Fixed Point Theory and Algorithms for Sciences and Engineering

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EISSN : 2730-5422
Total articles ≅ 11
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, Sümeyra Uçar, Necati Özdemir
Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2021, pp 1-13; doi:10.1186/s13663-021-00696-2

Abstract:
Nonlinear fractional differential equations have been intensely studied using fixed point theorems on various different function spaces. Here we combine fixed point theory with complex analysis, considering spaces of analytic functions and the behaviour of complex powers. It is necessary to study carefully the initial value properties of Riemann–Liouville fractional derivatives in order to set up an appropriate initial value problem, since some such problems considered in the literature are not well-posed due to their initial conditions. The problem that emerges turns out to be dimensionally consistent in an unexpected way, and therefore suitable for applications too.
Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2021, pp 1-31; doi:10.1186/s13663-021-00695-3

Abstract:
This paper proposes a stochastic approximation method for solving a convex stochastic optimization problem over the fixed point set of a quasinonexpansive mapping. The proposed method is based on the existing adaptive learning rate optimization algorithms that use certain diagonal positive-definite matrices for training deep neural networks. This paper includes convergence analyses and convergence rate analyses for the proposed method under specific assumptions. Results show that any accumulation point of the sequence generated by the method with diminishing step-sizes almost surely belongs to the solution set of a stochastic optimization problem in deep learning. Additionally, we apply the learning methods based on the existing and proposed methods to classifier ensemble problems and conduct a numerical performance comparison showing that the proposed learning methods achieve high accuracies faster than the existing learning method.
, Ariel Nisenbaum
Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2021, pp 1-21; doi:10.1186/s13663-021-00694-4

Abstract:
String-averaging is an algorithmic structure used when handling a family of operators in situations where the algorithm in hand requires to employ the operators in a specific order. Sequential orderings are well known, and a simultaneous order means that all operators are used simultaneously (in parallel). String-averaging allows to use strings of indices, constructed by subsets of the index set of all operators, to apply the operators along these strings, and then to combine their end-points in some agreed manner to yield the next iterate of the algorithm. String-averaging methods were discussed and used for solving the common fixed point problem or its important special case of the convex feasibility problem. In this paper we propose and investigate string-averaging methods for the problem of best approximation to the common fixed point set of a family of operators. This problem involves finding a point in the common fixed point set of a family of operators that is closest to a given point, called an anchor point, in contrast with the common fixed point problem that seeks any point in the common fixed point set. We construct string-averaging methods for solving the best approximation problem to the common fixed points set of either finite or infinite families of firmly nonexpansive operators in a real Hilbert space. We show that the simultaneous Halpern–Lions–Wittman–Bauschke algorithm, the Halpern–Wittman algorithm, and the Combettes algorithm, which were not labeled as string-averaging methods, are actually special cases of these methods. Some of our string-averaging methods are labeled as “static” because they use a fixed pre-determined set of strings. Others are labeled as “quasi-dynamic” because they allow the choices of strings to vary, between iterations, in a specific manner and belong to a finite fixed pre-determined set of applicable strings. For the problem of best approximation to the common fixed point set of a family of operators, the full dynamic case that would allow strings to unconditionally vary between iterations remains unsolved, although it exists and is validated in the literature for the convex feasibility problem where it is called “dynamic string-averaging”.
A. U. Bello, M. T. Omojola, J. Yahaya
Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2021, pp 1-22; doi:10.1186/s13663-021-00691-7

Abstract:
Let H be a real Hilbert space. Let $F:H\rightarrow 2^{H}$ F : H → 2 H and $K:H\rightarrow 2^{H}$ K : H → 2 H be two maximal monotone and bounded operators. Suppose the Hammerstein inclusion $0\in u+KFu$ 0 ∈ u + K F u has a solution. We construct an inertial-type algorithm and show its strong convergence to a solution of the inclusion. As far as we know, this is the first inertial-type algorithm for Hammerstein inclusions in Hilbert spaces. We also give numerical examples to compare the new algorithm with some existing ones in the literature.
Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2021, pp 1-15; doi:10.1186/s13663-021-00693-5

Abstract:
This paper studies uniqueness of solutions for a nonlinear Hadamard-type integro-differential equation in the Banach space of absolutely continuous functions based on Babenko’s approach and Banach’s contraction principle. We also include two illustrative examples to demonstrate the use of main theorems.
Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2021, pp 1-30; doi:10.1186/s13663-021-00692-6

Abstract:
In this paper, we introduce and study a modified multi-step Noor iterative procedure with errors for two Lipschitz strictly hemicontractive-type mappings in arbitrary Banach spaces and constitute its convergence and stability. The obtained results in this paper generalize and extend the corresponding result of Hussain et al. (Fixed Point Theory Appl. 2012:160, 2012) and some analogous results of several authors in the literature. Finally, a numerical example is included to illustrate our analytical results and to display the efficiency of our proposed novel iterative procedure with errors.
Konrawut Khammahawong, Poom Kumam, Parin Chaipunya, Somyot Plubtieng
Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2021, pp 1-20; doi:10.1186/s13663-021-00689-1

Abstract:
We propose Tseng’s extragradient methods for finding a solution of variational inequality problems associated with pseudomonotone vector fields in Hadamard manifolds. Under standard assumptions such as pseudomonotone and Lipschitz continuous vector fields, we prove that any sequence generated by the proposed methods converges to a solution of variational inequality problem, whenever it exits. Moreover, we give some numerical experiments to illustrate our main results.
Karim Chaira, Abderrahim Eladraoui, Mustapha Kabil, Abdessamad Kamouss
Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2021, pp 1-15; doi:10.1186/s13663-021-00690-8

Abstract:
In this paper, we investigate the existence and the uniqueness of a common fixed point of a pair of self-mappings satisfying new contractive type conditions on a modular metric space endowed with a reflexive digraph. An application is given to show the use of our main result.
Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2021, pp 1-9; doi:10.1186/s13663-020-00685-x

Abstract:
In the literature there are several methods for comparing two convergent iterative processes for the same problem. In this note we have in view mostly the one introduced by Berinde in (Fixed Point Theory Appl. 2:97–105, 2004) because it seems to be very successful. In fact, if IP1 and IP2 are two iterative processes converging to the same element, then IP1 is faster than IP2 in the sense of Berinde. The aim of this note is to prove this almost obvious assertion and to discuss briefly several papers that cite the mentioned Berinde’s paper and use his method for comparing iterative processes.
Zhiqun Xue, Guiwen Lv
Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2021, pp 1-13; doi:10.1186/s13663-021-00688-2

Abstract:
In this paper, we obtain a new convergence theorem for fixed points of weak contractions in Branciari type generalized metric spaces under weaker conditions. The proof process of the theorem is new and different from that of other authors. An illustrative example of this theorem is to show how the new conditions extend known results.
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