Resonant Problems by Quasilinearization

The purpose of this research is to establish conditions that guarantee the solvability of a resonant third-order boundary value problem (BVP) on the half-line. The resonant problem is subject to integral and m-point boundary conditions while, the kernel of the differential operator is equal to two. The coincidence degree theory will be applied and an illustrative example will be considered to demonstrate the result.

This article focuses on the creation of an existence theorem for a fully nonlinear Hadamard fractional boundary value problem subject to special three-point boundary conditions. By making use of the coincidence degree theory, it is proved that our governing problem makes resonance, that is, the linear part of the differential operator is non-invertible (equally, the corresponding linear problem has at least one nontrivial solution). Constructing some hypotheses on the linear part of the differential operator, nonlinearities and boundary conditions, we give an existence criterion for at least one solution of the fractional-order resonant boundary value problem under study. At the end, a numerical example is presented to illustrate the obtained theoretical results.

Multiple solutions for resonant semilinear elliptic problems in [formula omitted

Let us consider the resonant boundary value problem $$ \begin{align} - &u''(x) - u(x) + g(u(x)) = h(x), \quad x \in [0,\pi], \\ &u(0) = u(\pi) = 0, \end{align} $$ where $ g: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous and $ T$-periodic function with zero mean value, not identically zero, and $ h \in C[0,\pi].$ If each $ h \in C[0,\pi] $ is written as $ h(x) = a\sin x + \tilde{h}(x), $ where $ a \in \mathbb{R} $ and $\int_{0}^{\pi} \tilde{h}(x) \sin x \ dx = 0,$ then, it is shown that for each $ \tilde{h},$ there are real numbers $ a_{1}(\tilde{h}) < 0 < a_{2}(\tilde{h})$ (which depend continuously on $ \tilde{h}$), such that there is solution if and only if $ a\in [a_{1}(\tilde{h}),a_{2}(\tilde{h})].$ In relation to the multiplicity, it is proved that the number of solutions increases to infinity as $ a $ goes to zero. The proof combines different tools such as Liapunov-Schmidt reduction and upper-lower solutions notions, together with a careful analysis of the connected subsets of the solution set of the auxiliary equation in the alternative method, as well as a detailed study of the oscillatory behavior of some integrals associated to the bifurcation equation of the previous problem.

This paper discusses the quasilinearization process, that is, reduction of a given boundary value problem to a quasi-linear one. Quasi-linear BVP has a solution of definite type which is induced by oscillatory properties of the linear part of a system. Conditions are given for a solution of a modified (quasi-linear) BVP to satisfy also the original problem. It is shown that multiple quasilinearizations with different linear parts if possible lead to multiple solutions.

In this article, we deduce the existence of a solution to the weighted fractional differential equation with functional boundary data involving an ω-weighted fractional derivative with Riemann–Liouville settings, D0+α,ψ,ω of order α∈]n−1,n[, on certain weighted Banach spaces when the nonlinear term contains the proportional delay term and fractional derivatives of order (0,1). After carefully defining a few weighted spaces and building a few weighted projection operators, we use Mawhin’s coincidence theory to derive a number of existence results at resonance. Furthermore, our method generalizes some prior results because numerous fractional differential operators are specific instances of the operator D0+α,ψ,ω and represent functional boundary conditions in a highly generic way. Lastly, we illustrate and support our theoretical results with an example.

This paper investigates the problem of joint base station selection and resource allocation in an orthogonal frequency division multiple access (OFDMA) heterogeneous cellular network. The original throughput maximization problem is NP-hard and we propose solving it by using game theoretic stochastic learning approaches. To this end, we first transform the original problem into a tractable form, which has a weighted utility function. Then we prove that an exact potential game applies and it exists the best Nash equilibria which is a near optimal solution of the original problem when an efficient solution method of the weights is employed. To obtain the optimal solution, we redesign the utility function by leveraging a state space to formulate the original problem into an ordinal state based potential game, which is proved that it exists a recurrent state equilibrium point that maximizes system throughput. Furthermore, we propose two different variants of Max-logit learning algorithm based on these two games respectively: one is a simultaneous learning algorithm with less information exchange, which achieves the best Nash equilibria point of the exact potential game and the other is an efficient learning algorithm for the ordinal state based potential game, which can converge to the global optimization solution. Finally, numerical results are given to validate that theoretical findings.

Chapter 2 - The Method of Multipliers for Equality Constrained Problems

Subject.The canonical task of transport logistics in the distributed system "suppliers - consumers" is considered. Goal. Development of an accurate algorithm for solving this problem according to the probabilistic criterion in the assumption of the random nature of transportation costs has been done. Tasks. 1. Development of an exact method for solving the problem of finding a plan that minimizes the total cost of transportation in conditions when their costs are given by their distribution densities. 2. Development of a method for solving the problem when the distribution density of the cost of transportation is not known. Methods. A computational scheme for solving the problem is proposed, which is implemented by an iterative procedure for sequential improvement of the transportation plan. The convergence of this procedure is proved. In order to accelerate the convergence of the computational procedure to the exact solution, an alternative method is proposed based on the solution of a nontrivial problem of fractional nonlinear programming. The method reduces the original complex problem to solving a sequence of simpler problems. The original problem is supplemented by considering a situation that is important for practice when, in the conditions of a small sample of initial data, there is no possibility of obtaining adequate analytical descriptions for the distribution densities of the random costs of transportation. To solve the problem in this case, a minimax method is proposed for finding the best transportation plan in the most unfavorable situation, when the distribution densities of the random cost of transportation are the worst. To find such densities, the modern mathematical apparatus of continuous linear programming was used. Results. A mathematical model and a method for solving the problem of transport logistics in conditions of uncertainty of the initial data are proposed. The desired plan is achieved using the solution of the fractional nonlinear programming problem. Conclusions: The problem of forming a transportation plan is considered, provided that their costs are random values. Also, a method for solving the problem of optimization of transportation for a situation where the density of distribution of random cost cannot be correctly determined is considered.

We show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.

The Method of Fundamental Solution is applied to potential problems. The source and collocation points are supposed to coincide and are located along the boundary. The singularities due to the singularity of the fundamental solution are avoided by several techniques (regularization and desingularization). Both the monopole and the dipole formulations are investigated. The resulting algebraic systems have advantageous properties provided that pure Dirichlet or pure Neumann boundary condition is prescribed. Otherwise, the original problem is converted to a sequence of pure Dirichlet and pure Neumann subproblems, the solutions of which converge rapidly to the solution of the original mixed problem. The iteration is embedded to a multi-level context in a natural way. Thus, the computational cost can be significantly reduced, and the problem of large and ill-conditioned matrices is also avoided.

This paper considers transmit covariance matrix design for secrecy rate maximization problem in a multiple-input single-output (MISO) multicasting simultaneous wireless information and power transfer (SWIPT) system. In order to enhance the performance of the system, artificial noise (AN) is added to the transmit signal in the design for the following purposes: to reduce the received signal-to-noise ratio (SNR) at the eavesdroppers and increase the harvested energy. We assume that all the channel-state-information (CSI) is perfectly known at the transmitter and all legitimate users are capable of simultaneously receiving information and harvesting energy. In addition, all the eavesdroppers are passive and they can harvest energy only when they are not intercepting or eavesdropping the messages intended for the legitimate users. The original secrecy rate maximization problem is not convex in terms of transmit and artificial covariance matrices as well as the power splitting (PS) ratio. In order to circumvent this non-convexity issue, we exploit the Charnes-Cooper Transformation and semidefinite relaxation (SDR) to convert this original problem into a convex one. However, this convex problem does not always yield the rank-one transmit and AN covariance matrices to obtain the solution of the original problem. Therefore, we analyze the optimal conditions and utilize a Gaussian randomization (GR) method to construct the rank-one solutions from the non-rank one results. Simulation results have been provided to demonstrate the performance of the proposed transmit covariance matrices design for MISO multicasting SWIPT system.

This paper considers transmit covariance matrix design for secrecy rate maximization problem in a multiple-input single-output (MISO) multicasting simultaneous wireless information and power transfer (SWIPT) system. In order to enhance the performance of the system, artificial noise (AN) is added to the transmit signal in the design for the following purposes: to reduce the received signal-to-noise ratio (SNR) at the eavesdroppers and increase the harvested energy. We assume that all the channel-state-information (CSI) is perfectly known at the transmitter and all legitimate users are capable of simultaneously receiving information and harvesting energy. In addition, all the eavesdroppers are passive and they can harvest energy only when they are not intercepting or eavesdropping the messages intended for the legitimate users. The original secrecy rate maximization problem is not convex in terms of transmit and artificial covariance matrices as well as the power splitting (PS) ratio. In order to circumvent this non-convexity issue, we exploit the \emph{Charnes-Cooper} Transformation and semidefinite relaxation (SDR) to convert this original problem into a convex one. However, this convex problem does not always yield the rank-one transmit and AN covariance matrices to obtain the solution of the original problem. Therefore, we analyze the optimal conditions and utilize a Gaussian randomization (GR) method to construct the rank-one solutions from the non-rank one results. Simulation results have been provided to demonstrate the performance of the proposed transmit covariance matrices design for MISO multicasting SWIPT system.

A proposed approach is specialized for large scale vehicle routing problems(VRPs) and based on area segmentation and gradual area integration mechanisms so as to avoid combinatorial explosion. The purpose of the proposed approach is to deconstruct large scale problem into small size sub-problems and gradually restore these to original state. Firstly, an original large scale problem is divided into some small sub-areas and optimal solutions in each sub-area are derived. When a best incumbent solution remains unchanged for a certain period, subareas are gradually integrated and new optimal solutions in a new integrated sub-area are newly searched through use of the obtained solutions in previous sub area. This gradual integration and optimization are iterated until every sub-area are integrated into the one (the original problem), and the optimal solution of original problem can be obtained at this time. The proposed approach aims to deconstruct large scale problem into small size sub-problems and perform more efficient search. Through some typical test problems, it was demonstrated that our approach could derive better results more effectively than conventional approach.