Pure and Applied Mathematics http://repository.aust.edu.ng/xmlui/handle/123456789/364 This collection contains theses in Pure and Applied Mathematics from 2012-2022 2023-12-02T22:02:38Z Algorithms For Approximation of J-Fixed Points of Nonexpansive - Type Maps, Zeros of Monotone Maps, Solutions of Feasibility and Variational Inequality Problems http://repository.aust.edu.ng/xmlui/handle/123456789/5121 Algorithms For Approximation of J-Fixed Points of Nonexpansive - Type Maps, Zeros of Monotone Maps, Solutions of Feasibility and Variational Inequality Problems Nnakwe, Monday Ogudu It is well known that many physically significant problems in different areas of research can be transformed at equilibrium state into an inclusion problem of the form 0 ∈ Au, where A is either a multi-valued accretive map from a real Banach space into itself or a multi-valued monotone map from a real Banach space into its dual space. In several applications, the solutions of the inclusion problem, when the map A is monotone, corresponds to minimizers of some convex functions. It is known that the sub-differential of any convex function, say g, and denoted by ∂g is monotone, and for any vector, say v, in the domain of g, 0 ∈ ∂g(v) if and only if v is a minimizer of g. Setting ∂g ≡ A, solving the inclusion problem, is equivalent to finding minimizers of g. The method of approximation of solutions of the inclusion problem 0 ∈ Au, when the map A is monotone in real Banach spaces, was not known until in 2016 when Chidume and Idu  introduced J-fixed points technique. They proved that the J-fixed points correspond to zerosof monotone maps which are minimizers of some convex functions. In general, finding closed form solutions of the inclusion problem, where A is monotone is extremely difficult or impossible. Consequently, solutions are sought through the construction of iterative algorithms for approximating J-fixed points of nonlinear maps. In chapter three, four and seven of the thesis, we present a convergence result for approximating zeros of the inclusion problem 0 ∈ Au. Let H1 and H2 be real Hilbert spaces and K1, K2, · · · , KN , and Q1, Q2, · · · , QP , be nonempty, closed and convex subsets of H1 and H2, respectively, with nonempty intersections K and Q, respectively, that is, K = K1 ∩ K2 ∩ · · · ∩ KN ̸= ∅ and Q = Q1 ∩ Q2 ∩ · · · ∩ QP ̸= ∅. Let B : H1 → H2 be a bounded linear map, Gi : H1 → H1, i = 1, · · · , N and Aj : H2 → H2, j = 1, · · · , P be given maps. The common split variational inequality problem introduced by vi Censor et al.  in 2005, and denoted by (CSVIP), is the problem of finding an element u ∗ ∈ K for which ( ⟨u − u∗ , Gi(u∗)⟩ ≥ 0, ∀ u ∈ Ki, i = 1, 2, · · · , N, such that ∗ = Bu∗ ∈ Q solves ⟨v − v ∗ , Aj (v∗ )⟩ ≥ 0, ∀ v ∈ Qj , j = 1, 2, · · · , P. The motivation for studying this class of problems with N > 1 stems from a simple observation that if we choose Gi ≡ 0, the problem reduces to finding u ∗ ∈ ∩N i=1Ki , which is the known convex feasibility problem (CFP) such that Bu∗ ∈ ∩P j=1V I(Qj , Aj ). If the sets Ki are the fixed point sets of maps Si : H1 → H1, then, the convex feasibility problems (CFP) is the common fixed points problem(CFPP) whose image under B is a common solution to variational inequality problems (CSVIP). If we choose Gi ≡ 0 and Aj ≡ 0, the problem reduces to finding u ∗ ∈ ∩N i=1Ki such that the point Bu∗ ∈ ∩P j=1Qj which is the well known multiple-sets split feasibility problem or common split feasibility problem which serves as a model for many inverse problems where the constraints are imposed on the solutions in the domain of a linear operator as well as in the range of the operator. A lot of research interest is now devoted to split variational inequality problem and its gener-alizations.In chapter five and six of the thesis, we present convergence theorems for approximating solu-tions of variational inequalities and a convex feasibility problem; and solutions of split varia-tional inequalities and generalized split feasibility problems. Main Thesis 2019-07-05T00:00:00Z Approximation of Zeros of M-Accretive Operators; Solutions to Variational Inequality and Generalized Split Feasibility Problems http://repository.aust.edu.ng/xmlui/handle/123456789/5114 Approximation of Zeros of M-Accretive Operators; Solutions to Variational Inequality and Generalized Split Feasibility Problems Nnyaba, Ukamaka 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 Main Thesis 2019-06-06T00:00:00Z Algorithms for approximation of J-Fixed Points of Nonexpensive-Type Maps, Zeros of Monotone Maps, Solutions of Feasibility and Variational Inequality Problems. http://repository.aust.edu.ng/xmlui/handle/123456789/5112 Algorithms for approximation of J-Fixed Points of Nonexpensive-Type Maps, Zeros of Monotone Maps, Solutions of Feasibility and Variational Inequality Problems. Nnakwe, Monday It is well known that many physically significant problems in different areas of research can be transformed at an equilibrium state into an inclusion problem of the form 0 ∈ Au, where A is either a multi-valued accretive map from a real Banach space into itself or a multi-valued monotone map from a real Banach space into its dual space. In several applications, the solutions of the inclusion problem, when the map A is monotone, corresponds to minimizers of some convex functions. It is known that the sub-differential of any convex function, say g, and denoted by ∂g is monotone, and for any vector, say v, in the domain of g, 0 ∈ ∂g(v) if and only if v is a minimizer of g. Setting ∂g ≡ A, solving the inclusion problem, is equivalent to finding minimizers of g. The method of approximation of solutions of the inclusion problem 0 ∈ Au, when the map A is monotone in real Banach spaces, was not known until in 2016 when Chidume and Idu  introduced J-fixed points technique. They proved that the J-fixed points correspond to zeros of monotone maps which are minimizers of some convex functions. In general, finding closed form solutions of the inclusion problem, where A is monotone is extremely difficult or impossible. Consequently, solutions are sought through the construction of iterative algorithms for approximating J-fixed points of nonlinear maps. In chapter three, four and seven of the thesis, we present a convergence result for approximating zeros of the inclusion problem 0 ∈ Au. Let H1 and H2 be real Hilbert spaces and K1, K2, · · · , KN , and Q1, Q2, · · · , QP , be nonempty, closed and convex subsets of H1 and H2, respectively, with nonempty intersections K and Q, respectively, that is, K = K1 ∩ K2 ∩ · · · ∩ KN ̸= ∅ and Q = Q1 ∩ Q2 ∩ · · · ∩ QP ̸= ∅. Let B : H1 → H2 be a bounded linear map, Gi : H1 → H1, i = 1, · · · , N and Aj : H2 → H2, j = 1, · · · , P be given maps. The common split variational inequality problem introduced by Censor et al.  in 2005, and denoted by (CSVIP), is the problem of finding an element u∗ ∈ K for which(⟨u − u∗, Gi(u∗)⟩ ≥ 0, ∀ u ∈ Ki, i = 1, 2, · · · , N, such that v∗ = Bu∗ ∈ Q solves ⟨v − v∗, Aj (v∗)⟩ ≥ 0, ∀ v ∈ Qj , j = 1, 2, · · · , P. The motivation for studying this class of problems with N > 1 stems from a simple observation that if we choose Gi ≡ 0, the problem reduces to finding u ∗ ∈ ∩N i=1Ki , which is the known convex feasibility problem (CFP) such that Bu∗ ∈ ∩P j=1V I(Qj , Aj ). If the sets Ki are the fixed point sets of maps Si : H1 → H1, then, the convex feasibility problems (CFP) is the common fixed points problem(CFPP) whose image under B is a common solution to variational inequality problems (CSVIP). If we choose Gi ≡ 0 and Aj ≡ 0, the problem reduces to finding u ∗ ∈ ∩N i=1Ki such that the point Bu∗ ∈ ∩P j=1Qj which is the well known multiple-sets split feasibility problem or common split feasibility problem which serves as a model for many inverse problems where the constraints are imposed on the solutions in the domain of a linear operator as well as in the range of the operator. A lot of research interest is now devoted to split variational inequality problem and its generalizations. In chapter five and six of the thesis, we present convergence theorems for approximating solutions of variational inequalities and a convex feasibility problem; and solutions of split variational inequalities and generalized split feasibility problem Main Thesis 2019-07-05T00:00:00Z Iterative Algorithms for Split Equality Fixed Point Problems and Some Non-Linear Problems in Banach Spaces With Applications http://repository.aust.edu.ng/xmlui/handle/123456789/5104 Iterative Algorithms for Split Equality Fixed Point Problems and Some Non-Linear Problems in Banach Spaces With Applications Romanus, Ogonnaya Michael Let H1, H2, and H3 be real Hilbert spaces, T and S be non-linear maps defined on H1 and H2, respectively, and with non-empty fixed point sets, F ix(T) and F ix(S), respectively, A and B be linear maps respectively mapping from H1 and H2 to H3. The split equality fixed point problem (SEFPP) considered in this thesis is to find x ∈ F ix(T), y ∈ F ix(S) such that Ax = By. This problem has attracted the attention of numerous researchers due to its vast applications, for instance, in decomposition methods for partial differential equations (PDEs), applications in game theory, and in intensity-modulated radiation therapy, to mention but a few. Few iterative algorithms have been proposed in real Hilbert spaces for approx imating solutions of the SEFPP when they exist. However, the fact that these algorithms are confined in Hilbert spaces is a restriction, since the models of most real-life problems lives in spaces more general than Hilbert spaces. Besides, to guarantee the convergence of these algorithms, it is necessary to impose some compactness-type conditions on some of the involved mappings. In Chapter 3 of this thesis, we proposed the following iterative algorithm that approximates a solution of SEFPP in certain Banach spaces, in particular, in lp spaces 1 < p ≤ 2.  x1 ∈ E1, y1 ∈ E2, zn ∈ JE3 (Axn − Byn) xn+1 = J −1 E1 Main Thesis 2019-06-06T00:00:00Z