Smooth Real Banach Space Research Articles

Inertial Bilevel Variational Monotone Inclusion Problem in Banach Spaces

In this paper, we study accelerated Halpern-type iterative method for finding zero solution of sum of two monotone operators which solves variational inequality problem of inverse strongly monotone mapping in 2-uniformly convex and uniformly smooth real Banach spaces. The strong convergence of our proposed method is establish under some standard conditions imposed on parameters. Our theorems generalize many recently announced results in the literature.

Halpern inertial subgradient extragradient algorithm for solving equilibrium problems in Banach spaces

ABSTRACT In this work, we study an equilibrium problem involving a pseudomonotone bifunction in the context of uniformly smooth and 2-uniformly convex real Banach spaces. For the purpose of solving the fixed point problem of a finite family of multi-valued relatively nonexpansive mappings and the pseudomonotone equilibrium problem, we introduce an inertial subgradient extragradient method and establish its strong convergence. To increase the step size and enhance the performance of our iterative method, we use a new parameter that is independent of the inertial extrapolation method. We point out that our step size is chosen in a self-adaptive manner, making it simple to calculate our iterative algorithm without having to know the Lipschitz constants beforehand. Finally, we give some numerical results for the proposed algorithm and comparison with some other known algorithms.

APPROXIMATION METHOD FOR COMMON FIXED POINT OF A COUNTABLE FAMILY OF MULTI-VALUED QUASI-ϕ-NONEXPANSIVE MAPPINGS IN BANACH SPACES

In this paper, we study a new iterative algorithm to approximate a common fixed point of acountable family of multi-valued quasi-φ nonexpansive mappings in a uniformly smooth and uniformlyconvex real Banach space. We prove strong convergence theorem which improve and generalize recentlyannounced results in the literature. We also give some example to illustrate our theorem and we alsogive application to zero point problem of maximal monotone mappings

Inertial split projection and contraction method for pseudomonotone variational inequality problem in Banach spaces

In this article, we introduce a Halpern iterative method together with an inertial projection and contraction method for finding an approximate solution of variational inequality problem involving pseu- domonotone mapping which also solves split common fixed point problem of Bregman demigeneralized map- ping and Bregman strongly nonexpansive mapping in the framework of p-uniformly convex and uniformly smooth real Banach spaces. Using our iterative method, we establish a strong convergence result for approxi- mating the solution of the aforementioned problems and state some consequences of our main result. The result discussed in this article extends and complements many related results in the literature.

Inertial Method for Solving Pseudomonotone Variational Inequality and Fixed Point Problems in Banach Spaces

In this paper, we introduce a new iterative method that combines the inertial subgradient extragradient method and the modified Mann method for solving the pseudomonotone variational inequality problem and the fixed point of quasi-Bregman nonexpansive mapping in p-uniformly convex and uniformly smooth real Banach spaces. Under some standard assumptions imposed on cost operators, we prove a strong convergence theorem for our proposed method. Finally, we perform numerical experiments to validate the efficiency of our proposed method.

Inertial Halpern-type method for solving split feasibility and fixed point problems via dynamical stepsize in real Banach spaces

AbstractIn this paper, we introduce a modified Halpern inertial method for approximating solutions of split feasibility problem and fixed point problem of Bregman strongly nonexpansive mappings in the framework of p-uniformly convex and uniformly smooth real Banach spaces. We establish a strong convergence result for the sequence generated by our iterative scheme under some mild conditions without the computation of the operator norm. We state some consequences and present some examples to show the efficiency and implementation of our proposed method. The result discussed in this paper extends and generalizes many recent results in this direction. Our result extends and complements some related results in literature.

A Modified Algorithms for New Krasnoselskii's Type for Strongly Monotone and Lipschitz Mappings

Let E be a 2 uniformly smooth and convex real Banach space and let a mapping A: E → E∗ be lipschitz and strongly monotone such that A−1(0)≠∅. For an arbitrary ({x1}, {y1})∈E, we define the sequences {xn} and {yn} byyn = xn − θnJ−1(Axn), n≥1xn+1 = yn − λnJ−1(Ayn), n≥1where λn and θn are positive real number and J is the duality mapping of E. Letting (λn, θn)∈(0, 1), then xn and yn converges strongly to ρ∗, a unique solution of the equation Ax = 0. We also applied our algorithm in convex minimization and also proved the convergence of it in Lp, lp or Wm,p. At the end we proposed the algorithm of it in Lp(Ω) and its inverse Lq(Ω).

A generalized Halpern-type forward-backward splitting algorithm for solving variational inclusion problems

<abstract><p>In this paper, we investigate the problem of finding a zero of sum of two accretive operators in the setting of uniformly convex and $ q $-uniformly smooth real Banach spaces ($ q > 1 $). We incorporate the inertial and relaxation parameters in a Halpern-type forward-backward splitting algorithm to accelerate the convergence of its sequence to a zero of sum of two accretive operators. Furthermore, we prove strong convergence of the sequence generated by our proposed iterative algorithm. Finally, we provide a numerical example in the setting of the classical Banach space $ l_4(\mathbb{R}) $ to study the effect of the relaxation and inertial parameters in our proposed algorithm.</p></abstract>

AN ACCELERATED HALPERN-TYPE ALGORITHM FOR SOLVING VARIATIONAL INCLUSION PROBLEMS WITH APPLICATIONS

An inertial Halpern-type forward-backward iterative algorithm for approximating a zero of sum of two accretive operators is introduced and studied. Strong convergence theorem is established in a uniformly convex and q-uniformly smooth real Banach space. The convergence result obtained is applied to convex minimization and image restoration problems. Furthermore, numerical experiments are carried out on some classical test images and personal images degraded with motion blur and random noise. Finally, numerical illustrations in the Banach space, L5([−1, 1]) are presented to support the main theorem.

An iterative method involving a class of quasi-phi-nonexpansive mappings for solving split equality fixed point problems

"A new inertial iterative algorithm for approximating solution of split equality fixed point problem (SEFPP) for quasi-$\phi$- nonexpansive mappings is introduced and studied in $p$-uniformly convex and uniformly smooth real Banach spaces, $p>1$. A strong convergence theorem is proved without imposing any compactness-type condition on the mappings. Our theorems complement several important recent results that have been proved in 2-uniformly convex and uniformly smooth real Banach spaces. It is well known that these spaces do not include $L_p, l_p $ and the Sobolev spaces ${W^m}_p(\Omega)$, for $2< p < \infty$. Our theorems, in particular, are applicable in these spaces. Furthermore, application of our theorem to split equality variational inclusion problem is presented. Finally, numerical examples are presented to illustrate the convergence of our algorithms."

Self-adaptive forward–backward splitting algorithm for the sum of two monotone operators in Banach spaces

In this work, we prove the weak convergence of a one-step self-adaptive algorithm to a solution of the sum of two monotone operators in 2-uniformly convex and uniformly smooth real Banach spaces. We give numerical examples in infinite-dimensional spaces to compare our result with some existing algorithms. Finally, our results extend and complement several existing results in the literature.

Geometric inequalities in real Banach spaces with applications

"In this paper, new geometric inequalities are established in real Banach spaces. As an application, a new iterative algorithm is proposed for approximating a solution of a split equality fixed point problem (SEFPP) for a quasi-$\phi$-nonexpansive semigroup. It is proved that the sequence generated by the algorithm converges {\it strongly} to a solution of the SEFPP in $p$-uniformly convex and uniformly smooth real Banach spaces, $p>1$. Furthermore, the theorem proved is applied to approximate a solution of a variational inequality problem. All the theorems proved are applicable, in particular, in $L_p$, $l_p$ and the Sobolev spaces, $W_p^m(\Omega)$, for $p$ such that $2<p<\infty.$ "

On Birkhoff–James orthogonality preservers between real non-isometric Banach spaces

We present a short proof for the fact that if smooth real Banach spaces of dimension three or higher have isomorphic Birkhoff–James orthogonality structures, then they are (linearly) isometric to each other. This generalizes results of Koldobsky and of Wójcik. Moreover, in an arbitrary dimension, we construct examples of non-isometric pairs of non-smooth real Banach spaces that admit norm preserving homogeneous bicontinuous Birkhoff–James orthogonality preservers among them.

A Generalized Explicit Iterative Method for Solving Generalized Split Feasibility Problem and Fixed Point Problem in Real Banach Spaces

In this paper, we propose a generalized explicit algorithm for approximating the common solution of generalized split feasibility problem and the fixed point of demigeneralized mapping in uniformly smooth and 2-uniformly convex real Banach spaces. The generalized split feasibility problem is a general mathematical problem in the sense that it unifies several mathematical models arising in (symmetry and non-symmetry) optimization theory and also finds many applications in applied science. We designed the algorithm in such a way that the convergence analysis does not need a prior estimate of the operator norm. More so, we establish the strong convergence of our algorithm and present some computational examples to illustrate the performance of the proposed method. In addition, we give an application of our result for solving the image restoration problem and compare with other algorithms in the literature. This result improves and generalizes many important related results in the contemporary literature.

"A new Krasnoselskii’s type algorithm for zeros of strongly monotone and Lipschitz mappings"

"For q > 1, let E be a q-uniformly smooth real Banach space with dual space E∗. Let A : E → E∗ be a Lipschitz and strongly monotone mapping such that A^{−1}(0) ̸= ∅. For given x_1 ∈ E, let {x_n} be generated iteratively by the algorithm : x_{n+1} = x_n − λJ^{−1}(Ax_n), n ≥ 1, where J is the normalized duality mapping from E into E∗ and λ is a positive real number choosen in a suitable interval. Then it is proved that the sequence {xn} converges strongly to x∗, the unique point of A^{−1}(0). Our theorems are applied to the convex minimization problem. Futhermore, our technique of proof is of independent interest."

A self-adaptive Tseng extragradient method for solving monotone variational inequality and fixed point problems in Banach spaces

Abstract In this paper, we introduce a self-adaptive projection method for finding a common element in the solution set of variational inequalities (VIs) and fixed point set for relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth real Banach spaces. We prove a strong convergence result for the sequence generated by our algorithm without imposing a Lipschitz condition on the cost operator of the VIs. We also provide some numerical examples to illustrate the performance of the proposed algorithm by comparing with related methods in the literature. This result extends and improves some recent results in the literature in this direction.

Generalized Hybrid Viscosity-Type Forward-Backward Splitting Method with Application to Convex Minimization and Image Restoration Problems

Let E be a uniformly convex and q-uniformly smooth real Banach space. Let be an α- inverse strongly accretive mapping of order q, be a set-valued m- accretive mapping and be a nonexpansive mapping. In this article, a viscosity-type forward-backward splitting method for approximating a zero of (A + B) which is also a fixed point of S is introduced studied. Strong convergence theorem of the method is proved under suitable conditions. Furthermore, the convergence result obtained is applied to convex minimization and image restoration problems. Finally, numerical illustrations are presented to compare the convergence of the sequence of our algorithm and that of some recent important algorithms.

Approximation of solutions of split equality fixed point problems with applications

In this paper, we introduce and study an inertial algorithm for approximating solutions of split equality fixed point problem (SEFPP), involving quasi-phi-nonexpansive mappings in uniformly smooth and 2-uniformly convex real Banach spaces and establish a strong convergence theorem. We give applications of our result to split equality problem (SEP), split equality variational inclusion problem (SEVIP) and split equality equilibrium problem (SEEP). Our results extend, generalize and unify several recent inertial-type algorithms for approximating solutions of SEP and SEVIP. Moreover, to the best of our knowledge, our propose method which does not require any compactness type assumption on the operators is the first inertial algorithm for approximating solutions of SEFPP, SEP, SEVIP and SEEP in Banach spaces.

Remarks on a recent paper titled: \u201cOn the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces\u201d

In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition 0<kappa < frac{1}{sqrt{2}}, as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is E=l_{p}, 2leq p<infty . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition 0<kappa leq frac{1}{sqrt{2}}, then E is necessarily l_{2}, a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.

An algorithm for approximating a common solution of some nonlinear problems in Banach spaces with an application

In this paper, we construct a new Halpern-type subgradient extragradient iterative algorithm. The sequence generated by this algorithm converges strongly to a common solution of a variational inequality, an equilibrium problem, and a J-fixed point of a continuous J-pseudo-contractive map in a uniformly smooth and two-uniformly convex real Banach space. Also, the theorem is applied to approximate a common solution of a variational inequality, an equilibrium problem, and a convex minimization problem. Moreover, a numerical example is given to illustrate the implementability of our algorithm. Finally, the theorem proved complements, improves, and unifies some related recent results in the literature.