In this paper, we develop some new geometric inequalities in p-uniformly convex and uni-formly smooth real Banach spaces with p > 1. We use the inequalities as tools to obtain the strong convergence of the sequence generated by a subsgradient method to a solution that solves fixed point and variational inequality problems. Furthermore, the convergence theorem established can be applica-ble in, for example, Lp(& omega;), where & omega; C R is bounded set and lp(R) for p E (2, & INFIN;). Finally, numerical implementations of the proposed method in the real Banach space L5([-1,1]) are presented.

An inertial viscosity-type iterative method that approximates a solution of an inclusion problem and a fixed point problem is introduced and studied. Strong convergence theorem is proved in some Banach spaces. The theorem proved is applied to image restoration, convex minimization and signal processing problems. Finally, numerical illustrations are presented to support the main theorem and its applications.

Abstract 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.

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 $E=l_{p}$ , $2\leq 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 zeros of m-accretive operators is constructed in a uniformly smooth real Banach space. The sequence generated by the algorithm is proved to converge strongly to a zero of an m-accretive operator. In the case of a real Hilbert space, our theorem complements the celebrated proximal point algorithm of Martinet and Rockafellar for approximating zeros of maximal monotone operators. Furthermore, the convergence theorem proved is applied to approximate a solution of a Hammerstein integral equation. Finally, numerical experiments are presented to illustrate the convergence of our algorithm.

Let E be a real normed space. A new notion of quasi-boundedness for operators $$A:E\rightarrow 2^E$$ is introduced and the following general important result for accretive operators is proved: an accretive operator with zero in the interior of its domain is quasi-bounded. Using this result, a new strong convergence theorem for approximating a zero of an m-accretive operator is proved in a uniformly smooth real Banach space. This result complements the celebrated proximal point algorithm for approximating solutions of $$0\in Au$$ in a real Hilbert space where A is a maximal monotone operator. Furthermore, as an application of our theorem, a new strong convergence theorem for approximating a solution of a Hammerstein equation is proved. Finally, several numerical experiments are presented to illustrate the strong convergence of the sequence generated by our algorithm and the results obtained are compared with those obtained using some recent important algorithms.

Let E be a uniformly smooth and strictly convex real Banach space with dual space, E*. In this paper, we present a Krasnoselkii-type inertial algorithm and prove a strong convergence theorem for approximating a common fixed point for a countable family of generalized nonexpansive maps. Furthermore, we apply our theorem and prove a strong convergence theorem for approximating a common fixed point for a countable family of generalized-J-nonexpansive maps. Our theorem is an improvement of the results of Klin-earn et al. (Taiwanese J. of Maths. Vol. 16, No. 6, pp. 1971-1989, Dec. 2012), Chidume et al. (Advances in Fixed Point Theory, Vol. 7, No. 3 (2017), 413-431) and Dong et al. (Optimization Letters, 2017, DOI: 10.1007/s11590-016-1102-9). Finally, we give a numerical experiment to illustrate the efficiency and advantage of the inertial algorithm over an algorithm without inertial term.

Let K be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space with dual space E∗. In this paper, a new iterative algorithm of Halpern-type is constructed and used to approximate a common element of a generalized mixed equilibrium problem and a common fixed points for a countable family of generalized nonexpansive-type maps. Application of our theorem, in the case of real Hilbert spaces, complements, extends and improves several important recent results. Finally, we give numerical experiments to illustrate the convergence of our sequence.

An inertial iterative algorithm is proposed for approximating a solution of a maximal monotone inclusion in a uniformly convex and uniformly smooth real Banach space. The sequence generated by the algorithm is proved to converge strongly to a solution of the inclusion. Moreover, the theorem proved is applied to approximate a solution of a convex optimization problem and a solution of a Hammerstein equation. Furthermore, numerical experiments are given to compare, in terms of CPU time and number of iterations, the performance of the sequence generated by our algorithm with the performance of the sequences generated by three recent inertial type algorithms for approximating zeros of maximal monotone operators. In addition, the performance of the sequence generated by our algorithm is compared with the performance of a sequence generated by another recent algorithm for approximating a solution of a Hammerstein equation. Finally, a numerical example is given to illustrate the implementability of our algorithm for approximating a solution of a convex optimization problem.

For p ≥ 2, a new iterative algorithm is introduced and used to approximate a common element of the set of solutions of a split generalized mixed equality equilibrium problem and the set of solutions of a split equality fixed point problem for quasi-φ -nonexpansive mappings in p-uniformly convex and uniformly smooth real Banach spaces.A strong convergence theorem is proved without any compactness-type assumption on the mappings.Furthermore, our theorem, which is applicable, in particular, in L p , l p and the Sobolev spaces W m p (Ω) for 2 ≤ p < ∞, complements several important recent results that were established in 2-uniformly convex and uniformly smooth real Banach spaces.

An algorithm is constructed to approximate a zero of a maximal monotone operator in a uniformly convex anduniformly smooth real Banach space. The sequence of the algorithm is proved to converge strongly to a zeroof the maximal monotone map. In the case where the Banach space is a real Hilbert space, our theorem com-plements the celebrated proximal point algorithm of Martinet and Rockafellar. Furthermore, our convergencetheorem is applied to approximate a solution of a Hammerstein integral equation in our general setting. Finally,numerical experiments are presented to illustrate the convergence of our algorithm.

In this paper, iterative algorithms of Krasnoselkii-type and of Halpern-type for approximating an element of the set of common zeros of a countable family of inverse strongly monotone maps, common fixed points of a countable family of totally quasi- $$\phi $$ -asymptotically nonexpansive nonself multi-valued maps, and a solution of a system of generalized mixed equilibrium problems are constructed and studied. Strong convergence of the sequences generated by these algorithms is established in uniformly smooth and 2-uniformly convex real Banach spaces. The theorems obtained extend, improve and generalize several recent important results.

Let E be a uniformly convex and uniformly smooth real Banach space with dual space, E∗. Let F : E → E∗, K : E∗ → E be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation u+KF u = 0. This theorem is a significant improvement of some important recent results which were proved in Lp spaces, 1 < p ≤ 2 under the assumption that F and K are bounded. This restriction on K and F have been dispensed with even in the more general setting considered here. Finally, a numerical experiment is presented to illustrate the convergence of the sequence of the algorithm which is found to be much faster, in terms of the number of iterations and the computational time than the convergence obtained with existing algorithms.

Let and E* denote its dual space. Let and be bounded generalized -strongly, and -strongly monotone maps, respectively. Suppose the Hammerstein equation has a solution u* in E. An iteration sequence is constructed and proved to converge strongly to u*. Our technique of proof is also of independent interest.

Let X be a uniformly convex and uniformly smooth real Banach space with dual space $X^{*}$ . In this paper, a Mann-type iterative algorithm that approximates the zero of a generalized-Φ-strongly monotone map is constructed. A strong convergence theorem for a sequence generated by the algorithm is proved. Furthermore, the theorem is applied to approximate the solution of a convex optimization problem, a Hammerstein integral equation, and a variational inequality problem. This theorem generalizes, improves, and complements some recent results. Finally, examples of generalized-Φ-strongly monotone maps are constructed and numerical experiments which illustrate the convergence of the sequence generated by our algorithm are presented.

In this paper, an iterative algorithm that approximates solutions of split equality fixed point problems (SEFPP) for quasi-ϕ-nonexpansive mappings is constructed. Weak convergence of the sequence generated by this algorithm is established in certain real Banach spaces. The theorem proved is applied to solve split equality problem, split equality variational inclusion problem, and split equality equilibrium problem. Finally, some numerical examples are given to demonstrate the convergence of the algorithm. The theorems proved improve and complement a host of important recent results.

Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex real Banach space E with dual space $E^{*}$ . In this paper, a Krasnoselskii-type subgradient extragradient iterative algorithm is constructed and used to approximate a common element of solutions of variational inequality problems and fixed points of a countable family of relatively nonexpansive maps. The theorems proved are improvement of the results of Censor et al. (J. Optim. Theory Appl. 148:318–335, 2011).

Let E be a uniformly convex and uniformly smooth real Banach space with dual space \(E^*\) and C be a nonempty, closed and convex subset of E. Let \(A:E\rightarrow E^*\) be a generalized \(\Phi \)-strongly monotone and bounded map and let \(T_i:C\rightarrow E, i=1,2,3,\ldots , N\) be a finite family of quasi-\(\phi \)-nonexpansive maps such that \(\cap _{i=1}^{N} F(T_{i})\ne \emptyset \). Suppose \(VI(A,\cap _{i=1}^{N} F(T_{i}))\ne \emptyset \). A new iterative algorithm that converges strongly to a point in \(VI(A,\cap _{i=1}^{N} F(T_{i}))\) is constructed. Results obtained are applied to a convex optimization problem. Furthermore, the theorems proved complement, improve and unify several recent important results. Finally, we consider a family \(\{T_i\}_{i=1}^N\) of maps where for each \(i,\, T_i \) maps E into its dual space \(E^*\) and prove a strong convergence theorem for \(VI(A, \cap _{i=1}^{N} F(T_{i}))\), where \(F_J(T_i)\) is the set of J-fixed points introduced by Chidume and Idu.

ABSTRACT In this paper, an iterative algorithm that approximates solutions of split equality fixed point problems (SEFPP) for quasi-φ-nonexpansive maps is constructed. Strong convergence of the sequence generated by this algorithm is established in certain real Banach spaces without imposing any compactness-type condition on either the operators or the space considered. We applied our theorem to solve split equality problem, split equality variational inclusion problem and split equality equilibrium problem. Furthermore, some numerical example is given to demonstrate the implementability of our algorithm. Finally, our theorems improve and complement a host of important recent results.

Abstract Let E be a uniformly convex and uniformly smooth real Banach space, and let E* be its dual. Let A : E → 2E* be a bounded maximal monotone map. Assume that A−1(0) ≠ Ø. A new iterative sequence is constructed which converges strongly to an element of A−1(0). The theorem proved complements results obtained on strong convergence of the proximal point algorithm for approximating an element of A−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.