Nonlinear Convex Anal Research Articles

Numerical reckoning fixed points via new faster iteration process

In this paper, we propose a new iteration process which is faster than the leading S [J. Nonlinear Convex Anal. 8, no. 1 (2007), 61-79], Thakur et al. [App. Math. Comp. 275 (2016), 147-155] and M [Filomat 32, no. 1 (2018), 187-196] iterations for numerical reckoning fixed points. Using new iteration process, some fixed point convergence results for generalized α-nonexpansive mappings in the setting of uniformly convex Banach spaces are proved. At the end of paper, we offer a numerical example to compare the rate of convergence of the proposed iteration process with the leading iteration processes.

On the convergence of an iterative process for enriched Suzuki nonexpansive mappings in Banach spaces

<abstract><p>We study the existence and approximation of fixed points for the recently introduced class of mappings called enriched Suzuki nonexpansive mappings in the setting of Banach spaces. We use the modified $ K $-iteration process to establish the main results of the paper. The class of enriched Suzuki nonexpansive operators is an important class of nonlinear operators that includes properly the class of Suzuki nonexpansive operators as well as enriched nonexpansive operators. Various assumptions are imposed on the domain or on the operator to establish the main convergence theorems. Eventually, a numerical example of enriched Suzuki nonexpansive operators is used to show the effectiveness of the studied iteration scheme. The main outcome of the paper is new and essentially suggests a new direction for researchers who are working on fixed point problems in a Banach space setting. Our results improve and extend some main results due to Hussain et al. (<italic>J. Nonlinear Convex Anal.</italic> 2018, <italic>19</italic>, 1383–1393.), Ullah et al. (<italic>Axioms</italic> 2022, 1.) and others.</p></abstract>

Fixed point theorems for asymptotically regular semigroups equipped with generalized Lipschitzian conditions in metric spaces

In this paper, we prove some common fixed point theorems for one-parameter semigroups of asymptotically regular mappings which satisfy certain generalized Lipschitzian conditions in metric spaces. Our results do not assume the continuity of the mappings in the semigroups. The results extend some relevant common fixed point theorems in Gornicki (Colloq Math 64:55–57, 1993), Imdad and Soliman (Fixed Point Theory Appl 2010:692401, 2010), Imdad and Soliman (Bull Malays Math Sci Soc 2(35):687–694, 2012), Yao and Zeng (J Nonlinear Convex Anal 8:153–163, 2007).

Fixed point theorems for a new generalization of contractive maps in incomplete metric spaces and its application in boundary value problems

Abstract In this paper, we obtain some fixed point theorems for multivalued mappings in incomplete metric spaces. Moreover, as motivated by the recent work of Olgun, Minak and Altun [M. Olgun, G. Minak and I. Altun, A new approach to Mizoguchi–Takahashi type fixed point theorems, J. Nonlinear Convex Anal. 17 2016, 3, 579–587], we improve these theorems with a new generalization contraction condition for multivalued mappings in incomplete metric spaces. This result is a significant generalization of some well-known results in the literature. Also, we provide some examples to show that our main theorems are a generalization of previous results. Finally, we give an application to a boundary value differential equation.

Existence of Fixed Point Under Generalized Multivalued - -Contraction in Partial Metric Spaces

In this paper, we introduce the notion of generalized multivalued - -contraction in partial metric space endowed with an arbitrary binary relation and establish a fixed point theorem for this contraction mapping. Our result extends and generalize the result of Wardowski (Fixed Point Theory Appl. 2012:94 (2012)), Alam and Imdad (J. Fixed Point Theory Appl. 17 (4) (2015), 693–702) and Altun et al. (J. Nonlinear Convex Anal. 28 (16) (2015), 659-666). Also, we give an example to validate our result.

Fixed point results for nonlinear contractions with application to integral equations

We present a number of fixed and common fixed point theorems for a class of nonlinear contractions in metric spaces and metric spaces endowed with graphs. Our results complement, extend and generalize a number of fixed point theorems including a recent fixed point theorem of Kim et al. [Suzuki-type of common fixed theorem in metric spaces, J. Nonlinear Convex Anal. 16 (2015) 1779–1786]. We also discuss an application to a system of integral equations.

A new viscosity-type iteration for a finite family of split variational inclusion and fixed point problems between Hilbert and Banach spaces

In this paper, we introduce a new viscosity-type iteration process for approximating a common solution of a finite family of split variational inclusion problem and fixed point problem. We prove that the proposed algorithm converges strongly to a common solution of a finite family of split variational inclusion problems and fixed point problem for a finite family of type-one demicontractive mappings between a Hilbert space and a Banach space. Furthermore, we applied our results to study a finite family of split convex minimization problems, and also considered a numerical experiment of our results to further illustrate its applicability. Our results extend and improve the results of Byrne et al. (J. Nonlinear Convex Anal. 13:759–775, 2012), Kazmi and Rizvi (Optim. Lett. 8(3):1113–1124, 2014), Moudafi (J. Optim. Theory Appl. 150:275–283, 2011), Shehu and Ogbuisi (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 110(2):503–518, 2016), Takahashi and Yao (Fixed Point Theory Appl. 2015:87, 2015), Chidume and Ezeora (Fixed Point Theory Appl. 2014:111, 2014), and a host of other important results in this direction.

A viscosity iterative technique for split variational inclusion and fixed point problems between a Hilbert space and a Banach space

The main purpose of this paper is to introduce a viscosity-type iterative algorithm for approximating a common solution of a split variational inclusion problem and a fixed point problem. Using our algorithm, we state and prove a strong convergence theorem for approximating a common solution of a split variational inclusion problem and a fixed point problem for a multivalued quasi-nonexpansive mapping between a Hilbert space and a Banach space. Furthermore, we applied our results to study a split convex minimization problem. Also, a numerical example of our result is given. Our results extend and improve the results of Byrne et al. (J. Nonlinear Convex Anal. 13, 759–775, 2012), Moudafi (J. Optim. Theory Appl. 150, 275–283, 2011), Takahashi and Yao (Fixed Point Theory Appl. 2015, 87, 2015), and a host of other important results in this direction.

On the strong convergence of sequences of Halpern type in Hilbert spaces

ABSTRACTIn this paper, we introduce a concept of A-sequences of Halpern type where A is an averaging infinite matrix. If A is the identity matrix, this notion become the well-know sequence generated by Halpern's iteration. A necessary and sufficient condition for the strong convergence of A-sequences of Halpern type is given whenever the matrix A satisfies some certain concentrating conditions. This class of matrices includes two interesting classes of matrices considered by Combettes and Pennanen [J. Math. Anal. Appl. 2002;275:521–536]. We deduce all the convergence theorems studied by Cianciaruso et al. [Optimization. 2016;65:1259–1275] and Muglia et al. [J. Nonlinear Convex Anal. 2016;17:2071–2082] from our result. Moreover, these results are established under the weaker assumptions. We also show that the same conclusion remains true under a new condition.

Some Convergence Results for Multi-valued Mappings in Hyperbolic Spaces

In this paper, we modify the iteration process of Agarwal et al. (J. Nonlinear Convex Anal. 8(1) (2007), 61–79) to three multi-valued mappings and prove the strong and △-convergence theorems of this iteration in a hyperbolic space. Our theorems extend and improve some recent results announced in the current literature.

Approximating Fixed Points of Nonself Contractive Type Mappings in Banach Spaces Endowed with a Graph

AbstractLet K be a non-empty closed subset of a Banach space X endowed with a graph G. We obtain fixed point theorems for nonself G-contractions of Chatterjea type. Our new results complement and extend recent related results [Berinde, V., Păcurar, M., The contraction principle for nonself mappings on Banach spaces endowed with a graph, J. Nonlinear Convex Anal. 16 (2015), no. 9, 1925-1936; Balog, L., Berinde, V., Fixed point theorems for nonself Kannan type contractions in Banach spaces endowed with a graph, Carpathian J. Math. 32 (2016), no. 3 (in press)] and thus provide more general and flexible tools for studying nonlinear functional equations.

Some remarks on b-(E.A)-property in b-metric spaces.

In this paper we consider, discuss, improve and generalize recent b-(E.A)-property results for mappings in b-metric spaces established by Ozturk and Turkoglu (J Nonlinear Convex Anal 16(10):2059–2066, 2015). Thus, all our results are with much shorter proofs. One example is given to support the result.

Best proximity point theorems via fixed point theorems for multivalued mappings

AbstractIt is well known that the concept of a best proximity point includes that of a fixed point as a special case. In this paper, we show that the best proximity point theorems of Basha and Shahzad (Fixed Point Theory Appl. 2012:42, 2012) and of Fernández-León (J. Nonlinear Convex Anal. 15(2):313-324, 2014) can be regarded as a fixed point theorem for multivalued mappings which is modified as regards the results of Mizoguchi and Takahashi (J. Math. Anal. Appl. 141(1):177-188, 1989) and of Kada et al. (Math. Jpn. 44(2):381-391, 1996).

Viscosity approximation method for generalized asymptotically quasi-nonexpansive mappings in a convex metric space

A general viscosity iterative method for a finite family of generalized asymptotically quasi-nonexpansive mappings in a convex metric space is introduced. Special cases of the new iterative method are the viscosity iterative method of Chang et al. (Appl. Math. Comput. 212:51-59, 2009), an analogue of the viscosity iterative method of Fukhar-ud-din et al. (J. Nonlinear Convex Anal. 16:47-58, 2015) and an extension of the multistep iterative method of Yildirim and Ozdemir (Arab. J. Sci. Eng. 36:393-403, 2011). Our results generalize and extend the corresponding known results in uniformly convex Banach spaces and $\operatorname{CAT}(0)$ spaces simultaneously.

Recent results on the topological fixed point theory of multivalued mappings: a survey

AbstractIn this survey, we present current results from the topological fixed point theory of multivalued mappings which were obtained by ourselves in the last five years (see Andres and Górniewicz in Fixed Point Theory 12(2):255-264, 2011; Topol. Methods Nonlinear Anal. 40:337-358, 2012; Libertas Mathematica 33(1):69-78, 2013; Int. J. Bifurc. Chaos 24(11):1450148, 2014; J. Nonlinear Convex Anal. 16(6):1013-1023, 2015; Int. J. Bifurc. Chaos, 2015, to appear). Some abstract theorems are applied to differential inclusions and multivalued fractals. A part of the deterministic theory is randomized, including the applications to random differential inclusions.

Nonconvex Sweeping Process with a Moving Set Depending on the State

Recently, a great advance has been made in the study of sweeping process variational inequalities with the papers (Chemetov, Monteiro Marques Set-Valued Anal. 15, 209–221, 2007 Castaing et al. J. Nonlinear Convex Anal. 10, 1–20, 2009 Haddad J. Optim. Theory Appl. 159, 386–398, 2013 Azzam-Laouir et al. Set-Valued Var. Anal. 22, 271–283, 2014 where, for a prox-regular moving set depending on both the time and the state, several existence results are provided. Those authors also studied the case where such a differential inclusion is perturbed by multimapping. The present paper establishes the existence of solutions for such perturbed differential inclusions in some context not considered in the previous papers.

On the convergence of fixed points for Lipschitz type mappings in hyperbolic spaces

In this paper, we prove strong and Δ-convergence theorems for a class of mappings which is essentially wider than that of asymptotically nonexpansive mappings on hyperbolic space through the S-iteration process introduced by Agarwal et al. (J. Nonlinear Convex Anal. 8:61-79, 2007) which is faster and independent of the Mann (Proc. Am. Math. Soc. 4:506-510, 1953) and Ishikawa (Proc. Am. Math. Soc. 44:147-150, 1974) iteration processes. Our results generalize, extend, and unify the corresponding results of Abbas et al. (Math. Comput. Model. 55:1418-1427, 2012), Agarwal et al. (J. Nonlinear Convex Anal. 8:61-79, 2007), Dhompongsa and Panyanak (Comput. Math. Appl. 56:2572-2579, 2008), and Khan and Abbas (Comput. Math. Appl. 61:109-116, 2011). MSC:47H10.

Common fixed points of two multivalued nonexpansive maps in Kohlenbach hyperbolic spaces

In this paper, we introduce an iteration scheme for two multivalued maps in Kohlenbach hyperbolic spaces. This extends the single-valued iteration process due to Agarwal et al. (J. Nonlinear Convex Anal. 8(1):61-79, 2007). Using this new algorithm, we approximate common fixed points of two multivalued mappings through � -convergence and strong convergence under some weaker conditions. A necessary and sufficient condition is given for strong convergence. MSC: Primary 47A06; 47H09; 47H10; secondary 49M05

Results on n-tupled fixed points in metric spaces with uniform normal structure

In this paper, we introduce the concept of new notions related to n-tupled fixed point and prove some related results for an asymptotically regular one-parameter semigroup ℑ={F(t):t∈G,where G is an unbounded subset of [0,∞)} of Lipschitzian self-mappings on ∏ i = 1 n X in the case when (X,d) is a complete bounded metric space with uniform normal structure. Our results extend the results due to Yao and Zeng (J. Nonlinear Convex Anal. 8(1):153-163, 2007) and Soliman (Fixed Point Theory Appl. 2013:346, 2013; J. Adv. Math. Stud. 7(2):2-14, 2014).

On Nondifferentiable Higher-Order Symmetric Duality in Multiobjective Programming Involving Cones

In this paper, we point out some deficiencies in a recent paper (Lee and Kim in J. Nonlinear Convex Anal. 13:599–614, 2012), and we establish strong duality and converse duality theorems for two types of nondifferentiable higher-order symmetric duals multiobjective programming involving cones.