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This paper is devoted to finding out some realization of the concept of b-metric like space. First, we attain a fixed point for two fuzzy mappings satisfying a suitable requirement of contractiveness. Subsequently, we apply such a result to graphic contractions. Also, we attain a unique solution for a system of integral equations, and lastly we give an application to ensure that there exists a common bounded solution of a suitable functional equation in dynamic programming.

The purpose of this paper is to study the solution of two systems of nonlinear integral equations via fixed point results in a complete dislocated b-metric space. Also the notion of graphic contractions on a closed set for two families of graph dominated multivalued mappings is introduced. Our results generalize some previous results in the existing literature.

In this article, we study the approximation of common zeros of non-self inverse strongly monotone operators defined on a closed convex subset C of a Hilbert space H. For a non-self family of operators, we introduce an iterative algorithm without relying on projections. Approximation of common fixed points for finite families of non-self strict pseudo-contractions in the sense of Browder-Petryshyn is also obtained. The novelty of our algorithm is that the coefficients are not given a priori and no assumptions are made on them, but they are constructed step by step in a natural way.

Let H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If \(T:C\rightarrow H\) is a non-self and k-strict pseudocontractive mapping, we can define a map \(v:C\rightarrow \mathbb {R}\) by \(\, v(x):=\inf \{\lambda \ge 0:\lambda x+(1-\lambda )Tx\in C\}.\) Then, for a fixed \(x_{0}\in C\) and for \(\alpha _{0}:=\max \{k,v(x_{0})\},\) we define the Krasnoselskii–Mann algorithm \(x_{n+1}=\alpha _{n}x_{n}+(1-\alpha _{n})Tx_{n},\) where \(\alpha _{n+1}=\max \{\alpha _{n},v(x_{n+1})\}.\) So, here the coefficients \(\alpha _{n}\) are not chosen a priori, but built step by step. We prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping.

In this paper, inspired by Hussain et al. (Fixed Point Theory Appl. 2015:17, 2015), we study a modified Mann method to approximate strongly fixed points of strict pseudo-contractive mappings. In (Hussain et al. in Fixed Point Theory Appl. 2015:17, 2015) it is shown that the same algorithm converges strongly to a fixed point of a nonexpansive mapping under suitable hypotheses on the coefficients. Here the assumptions on the coefficients are different, as well as the techniques of the proof.

In this paper, we introduce a Halpern’s type method to approximate common fixed points of a nonexpansive mapping T and a strongly quasi-nonexpansive mappings S, defined in a Hilbert space, such that I − S is demiclosed at 0. The result shows as the same algorithm converges to different points, depending on the assumptions of the coefficients. Moreover, a numerical example of our iterative scheme is given.

An equilibrium theory is developed in Hadamard manifolds. The existence of equilibrium points for a bifunction is proved under suitable conditions, and applications to variational inequality, fixed point and Nash equilibrium problems are provided. The convergence of Picard iteration for firmly nonexpansive mappings along with the definition of resolvents for bifunctions in this setting is used to devise an algorithm to approximate equilibrium points.

In this paper, we present a new multi-step iterative method. We prove the strong convergence of the method to a common fixed point of a finite number of nonexpansive mappings that also solves a suitable equilibrium problem.

In this paper, we present iteration schemes to weakly and strongly approximate common fixed points of a finite family of a class of strict pseudocontractions.

Let H be a Hilbert space. Consider on H a sequence of nonexpansive mappings { T n } with common fixed points, an equilibrium function G , a contraction f with coefficient 0 α 1 and a strongly positive linear bounded operator A with coefficient γ ¯ > 0 . Let 0 γ γ ¯ α . We define a suitable Mann type algorithm which strongly converges to the unique solution of the minimization problem min x ∈ C 1 2 〈 A x , x 〉 − h ( x ) , where h is a potential function for γ f and C is the intersection of the equilibrium points and the common fixed points of the sequence { T n } .

We propose a modified hybrid projection algorithm to approximate a common fixed point of a -strict pseudocontraction and of two sequences of nonexpansive mappings. We prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.

In this paper, we suggest and analyze a hybrid algorithm for pseudo-contractive mappings in Hilbert spaces. Further, we prove the strong convergence of the proposed iterative algorithm for Lipschitz pseudo-contractive mappings in Hilbert spaces.

By using a modification of the hybrid method in mathematical programming, we give some new algorithm to find out a fixed point of mappings which are in some sense non-expansive. Our results hold in reflexive, strictly convex, smooth Banach spaces with the Kadec˘–Klee property.

Let H be a Hilbert space. Consider on H a sequence of nonexpansive mappings { T n } with common fixed points, a finite family of equilibrium functions { G i } i = 1 , … , K , a contraction f with coefficient 0 α 1 and a strongly positive linear bounded operator A with coefficient γ ¯ > 0 . Let 0 γ γ ¯ / α . Assuming there are common equilibrium points of the family { G i } i = 1 , … , K which are also fixed points for { T n } , we define a suitable sequence which strongly converges to the unique such point which also satisfies the variational inequality 〈 ( A − γ f ) x ∗ , x − x ∗ 〉 ≥ 0 for all the x in the intersection of the equilibrium points and the common fixed points of the sequence { T n } .

We give new sharper valuations for the optimal retraction constant in some Banach spaces.

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ -strict pseudo-contraction on C with a fixed point, for some 0 ⩽ κ 1 . Given an initial guess x 0 ∈ C and given also a real sequence { α n } in ( 0 , 1 ) . The Mann's algorithm generates a sequence { x n } by the formula: x n + 1 = α n x n + ( 1 − α n ) T x n , n ⩾ 0 . It is proved that if the control sequence { α n } is chosen so that κ α n 1 and ∑ n = 0 ∞ ( α n − κ ) ( 1 − α n ) = ∞ , then { x n } converges weakly to a fixed point of T . However this convergence is in general not strong. We then modify Mann's algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence. This result extends a recent result of Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] from nonexpansive mappings to strict pseudo-contractions.

In the setting of Hilbert spaces, we study Mann’s type method to approximate strong solutions of variational inequalities. We show that these solutions are fixed points of a nonexpansive mapping and/or a strongly quasinonexpansive mapping, depending on the coefficients involved in the algorithm.

Let H be a real Hilbert space. Consider on H a nonexpansive mapping T with a fixed point, a contraction f with coefficient 00. Let 0

Weak and strong convergence for some generalized proximal point algorithms are proved. These algorithms include the Eckstein and Bertsekas generalized proximal point algorithm, a contraction-proximal point algorithm, and inexact proximal point algorithms. Convergence rate is also considered.

In the setting of Hilbert spaces, inspired by Iemoto and Takahashi (2009), we study a Mann’s method with viscosity to approximate strongly (common) fixed points of a nonexpansive mapping and a nonspreading mapping. A crucial tool in our results is the nonspreading-average type mapping.

Properties strict (M) and uniform (M) are introduced. It is shown that if X has property (M) and is uniformly convex in every direction, then X has both strict (M) and uniform (M). It is also shown that if X* is separable, then strict (M) implies uniform (M) and property (M) implies weak uniform normal structure. Relations with other geometrical properties of Banach spaces are also discussed.

We prove the equivalence and the strong convergence of (1) the modified Mann iterative process and (2) the modified Ishikawa iterative process for asymptoticallyϕ-strongly pseudocontractive mappings in a uniformly smooth Banach space.

A common method in solving ill-posed problems is to substitute the original problem by a family of well-posed (i.e., with a unique solution) regularized problems. We will use this idea to define and study a two-step algorithm to solve hierarchical fixed point problems under different conditions on involved parameters.

We introduce two iterative algorithms for nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to a fixed point of a nonexpansive mapping .

We give some fixed points theorems for multivalued nonexpansive and pseudocontractive mappings defined on closed convex unbounded subsets of a Banach space. Some of such results extend to the multivoque case a few of the theorems already known for the univoque case, while some other results improve a few of the theorems already known for the multivoque case.

We introduce a new iterative method for finding a common element of the set of solutions of a generalized equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings in a Hilbert space and then prove that the sequence converges strongly to a common element of the two sets. Using this result, we prove several new strong convergence theorems in fixed point problems, variational inequalities, and equilibrium problems.

We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.

Let H be a Hilbert space. Let $(W_{n})_{n\in\mathbb{N}}$ be a suitable family of mappings. Let S be a nonexpansive mapping and D be a strongly monotone operator. We study the convergence of the general scheme $x_{n+1}=W_{n}(\alpha_{n}Sx_{n}+(1-\alpha_{n})(I-\mu_{n}D)x_{n}) $ in dependence on the coefficients $(\alpha_{n})_{n\in\mathbb{N}}$ , $(\mu_{n})_{n\in\mathbb{N}}$ .

An iterative algorithm is introduced for the construction of the minimum-norm fixed point of a pseudocontraction on a Hilbert space. The algorithm is proved to be strongly convergent. MSC:47H05, 47H10, 47H17.