Algorithms for Approximation of Solutions of Equations Involving Nonlinear Monotone-Type and Multi-Valued Mappings.
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 ∪ {+∞} is a convex, Gˆateaux differentiable function and x ∗ is a minimizer of f, then f 0 (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→ |x| 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 differentiable, 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 subdifferentiability allows us to recover the above result for non differentiable maps. For a convex lower semi-continuous function which is not identically +∞, the subdifferential of f at x is given by ∂f(x) = {x ∗ ∈ E ∗ : hx ∗ , y − xi ≤ f(y) − f(x) ∀ y ∈ E}. (0.0.2) Observe that ∂f maps E into the power set of its dual space, 2E∗ .
Clearly, 0 ∈ ∂f(x) if and only if x minimizes f. If we set A = ∂f, then the inclusion problem becomes 0 ∈ 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 < p < ∞, where the operator A vi Abstract vii is Lipschitz strongly monotone and generalised Φ-strongly monotone and bounded maps respectively.
INTRODUCTION
The contents of this thesis fall within the general area of nonlinear functional analysis, an area which has attracted the attention of prominent mathematicians due to its diverse applications in numerous fields of sciences. The contributions of this thesis concentrate mainly on the following three important topics.
Namely; • Approximation of zeros of nonlinear monotone mappings in classical Banach spaces. • Approximation of fixed points of a finite family of k-strictly pseudo-contractive mappings in CAT(0) spaces, and a countable family of k-strictly pseudocontractive maps in Hilbert spaces. •
Approximating solutions of Integral equations of Hammerstein-type with monotone operators in Banach spaces.
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, (1.1.1) where A is a nonlinear monotone operator defined on a real Banach space E.
Let H be a real inner product space. A mapping A : D(A) ⊂ H → H is called monotone if for each x, y ∈ D(A), the following inequality holds: hAx − Ay, x − yi ≥ 0, 1 General Introduction 2 and is called strongly monotone if there exists k ∈ (0, 1) such that for all x, y ∈ D(A), the following inequality holds: hAx − Ay, x − yi ≥ kkx − yk 2 .
Monotone mappings were studied in Hilbert spaces by Zarantonello [118], Minty [83], Kaˇcurovskii [69] and a host of other authors.
Interest in such mappings stems mainly from their usefulness in numerous applications. Consider, for example, the following: Let f : H → R ∪ {∞} be a proper convex function. The sub-differential of f at x ∈ H is defined by ∂f(x) = x ∗ ∈ H : f(y) − f(x) ≥ y − x, x∗ ∀ y ∈ H .
It is easy to check that ∂f : H → 2 H is a monotone operator on H, and that 0 ∈ ∂f(x) if and only if x is a minimizer of f. Setting ∂f ≡ A, it follows that solving the inclusion 0 ∈ Au, in this case, is solving for a minimizer of f. In a case where the operator A is single valued, the inclusion 0 ∈ Au reduces to equation (1.1.1).
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