Project Materials

Algorithms for Approximation of Solutions of Equations Involving Nonlinear Monotone-Type and Multi-Valued Mappings

ABSTRACT

It is well known that many physically significant problems in different areas of research can be transformed into an equation of the form Au = 0; (0.0.1) where A is a nonlinear monotone operator from a real Banach space E into its dual E. For instance, in optimization, if f : E 􀀀! R [ f+1g is a convex, G^ateaux differentiable function and x is a minimizer of f, then f0(x) = 0.

This gives a criterion for obtaining a minimizer of f explicitly. However, most of the operators that are involved in several significant optimization problems are not differentiable.

For instance, the absolute value function x 7! jxj has a minimizer, which, in fact, is 0. But, the absolute value function is not differentiable at 0. So, in a case where the operator under consideration is not dierentiable, it becomes difficult to know a minimizer even when it exists. Thus, the above characterization only works for
differentiable operators.

A generalization of differentiability called sub-differentiability allows us to recover the above result for non differentiable maps.

For a convex lower semi-continuous function which is not identically +1, the sub-differential of f at x is given by
@f(x) = fx 2 E : hx; y 􀀀 xi f(y) 􀀀 f(x) 8 y 2 Eg: (0.0.2)

Observe that @f maps E into the power set of its dual space, 2E. Clearly, 0 2 @f(x) if and only if x minimizes f. If we set A = @f, then the inclusion problem becomes 0 2 Au which also reduces to (0.0.1) when A is single-valued. In this case, the operator maps E into E. Thus, in this example, approximating zeros of A, is equivalent to the approximation of a minimizer of f.

In chapter three and four of the thesis, we give convergence results for approximating zeros of equation (0.0.1) in Lp spaces, 1 0 such that if n 0 for all n 1.
Then, the sequence fxng1 n=1 converges strongly to a solution of the equation
Ax = 0:
Let E = Lp; 2 p 0 such that if n 0, the sequence
fxng1 n=1 converges strongly to a solution of the equation Ax = 0:

Let K be a nonempty closed convex subset of a complete CAT(0) space X. Let Ti : K ! CB(K); i = 1; 2; : : : ; m; be a family of semi-contractive mappings with constants ki 2 (0; 1); i = 1; 2; : : : ;m, such that
Tm
i=1 F(Ti) 6= ;. Suppose
that Ti(p) = fpg for all p 2
Tn
i=1 F(Ti). For arbitrary x1 2 K, dene a
sequence fxng by
xn+1 = 0xn 1y1n
2y2n
mym
n ; n 1;
where yin
2 Tixn; i = 1; 2; : : : ; m; 0 2 (k; 1); i 2 (0; 1); i = 1; 2; : : : ; m; such
that
Pm
i=0 i = 1, and k := maxfki; i = 1; 2; : : : ;mg. Then, lim
n!1
fd(xn; p)g
exists for all p 2
Tn
i=1 F(Ti), and lim
n!1
d(xn; Tixn) = 0 for all i = 1; 2; : : : ;m.
Let K be a nonempty closed and convex subset of a real Hilbert space H, and
Ti : K ! CB(K) be a countable family of multi-valued ki-strictly pseudo-contractive mappings; ki 2 (0; 1); i = 1; 2; ::: such that
T1
i=1 F(Ti) 6= ;; and
supi1 ki 2 (0; 1). Assume that for p 2
T1
i=1 F(Ti), Ti(p) = fpg: Let fxng1 n=1
be a sequence dened iteratively for arbitrary x0 2 K by
xn+1 = 0xn +
1X
i=1
iyin
;
Abstract ix
where yin
2 Tixn; n 1 and 0 2 (k; 1);
P1
i=0 i = 1 and k := supi1 ki.
Then, limn!1 d(xn; Tixn) = 0, i = 1; 2; ::::
Let E = Lp; 1 0 such
that if n 0 for all n 1, the sequences fung1 n=1 and fvng1 n=1 converge
strongly to u and v, respectively, where u is the solution of u + KFu = 0
with v = Fu.
Let E = Lp; 2 p 0 such that if n 0 for all n 1, the sequences fung1 n=1 and fvng1 n=1 converge strongly to u and v respectively, where u is the solution of u + KFu = 0 with v = Fu

TABLE OF CONTENTS

Dedication iii
Acknowledgements iv
Abstract vi
1 General introduction 1
General Introduction 1
1.1 Some Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Approximation of zeros of nonlinear mappings of monotonetype
in classical Banach spaces . . . . . . . . . . . . . . . . . 1
1.2 Approximation Methods for the Zeros of Nonlinear Mappings of
Accretive-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Iterative methods for zeros of monotone-type mappings . . . . . . . 7
1.4 Approximation of xed points of a nite family of k-strictly pseudo-contractive
mappings in CAT(0) spaces . . . . . . . . . . . . . . . . 8
1.5 Fixed point of multivalued maps . . . . . . . . . . . . . . . . . . . . 10
1.5.1 Game Theory and Market Economy . . . . . . . . . . . . . . 10
1.5.2 Non-smooth Differential Equations . . . . . . . . . . . . . . . 11
1.6 Iterative methods for xed points of some nonlinear multi-valued
mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Hammerstein Integral Equations . . . . . . . . . . . . . . . . . . . . 14
1.8 Approximating solutions of equations of Hammerstein-type . . . . . 16
2 Preliminaries 19
2.1 Duality Mappings and Geometry of Banach Spaces . . . . . . . . . . 19
2.2 Some Nonlinear Functionals and Operators . . . . . . . . . . . . . . 23
2.3 Some Important Results about Geodesic Spaces . . . . . . . . . . . . 27

Abstract xiii
3 Krasnoselskii-Type Algorithm For Zeros of Strongly Monotone
Lipschitz Maps in Classical Banach Spaces 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Convergence in LP spaces, 1