Weak and strong convergence theorems for nonspreading type mapping in a hilbert spaces

Thesis Overview

<p> </p><p>INTRODUCTION<br>1.1 General Introduction<br>The content of this thesis falls within the general area of functional analysis, in particular, nonlinear<br>operator theory; an area which has attracted the attention of prominent mathematicians due to its<br>diverse applications in numerous elds of science. In this thesis, we concentrate on an important<br>topic in this area { Weak and strong convergence theorems of nonspreading type mappings in a<br>Hilbert space.<br>In this chapter, the background of our research work will be given; this will reveal how relevant our<br>work is. Then in chapter two, we shall review the research work carried out in the area of research<br>described in this thesis. Some basic denitions and fundamental tools we used in our work will be<br>given in chapter three as preliminaries, while our main results will be presented in chapter four.<br>In chapter ve, conclusions will be given.<br>1.2 Background of study<br>Fixed point theory has been an important area of mathematics due to its applications in several<br>areas of research such as in Optimization, Economics, and Evolution Equations, to mention but a<br>few. For example, consider the problem of nding the equilibrium points of the system described<br>by the following equation<br>du<br>dt<br>+ Au(t) = 0 (1.1)<br>1<br>where A : D(A) H ô€€€! H is a nonlinear map and H a real Hilbert space. At equilibrium,<br>du<br>dt = 0. Thus, the original problem is reduced to the problem of nding solutions of the equation:<br>Au = 0 (1.2)<br>i.e, nding the zeros of A. Several problems arising in Reservoir Engineering, Economics, Physics<br>to mention but a few, can be modelled in the form of equation (1.2). Since generally A is nonlinear,<br>there is no closed form solution of this equation. To solve equation (1.2), where A is a multi-valued<br>monotone map, Felix Browder dened an operator T : H ô€€€! H by T := I ô€€€ A, where I is the<br>identity map on H. He called such T a pseudocontraction. It is easy to see that xed points of<br>T correspond to zeros of A which in turn correspond to equilibrium points of dynamical system<br>described by equation(1.1). As a result of this, the study of xed point theory of pseudocontractive<br>maps and their types has attracted the interest of numerous scientists and researchers.<br>Kohsaka and Takahashi introduced an important class of mappings called the class of nonspread-<br>ing mappings. They obtained a xed point theorem for a single valued nonspreading mapping<br>in Banach space. Furthermore, they obtained a common xed point theorem for a commutative<br>family of nonspreading mappings in Banach space.<br>Let C be a nonempty closed and convex subset of a real smooth, strictly convex and re exive<br>real Banach space, a map T : C ô€€€! C is nonspreading if 8 x; y 2 C<br>(Tx; Ty) + (Ty; Tx) (Tx; y) + (Ty; x): (1.3)<br>where (x; y) = kxk2 ô€€€ 2hx; j(y)i + kyk2; 8 x; y 2 E and J : E ô€€€! 2E<br>dened by<br>Jx = fj(x) 2 E : hx; j(x)i = kxkkj(x)kg; 8x 2 E, where h:; :i is the duality pairing between<br>x 2 E and j(x) 2 E, so that hy; j(x)i = (j(x))(y).<br>We observe that if E is a real Hilbert space, then j is the identity and<br>(x; y) = kxk2 ô€€€ 2hx; yi + kyk2 = kx ô€€€ yk2.<br>Thus, if C is a nonempty closed and convex subset of a real Hilbert space, then<br>T : C ô€€€! C is nonspreading if<br>2kTx ô€€€ Tyk2 kTx ô€€€ yk2 + kTy ô€€€ xk2; 8 x; y 2 C: (1.4)<br>It is shown in Lemma 3.22 that inequality 1.4 is equivalent to<br>kTx ô€€€ Tyk2 kx ô€€€ yk2 + 2hx ô€€€ Tx; y ô€€€ Tyi 8x; y 2 C.<br>2<br>S. Lemoto and W. Takahashi obtained some fundamental properties for nonspreading mapping<br>in a Hilbert space. Furthermore, they studied the approximation of common xed points of non-<br>expansive mappings and nonspreading mappings in a Hilbert space.<br>Y. Kurokawa and W. Takahashi obtained a weak convergence theorem of Bailon’s type for non-<br>spreading mapping in Hilbert space, using an idea of mean convergence they proved a strong<br>convergence theorem for nonspreading mapping in a Hilbert space.<br>Osilike and Isiogugu [Osilike and Isiogugu, 2011] introduced a new class of nonspreading type<br>mappings which is more general than the class studied by Kurokawa and Takahashi. Following the<br>terminology of Browder and Petryshn they called a mapping<br>T : C ô€€€! C k ô€€€ strictly pseudononspreading if there exists k 2 [0; 1) such that<br>kTx ô€€€ Tyk2 kx ô€€€ yk2 + kkx ô€€€ Tx ô€€€ (y ô€€€ Ty)k2 + 2hx ô€€€ Tx; y ô€€€ Tyi 8x; y 2 C (1.5)<br>Our main focus in this thesis, is to review the work done by Osilike and Isiogugu in their paper<br>titled “Weak and strong convergence theorem for nonspreading type mapping in Hilbert space”<br>which appeared in Nonlinear Analysis, Vol 74 (2011), 1814-1822.<br>3</p><p><b>GET THE COMPLETE PROJECT»</b></p> <br><p></p>