Fixed point results for non-self nonlinear graphic contractions in complete metric spaces with applications
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Journal of Fixed Point Theory and Applications
Fixed point results for non-self nonlinear graphic contractions in complete metric spaces with applications Cristian Chifu , Adrian Petru¸sel and Gabriela Petru¸sel Abstract. In this paper, we will present some fixed point results for the case of non-self operators satisfying a nonlinear graphic contraction condition in complete metric spaces. Properties of the corresponding fixed point equation are established and some applications are given. Mathematics Subject Classification. 47H10, 54H25. Keywords. Ordered metric space, fixed point, nonlinear graphic (orbital) contraction, data dependence, well-posedness, Ulam–Hyers stability, applications.
1. Introduction Let (X, d) be a metric space and r > 0. Then, the open ball centered at x0 ∈ X of radius r is denoted by B(x0 ; r), while for the closed ball centered ˜ 0 ; r). Throughout this at x0 ∈ X of radius r, we will use the symbol B(x ∗ paper, we denote N := {0, 1, 2, . . .} and N := N\{0}. We also set R for the set of all real numbers and R+ := [0, +∞) for the set of all nonnegative real numbers. Definition 1. A function ϕ : R+ → R+ is called a comparison function if it satisfies: (i)ϕ ϕ is increasing; (ii)ϕ (ϕn (t))n∈N converges to 0 as n → ∞, for all t ∈ R+ . If the condition (ii)ϕ is replaced by the condition: ∞ (iii)ϕ ϕk (t) < ∞, for any t > 0, k=0
then ϕ is called a strong comparison function. Lemma 1. If ϕ : R+ → R+ is a comparison function, then the following holds: 0123456789().: V,-vol
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(i) ϕ(t) < t, for any t > 0; (ii) ϕ(0) = 0; (iii) ϕ is continuous at 0. Lemma 2. If ϕ : R+ → R+ is a strong comparison function, then the following holds: (i) ϕ is a comparison function; (ii) the function s : R+ → R+ , defined by: s(t) := ϕk (t), t ∈ R+ , (1) k≥0
is increasing and continuous at 0; ∞ (iii) there exist k0 ∈ N, a ∈ (0, 1) and a convergent series k=1 vk with nonnegative terms, such that: ϕk+1 (t) ≤ aϕk (t) + vk , for k ≥ k0 and any t ∈ R+ . Remark 1. The concept of (c)-comparison function (which appears in some papers) coincides with that of strong comparison function, see A. Magda¸s [9]. Example 1. The following functions ϕ : R+ → R+ are comparison functions: (1) ϕ(t) = kt, where k ∈ [0, 1). (2) ϕ(t) = ln(1 + t), t , where a ∈ [1, ∞). (3) ϕ(t) = t+a Moreover, the first example is also a strong comparison function, while the second is a comparison function which is not a strong comparison function. The third example is a strong comparison function if and only if a ∈ (1, ∞) (see Magda¸s [8,9]). For more considerations on comparison functions, see [30,33] and the references therein. See also [4,27]. If f : X → X is an operator, then x ∈ X is called fixed point for f if and only if x = f (x). The set Ff := {x ∈ X|x = f (x)} denotes the fixed point set of f , while the symbol Graph(f ) := {(x, y) ∈ X × X : y = f (x)} is the graph of the operator f . Finally, we will denote the attraction basin of x∗ ∈ Ff by: (AB)f (x∗ ) := {x ∈ X| f n (x) → x∗ as n → ∞}. Definition 2. Let (X, d) be
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