The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo gH-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and uniqueness of two kinds of gH-weak solutions of the coupled system for fuzzy fractional partial differential equations under Lipschitz conditions. Then we give an example to illustrate the correctness of the existence and uniqueness results. Furthermore, because of the coupling in the initial value problems, we develop Gronwall inequality of the vector form, and creatively discuss continuous dependence of the solutions of the coupled system for fuzzy fractional partial differential equations on the initial values and ε-approximate solution of the coupled system. Finally, we propose some work for future research.
It is meaningful and valuable to find common fixed points of different nonexpansive-type operators, which are associated with variational inequalities, integral equations, image process and other optimization problems in real life. The purpose of this paper is to suggest and consider a class of general semi-implicit iterative methods involving semi-implicit rule and inaccurate computing errors, which extend the iterative algorithm introduced by Ali et al. in 2020. Using Liu’s lemma, we analyze convergence and stability of the new iterative approximations for common fixed points of three different nonexpansive-type operators. Furthermore, we provide convergence rates of the new iterations and some numerical examples to illustrate the efficiency and stability of the new iterative schemes. As an application of our main results presented in this paper, we use the proposed iterative schemes to solve the known Stampacchia variational inequality.
Traveling salesman problems (TSPs) are well-known combinatorial optimization problems, and most existing algorithms are challenging for solving TSPs when their scale is large. To improve the efficiency of solving large-scale TSPs, this work presents a novel adaptive layered clustering framework with improved genetic algorithm (ALC_IGA). The primary idea behind ALC_IGA is to break down a large-scale problem into a series of small-scale problems. First, the k-means and improved genetic algorithm are used to segment the large-scale TSPs layer by layer and generate the initial solution. Then, the developed two phases simplified 2-opt algorithm is applied to further improve the quality of the initial solution. The analysis reveals that the computational complexity of the ALC_IGA is between O(nlogn) and O(n2). The results of numerical experiments on various TSP instances indicate that, in most situations, the ALC_IGA surpasses the compared two-layered and three-layered algorithms in convergence speed, stability, and solution quality. Specifically, with parallelization, the ALC_IGA can solve instances with 2×105 nodes within 0.15 h, 1.4×106 nodes within 1 h, and 2×106 nodes in three dimensions within 1.5 h.
As is known to all, Lipschitz condition, which is very important to guarantee existence and uniqueness of solution for differential equations, is not frequently satisfied in real-world problems. In this paper, without the Lipschitz condition, we intend to explore a kind of novel coupled systems of fuzzy Caputo Generalized Hukuhara type (in short, gH-type) fractional partial differential equations. First and foremost, based on a series of notions of relative compactness in fuzzy number spaces, and using Schauder fixed point theorem in Banach semilinear spaces, it is naturally to prove existence of two classes of gH-weak solutions for the coupled systems of fuzzy fractional partial differential equations. We then give an example to illustrate our main conclusions vividly and intuitively. As applications, combining with the relevant definitions of fuzzy projection operators, and under some suitable conditions, existence results of two categories of gH-weak solutions for a class of fire-new fuzzy fractional partial differential coupled projection neural network systems are also proposed, which are different from those already published work. Finally, we present some work for future research.
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