Abstract
It is well-known in the literature that many analytical techniques are introduced in order to find a solution for problems such as functional, differential, and integral equations. These analytical techniques sometimes fail to solve such problems, thus prompting the proposal of numerical methods for approaching their approximate solutions. This paper suggests a multi-valued version of an efficient iterative procedure called the F iterative procedure in Banach space and establishes its weak and strong convergence to fixed points of certain proximally quasi-nonexpansive operators. To support these results and to suggest the high accuracy of this procedure, we develop an example of a proximally quasi-nonexpansive operator and perform a comparative numerical experiment. As an application, we solve a two-point boundary value problem (BVP) in Banach space. Our results are new and extend some results from the literature for the new setting of mappings.
Highlights
Introduction and PreliminariesLet E be a subset of a normed vector space X
We use an iterative method, which converges in the fixed point set of the fixed point equation, and in the solution set of the original equation as well
If the operator T : E → PE is proximally quasi-nonexpansive on a convex nonempty closed subset E of a uniformly convex Banach space (UCBS) X and the set Fix(T ) = ∅, the sequence of iterates (2) has the property limn→∞ ||an − q|| that exists for each fixed point q of T
Summary
Introduction and PreliminariesLet E be a subset of a normed vector space X. Banach [1] was the first to introduce these mappings and proved that such mappings in a closed subset of Banach space always posses a unique fixed point, that is, T q = q for some q. When the space X is restricted to the case of uniformly convex Banach space (UCBS), and E is closed bounded and convex in X, each nonexpansive map T : E → E has a fixed point (e.g., see Browder [2] and Gohde [3]). Note that it is sometimes possible that a given equation (functional, differential, integral, etc.) has a solution, but its value cannot be found by applying available analytical methods. In such cases, one needs the approximate value of such solutions. We use an iterative method, which converges in the fixed point set of the fixed point equation, and in the solution set of the original equation as well
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
