We introduce a novel class of asymptotically alpha- hemicontractive mappings and demonstrate its relationship with the existing related families of mappings. We establish certain interesting properties of the fixed point set of the new class of mappings. Furthermore, we propose and investigate a new iterative algorithm for solving split common fixed point problem for the new class of mappings. In particular, weak and strong convergence theorems for solving split common fixed point problem for our new class of mappings in Hilbert spaces are proved. Moreover, using our method, we require no prior knowledge of norm of the transfer operator. The results presented in the paper extend and improve the results of Censor and Segal [Censor, Y.; Segal, A. The split common fixed point problem for directed operators. J. Convex Anal. 16 (2009), no. 2, 587-600.], Moudafi [Moudafi, A. The split common fixed -point problem for demicontractive mappings. Inverse Problems 26 (2010), no. 5:055007.; Moudafi, A. A note on the split common fixed -point problem for quasi-nonexpansive operators. Nonlinear Anal. 74 (2011), no. 12, 4083-4087.], Chima and Osilike [Chima, E. E.; Osilike, M. O. Split common fixed point problem for class of asymptotically hemicontractive mappings. J. Nigerian Math. Soc. 38 (2019), no. 3, 363-390.], Fan et al [Fan, Q.; Peng, J.; He, H. Weak and strong convergence theorems for the split common fixed point problem with demicontractive operators. Optimization 70 (2021), no. 5-6, 1409-1423.] and host of other related results in literature.

We study a perturbed inertial Krasnoselskii-Mann-type algorithm and prove that the algorithm is an approximate fixed point sequence for Lipschitz pseudocontractive maps in arbitrary real Banach spaces. Strong convergence results are then established for our inertial iteration scheme for approximation of fixed points of Lipschitz pseudocontractive maps and solutions of certain important accretive-type operator equations in certain real Banach spaces. Implementation of our algorithm is illustrated using numerical examples in both finite and infinite dimensional Banach spaces. Our results improve rate of convergence and extend several related recent results.

We study a Halpern-type algorithm with both inertial and error terms for the approximation of fixed points of strictly pseudocontractive mappings and zeros of inverse strongly monotone operators in real Hilbert spaces.Our algorithm is illustrated via numerical examples in both finite and infinite dimensional real Hilbert spaces.Our results extend recent results of [Y.Shehu, O.S. Iyiola, F.U. Ogbuisi, Iterative method with inertial terms for nonexpansive mappings: applications to compressed sensing, Numer.Algor.83 (2020Algor.83 ( ), 1321Algor.83 ( -1347] ] from the class of nonexpansive mappings to the much more general class of strictly pseudocontractive mappings.

We introduce and study a Halpern-type averaging algorithm with both inertial and error terms for the approximation of fixed points of asymptotically nonexpansive maps in real Hilbert spaces. Implementation of our algorithm is illustrated using numerical examples in both finite and infinite dimensional real Hilbert spaces. Our results extend recent results of Yekini, Iyiola and Ogbuisi, Numer Algor (2019), https://doi.org/10.1007/s11075-019-00727-5 from the important class of nonexpansive maps to the much more general class of asymptotically nonexpansive maps. Furthermore, our preliminary lemma is of independent interest.

Let $H_1$ and $H_2$ be two real Hilbert spaces. $T:H_1\rightarrow H_1$ and $S:H_2\rightarrow H_2$ two asymptotically hemicontractive maps. Let $A:H_1\rightarrow H_2$ be a bounded linear operator. The split common fixed point problem (SCFP) for $T$ and $S$, which is to find a fixed point $x^*\in F(T)$ such that $Ax^*\in F(S)$ is studied. We proved that the set of fixed points of a class of asymptotically hemicontractive maps is closed and convex. We then obtain strong convergence results for the SCFP involving two asymtotically hemicontractive maps using new averaging iterative scheme.

Abstract We extend the notion of k-strictly pseudononspreading mappings introduced in Nonlinear Analysis 74 (2011) 1814-1822 to the notion of the more general pseudononspreading mappings. It is shown with example that the class of pseudononspreading mappings is more general than the class of k-strictly pseudonon-spreading mappings. Furthermore, it is shown with explicit examples that the class of pseudononspreading mappings and the important class of pseudocontractive mappings are independent. Some fundamental properties of the class of pseudononspreading mappings are proved. In particular, it is proved that the fixed point set of certain class of pseudononspsreading selfmappings of a nonempty closed and convex subset of a real Hilbert space is closed and convex. Demiclosedness property of such class of pseudonon-spreading mappings is proved. Certain weak and strong convergence theorems are then proved for the iterative approximation of fixed points of the class of pseudononspreading mappings.

Let $C$ be a nonempty closed convex subset of a real Hilbert space, $H$ and let $ T: C \rightarrow C $ be an asymptotically $k$-strictly pseudo-contractive mapping with a nonempty fixed-point set, $F(T)=\{x\in C: Tx=x\}$. Let $\{t_n\}$, $\lbrace\alpha_{n}\rbrace$~ and $\lbrace\beta_{n}\rbrace$~ be real ~sequences in $( 0, 1)$. We consider the sequence $\lbrace x_{n}\rbrace$, ge nerated from an arbitrary $ x_{1} \in C $, by either I. \hskip 3.0cm $x_{n+1} = P_C[\left( 1-\alpha_{n} - \beta_{n}\right) x_{n}+ \beta_{n}T^{n}x_{n}], \; n\geq 1,$ or II. $\left\{\begin{array}{ll} \nu_n=P_C((1-t_n)x_n) x_{n+1}=(1-\alpha_n)\nu_n+\alpha_nT^n\nu_n, \; n\geq 1\end{array}\right.$ We prove that under some mild conditions on the real sequences $\lbrace\alpha_{n}\rbrace$ and $\lbrace\beta_{n}\rbrace$, the sequence $ \lbrace x_{n}\rbrace$ generated by I converges strongly to a fixed point of $T$. Furthermore, under some mild conditions on the sequences $\{t_n\}$ and $\{\alpha_n\}$, the sequence generated by II converges strongly to the least norm element of the fixed point set of $T$. Some examples are used to compare the convergence rates of these two iteration schemes. Our results compliment and extend several strong convergence results in the literature to the class of mappings considered in our work.

Let H be a real Hilbert space. Weak and strong convergence theorems forapproximation of fixed points of k-strictly assymptotically pseudocontractive mapping,T : H → H are proved using an averaging hybrid iterative scheme { x n } ∞ n = 1 . Furthermore,if H is replaced with an arbitrary Banach space E , necessary and sufficient conditionsthat guarantee the strong convergence of our iterative scheme, { x n } to a fixed point of Tin E are given. Our results extend recent results of Osilike, Isiogugu and Nwokoro ( J.Nigerian Math. Soc., 27 (2008), 91-108)) which are themselves extensions andgeneralizations of results of Wang [Fixed Point Theory and Applications Vol 2007, ID28619 (2007), 1-8, http://fixedpointtheory and application.springeropen.com/article/10.1155/ 2007/28619] from the class of strictly pseudocontractive mappings of Browder-Petryshyn type to the class of k -strictly asymptotically pseudocontractive mappings.

We study the split common fixed point problem (SCFP) for a class of total asymptotically pseudocontractive mappings. We obtain some important properties of our class of mappings including the demiclosedness property and the closedness and convexity of the fixed point set. We then propose an algorithm and prove weak and strong convergence theorems for the approximation of solutions of the SCFP for certain class of these mappings.

A sufficient condition that guarantees a demiclosedness property for multi-valued nonexpansive mappings in a real Banach space is introduced. It is also proved that under this condition, the Mann sequence converges weakly to a fixed point of a multi-valued nonexpansive and quasi-nonexpansive mappings in a Hilbert space without the condition that the fixed point set of T is strict. The results obtained give a partial answer to the open problem of removing the condition that the fixed point set of T is strict. Thus, the results extend, complement and improve the results on multi-valued and single-valued nonexpansive mappings in the contemporary literature.

A new class of α-hemicontractive maps T for which the strong convergence of the Ishikawa iteration algorithm to a fixed point of T is assured is introduced and studied. The study is a continuation of a recent study of a new class of α-demicontractive mappings T by L. Mărușter and Ș. Mărușter, Mathematical and Computer Modeling 54 (2011) 2486-2492 in which they proved strong convergence of the Mann iteration scheme to a fixed point of T. Our class of α-hemicontractive maps is more general than the class of α-demicontractive maps. No compactness assumption is imposed on the operator or it’s domain, and no additional requirement is imposed on the set of fixed points.

It is proved that a recent Theorem of Qihou(1) concerning the iterative approximation of fixed points of demicontractive mappings in Hilbert spaces can be extended to the much more general q-uniformly smooth Banach spaces, 1 < q < infinity, and to a more general iteration method.

Zhenhua He and Wei-Shih Du, Fixed Point Theory and Applications 2011, 2011:33 introduced a new method of finding a common element in the intersection of the set of solutions of a finite family of equilibrium problems and the set of fixed points of a nonexpansive mapping in real Hilbert spaces. In this paper we modify the algorithm of He and Du and prove strong convergence results for finding a common element in the intersection of the set of solutions of a finite family of equilibrium problems and the set of fixed points of an asymptotically nonexpansive mapping in real Hilbert spaces.

Weak and strong convergence theorems are proved in Hilbert spaces for new classes of multivalued demicontractive-type and hemicontractive-type mappings which are related to the class of multivalued pseudocontractive-type mappings studied by Isiogugu (Fixed Point Theory Appl. 2013:61, 2013). Thus our results extend and improve several corresponding results in the contemporary literature.

In this paper, we combine the gradient projection algorithm and the hybrid steepest descent method and prove the strong convergence to a common element of the equilibrium problem; the null space of an inverse strongly monotone operator; the set of fixed points of a continuous pseudocontractive mapping and the minimizer of a convex function. This common element is proved to be the unique solution of a variational inequality problem.

Let C be a nonempty closed convex subset of a real Hilbert space and let T : C → C be an asymptotically nonexpansive mapping with F(T ) = {x ∈ C : T x = x} = ∅. Let {αn}∞n=1, and {tn}∞n=1 be real sequences in (0, 1). Let {xn}∞n=1 be the sequence generated from an arbitrary x1 ∈ C by νn := PC (1 − tn)xn, n ≥ 1 xn+1 := (1 − αn)νn + αnT nνn, n ≥ 1,where PC : H → C is the metric projection. Under some appropriate mild conditions on {αn}∞n=1 and {tn}∞n=1, we prove that{xn}∞n=1 converges strongly to a fixed point of T . No compactness assumption is imposed on T or C and no further requirement is imposed on F(T ).

Let C be a nonempty closed convex subset of a real Hilbert space, and let be an asymptotically k -strictly pseudocontractive mapping with . Let and be real sequences in . Let be the sequence generated from an arbitrary by where is the metric projection. Under some appropriate mild conditions on and , we prove that converges strongly to a fixed point of T . Furthermore, if is uniformly L -Lipschitzian and asymptotically pseudocontractive with , we first prove that is demiclosed at 0, and then prove that under some suitable conditions on the real sequences , and in , the sequence generated from an arbitrary by converges strongly to a fixed point of T . No compactness assumption is imposed on T or C and no further requirement is imposed on . MSC: 47H09, 47J25, 65J15.

Let H be a real Hilbert space and let T : H → H be a Lipschitz pseudocontractive mapping. We introduce a modied Ishikawa iterative algorithm and prove that if F(T) = {x ∈ H : Tx = x} ∅, then our proposed iterative algorithm converges strongly to a xed point of T. No compactness assumption is imposed on T and no further requirement is imposed on F(T).

We prove fixed point theorems for nonspreading-type mappings and obtain convergence theorems for approximation of common fixed points of k-strictly pseudocontractive mappings of Browder-Petryshyn type and β-strictly pseudonon-spreading mapping

Weak and strong convergence theorems are proved in real Hilbert spaces for a new class of nonspreading-type mappings more general than the class studied recently in Kurokawa and Takahashi [Y. Kurokawa, W. Takahashi, Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal. 73 (2010) 1562–1568]. We explored an auxiliary mapping in our theorems and proofs and this also yielded a strong convergence theorem of Halpern’s type for our class of mappings and hence resolved in the affirmative an open problem posed by Kurokawa and Takahashi in their final remark for the case where the mapping T is averaged.