Abstract In this paper, we examine the numerical solution of delay Volterra integral equations (DVIEs). Here, a hybrid stable and convergent numerical method is presented to solve a class of DVIEs. First, by combining the method of successive approximations with the trapezoidal integration rule and cubic spline interpolation, we obtain an efficient scheme for numerical solving of the given equation. Then, for the given type of equation, we perform an error analysis and prove the convergence and numerical stability of the method. Finally, some numerical examples are provided to confirm the theoretical results.
Abstract In this paper, we investigate convergence in measure for a sequence of generalized Sugeno integrals with respect to monotone measures. While the existing results typically require continuity of the underlying measure or provide only sufficient conditions, we establish new necessary and sufficient conditions that differ from the assumptions previously considered in the literature. We prove that the autocontinuity of a monotone measure completely characterizes convergence in measure for a sequence of generalized Sugeno integrals induced by a continuous semicopula. Our results are first established for the case where the range of the limiting function is finite, and are subsequently extended to classes of functions with infinite range under an additional assumption. In particular, the proof techniques developed in this paper are entirely different from those employed in the existing literature.
Abstract In this paper, we present upper and lower bounds for eigenvalues of quaternionic Hermitian and non-Hermitian matrices under structured (especially rank-one) perturbations. Then we derive rigorous perturbation bounds for right eigenvalues of quaternionic matrices, extending classical results to the quaternions with particular attention to similarity classes of right eigenvalues. Numerical experiments demonstrate the superiority of our bounds in quaternionic continuation methods, with better improvements over existing approaches.
In this paper some enriched rational type contractive mappings are introduced and related fixed point theorems are proved by using Krasnselskii iteration procedure. Some examples are cited in support of our theorems to show that our theorems are more generalized. Finally, two different applications of our proved theorems are given to ensure the existence of a unique solution to a nonlinear boundary value problem and also to get solutions for some functional equations relating to the problem of dynamical programming.
