# Approximation of Zeros of M-Accretive Operators; Solutions to Variational Inequality and Generalized Split Feasibility Problems

Main Thesis

Thesis

In this thesis, the problem of solving the equation of the form Au = 0, (0.0.1) where A is a nonlinear map (either mapping a Banach space, E to itself or mapping E to its dual, E ∗ ), is considered. This problem is desirable due to its enormous applications in optimization theory,ecology, economics, signal and image processing, medical imaging, finance, agriculture, engineering, etc. Solving equation (0.0.1) is connected to solving the following problems. • In optimization theory, it is always desirable to find the minimizer of functions. Let f : E → R be a convex and proper function. The subdif ferential associated to f, ∂f : E → 2 E∗ defined by ∂f(x) = {u ∗ ∈ E ∗ : hu ∗ , y − xi ≤ f(y) − f(x) ∀ y ∈ E}. It is easy to check that the subdifferential map ∂f is monotone on E 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 is equivalent to solving for a minimizer of f. In the case where the operator A is single-valued, the inclusion 0 ∈ Au reduces to equation (0.0.1). • The differential equation, du dt +Au = 0, where A is an accretive-type map, describes the evolution of many physical phenomena that generate over time. At equilibrium state, du dt = 0, thus the differential equation reduces to equation (0.0.1). Thus, solution of equation (0.0.1) correspond to equilibrium state of some dynamical system. Moreover, such equilibrium states are very desirable in many applications, e.g., economics, physics, agriculture and so on. • In nonlinear integral equations, the Hammerstein integral equation which is of the form u(x) + ZΩk(x, y)f(y, u(y))dy = w(x), (0.0.2) where Ω ⊂ Rn is bounded, k : Ω × Ω → R and f : Ω × R → R are measurable real-valued functions, and the unknown function u and in homogeneous function w lie in a Banach space E of measurable real-valued functions, can be transformed into the form u+KF u = 0, without loss of generality. Thus, setting A := I + KF, where I is the identity map, will reduce to equation (0.0.1). Interest in Hammerstein integral equations stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear part posses Green’s function can, as a rule, be transformed into the form (0.0.2). Our objectives in this thesis are: studying and constructing new iterative algo rithms; proving that the sequences generated by these algorithms approximate solutions of some nonlinear problems, such as, variational inequality problems, equilibrium problem, convex split feasibility problems, convex minimization problems and so on, and conducting numerical experiments to show the effi ciency of our algorithms. In particular, the following results are proved in this thesis. • Let E be a uniformly smooth and uniformly convex real Banach space and let A : E → 2 E be a multi-valued m-accretive operator with D(A) = E such that the inclusion 0 ∈ Au has a solution. For arbitrary x1 ∈ E, define a sequence {xn} by xn+1 = xn − λnun − λnθn(xn − x1), un ∈ Axn, n ≥ 1. Then the sequence {xn} converges strongly to a solution of the inclusion 0 ∈ Au. • Let E be a uniformly convex and uniformly smooth real Banach space and E ∗ be its dual. Let A : E → E ∗ be a generalized Φ-strongly mono tone and bounded map and let Ti : E → E, i = 1, 2, 3, ..., N be a finite family of quasi-φ-nonexpansive maps such that Q := ∩ Ni=1F(Ti) 6= ∅. Let {xn} be a sequence in E defined iteratively by x1 ∈ E, xn+1 = J −1 (J(T[n]xn) − θnA(T[n]xn)), ∀ n ≥ 1, where T[n] := Tn mod N . Assume V I(A, Q) 6= ∅, then {xn} converges strongly to some x ∗ ∈ V I(A, Q). • Let E be a uniformly smooth and 2-uniformly convex real Banach space. Let C be closed and convex subset of E. Suppose Ai : C → E ∗, i = 1, 2, ..., N is a finite family of monotone and L-Lipschitz continuous maps and the solution set F is nonempty. Then, the sequences {xn}, {y in}, {zin} generated by x0 ∈ E, 0 < λ < 1 L, C0 = C,y i n = ΠCJ−1 (Jxn − λAi(xn)), i = 1, ..., N,Ti n = {v ∈ E : h(Jxn − λAi(xn)) − Jyin, v − y in i ≤ 0},z i n = ΠTin J −1(Jxn − λAi(yin )), i = 1, ..., N, in = argmax{||z i n − xn|| : i = 1, ..., N}, z¯n := zinn , Cn+1 = {v ∈ Cn : φ(v, z¯n) ≤ φ(v, xn)}, xn+1 = ΠCn+1 (x0), n ≥ 0. converge strongly to ΠF x0. • Let K be a closed convex subset of E1. Let E1 and E2 be uniformly smooth and 2-uniformly convex real Banach spaces, and E∗1, E∗2 be their dual spaces respectively. Let A : E1 → E2 be a bounded linear operator whose adjoint is denoted by A∗ and S : E2 → E2 be a nonexpansive map such that F(S) 6= ∅ and T : K → K be a relatively nonexpansive map such that F(T) 6= ∅. Let B : E1 → 2E∗1 be a maximal monotone mapping such that B−10 6= ∅. Then the sequence generated by the following algorithm: for x1 ∈ K arbitrary and βn ∈ (0, 1), yn = J−1 E1 monotone functions. Suppose W := ∩∞ i=1 F(Gi) ∩ ∩∞i=1 A−1i(0) ∩ ∩∞i=1 GMEP(hi, Φi, Bi) 6= ∅ and the sequence {xn} in K is defined iteratively as follows: x0 ∈ K0 = K, yn = ΠKJ −1 (Jxn − λξin ), (ξin ∈ Ain xn), zn = J−1