On the Semi-Local Convergence of a Traub-Type Method for Solving Equations

The celebrated Traub’s method involving Banach space-defined operators is extended. The main feature in this study involves the determination of a subset of the original domain that also contains the Traub iterates. In the smaller domain, the Lipschitz constants are smaller too. Hence, a finer analysis is developed without the usage of additional conditions. This methodology applies to other methods. The examples justify the theoretical results.

On the Ostrowski Method for Solving Equations

In this paper, we revisited the Ostrowski's method for solving Banach space valued equations. We developed a technique to determine a subset of the original convergence domain and using this new Lipschitz constants derived. These constants are at least as tight as the earlier ones leading to a finer convergence analysis in both the semi-local and the local convergence case. These techniques are very general, so they can be used to extend the applicability of other methods without additional hypotheses. Numerical experiments complete this study.

Numerical Method for Fractional Fuzzy Integral Equations

In the present work we construct an iterative method for the numerical solution of fuzzy fractional Volterra integral equations, by using the technique of fuzzy product integration. The existence and uniqueness of the solution and the uniform boundedness of the terms of the Picard iterations are proved. The convergence of the iterative algorithm is obtained and the apriori error estimate is given in terms of the Lipschitz constants. A numerical example illustrates the accuracy of the method.

Generalized Reich–Ćirić–Rus-Type and Kannan-Type Contractions in Cone b -Metric Spaces over Banach Algebras

In this paper, we firstly introduce the generalized Reich‐Ćirić‐Rus-type and Kannan-type contractions in cone b -metric spaces over Banach algebras and then obtain some fixed point theorems satisfying these generalized contractive conditions, without appealing to the compactness of X . Secondly, we prove the existence and uniqueness results for fixed points of asymptotically regular mappings with generalized Lipschitz constants. The continuity of the mappings is deleted or relaxed. At last, we prove that the completeness of cone b -metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in X . Our results greatly extend several important results in the literature. Moreover, we present some nontrivial examples to support the new concepts and our fixed point theorems.

Extending the Applicability and Convergence Domain of a Fifth-Order Iterative Scheme under Hölder Continuous Derivative in Banach Spaces

The most significant contribution made by this study is that the applicability and convergence domain of a fifth-order convergent nonlinear equation solver is extended. We use Hölder condition on the first Fréchet derivative to study the local analysis, and this expands the applicability of the formula for such problems where the earlier study based on Lipschitz constants cannot be used. This study generalizes the local analysis based on Lipschitz constants. Also, we avoid the use of the extra assumption on boundedness of the first derivative of the involved operator. Finally, numerical experiments ensure that our analysis expands the utility of the considered scheme. In addition, the proposed technique produces a larger convergence domain, in comparison to the earlier study, without using any extra conditions.

Adaptive operator extrapolation method

This paper is devoted to the study of nоvel algorithm with Bregman projection for solving variational inequalities in Hilbert space. Proposed algorithm is an adaptive version of the operator extrapolation method, where the used rule for updating the step size does not require knowledge of Lipschitz constants and the calculation of operator values at additional points. An attractive feature of the algorithm is only one computation at the iterative step of the Bregman projection onto the feasible set.

Explicit Solutions of the Extended Skorokhod Problems in Affine Transformations of Time-Dependent Strata

The goal of this paper is to expand the explicit formula for the solutions of the Extended Skorokhod Problem developed earlier for a special class of constraining domains in ℝ n with orthogonal reflection fields. We examine how affine transformations convert solutions of the Extended Skorokhod Problem into solutions of the new problem for the transformed constraining system. We obtain an explicit formula for the solutions of the Extended Skorokhod Problem for any ℝ n - valued càdlàg function with the constraining set that changes in time and the reflection field naturally defined by any basis. The evolving constraining set is a region sandwiched between two graphs in the coordinate system generating the reflection field. We discuss the Lipschitz properties of the extended Skorokhod map and derive Lipschitz constants in special cases of constraining sets of this type.

Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $${\mathbb {H}}^n$$ H n , $$n\in {\mathbb {N}}$$ n ∈ N . For $$1\leqslant k\leqslant n$$ 1 ⩽ k ⩽ n , we show that every intrinsic L-Lipschitz graph over a subset of a k-dimensional horizontal subgroup $${\mathbb {V}}$$ V of $${\mathbb {H}}^n$$ H n can be extended to an intrinsic $$L'$$ L ′ -Lipschitz graph over the entire subgroup $${\mathbb {V}}$$ V , where $$L'$$ L ′ depends only on L, k, and n. We further prove that 1-dimensional intrinsic 1-Lipschitz graphs in $${\mathbb {H}}^n$$ H n , $$n\in {\mathbb {N}}$$ n ∈ N , admit corona decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that were known previously only in the first Heisenberg group $${\mathbb {H}}^1$$ H 1 . The main difference to this case arises from the fact that for $$1\leqslant k<n$$ 1 ⩽ k < n , the complementary vertical subgroups of k-dimensional horizontal subgroups in $${\mathbb {H}}^n$$ H n are not commutative.

Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

The objective of this paper is to introduce an iterative method with the addition of an inertial term to solve equilibrium problems in a real Hilbert space. The proposed iterative scheme is based on the Mann-type iterative scheme and the extragradient method. By imposing certain mild conditions on a bifunction, the corresponding theorem of strong convergence in real Hilbert space is well-established. The proposed method has the advantage of requiring no knowledge of Lipschitz-type constants. The applications of our results to solve particular classes of equilibrium problems is presented. Numerical results are established to validate the proposed method’s efficiency and to compare it to other methods in the literature.