Perturbed Operators on Banach Spaces

Glasgow Mathematical Journal, 1982

1. Let X be a Banach space. Then a self-mapping A of X is said to be nonexpansive provided that ‖AX − Ay‖≤‖X − y‖ holds for all x, y ∈ X. The class of nonexpansive mappings includes contraction mappings and is properly contained in the class of all continuous mappings. Keeping in view the fixed point theorems known for contraction mappings (e.g. Banach Contraction Principle) and also for continuous mappings (e.g. those of Brouwer, Schauderand Tychonoff), it seems desirable to obtain fixed point theorems for nonexpansive mappings defined on subsets with conditions weaker than compactness and convexity. Hypotheses of compactness was relaxed byBrowder [2] and Kirk [9] whereas Dotson [3] was able to relax both convexity and compactness by using the notion of so-called star-shaped subsets of a Banach space. On the other hand, Goebel and Zlotkiewicz [5] observed that the same result of Browder [2] canbe extended to mappings with nonexpansive iterates. In [6], Goebel-Kirk-Shimi obtainedfix...

Acta Mathematica Sinica, English Series, 2001

In this paper, we first prove the following best approximation theorems: Let E be a Hausdorff locally convex space and W be a wedge in E. Let D be an open subset of E with QED such that the closure D of D is convex. Suppose that f: B,-t CK(W) is a continuous condensing mapping. Then there exists e0 E B, such that dPD (f(eo), e,) = d,, (f(eo), D,), where pow denotes the Minkowskii function of 0; in E. Moreove:, if d,,, (/(e,,), 6,) > 0, then es E aD,. As a direct consequence, we improve and general&e the main results of Fan, Lin, and Sehgal and Singh. Next, we show several best approximation theorems and fixed point theorems for multivalued k-set-contractive mapping defined on the closed balls and annulus in cones of Banach spaces which generalize the recent results of Lin and Sehgal and Singh.

2021

Let X be a real Banach space with its dual X∗ and G be a nonempty, bounded and open subset of X with 0 ∈ G. Let T : X ⊇ D(T ) → 2 be an m-accretive operator with 0 ∈ D(T ) and 0 ∈ T (0), and let C be a compact operator from X into X with D(T ) ⊆ D(C). We prove that f ∈ R(T ) + R(C) if C is multivalued and f ∈ R(T + C) if C is single-valued, provided Tx+ Cx+ εx 6∋ f for all x ∈ D(T ) ∩ ∂G and ε > 0. The surjectivity of T +C is proved if T is expansive and T +C is weakly coercive. Analogous results are given if T has compact resolvents and C is continuous and bounded. Various results by Kartsatos, and Kartsatos and Liu are improved, and a result by Morales is generalized. 1. Preliminaries In what follows, X and Y denote real Banach spaces, and X denotes the dual space of X. The norm of X will be denoted by ‖ · ‖, and for each x ∈ X and x ∈ X, we denote the value of x at x by either 〈x, x〉 or by 〈x, x〉. These pairings will be understood based on the context in which they are used. W...

2007

In this paper, we obtain some results on Banach operator pair better than those given by J.

Indian Journal of Mathematics

Nonlinear Analysis-theory Methods & Applications, 1980

LET X BE a (real) Banach space and J : X + 2'* the duality mapping defined by J(x) = {jEX*:(x,j) = Ilj11' = 11x11'}.

2020

Let X be a Banach space, B a closed ball centred at origin in X, f : B → X a pseudo contractive mapping i.e. (α−1)‖x− y‖ ≤ ‖(αI−f)(x)− (αI−f)(y)‖ for all x and y in B and α > 1. Here we shown that Mapping f satisfies the property that f(x) = −f(−x) ∀ x in ∂B called antipodal boundary condition assures existence of fixed point of f in B provided that ball B has a fixed point property with respect to non expansive self mapping. Also included some fixed point theorems which involve the Leray-Schauder condition.

2004

Let C((0,T),X) be the Banach space of continuous functions defined in (0,T) and taking values in a Banach space X and let Q be a positive linear operator acting on C((0,T),R). In this article we deal with nonlinear operators A acting on a subset M C((0,T),X) that satisfies the operator Lipschitz condition kAx1(t) Ax2(t)k Qkx1(t) x2(t)k (0 t T,x1,x2 2 M). We are interested in the case when A is a nonlinear integral operator and the operator Lipschiz coecient Q is itself a linear integral operator. If the spectral radius of Q is strictly less than 1, we can use a generalized contraction principle to achieve the existence of an attractive fixed point for A. In particular we focus on the case when Q is a linear Volterra operator, where eective estimates for the spectral radius of Q

The Journal of Nonlinear Sciences and Applications

In this paper, an equilibrium problem which is also known as the Ky Fan inequality is investigated based on a fixed point method. Strong convergence theorems for solutions of the equilibrium problem are established in the framework of reflexive Banach spaces. Applications are also provided to support the main results.

Arabian Journal of Mathematics, 2014

In this paper, we investigate some properties of semi-Fredholm operators on Banach spaces. These results are applied to the determination of the stability of various essential spectra of closed densely defined linear operators. Also, we generalize some results in the literature and we extend and unify those obtained