point theorems we prove the existence of fixed points for self and nonself multivalued weakly w-contractive maps. Consequently, Our results either improve or generalize the corresponding fixed point results due to Latif (2007), Bae (2003), Suzuki, and Takahashi (1996) and others. [ABSTRACT FROM AUTHOR]

We study a class of fractional stochastic integrodifferential equations considered in a real Hilbert space. The existence and uniqueness of the Mild solutions of the considered problem is also studied.We also give an application for stochastic integropartial differential equations of fractional order. [ABSTRACT FROM AUTHOR]

Let E be a real Banach space, and K a closed convex nonempty subset of E. Let T1, T2, … , Tm : K → K be m total asymptotically nonexpansive mappings. A simple iterative sequence {xn}n≥1 is constructed in E and necessary and sufficient conditions for this sequence to converge to a common fixed point of {Ti}i=1m are given. Furthermore, in the case that E is a uniformly convex real Banach space, strong convergence of the sequence {xn}n=1∞ to a common fixed point of the family {Ti}i=1m is proved. Our recursion formula is much simpler and much more applicable than those recently announced by several authors for the same problem. [ABSTRACT FROM AUTHOR]

We prove a new strong convergence theorem for an element in the intersection of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of some variational inequality problems, and the set of solutions of some equilibrium problems using a new iterative scheme. Our theorem generalizes and improves some recent results. [ABSTRACT FROM AUTHOR]

An existence result of a renormalized solution for a class of nonlinear elliptic equations is established. The diffusion functions a(x, u,∇u) may not be in (Lloc1(Ω))N for a finite value of the unknown and the data belong to L1(Ω). [ABSTRACT FROM AUTHOR]

Existence of common fixed point for two mappings which satisfy a generalized weak contractive condition is established. As a consequence, a common fixed point result for mappings satisfying a contractive condition of integral type is obtained. Our results generalize, extend, and unify several well-known comparable results in literature. [ABSTRACT FROM AUTHOR]

Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, F = {T(h) : h ≥ 0} a generalized asymptotically nonexpansive self-mapping semigroup of K, and f : K → K a fixed contractive mapping with contractive coefficient β ϵ (0, 1). We prove that the following implicit and modified implicit viscosity iterative schemes {xn} defined by xn = αnf(xn) + (1 - αn)T(tn)xn and xn = αnyn + (1 - α)T(tn)xn, yn = ≈nf(xn-1) + (1 - βn)xn-1 strongly converge to p ϵ F as n → ∞ and p is the unique solution to the following variational inequality: ⟨f(p) - p, j(y - p)⟨≤ = 0 for all y ϵ F. [ABSTRACT FROM AUTHOR]

Given an arbitrary measure μ, this study shows that the set of norm attaining multilinear forms is not dense in the space of all continuous multilinear forms on L1(μ). However, we have the density if and only if μ is purely atomic. Furthermore, the study presents an example of a Banach space X in which the set of norm attaining operators from X into X* is dense in the space of all bounded linear operators L(X,X*). In contrast, the set of norm attaining bilinear forms on X is not dense in the space of continuous bilinear forms on X. [ABSTRACT FROM AUTHOR]