Stability of Generalized Functional Equations in Nonlinear Locally Convex Spaces via Fixed Point Techniques
DOI:
https://doi.org/10.31642/JoKMC/2018/130204Keywords:
Boyd–Wong contractions, Fixed point techniques, Hyers–Ulam–Rassias stability, Locally convex spaces, Nonlinear functional equationsAbstract
The given paper explores the stability of a new type of generalized functional equations, which are defined in nonlinear locally convex spaces and generalizes classical theories of Hyers-Ulam and Hyers-Ulam-Rassias stability in the non-linear case including linear and normed functional spaces. We introduce a generalized nonlinear functional equation in the form of operator dependent perturbation terms and we develop an appropriate locally convex topology employing a directed family of seminorms which reflects the nonlinear geometry of the underlying space. Through the application of a variety of fixed point methods, such as Banach contractions, and nonlinear contraction methods such as Zamfirescu and Boyd-Wong operators, we prove a number of new stability theorems of generalized functional equations. The findings consolidate and vastly generalize a variety of stability findings in Banach and Frechet spaces which demonstrate that classical conditions of linearity and normability are not required to get stability. Moreover, we give some examples in function space, such as integral and nonlinear transformation operators, of the applicability and sharpness of the stability bounds obtained. These results add significant extension to the theory of functional equations stability, and furnish a fixed point-oriented basis to future research of nonlinear analysis, operator theory, and applied functional equations.
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