Fixed points of some nonlinear operators in spaces of multifunctions and the Ulam stability

We prove a fixed point theorem for nonlinear operators, acting on some function spaces (of set-valued maps), which satisfy suitable inclusions. We also show some applications of it in the Ulam type stability.


Introduction
The question when we can replace an approximate solution to an equation by an exact solution to it (or conversely) and what error we thus commit seems to be very natural. Some convenient tools to study such issues are provided by the theory of Ulam's (often also called the Hyers-Ulam) type stability. For some updated information and further references concerning the Ulam stability, we refer to [1,4,5]. Let us only mention that the investigation of that problem started with a question raised by Ulam in 1940 and an answer to it given by Hyers in [3].
It has been noticed in numerous papers that there are strict connections between some fixed point theorems and the results concerning the Ulam stability of various (differential, difference, functional, and integral) equations; for a suitable survey we refer to [2]. In this paper we continue those investigations by proving a fixed point result for a class of nonlinear operators acting on some spaces of set-valued mappings and showing several of its consequences.
Through this paper, we assume that K is a nonempty set and (Y, d) is a complete metric space. We denote by n(Y ) the family of all nonempty subsets of Y , by bd(Y ) the family of all nonempty and bounded subsets of JFPTA Y , and by bcl(Y ) the family of all closed sets from bd(Y ). Moreover, h is the Hausdorff distance induced by the metric in Y and given by It is well known that h is a metric if restricted to bcl(Y ).
The number (possibly also ∞) is said to be the diameter of A ∈ n(Y ). For F : K → n(Y ) and g : K → Y , we denote by cl F and g the multifunctions defined by We write a 0 (x) = x for x ∈ K and a n+1 = a n • a for a : K → K, n ∈ N 0 (N 0 stands for the set of nonnegative integers). We present a theorem, concerning fixed points of some operators acting on set-valued functions, and several of its consequences. To do this, we need to introduce some notations. Namely, given functions a, b ∈ R K (as usually, B A denotes the family of all functions mapping a set We say that Λ : We always assume the Tichonoff topology (of pointwise convergence) in bcl(Y ) K , with the Hausdorff metric in bcl(Y ).
We write for each sequence (H n ) n∈N in bd(Y ) K , such that the sequences (cl H n ) n∈N and (cl (αH n )) n∈N are convergent in bcl(Y ) K . We also need the following hypothesis for operators α : bd Vol. 19 (2017) Fixed points of some nonlinear operators 2443 Clearly, (H) is somewhat complementary to (1). Finally, δ : bd(Y ) K → R + K is given by the formula and, for every t ∈ R + and a ∈ R K + , we define the mapping ta ∈ R K + by (ta)(x) := ta(x) for x ∈ K.

Main results
are given. We consider functions F ∈ bd(Y ) K that satisfy the equation: We use the following contraction condition on α: Now, we are in a position to present the main result of this paper.

Theorem 1. Assume that Λ is non-decreasing, α is i.p. and satisfies
Suppose that α is l.p. or (H) is valid. Then, there exists a function f : K → Y , such that f is a fixed point of the operator α (i.e., α f = f ) and then Since α is i.p., by (2), we get for every n ∈ N 0 (nonnegative integers). Hence Therefore, for k ∈ N, n ∈ N 0 , we have Furthermore, by (4), we get Moreover, whence (cl α n F (x)) n∈N0 is a Cauchy sequence of closed and bounded sets and, as the space (bcl(Y ), h) is complete, there exists the limit Furthermore, by (3) and (7), we have and (Λ n ( δF )(x)) n∈N0 is convergent to 0 as n → ∞. Therefore, the set ρ(x) has exactly one element for each x ∈ K and we denote that element by f (x). If α is l.p., it is clear that Thus, α f = f . If (H) holds, then whence again, α f = f . Next, by (6), we have for n ∈ N, and consequently, with n → ∞, Vol. 19 (2017) Fixed points of some nonlinear operators 2445 It remains to show the statement on the uniqueness of f . Therefore, fix G ∈ bd(Y ) K and μ ∈ R K + , such that (5) holds, G ⊂ αG, and h(G(x), F (x)) ≤ μ(x) for x ∈ K. Define the multifunction B F : K → n(Y ) by Then, it is easily seen that F, G ⊂ B F , and consequently α n F, α n G ⊂ α n B F , n ∈ N.
Next, for each n ∈ N, we have G ⊂ α n G, whence Note that for every x ∈ K, y, z ∈ B F (x) and w 1 , w 2 ∈ F (x), we have This completes the proof in view of (5).

Some consequences
The next simple theorems show some direct applications of Theorem 1; they correspond to the results on stability of functional equations (for the setvalued mappings) in [6][7][8][9][10].
Then, it is easily seen that it is i.p. Next, let (H n ) n∈N be a sequence in bd(Y ) K , such that there exist H L := lim n→∞ cl H n ∈ bcl(Y ) K and lim n→∞ cl (αH n ) ∈ bcl(Y ) K . Clearly, on account of (9),