P-uniformly Convex Research Articles

Inertial-Type Algorithm for Solving Split Common Fixed Point Problems in Banach Spaces

In this paper, motivated by the works of Kohsaka and Takahashi (SIAM J Optim 19:824–835, 2008) and Aoyama et al. (J Nonlinear Convex Anal 10:131–147, 2009) on the class of mappings of firmly nonexpansive type, we explore some properties of firmly nonexpansive-like mappings [or mappings of type (P)] in p-uniformly convex and uniformly smooth Banach spaces. We then study the split common fixed point problems for mappings of type (P) and Bregman weak relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We propose an inertial-type shrinking projection algorithm for solving the two-set split common fixed point problems and prove a strong convergence theorem. Also, we apply our result to the split monotone inclusion problems and illustrate the behaviour of our algorithm with several numerical examples s. The implementation of the algorithm does not require a prior knowledge of the operator norm. Our results complement many recent results in the literature in this direction. To the best of our knowledge, it seems to be the first to use the inertial technique to solve the split common fixed point problems outside Hilbert spaces.

Generalized split feasibility problem for multi-valued Bregman quasi-nonexpansive mappings in Banach spaces

In this paper, the notion of generalized split feasibility problem (GSFP) is studied in p-uniformly convex Banach spaces. Some special cases of the GSFP are highlighted. A self-adaptive step-size iterative algorithm which converges strongly to solution of the GSFP is proved. The implementation of the method is demonstrated with two numerical examples. Our method does not require prior information of operator norms. Our results extend, improve and enrich recently announced related results in the literature.

A Self-Adaptive Shrinking Projection Method with an Inertial Technique for Split Common Null Point Problems in Banach Spaces

In this paper, we present a new self-adaptive inertial projection method for solving split common null point problems in p-uniformly convex and uniformly smooth Banach spaces. The algorithm is designed such that its convergence does not require prior estimate of the norm of the bounded operator and a strong convergence result is proved for the sequence generated by our algorithm under mild conditions. Moreover, we give some applications of our result to split convex minimization and split equilibrium problems in real Banach spaces. This result improves and extends several other results in this direction in the literature.

Solving a General Split Equality Problem Without Prior Knowledge of Operator Norms in Banach Spaces

In this paper, using Bregman distance, we introduce an iterative algorithm for approximating a common solution of Split Equality Fixed Point Problem and Split Equality Equilibrium Problem in p-uniformly convex and uniformly smooth Banach spaces that are more general than Hilbert spaces. The advantage of the algorithm is that it is done without the prior knowledge of Bregman Lipschitz coefficients and operator norms. The strong convergence of the algorithm is established under mild assumptions. As special cases, we shall utilize our results to study the Split Equality Null point Problems and Split Equality Variational Inequality Problems. A numerical example is given to demonstrate the convergence of the algorithm. Our results complement and extend some related results in the literature.

A common solution of split equality monotone inclusion problem and split equality fixed point problem in real Banach spaces

In this paper, we introduce an iterative algorithm for approximating a common solution of Split Equality Monotone Inclusion Problem (SEMIP) and Split Equality Fixed Point Problem (SEFPP) in p-uniformly convex Banach spaces which are also uniformly smooth. Under standard and mild assumptions of monotonicity and right Bregman strongly condition of the SEMIP- and SEFPP-associated mappings, we establish the strong convergence of the scheme. Finally, we applied our result to study the convex minimization problem (CMP) and Equilibrium Problem. Our result complements and extends some recent results in literature.

Convergence theorems for generalized hemicontractive mapping in p-uniformly convex metric space

Abstract In this paper, we introduce and study an Ishikawa-type iteration process for the class of generalized hemicontractive mappings in 𝑝-uniformly convex metric spaces, and prove both Δ-convergence and strong convergence theorems for approximating a fixed point of generalized hemicontractive mapping in complete 𝑝-uniformly convex metric spaces. We give a surprising example of this class of mapping that is not a hemicontractive mapping. Our results complement, extend and generalize numerous other recent results in CAT(0) spaces.

Δ-Convergence of Products of Operators in p-Uniformly Convex Metric Spaces

In this paper, we first introduce the new notion of p-strongly quasi-nonexpansive maps on p-uniformly convex metric spaces, and then we study the Δ (weak)-convergence of products of p-strongly quasi-nonexpansive maps on p-uniformly convex metric spaces. Furthermore, using the result, we prove the Δ -convergence of the weighted averaged method for projection operators.

Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces

In this paper, we propose and study a multi-step iterative algorithm that comprises of a finite family of asymptotically \begin{document}$ k_i $\end{document} -strictly pseudocontractive mappings with respect to \begin{document}$ p, $\end{document} and a \begin{document}$ p $\end{document} -resolvent operator associated with a proper convex and lower semicontinuous function in a \begin{document}$ p $\end{document} -uniformly convex metric space. Also, we establish the \begin{document}$ \Delta $\end{document} -convergence of the proposed algorithm to a common fixed point of finite family of asymptotically \begin{document}$ k_i $\end{document} -strictly pseudocontractive mappings which is also a minimizer of a proper convex and lower semicontinuous function. Furthermore, nontrivial numerical examples of our algorithm are given to show its applicability. Our results complement a host of recent results in literature.

An iterative method for solving multiple-set split feasibility problems in Banach spaces

"In this paper, we study generalized multiple-set split feasibility problems (in short, GMSSFP) in the frame workof p-uniformly convex real Banach spaces which are also uniformly smooth. We construct an iterative algo-rithm which is free from an operator norm and prove its strong convergence to a solution of GMSSFP, thatis, a solution of convex problem and a common fixed point of a countable family of Bregman asymptoticallyquasi-nonexpansive mappings without requirement for semi-compactness on the mappings. We illustrate ouralgorithm and convergence result by a numerical example. "

Proximal point algorithm involving fixed point of nonexpansive mapping in 𝑝-uniformly convex metric space

AbstractWe prove some important properties of the p-resolvent mapping recently introduced by B. J. Choi and U. C. Ji, The proximal point algorithm in uniformly convex metric spaces, Commun. Korean Math. Soc. 31 2016, 4, 845–855, in p-uniformly convex metric space. Furthermore, we introduce a modified Mann-type PPA involving nonexpansive mapping and prove that the sequence generated by the algorithm converges to a common solution of a finite family of minimization problems, which is also a fixed point of a nonexpansive mapping in the framework of a complete p-uniformly convex metric space.

A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces

In this paper, we introduce a modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and split-equality fixed-point problem for Bregman quasi-nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We introduce a generalized step size such that the algorithm does not require a prior knowledge of the operator norms and prove a strong convergence theorem for the sequence generated by our algorithm. We give some applications and numerical examples to show the consistency and accuracy of our algorithm. Our results complement and extend many other recent results in this direction in literature.

A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems

In this work, we study the split feasibility problem (SFP) in the framework of p-uniformly convex and uniformly smooth Banach spaces. We propose an iterative scheme with inertial terms for seeking the solution of SFP and then prove a strong convergence theorem for the sequences generated by our iterative scheme under implemented conditions on the step size which do not require the computation of the norm of the bounded linear operator. We finally provide some numerical examples which involve image restoration problems and demonstrate the efficiency of the proposed algorithm. The obtained result of this paper complements many recent results in this direction and seems to be the first one to investigate the SFP outside Hilbert spaces involving the inertial technique.

An iterative method with residual vectors for solving the fixed point and the split inclusion problems in Banach spaces

In this paper, we propose an iterative technique with residual vectors for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of a split inclusion problem (SIP) with a way of selecting the stepsizes without prior knowledge of the operator norm in the framework of p-uniformly convex and uniformly smooth Banach spaces. Then strong convergence of the proposed algorithm to a common element of the above two sets is proved. As applications, we apply our result to find the set of common fixed points of a family of mappings which is also a solution of the SIP. We also give a numerical example and demonstrate the efficiency of the proposed algorithm. The results presented in this paper improve and generalize many recent important results in the literature.

Auxiliary Principle for a System of Generalized Set-Valued Nonlinear Mixed Variational-Like Inequalities in Banach Spaces

In this paper, the auxiliary principle technique is extended to study a system of generalized nonlinear mixed variational-like inequalities problem for set-valued mappings without compact values in Banach spaces with p-uniformly convex bidual spaces. First, the existence of the solutions of the related auxiliary problem is proved. Then, a new iterative algorithm based on the system of auxiliary variational inequalities is constructed. Finally, both the existence of the solutions of the original problem and the convergence of the iterative sequences generated by the algorithm are proved. And we also present a numerical example to demonstrate the result. Our results improve and extend some known results.

Convex feasibility problems on uniformly convex metric spaces

ABSTRACTIn this paper, we extend two important notions, weighted average method and Mann's iterative method, in the study of convex feasibility problem for general maps defined on p-uniformly convex metric spaces. Then we prove the Δ-convergence theorems for the weighted average sequence and the Mann's alternating sequence in p-uniformly convex metric spaces.

Proximal-type algorithms for split minimization problem in P-uniformly convex metric spaces

In this paper, we study strong convergence of some proximal-type algorithms to a solution of split minimization problem in complete p-uniformly convex metric spaces. We also analyse asymptotic behaviour of the sequence generated by Halpern-type proximal point algorithm and extend it to approximate a common solution of a finite family of minimization problems in the setting of complete p-uniformly convex metric spaces. Furthermore, numerical experiments of our algorithms in comparison with other algorithms are given to show the applicability of our results.

A halpern-type iteration for solving the split feasibility problem and the fixed point problem of Bregman relatively nonexpansive semigroup in Banach spaces

We study the split feasibility problem (SFP) involving the fixed point problems (FPP) in the framework of p-uniformly convex and uniformly smooth Banach spaces. We propose a Halpern-type iterative scheme for solving the solution of SFP and FPP of Bregman relatively nonexpansive semigroup. Then we prove its strong convergence theorem of the sequences generated by our iterative scheme under implemented conditions. We finally provide some numerical examples and demonstrate the efficiency of the proposed algorithm. The obtained result of this paper complements many recent results in this direction.

Convergence analysis of an iterative method for solving multiple-set split feasibility problems in certain Banach spaces

Our purpose in this paper is to introduce an iterative scheme for solving multiple-set split feasiblity problems in p-uniformly convex Banach spaces which are also uniformly smooth using Bregman distance techniques. We further obtain a strong convergence result for approximating solutions of multiple-set split feasiblity problems in the framework of p-uniformly convex Banach spaces which are also uniformly smooth.

A simultaneous Bregman projection methods for solving mixed split equality problems and fixed point problems in p-uniformly convex and uniformly smooth Banach spaces

A simultaneous Bregman projection methods for solving mixed split equality problems and fixed point problems in p-uniformly convex and uniformly smooth Banach spaces

A Modified Iterative Algorithm for Split Feasibility Problems of Right Bregman Strongly Quasi-Nonexpansive Mappings in Banach Spaces with Applications

In this paper, we present a new iterative scheme for finding a common element of the solution set F of the split feasibility problem and the fixed point set F ( T ) of a right Bregman strongly quasi-nonexpansive mapping T in p-uniformly convex Banach spaces which are also uniformly smooth. We prove strong convergence theorem of the sequences generated by our scheme under some appropriate conditions in real p-uniformly convex and uniformly smooth Banach spaces. Furthermore, we give some examples and applications to illustrate our main results in this paper. Our results extend and improve the recent ones of some others in the literature.