A refinement of the Browder–Göhde–Kirk fixed point theorem and some applications

The following generalization of the Browder–Göhde–Kirk fixed point theorem is proved: ifCis a nonempty bounded closed and convex subset of a uniformly convex normed spaceXandTis a self-mapping ofCsuch thatTx-Ty≤βx-y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\| Tx-Ty\right\| \le \beta \left( \left\| x-y\right\| \right) $$\end{document} for all x,y∈C,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y\in C,$$\end{document}x≠y,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ne y,$$\end{document}where a functionβ:0,∞→0,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta :\left( 0,\infty \right) \rightarrow \left[ 0,\infty \right) $$\end{document}is such thatlimt→0+βtt=1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lim _{t\rightarrow 0+}\frac{\beta \left( t\right) }{t}=1,$$\end{document}thenThas a fixed point. Two modifications of this theorem as well as some accompanying results on Lipschitz-type mappings are given. An application in the theory of Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}$$\end{document}-solutions of an iterative functional equation, and some refinements of the Radamacher theorem are proposed.


Introduction
Basing on an observation that every continuous map of a convex set satisfying a restricted Lipschitz condition must be Lipschitz continuous (see [12]), we present some generalizations of Browder-Göhde-Kirk fixed point (Browder [1], Göhde [7], Kirk [9], see also [4,6,15,16]), and propose their applications, including an extension of the classical Radamacher theorem. Section 2 contains the auxiliary results characterizing the Lipschitz continuous functions with the aid of some weaker conditions. In Sect. 3, we present two fixed point theorems. Let X be a uniformly convex Banach space, C ⊂ X a nonempty bounded convex closed set, and T a selfmapping of C. Theorem 1 says that T has a fixed point, if for some function β : (0, ∞) → [0, ∞) satisfying the conditions we have, for all n ∈ N and for all x, y ∈ C, In Sect. 4, we use Theorem 1 to get a result on the existence and uniqueness of L p -solutions (1 < p < +∞) of the iterative functional equation In Sect. 5, we give some refinements of the classical Radamacher theorem on the differentiability of the Lipschitz mappings.

Some auxiliary results on Lipschitz continuity
We begin with the following then Proof. Note that conditions (2) and (1) imply that T is continuous. Indeed, from (2) there are some real positive M and δ such that Take arbitrary x, y ∈ C, x = y. By (3), for every ε > 0 there is a t ε > 0 and a unique n = n ε ∈ N 0 such that and lim ε→0 t ε = 0. Put Then, by the convexity of C, and, by (5), Hence, applying in turn: the triangle inequality, condition (1), some obvious identities, (4), (7) and (5), we get Since the continuity of T and the conditions (6) and (8) imply that letting ε → 0 in the above inequality, we obtain which completes the proof.  [11]). In this case, instead of (3) it is enough to assume that L < +∞.

Remark 2.
The reasoning in the proof of Lemma 1 simplifies, if Indeed, if this condition holds, then for every ε > 0 there is a δ > 0 such that Take arbitrary x, y ∈ C, x = y, choose n ∈ N such that and put Of course and, by the convexity of C, Applying the triangle inequality and (1), we hence get A much weaker necessary and sufficient condition for a continuous map to be Lipschitz continuous gives the following Lemma 2. Let X and Y be real normed spaces and C ⊂ X a bounded convex set. Suppose that T : C → Y is continuous. If there are a nonnegative real L and two positive sequences (t n ) , (c n ) , such that for every n ∈ N and for all x, y ∈ C, then T is Lipschitz continuous, and Vol. 24 (2022) Browder-Göhde-Kirk fixed point theorem Page 5 of 12 70 Proof. Take arbitrary x, y ∈ C, x = y. For every n ∈ N, there is a unique m n ∈ N ∪ {0} such that m n t n ≤ x − y < (m n + 1) t n . Put Since 0 ≤ m n t n y − x ≤ 1, and, for each k = 0, 1, . . . , m n , the convexity of C implies that z k ∈ C, k = 0, 1, . . . , m n .
Moreover, by (10), and, for k = m n , we have From (11) and (9), we get T z k − T z k+1 ≤ c n t n , k = 0, 1, . . . , m n − 1, so, by the triangle inequality, whence, taking into account that m n t n ≤ x − y , by (12), we get Since, by (12), z mn − y < t n , we have Hence, letting n → ∞ in (13), and taking into account that lim n→∞ c n = L, we conclude that which was to be shown.

Fixed-point theorems
Recall that a real normed vector space (X, · ) is called uniformly convex, if for every ε ∈ (0, 2] there is some δ > 0 such that for any two vectors x, y ∈ X with x = y = 1, the condition x − y ≥ ε implies that x+y 2 ≤ 1 − δ (Goebel and Reich [6]; see also [13] for a generalization).
Applying Lemma 1 with L = 1 we obtain the following generalization of the Browder-Göhde-Kirk theorem.

then T has a fixed point in C.
Proof. Applying Lemma 1 with L = 1, we get that is T is nonexpansive, and the result follows from the original version of the Browder-Göhde-Kirk theorem.
In particular, the thesis of Browder-Göhde-Kirk theorem remains true, if the nonexpansivity of the mapping T is replaced for instance, by the inequality This proposition improves the relevant result in [11] where the uniform continuity of T is assumed.
The main result of this section reads as follows.
Theorem 2. Let X be a uniformly convex Banach space and C ⊂ X a nonempty bounded closed and convex set. Suppose that T : C → C is continuous. If there exist a function β : (0, ∞) → [0, ∞) and a sequence of positive real (t n ) , lim n→∞ t n = 0 satisfying the condition such that for every n ∈ N and for all x, y ∈ C, then T has a fixed point.
Proof. Setting c n := β(tn) tn we have lim n→∞ c n = 1 and for every n ∈ N and for all x, y ∈ C, if x − y = t n , then and the result follows from Proposition 1.

An application in the theory of iterative functional equations
For a measure space (Ω, Σ, μ) and a real p > 1, denote by L p (Ω) , · p the Banach space of all (equivalence classes with respect to the μ-a.e. equality) of Σ-measurable functions ϕ : Ω → R such that |ϕ| p is μ-integrable, and It is well known that L p (Ω) , · p is a uniformly convex Banach space (Clarkson [3]).
In this section, we consider solutions ϕ ∈ L p (Ω) of the iterative-type functional equation We assume that (Ω, Σ, μ) the given functions f and h satisfy the following conditions: (i) k ∈ N; Ω ⊂ R k is an open set; μ is the Lebesgue measure, μ (Ω) = 1; and f : Ω → Ω, f = (f 1 , . . . , f k ) , is a locally Lipschitzian homeomorphic mapping; (ii) h : Ω × R → R is such that: for every y ∈ R the function Ω x −→ h (x, y) is Lebesgue measurable, and R y −→ h (x, y) is continuous for almost all x ∈ Ω (with respect to μ); (iii) p ∈ R, p > 1, and there are g 1 , g 2 ∈ L p (Ω) , g 1 ≤ g 2 a.e. in Ω such that for all x ∈ Ω and y ∈ R, the following implication holds true: Applying Theorem 1, we prove the following: where and Then there exists ϕ ∈ L p (Ω) , g 1 ≤ ϕ ≤ g 2 a.e. in Ω, such that moreover, if β (t n ) = t n for a sequence of t n > 0, lim n→∞ t n = 0, then such ϕ is unique, and for any ϕ 0 ∈ L p (Ω) , g 1 ≤ ϕ 0 ≤ g 2 a.e. in Ω, the sequence (ϕ n ) defined recursively by ϕ n (x) = h (x, ϕ n−1 [f (x)]) a.e. for x ∈ Ω; n ∈ R, converges to ϕ in the norm · p .
It is easy to see that C is a nonempty, convex and closed subset of L p (Ω) . Define the mapping T on C by Take an arbitrary ϕ ∈ C. Then, in view of Carathéodory theorem [2], conditions (ii) imply that the function T (ϕ) is Lebesgue measurable. Since g 1 ≤ ϕ ≤ g 2 a.e. in Ω we have, for a.e. x ∈ Ω whence, in view of condition (iii), for a.e. x ∈ Ω, that is T (ϕ) ∈ C, which proves that T maps C into itself. Take arbitrary ϕ 1 , ϕ 2 ∈ C. Making use in turn of: the definition of T ; (14) (we use here the measurability of α and β); (15); the theorem on change of the variables under integral (see Lojasiewicz [10]), the inclusion f (Ω) ⊂ Ω; an obvious equality; the assumption that the Lebesgue measure of Ω is 1 and