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 re- search 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, Gˆteaux a differentiable function and x∗ is a minimizer of f , then f (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 → |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 sub-differentiability allows us to recover the above result for non differentiable maps. For a convex lower semi-continuous function which is not identically +∞, the sub-differential of f at x is given by ∂f (x) = {x∗ ∈ E ∗ : x∗ , y − x ≤ 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 is Lipschitz strongly monotone and generalised Φ-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 nonexpansive in nature. A more general class of nonexpansive operators is the class of k-striclty pseudo-contractive maps. In chapter five of this thesis, we prove some convergence results for a fixed point of finite family of k-striclty pseudo-contractive maps in CAT (0) spaces. We also prove a convergence result for a countable family of k-striclty pseudo-contractive maps in Hilbert spaces in chapter six of the thesis. Let Ω ⊂ Rn be bounded. Let k : Ω × Ω → R and f : Ω × R → R be measurable functions. An integral equation of Hammerstein has the form k(x, y)f (y, u(y))dy = w, u(x) + (0.0.3) Ω 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 + AF u = 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 finite 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 fixed point theory. In chapter seven of this thesis, we give convergence results for approximating solutions of Hammerstein-type equations in LP spaces, 1 < p < ∞. 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 ∈ E arbitrary, let the sequence {xn } be defined by: xn+1 = J −1 (Jxn − λAxn ), n ≥ 0, where λ ∈ 0, δ . Then, the sequence {xn } converges strongly to x∗ ∈ A−1 (0) and x∗ is unique. • Let E= Lp , 2 ≤ p < ∞. Let A : E → E ∗ be a Lipschitz map. Assume that there exists a constant k ∈ (0, 1) such that A satisfies the following condition: Ax − Ay, x − y ≥ k x − y p p−1, and that A−1 (0) = ∅. For arbitrary x0 ∈ E, define the sequence {xn } iteratively by: xn+1 = J −1 (Jxn − λAxn ), n ≥ 0, where λ ∈ (0, δp ). Then, the sequence {xn } converges strongly to the unique solution of the equation Ax = 0. • Let E = Lp , 1 < p < 2. Let A : E → E ∗ be a generalized Φ-strongly monotone and bounded map with A−1 (0) = ∅. For arbitrary x1 ∈ E, define a sequence {xn } iteratively by: xn+1 = J −1 (Jxn − αn Axn ), n ≥ 1, ∞ where {αn }∞ ⊂ (0, 1) satisfies the following conditions: n=1 n=1 αn = ∞ and ∞ 2 < ∞. Suppose there exists γ > 0 such that if α ≤ γ for all n ≥ 1.0 n 0 n=1 αn Then, the sequence {xn }∞ converges strongly to a solution of the equation n=1 Ax = 0. • Let E = Lp , 2 ≤ p < ∞. Let A : E → E ∗ be a generalized Φ-strongly monotone and bounded map with A−1 (0) = ∅. For arbitrary x1 ∈ E, define a sequence {xn } iteratively by: xn+1 = J −1 (Jxn − αn Axn ), n ≥ 1, where {αn }∞ ⊂ (0, 1) satisfies the following conditions: ∞ n=1 n=1 αn ∞ γ0 , the sequence < ∞. Then, there exists γ0 > 0 such that if αn ≤ = 0. n=1 αn {xn }∞ converges strongly to a solution of the equation Ax n=1 p p−1= ∞ and • 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 demi-contractive mappings with constants ki ∈ (0, 1), i = 1, 2, . . . , m, such that m F (Ti ) = ∅. Suppose i=1 that Ti (p) = {p} for all p ∈ n F (Ti ). For arbitrary x1 ∈ K, define a i=1 sequence {xn } by 1 2 m xn+1 = α0 xn ⊕ α1 yn ⊕ α2 yn ⊕ · · · ⊕ αm yn , n ≥ 1, i where yn ∈ Ti xn , i = 1, 2, . . . , m, α0 ∈ (k, 1), αi ∈ (0, 1), i = 1, 2, . . . , m, such m that i=0 αi = 1, and k := max{ki , i = 1, 2, . . . , m}. Then, lim {d(xn , p)} exists for all p ∈ n i=1 F (Ti ), n→∞ and lim d(xn , Ti xn ) = 0 for all i = 1, 2, . . . , m. n→∞ • 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 ∈ (0, 1), i = 1, 2, ... such that ∞ F (Ti ) = ∅; and i=1 supi≥1 ki ∈ (0, 1). Assume that for p ∈ ∞ F (Ti ), Ti (p) = {p}. Let {xn }∞ n=1 i=1 be a sequence defined iteratively for arbitrary x0 ∈ K by ∞ i λ i yn , xn+1 = λ0 xn + i=1 where yn ∈ Ti xn , n ≥ 1 and λ0 ∈ (k, 1); Then, limn→∞ d(xn , Ti xn ) = 0, i = 1, 2, ....∞ i=0 λi = 1 and k := supi≥1 ki . • Let E = Lp , 1 < p < 2. Let F : E → E ∗ and K : E ∗ → E be strongly monotone and bounded maps. For (u0 , v0 ) ∈ E × E ∗ , define the sequences {un } and {vn } in E and E ∗ respectively by un+1 = J −1 (Jun − αn (F un − vn )), n ≥ 0, −1 vn+1 = J∗ (J∗ vn − αn (Kvn + un )), n ≥ 0, ∞ n=1 αn where {αn }∞ ⊂ (0, 1) satisfies the following conditions: n=1 ∞ 2 n=1 αn q q−1 ∞ n=1 αn 1 p = ∞,1 q < ∞ and < ∞, where q is such that + = 1. Assume that the equation u + KF u = 0 has a solution. Then, there exists γ0 > 0 such that if αn ≤ γ0 for all n ≥ 1, the sequences {un }∞ and {vn }∞ converge n=1 n=1 strongly to u∗ and v ∗ , respectively, where u∗ is the solution of u + KF u = 0 with v ∗ = F u∗ . • Let E = Lp , 2 ≤ p < ∞. Let F : E → E ∗ and K : E ∗ → E be strongly monotone and bounded maps. For (u0 , v0 ) ∈ E × E ∗ , define the sequences {un } and {vn } in E and E ∗ , respectively, by un+1 = J −1 (Jun − αn (F un − vn )), n ≥ 0, −1 vn+1 = J∗ (J∗ vn − αn (Kvn + un )), n ≥ 0, where {αn }∞ ⊂ (0, 1) satisfies the following conditions: n=1 p p−1 ∞ n=1 αn = ∞, < ∞ and ∞ αn < ∞. Assume that the equation u + KF u = 0 n=1 has a solution. Then, there exists γ0 > 0 such that if αn ≤ γ0 for all n ≥ 1, the sequences {un }∞ and {vn }∞ converge strongly to u∗ and v ∗ respectively, n=1 n=1 where u∗ is the solution of u + KF u = 0 with v ∗ = F u∗ .