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

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 [ 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 < p < 1, where the operator A Abstract is Lipschitz strongly monotone and generalised -strongly monotone and bounded maps respectively. As remarked by Charles Byrne [23], most of the maps that arise in image reconstruction and signal processing are non-expansive in nature. A more general class of non-expansive operators is the class of k-strictly pseudo-contractive maps. In chapter of this thesis, we prove some convergence results for a xed point of nite family of k-strictly pseudo-contractive maps in CAT(0) spaces. We also prove a convergence result for a countable family of k-strictly pseudo-contractive maps in Hilbert spaces in chapter six of the thesis. Let Rn be bounded. Let k : ! R and f : R ! R be measurable functions. An integral equation of Hammerstein has the form u(x) + Z k(x; y)f(y; u(y))dy = w; (0.0.3) where the unknown function u and inhomogeneous function w lie in the function space E. In abstract form, the equation (0.0.3) can be written in the form u + AFu = w (0.0.4) where A : E ! E and F : E ! E are monotone operators. In general, every elliptic boundary value problem whose linear part posses a Green’s function (e.g., the problem of forced oscillation of nite amplitude pendulum) can be transformed into an equation of Hammerstein type. Thus, approximating zeros of the Hammerstein-type equation in (0.0.4) (when w = 0) is equivalent to the approximation of solutions of some boundary value problems. Hammerstein equations also play crucial role in variational calculus and xed point theory. In chapter seven of this thesis, we give convergence results for approximating solutions of Hammerstein-type equations in LP spaces, 1 < p < 1. In particular, we prove the following results in this thesis. Let E = Lp; 1 < p < 2. Let A : E ! E be a strongly monotone and Lipschitz map. For x0 2 E arbitrary, let the sequence fxng be dened by: xn+1 = J?1(Jxn ? Axn); n 0; where 2 0; Then, the sequence fxng converges strongly to x 2 A?1(0) and x is unique. Let E= Lp; 2 p < 1. Let A : E ! E be a Lipschitz map. Assume that there exists a constant k 2 (0; 1) such that A satisfies the following condition: Ax ? Ay; x ? y kkx ? yk p p?1 ; Abstract viii and that A?1(0) 6= ;: For arbitrary x0 2 E, dene the sequence fxng iteratively by: xn+1 = J?1(Jxn ? Axn); n 0; where 2 (0; p). Then, the sequence fxng converges strongly to the unique solution of the equation Ax = 0: Let E = Lp; 1 < p < 2. Let A : E ! E be a generalized -strongly monotone and bounded map with A?1(0) 6= ;. For arbitrary x1 2 E, dene a sequence fxng iteratively by: xn+1 = J?1(Jxn ? nAxn); n 1; where fng1 n=1 (0; 1) satises the following conditions: P1 P n=1 n = 1 and 1 n=1 2n < 1. Suppose there exists 0 > 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 < 1. Let A : E ! E be a generalized -strongly monotone and bounded map with A?1(0) 6= ;. For arbitrary x1 2 E, dene a sequence fxng iteratively by: xn+1 = J?1(Jxn ? nAxn); n 1; where fng1 n=1 (0; 1) satises the following conditions: P1 n=1 n = 1 and P1 n=1 p p?1 n < 1. Then, there exists 0 > 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 < p < 2. Let F : E ! E and K : E ! E be strongly monotone and bounded maps. For (u0; v0) 2 E E, dene the sequences fung and fvng in E and E respectively by un+1 = J?1(Jun ? n(Fun ? vn)); n 0; vn+1 = J?1 (Jvn ? n(Kvn + un)); n 0; where fng1 n=1 (0; 1) satises the following conditions: P1 n=1 n = 1, P1 n=1 2n < 1 and P1 n=1 q q?1 n < 1, where q is such that 1 p + 1 q = 1. Assume that the equation u+KFu = 0 has a solution. Then, there exists 0 > 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 < 1. Let F : E ! E and K : E ! E be strongly monotone and bounded maps. For (u0; v0) 2 E E, dene the sequences fung and fvng in E and E, respectively, by un+1 = J?1(Jun ? n(Fun ? vn)); n 0; vn+1 = J?1 (Jvn ? n(Kvn + un)); n 0; where fng1 n=1 (0; 1) satises the following conditions: P1 n=1 n = 1, P1 n=1 2n < 1 and P1 n=1 p p?1 n < 1. Assume that the equation u+KFu = 0 has a solution. Then, there exists 0 > 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
xii
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 < p < 2 . . . . . . . . . . . . . . . . . . 32 3.3 Convergence in Lp spaces, 2 p < 1. . . . . . . . . . . . . . . . . . 33 4 An Algorithm for Computing Zeros of Generalized Phi-Strongly Monotone and bounded Maps in Classical Banach Spaces 35 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Convergence Theorems in Lp spaces, 1 < p < 2 . . . . . . . . . . . . 36 4.3 Convergence Theorems in Lp spaces, 2 p < 1 . . . . . . . . . . . . 38 5 Strong and -Convergence Theorems for Common Fixed Point of a Finite Family of Multivalued Demi-Contractive Mappings in CAT(0) Spaces 41 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6 Convergence Theorem for a Countable Family of Multi-Valued Strictly Pseudo-Contractive Mappings in Hilbert Spaces 47 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7 Approximation of Solutions of Hammerstein Equations with Strongly Monotone and Bounded Operators in Classical Banach Spaces 53 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.2 Convergence Theorems in Lp spaces, 1 < p < 2 . . . . . . . . . . . . 54 7.3 Convergence Theorems in Lp spaces, p 2 . . . . . . . . . . . . . . . 57