Generalized trend constants of Lipschitz mappings

In 2015, Goebel and Bolibok defined the initial trend coefficient of a mapping and the class of initially nonexpansive mappings. They proved that the fixed point property for nonexpansive mappings implies the fixed point property for initially nonexpansive mappings. We generalize the above concepts and prove an analogous fixed point theorem. We also study the initial trend coefficient more deeply.


Introduction.
Let X be a Banach space, C be a nonempty subset of X, and T be a mapping from C into itself.The mapping T is known as k-Lipschitz (k ≥ 0) if T x − T y ≤ k x − y for every x, y ∈ C. The minimal k, for which the above condition holds, is called the Lipschitz constant of T and is denoted by k (T ).If k (T ) < 1 (resp.k (T ) ≤ 1), then T is said to be a contraction (resp.a nonexpansive mapping).By L C (k) (or L (k) in short) we denote the set of all k-Lipschitz mappings from C into itself.The mapping T is Lipschitz if it is k-Lipschitz for some k.
In [1] (see also [3]) the following coefficients were defined where ∂ + and ∂ − denote one-sided derivatives.The expression ∂ + ψ x, y (0) can be seen as the directional right derivative of the norm at the point x − y along the vector T x − T y − (x − y).The coefficients ι (T ) and τ (T ) are known as the initial and the final trend constants of T , respectively.The mapping T is said to be an initial contraction if ι (T ) < 0, and initially nonexpansive if ι (T ) ≤ 0. We extend the above notion in the following way.The trend constant (more precisely the right trend constant) of the mapping T at a point α ∈ R is given by the formula We say that T : C → C is a pre-initial contraction (resp.pre-initially nonexpansive) if it is a k-Lipschitz mapping, where k > 1, and there exists α ∈ −1 k−1 , 0 such that ι α (T ) < 0 (resp.ι α (T ) ≤ 0).Note that the initial trend coefficient of T is equal to ι 0 (T ).
The mapping T is said to be firmly nonexpansive if for all x, y ∈ C the function ϕ x, y (t) is nonincreasing on the interval [0, 1].The mapping T is firmly nonexpansive if and only if τ (T ) ≤ 0.
The fixed point set for T is defined as

General trend of mappings.
In this section, we generalize results obtained in [1].Let u, v ∈ X.The function t → G u, v (t) is convex, so it is a semi-differentiable function at every real number t. Assume that α < β.
The following inequalities are obvious Proof.Putting w (s) = (1 − s) u + sv, where s ∈ R, we obtain Let C be a nonempty closed convex and bounded subset of X. Assume that T : Since the right hand side is a k A -Lipschitz mapping and k A < 1, this equation has a unique solution.Denoting this solution by F x, we obtain the function Rearranging the above equality, we obtain (2.4)

and
(2.5) Observe that F α x belongs to the line segment [x, F x], which is a subset of C, so F α is a mapping from C into itself.Let x 0 ∈ C. If F x 0 = x 0 , then from (2.3) we obtain T x 0 = x 0 .Conversely, if T x 0 = x 0 , then This is an equality of the (2.2) form.We know that the equality (2.2) has a unique solution, so F x 0 = x 0 .Assume that F α x 0 = x 0 .It is equivalent to Applying the equality T F x 0 = AF x 0 − (A − 1) x 0 , we obtain which is equivalent to F x 0 = x 0 .We have proved that FixF α = FixF = FixT .
Given distinct x, y ∈ C, we have x − y As a special case, if α = 0, we obtain , From the above consideration, we obtain the following corollaries.
We say that a subset C ⊂ X has the fixed point property for nonexpansive mappings if every nonexpansive mapping T : C → C has a fixed point.The space X has the fixed point property for nonexpansive mappings if all nonempty closed convex and bounded subsets of X have this property.Similarly we define the fixed point properties for pre-initially nonexpansive mappings.

Corollary 2.3. If a nonempty closed convex and bounded set C ⊂ X has the fixed point property for nonexpansive mappings, then it has the fixed point property for pre-initially nonexpansive mappings.
3. Formulas for trend constants.In this section, we provide a few formulas for trend constants of a mapping T : C → C, where C is a nonempty subset of a Banach space X.Let us recall some basic facts about the subdifferential of the norm.Let x, y ∈ X.The function R t → x + ty is convex, so the following limits exist The subdifferential of the norm is defined by We have also the following formulas and on the right at t 0 ∈ [a, b).Moreover, we have the following chain rules For further details about subdifferentials of norms, see for example [2]. Let Then, by (3.1), for every α ∈ R. As a consequence, we obtain a new formula for the trend constant of T at α.

Corollary 3.2.
For every α ∈ R, where the supremum is taken over all distinct vectors x, y ∈ C and functionals In case of the initial trend coefficient, this formula takes the following form Theorem 3.3.Let x, y ∈ C be distinct vectors, and α ∈ R. If Proof.Let t 0 > t 1 > t 2 = α, and let E (t) = E x−y, T x−T y (t) for t ≥ α.We obtain The equality (3.4) follows from the above inequalities and the following equalities From the above theorem we obtain another formula for the trend constant of the mapping T at α.
In [1] the formula for the initial trend coefficient for Hilbert spaces is given.Using the equality (3.3), we can calculate formulas for this coefficient in some spaces.Here we will deal with the space C [a, b].In order to prove such a result, we can use the characterization of the subdifferential of the norm in C [a, b] given in [2].Note that in the literature, one can find similar characterizations for some other spaces (see for example [2] and [4]).Another approach is to apply a formula for the directional right derivative of the norm.Such a formula for C [a, b] is given in [5].Using one of the above methods, we obtain the following claim.where the supremum is taken over all distinct vectors x, y ∈ C and all s in M 0 (x − y).

Remarks about the initial trend coefficient. We say that the norm
• of a Banach space X is Gâteaux differentiable at a point x ∈ X if for every h ∈ X the limit lim t→0 x + th − x t exists.If, moreover, this limit is uniform for x, h ∈ S X , then we say that the norm is uniformly Fréchet differentiable.We say that the norm is Gâteaux differentiable if it is Gâteaux differentiable at every point x ∈ S X .Theorem 4.1.Let X be a Banach space whose norm is uniformly Fréchet differentiable and C be a nonempty and convex subset of X.If T : C → C is a Lipschitz mapping such that ι (T ) < 0, then there exists δ ∈ (0, 1) such that the mapping (1 − t 0 ) I + t 0 T is a contraction, where I is the identity on C and t 0 ∈ (0, δ).
. Since the norm of X is uniformly Fréchet differentiable, there exists τ > 0 such that for every t ∈ (0, τ ) and z, h ∈ S X , where z * ∈ ∂ z .Note that ∂ z is a one-element set, because the norm of X is Gâteaux differentiable.We choose t 0 ∈ 0, τ 1+k , and define the mapping T t 0 : C → C by the formula Given distinct elements x, y ∈ C, we have x − y = T x − T y.Otherwise, ψ x, y (t) = 1 for t ∈ [0, 1], and ι (T ) ≥ ∂ + ψ x, y (0) = 0, which contradicts our assumption.Put u = x − y, and v = T x − T y.Note that

Using this inequality and putting
, we obtain where z * ∈ ∂ u u .Since ∂ u u = ∂ x − y , applying Corollary 3.2, we get Let C be a nonempty subset of a Banach space X.Consider a Lipschitz mapping T : The function µ is convex.We define the coefficient which is similar to the initial trend coefficient (1.1), but the derivative and the supremum are swapped.This coefficient is greater than or equal to the initial trend coefficient of T .Indeed, given ε > 0, there exist distinct elements x, y ∈ C such that and therefore Since ε > 0 is arbitrary, ι (T ) ≤ κ (T ).In the linear case we have the equality.
Proof.Observe that, by the assumptions of the theorem, and according to Corollary 3.2, In case of T = 0, we have ι (T ) = κ (T ) = −1.Now we can assume that T = 0. Let ε > 0 and t 0 ∈ 0, min . There exists an element x 0 ∈ S X such that We obtain Since ε is an arbitrary positive number, κ (T ) ≤ ι (T ).
The next claim gives us the characterization of mappings such that κ (T ) < 0. Since for a linear bounded self-mappings T we have ι (T ) = κ (T ), this claim also applies to the linear initial contractions.Proof.There exists δ > 0 such that For every x, y ∈ C, x = y we get Putting Rx = (1 − δ) x + δT x for x ∈ C, we obtain where 1 + κ(T )δ 2 < 1, so R is a contraction.
Let C be a subset of the space X = C [0, 1].We define the Hammerstein operator T : C → X by the following formula where k, f are continuous functions on [0, 1] 2 , t → f (t 0 , t) is a k f (t 0 )-Lipschitz mapping on the interval [0, 1], and the function Proof.The inequality ι (T ) ≤ k (T ) − 1 is true in general (see [1]), so it is enough to prove the opposite inequality.
At the end, we will study some examples from [1].

Claim 4 . 3 .
Let T : C → C be such that κ (T ) < 0. Then there exist a contraction R : C → C and δ > 0 such that T x = 1 δ Rx + 1 − 1 δ x for every x ∈ C.