PhD Theses
http://repository.aust.edu.ng/xmlui/handle/123456789/353
This sub-community contains all PhD thesis of the five streams offered at AUST2023-10-01T07:55:17ZAlgorithms 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 [52] 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. [32] 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
2019-07-05T00:00:00ZApproximation 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
2019-06-06T00:00:00ZBiomass Valorization: Assessment and Characterization of Biomass Waste for Valuable Products.
http://repository.aust.edu.ng/xmlui/handle/123456789/5113
Biomass Valorization: Assessment and Characterization of Biomass Waste for Valuable Products.
Ezealigo, Uchechukwu
Research in the use of biomass residues has a huge interest as their potentials span a wide range of applications. Processed residues are useful as green energy such as biofuel (pellets and briquettes), animal feed, antioxidants, and activated charcoal for filtration and even carbon capture. With this in mind, my doctoral research covers the assessment of biomass residues generated in Nigeria for bioenergy. Also, Ficus benjamina fruit, identified as a biomass waste, was characterized for its value addition in bioenergy application. The latter fruit was further characterized for its value as a potential feed substrate for animals as well as the chemical source for industrial applications. The results from the research within this framework include the following. First, a proper bio-resource assessment, particularly, biomass residues availability and potential were investigated. This is a key requirement for an efficient and functional bioenergy sector in Nigeria, proposing to generate biofuel from agro-waste materials. In this study, computational
and analytical approaches with mild assumptions were employed to evaluate the bioenergy potential in agricultural residues, including municipal solid and liquid waste. This assessment was performed using data from 2008 to 2018. The available technical potential of 84 Mt yielded cellulosic ethanol and biogas of 14,766 ML/yr (8 Mtoe) and 15,014 Mm3/yr (13 Mtoe), respectively. The residues gave more biogas than cellulosic ethanol from the same amount of residue potential. The energy potential from residues in Nigeria may be tailored towards biogas production for diverse applications ranging from heat to electric power generation and therefore holds great potential in solving the current electricity crisis in Nigeria. It will also position the nation towards achieving the 7th sustainable development goal (SDG 7) on clean and affordable energy Secondly, having identified that some residues may be limited in supply due to seasonality and
multiple applications for various purposes, there is a need to continue a search for more plant waste that is resourceful as a potential feedstock. Ficus benjamina (FB) is an ornamental plant that produces nonedible fruits considered as waste. These fruits have no defined application, hence, identifying the potential in these fruits for possible valorization is necessary. Detailed preliminary characterization was performed to determine its suitability as a biofuel feedstock. The whole fruit (pulverized) was characterized by scanning electron microscopy (SEM), energy dispersive X-ray (EDS), thermogravimetric analysis (TGA), Fourier transform infrared
spectroscopy (FTIR), X-ray diffraction (XRD), and bomb calorimeter. In addition, the physical, thermal, and chemical properties of FB fruits for potential biofuel application was determined using the proximate and ultimate analyses. Pulverized Ficus benjamina fruits
(PFB) have a porous morphology that makes them less dense and a crystallinity index of 25.5%. The moisture, ash, volatile matter, and fixed carbon contents were 9.29, 6.26, 64.35, and 20.10%, respectively. The higher and lower heating values are 19.74 and 18.55 MJ/kg, respectively, and are comparable to other biomass feedstock. The results establish the possibility of using PFB as a solid biofuel.
Thirdly, another possible approach in valorizing FB fruit focuses on other value products and benefits for livelihood. On this basis, the nutritional analysis, as well as the identification and quantification of micro and macro-nutrients and amino acid profile, were performed. HPLC and GC-MS were used to investigate the sugar profile of the water extract and the chemical content on the extracts obtained with solvents (ethanol, n-hexane, and ethyl-acetate), respectively. Found in FB fruits were: eighteen (18) amino acids, diverse micro- and macro mineral content, metabolizable sugars (such as galactose and glucose), and other chemicals,
including phytochemicals. In addition, these fruits showed low anti-nutritional factors such as phytate and tannins. From these findings, FB fruits offer diverse biological potential and functions and may be a prospective bio-resource for animal feed. The high fiber content reveals rich lignocellulose for bowel bulkiness. This result indicates that the fruits of FB can offer health benefits and can serve as a biomaterial. Thus, FB fruits may possess the potential as an additive material for animal feed, and phytochemicals for industrial and pharmaceutical uses.
Main Thesis
2022-04-15T00:00:00ZAlgorithms 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 [52] 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. [32] 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