An Algorithm for Solutions of Hammerstein Integral Equations with Monotone Operators

An Algorithm for Solutions of Hammerstein Integral Equations with Monotone Operators

ABSTRACT

Let X be a uniformly convex and uniformly smooth real Banach space with dual space X. Let F : X ! X and K : X ! X be bounded monotone mappings such that the Hammerstein equation u + KFu = 0 has a solution in X. An explicit iteration sequence is constructed and proved to converge strongly to a solution of the equation. This is achieved by combining geometric properties of uniformly convex and uniformly smooth real Banach spaces recently introduced by Alber with our method of proof which is also of independent interest.

TABLE OF CONTENTS

certification ii

1 Introduction and literature review 2

1.0.1 Hammerstein equations . . . . . . . . . . . . . . . . . . . . . . 3

1.0.2 Approximation of solutions of Hammerstein integral equations 10

2 PRELIMINARIES 13

2.1 Definition of some terms and concepts. . . . . . . . . . . . . . . . . . 13

2.2 Results of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Some interesting properties of Normalized Duality map . . . . . . . . 19

3 A Strong convergence theorem 21

3.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

CHAPTER ONE

Introduction and literature review

The contents of this thesis fall within the general area of nonlinear operator theory, a flourishing area of research for numerous mathematicians. In this thesis, we concentrate on an important topic in this area-approximation of solutions of nonlinear integral equations of Hammerstein type involving monotone-type mappings.

Let H be a real inner product space. A map A : H ! 2H is called monotone if for each x; y 2 H,

? ; x ? y

0; 8 2 Ax; 2 Ay: (1.1)
If A is single-valued, the map A : H ! H is monotone if

Ax ? Ay; x ? y

0 8 x; y 2 H: (1.2)

Monotone mappings were rst studied in Hilbert spaces by Zarantonello [50], Minty [42], Kacurovskii [37] and a host of other authors. Interest in such mappings stems mainly from their usefulness in numerous applications. Consider, for example, the following:

Example 1. Let g : H ! R [ f1g be a proper convex function. The sub-differential
of g at x 2 H, @g : H ! 2H , is dened by
@g(x) =

x 2 H : g(y) ? g(x)

y ? x; x
8 y 2 H

:
It is easy to check that @g is a monotone operator on H, and that 0 2 @g(u) if and only if u is a minimizer of g. Setting @g A, it follows that solving the inclusion 0 2 Au, in this case, is solving for a minimizer of g. Example 2. Again, let A : H ! H be a monotone map. Consider the evolution equation
du
dt
+ Au = 0: (1.3)
At equilibrium state, du
dt = 0 so that
Au = 0: (1.4)

Consequently, solving the equation Au = 0, in this case, corresponds to solving for the equilibrium state of the system described by (1.3).

Monotone maps also appear in Hammerstein equations. Since this thesis focuses on this class of equations, we give a brief review.
2
3

Chapter 1. Introduction and literature review

1.0.1 Hammerstein equations

Let
Rn be bounded. Let k :

! R and f :
R ! R be measurable real valued functions. An integral equation (generally nonlinear) of Hammerstein-type has the form u(x) +
Z

k(x; y)f(y; u(y))dy = w(x); (1.5)
where the unknown function u and inhomogeneous function w lie in a Banach space
E of measurable real-valued functions. If we dene F : F(
;R) ! F(
;R) and
K : F(
;R) ! F(
;R) by
Fu(y) = f(y; u(y)); y 2
;
and
Kv(x) =
Z

k(x; y)v(y)dy; x 2
;
respectively, where F(
;R) is a space of measurable real-valued functions dened from
to R, then equation (1.5) can be put in the abstract form u + KFu = 0: (1.6)
where, without loss of generality, we have assumed that w 0. The operators K and F are generally of the monotone-type. A closer look at equation (1.6) reveals that it is a special case of (1.4), where
A := I + KF:

Interest in (1.6) stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Green’s function can, as a rule, be transformed into the form (1.5) (see e.g., Pascali and Sburlan [43], chapter IV, p. 164). Among these, we mention the problem of the forced oscillation of nite amplitude of a pendulum.

Consider the problem of the pendulum:
8< : d2v(t) dt2 + a2 sin v(t) = z(t); t 2 [0; 1]; v(0) = v(1) = 0; (1.7) where the driving force z is odd. The constant a (a 6= 0) depends on the length of the pendulum and gravity. Since Green’s function for the problem v00(t) = 0; v(1) = v(0) = 0; is the triangular function K(t; x) = 8< : t(1 ? x); if 0 t x x(1 ? t); if x t 1; (1.8) 4 it follows that problem (1.7) is equivalent to the nonlinear integral equation v(t) = ? Z 1 0 K(t; x)[z(x) ? a2 sin v(x)]dx: (1.9) If g(t) = ? R 1 0 K(t; x)z(x)dx and v(t) + g(t) = u(t); then (1.9) can be written as the Hammerstein equation u(t) = ? Z 1 0 K(t; x)f(x; u(x))dx = 0; (1.10) where f(x; u(x)) = a2sin[u(x) ? g(x)]; (see e.g., [43]) Equations of Hammerstein-type also play a crucial role in the theory of optimal control systems and in automation and network theory (see e.g., Dolezale [33]). Several existence results have been proved for equations of Hammerstein-type (see e.g., Brezis and Browder [4, 5, 6], Browder [7], Browder, De Figueiredo and Gupta [8]). The concept of monotone maps has been extended to arbitrary real normed spaces. There are two well-studied extensions of Hilbert-space monotonicity to arbitrary normed spaces. We brie y explore the two. The rst is the class of accretive operators.




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