We study the generalized Ulam-Hyers stability, the well-posedness, and the limit shadowing of the fixed point problem for new type of generalized contraction mapping, the so-called α-β-contraction mapping. Our results in this paper are generalized and unify several results in the literature as the result of Geraghty (1973) and the Banach contraction principle.

where 0 < k < 1. The Banach Contraction Mapping Principle appeared in explicit form in Banach’s thesis in 1922 1 . For its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many different directions; see 2–6 . In 1997Alber andGuerreDelabriere 7 introduced the con...

Fixed point theory is one of fundamental tools of nonlinear functional analysis. Since the fixed point theory has a wide application area in almost all quantitative sciences, many authors have been working on this field. One of the impressive initial results in this direction was given by Banach 1 , known as Banach Contraction Mapping Principle. It states that each contraction in a complete met...

in a fuzzy metric space (x;m; *), where * is a continuous t-norm,a locally fuzzy contraction mapping is de ned. it is proved that any locally fuzzy contraction mapping is a global fuzzy contractive. also, if f satis es the locally fuzzy contractivity condition then it satis es the global fuzzy contrac-tivity condition.

In this paper, we consider the second-order nonlinear and the nonlinear neutral functional differential equations (a(t)x′(t))′ + f(t, x(g(t))) = 0, t ≥ t0 (a(t)(x(t)− p(t)x(t− τ))′)′ + f(t, x(g(t))) = 0, t ≥ t0 . Using the Banach contraction mapping principle, we obtain the existence of throughout positive solutions for the above equations.