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

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Algorithms for of Solutions of Equations Involving

ABSTRACT

It is well know 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 , 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

OF CONTENTS

Dedication iii
iv
Abstract vi
1 General introduction 1
General Introduction 1
1.1 Some Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 of zeros of nonlinear mappings of monotonetype
in classical . . . . . . . . . . . . . . . . . 1
1.2 Methods for the Zeros of Nonlinear Mappings of
Accretive-type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Iterative methods for zeros of monotone-type mappings . . . . . . . 7
1.4 of xed points of a nite family of k-strictly pseudo-contractive
mappings in CAT(0) spaces . . . . . . . . . . . . . . . . 8
1.5 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 Equations . . . . . . . . . . . . . . . . . . . . 14
1.8 Approximating solutions of equations of Hammerstein-type . . . . . 16
2 Preliminaries 19
2.1 Duality Mappings and Geometry of Banach s . . . . . . . . . . 19
2.2 Some Nonlinear Functionals and . . . . . . . . . . . . . . 23
2.3 Some Important Results about Geodesic s . . . . . . . . . . . . 27
xii
Abstract xiii
3 Krasnoselskii-Type Algorithm For Zeros of Strongly Monotone
Lipschitz Maps in Classical Banach s 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Convergence in LP spaces, 1

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