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In this paper, we constructed a novel fixed point iterative scheme called the Modified-JK iterative scheme. This iteration process is a modification of the JK iterative scheme. Our scheme converged weakly to the fixed point of a nonexpansive mapping and strongly to the fixed point of a mapping satisfying condition (E). We provided some examples to show that the new scheme converges faster than some existing iterations. Stability and data dependence results were proved for this iteration process. To substantiate our results, we applied our results to solving delay differential equations. Furthermore, the newly introduced scheme was applied in approximating the solution of a class of third order boundary value problems (BVPs) by embedding Green's functions. Moreover, some numerical examples were presented to support the application of our results to BVPs via Green's function. Our results extended and generalized other existing results in literature.

In this paper, we introduce a new fixed point iterative scheme called the AG iterative scheme that is used to approximate the fixed point of a contraction mapping in a uniformly convex Banach space. The iterative scheme is used to prove some convergence result. The stability of the new scheme is shown. Furthermore, weak convergence of Suzuki's generalized non-expansive mapping satisfying condition (C) is shown. The rate of convergence result is proved and it is demonstrated via an illustrative example which shows that our iterative scheme converges faster than the Picard, Mann, Noor, Picard-Mann, M and Thakur iterative schemes. Data dependence results for the iterative scheme are shown. Finally, our result is used to approximate the solution of a nonlinear fractional differential equation of Caputo type.

In this paper, we prove the existence and uniqueness of common fixed point of tower type contractive mappings in complete metric (pseudo)modular spaces involving the theoretic relation. However, the newly introduced contraction in this paper further characterize and includes in their full strength several existing results in metrical fixed point theory. Some nontrivial supportive examples were given to justify our result. Our results generalize, improve, and unify some existing results.

In this paper, we extend the notion of weakly commuting mappings results in modular metric spaces to setting of modular ωG-metric spaces and prove the existence of unique common fixed point of three pairs of weakly commuting self-maps in modular ωG-metric spaces. Moreover, we apply our results in solving certain nonlinear three dimensional integral equations. The results proved in this paper extend, improve and generalize several existing results.

In this paper, we constructed a new and robust fixed point iterative scheme called the UO iterative scheme for the approximation of a contraction mapping. The scheme converges strongly to the fixed point of a contraction mapping. A rate of convergence result is shown with an example, and our scheme, when compared, converges faster than some existing iterative schemes in the literature. Furthermore, the stability and data dependence results are shown. Our new scheme is applied in the approximation of the solution to the oxygen diffusion model. Finally, our results are applied in the approximation of the solution to the boundary value problems using Green’s functions with an example.

The purpose of this article is to study the convergence, stability and data dependence results of an iterative method for contractive-like mappings. The concept of stability considered in this study is known as $ w^2 $-stability, which is larger than the simple notion of stability considered by several prominent authors. Some illustrative examples on $ w^2 $-stability of the iterative method have been presented for different choices of parameters and initial guesses. As an application of our results, we establish the existence, uniqueness and approximation results for solutions of a nonlinear Volterra integral equation with delay. Finally, we provide an illustrative example to support the application of our results. The novel results of this article extend and generalize several well known results in existing literature.

Abstract In this article, we propose an iterative method, called the GA iterative method, to approximate the fixed points of generalized α-nonexpansive mappings in uniformly convex Banach spaces. Further, we obtain some convergence results of the new iterative method. Also, we provide a nontrivial example of a generalized α-nonexpansive mapping and with the example, we carry out a numeral experiment to show that our new iterative algorithm is more efficient than some existing iterative methods. Again, we present an interesting strategy based on the GA iterative method to solve nonlinear third-order boundary value problems (BVPs). For this, we derive a sequence named the GA–Green iterative method and show that the sequence converges strongly to the fixed point of an integral operator. Finally, the approximation of the solution for a nonlinear integrodifferential equation via our new iterative method is considered. We present some illustrative examples to validate our main results in the application sections of this article. Our results are a generalization and an extension of several prominent results of many well-known authors in the literature.

Purpose – This paper aims to prove some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results of this paper generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. Furthermore, the authors produce an example to demonstrate the applicability of the results. Design/methodology/approach – The results of this paper are theoretical and analytical in nature. Findings – The authors established some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. An example was constructed to demonstrate the applicability of the results. Research limitations/implications – Analytical and theoretical results. Practical implications – The results of this paper can be applied in science and engineering. Social implications – The results of this paper is applicable in certain social sciences. Originality/value – The results of this paper are new and will open up new areas of research in mathematical sciences.

Abstract We prove the existence and uniqueness of fixed points of some generalized contractible operators defined on modular G-metric spaces and also prove the modular G-continuity of such operators. Furthermore, we prove that some generalized weakly compatible contractive operators in modular G-metric spaces have a unique fixed point. Our results extend, generalize, complement and include several known results as special cases.

Purpose – The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. The authors apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and include several known results as special cases. Design/methodology/approach – The results of this paper are theoretical and analytical in nature. Findings – The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend, generalize, compliment and include several known results as special cases. Research limitations/implications – The results are theoretical and analytical. Practical implications – The results were applied to solving nonlinear integral equations. Social implications – The results has several social applications. Originality/value – The results of this paper are new.

The purpose of this paper is to extend the recent results of Okeke et al. (2018) to the class of multivalued ρ-quasi-contractive mappings in modular function spaces. We approximate fixed points of this class of nonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts of T-stability, almost T-stability and summably almost T-stability to modular function spaces and give some results.

It is our purpose in this paper to prove some fixed point results and Fej´er monotonicity of some faster fixed point iterative sequences generated by some nonlinear operators satisfying rational inequality in complex valued Banach spaces. We prove that results in complex valued Banach spaces are valid in cone metric spaces with Banach algebras. Furthermore, we apply our results in solving certain mixed type VolterraFredholm functional nonlinear integral equation in complex valued Banach spaces.

Abstract The purpose of this paper is to define a new random operator called the generalized ϕ-weakly contraction of the rational type. This new random operator includes those studied by Khan et al. (Filomat 31(12):3611–3626, 2017) and Zhang et al. (Appl. Math. Mech. 32(6):805–810, 2011) as special cases. We prove some convergence, existence, and stability results in separable Banach spaces. Moreover, we produce some numerical examples to demonstrate the applicability of our analytical results. Furthermore, we apply our results in proving the existence of a solution of a nonlinear integral equation of the Hammerstein type.

We approximate the fixed points of contraction mappings using the Picard–Krasnoselskii hybrid iterative process, which is known to converge faster than all of Picard, Mann and Ishikawa iterations in complex valued Banach spaces. Moreover, we prove analytically and with a numerical example that the Picard–Mann hybrid iteration and the Picard–Krasnoselskii hybrid iteration have the same rate of convergence. Furthermore, we apply our results in finding solutions of delay differential equations in complex valued Banach spaces. Keywords: Complex valued Banach spaces, Picard–Krasnoselskii hybrid iterative process, Delay differential equations, Picard–Mann hybrid iterative process, Stability, Data dependence, Mathematics Subject Classification: 47H09, 47H10, 49M05, 54H25

We prove some theorems on the existence and uniqueness of fixed point for Reich-type contraction mappings and Geraghty-type mappings satisfying rational inequalities in modular metric spaces. Our results include the results of [1] and [2] as special cases. Furthermore, we apply our main results in proving the existence and uniqueness of solution of nonlinear Barbashin-type integrodifferential equation satisfying a given initial value problem in modular metric space Xω.

In this paper, we approximate the fixed points of multivalued quasi-nonexpansive mappings via a faster iterative process and propose a faster fixed-point iterative method for finding the solution of two-point boundary value problems. We prove analytically and with series of numerical experiments that the Picard–Ishikawa hybrid iterative process has the same rate of convergence as the CR iterative process.

The aim of this paper is to propose a new iterative algorithm to approximate the solution for a variational inequality problem in real Hilbert spaces. A strong convergence result for the above problem is established under certain mild conditions. Our proposed method requires the computation of only one projection onto the feasible set in each iteration. Some numerical examples are presented to support that our proposed method performs better than some known comparable methods for solving variational inequality problems.

We propose two new iterative algorithms for solving K-pseudomonotone variational inequality problems in the framework of real Hilbert spaces. These newly proposed methods are obtained by combining the viscosity approximation algorithm, the Picard Mann algorithm and the inertial subgradient extragradient method. We establish some strong convergence theorems for our newly developed methods under certain restriction. Our results extend and improve several recently announced results. Furthermore, we give several numerical experiments to show that our proposed algorithms performs better in comparison with several existing methods.

Recently, Khan and Abbas initiated the study of approximating fixed points of multivalued nonlinear mappings in modular function spaces. It is our purpose in this study to continue this recent trend in the study of fixed point theory of multivalued nonlinear mappings in modular function spaces. We prove some interesting theorems for ρ-quasi-nonexpansive mappings using the Picard-Krasnoselskii hybrid iterative process. We apply our results to solving certain initial value problem.