Three Existence Results in the Fixed Point Theory

: In the present paper, we obtain three results on the existence of a fixed point for nonexpan-sive mappings. Two of them are generalizations of the result for F -contraction, while third one is a generalization of a recent result for set-valued contractions


Introduction
The work referenced here [1] was the starting point of the fixed point theory of nonexpansive maps, which is a growing field of research; see [2][3][4][5][6][7].In particular, the convergence of iterated Bregman projections and of the alternating algorithm were studied in [2], fixed point theorems under Mizoguchi-Takahashi-type conditions were obtained in [3], Ciric-type results were proved in [4], fixed point theory in modular function spaces was discussed in [5] and cyclic contractions were analyzed in [6].Note that the research on nonexpansive maps in spaces with graphs is of great importance; see [8][9][10][11][12] and the references mentioned therein.In particular, fixed point results on metric spaces with a graph are obtained in [9,11], Reich-type contractions were studied in [8], hybrid methods were studied in [10] and the convergence of fixed points in graphical spaces was considered in [12].In [13], D. Wardowski introduced an interesting class of mappings which contains Banach contractions and showed the existence of fixed points for these mappings.More precisely, we can assume that (X, ρ) is a complete metric space, τ > 0, F : (0, ∞) → R 1 is a strictly increasing function and that T : X → X is a mapping such that for every pair of points u, v ∈ X satisfying u ̸ = v, F(ρ(T(u), T(v))) + τ ≤ F(ρ(u, v)).
Assuming two additional assumptions on F, D. Wardowski showed that the mapping of T has a unique fixed point.In the subsequent research it was shown that these two additional assumptions are not necessary.One of the examples of F is the function ln(•), and in this case the Wardowski contraction is a strict contraction.Wardowski-type contractions were studied in [14][15][16].In this work, we obtain three results on the existence of a fixed point for nonexpansive mappings in a complete metric space.Two of them are generalizations of the result by D. Wardowski for F-contraction, while the third one is a generalization of a recent result by S.-H.Cho for set-valued contractions [17].
Assume that (X, ρ) is a complete metric space.Let N be the set of all natural numbers.We assume that the sum over empty set is zero.For every element x ∈ X and every real number r > 0, set B(x, r) = {y ∈ X : ρ(x, y) ≤ r}.

The First Result
Assume that (X, ρ) is a complete metric space where ρ is a metric, K ⊂ X is a nonempty closed set, τ > 0, and F : (0, ∞) → R 1 is an increasing function satisfying for each s > t > 0 and that T : K → X is an operator such that for every pair of points x, y ∈ K satisfying x ̸ = y, F(ρ(T(x), T(y))) + τ ≤ F(ρ(x, y)). (1) Wardowski in [13] proved the existence of a fixed point of T in the case when K = X 0 , assuming that F is strictly increasing.Here, we assume that F is merely increasing in general.
Since the function F is increasing, the following proposition holds [18].
Proposition 1.There is a countable set E ⊂ (0, ∞) such that the function F is continuous at every element z ∈ (0, ∞) \ E.
Theorem 1. Assume that K 0 ⊂ X is a nonempty bounded set and for each integer n ≥ 1 there is an element (2) Proposition 2 and (2) imply that for each n, m ∈ N, Let ϵ ∈ (0, 1).We show that the following property holds: (P1) There exists a natural number n 0 such that for each integer n > n 0 and each Choose an integer: Let n > n 0 be an integer.In order to prove that (P1) holds in view of Proposition 2, it is enough to prove that Assume the contrary.Then, according to Proposition 2, Equations ( 1) and (5) imply that for every i ∈ {0, . . ., n 0 − 1} \ {n 0 − 1}, It follows from (3), ( 5) and ( 6) that This contradicts inequality (4) and proves property (P1).We show that the following property holds: Choose an integer: Let n 1 , n 2 ≥ n 0 be integers.In view of Proposition 2, in order to show that property (P2) holds, it is enough to show that Assume the contrary.Then, Proposition 2 and (1), (2) and (8) imply that for each i ∈ {0, . . ., n 0 − 1}, According to ( 8) and (9), This contradicts (7).Therefore, (P2) holds.We will prove that the following property is fulfilled: According to Proposition 1, we may assume that F is continuous at ϵ. Assume that property (P3) does not hold.Then, for any k ∈ N, there are integers i k,1 , i k 2 , n k for which In view of property (P1), we may assume that for every integer k ≥ 1, Let k ≥ 1.According to ( 11) and ( 12), and we may assume that Equations ( 1) and ( 13) imply that Property (P1) and ( 10), ( 14) imply that Clearly, for each k ∈ N, According to ( 1) and ( 11), for each integer k ≥ 1, In view of Proposition 2 and (15), we may assume that there exists It follows from ( 15), ( 17) and ( 18) that Property (P1) and ( 15), ( 18) and (19) imply that Since the function F is continuous at ϵ, Equations ( 15) and (21) imply that This contradicts (17) and proves property (P3).Let ϵ > 0. Property (P3) implies that there exists n 0 ∈ N such that for any triplet of integers i 1 , i 2 , n satisfying Property (P2) implies that there exists a natural number m 0 ≥ n 0 such that for any triplet of integers i, n 1 , n 2 satisfying n 1 , n 2 , i ≥ m 0 , i ≤ n j , j = 1, 2 we have Assume that integers n 1 , n 2 , i 1 , i 2 satisfy According to ( 22)-(24), These relations imply that Thus, we have shown that the following property holds: For (P4), if integers i 1 , i 2 , n 1 , n 2 satisfy (24), then (25) holds.
In view of (P4), the sequences In view of Proposition 2, x * = T(x * ).Theorem 1 is proved.
The following example illustrates Theorem 1. Assume that X is the collection of all continuous functions on [0, 1], Clearly, T is a Wardowski contraction with F(t) = ln(t), t > 0 and τ = ln(2).The formula above defines T for all x ∈ X, but we consider T to be a mapping from K to X. Evidently, T does not have a fixed point in T. In view of Theorem 1, n ∈ N can be found, such that T n (x) ̸ ∈ K for any x ∈ K.A direct calculation shows that n = 3.
Let us consider the same mapping T with the same F and τ and It is not difficult to see that T(K) ̸ ⊂ K but T has a fixed point in K.

The Second Result
Assume that (X, ρ) is a complete metric space, T : X → 2 X \ {∅}; for each x ∈ X, the set T(x) is closed and a function In [17], it was shown that the set-valued T has a fixed point if for each (x, y) ∈ X 2 and each u ∈ T(x) there is v ∈ T(y), such that Examples of such mappings are considered in [17].Here, we show that T possesses a fixed point under a weaker assumption.Namely, we assume that the following assumption holds: (A) For each x, y ∈ X and each u ∈ T(x), (This sequence exists according to assumption (A)).Then, the sequence {x i } ∞ i=0 converges to a fixed point of T.
Let ϵ > 0. According to (35), there exists a natural number n 0 such that for each integer Equations ( 28) and (35) imply that there exists a natural number According to (32), ( 37) and (38), for any integer n > n 1 ,
Assume that a sequence {x i } ∞ i=2 is as in Assertion 1.Then, there exists and in view of ( 34) and ( 41)-( 43), Assertion 2 is proved.
Applying Theorem 3 by induction, we can obtain the following result.
Our results from this section can be applied to the following problem, considered in [17].Assume that a < b are real numbers, X is the space C([a, b]) of all real-valued continuous functions and that We consider a Fredholm-type integral inclusion: It was shown in [17] that the study of this problem is reduced to the analysis of a fixed point problem and that for mapping T, all the assumptions made in this section hold.Therefore, all the results can be applied for T.

The Third Result
Assume that (X, ρ) is endowed with a graph G. Let V(G) be the set of its vertices, E(G) be the set of its edges and let Assume that τ > 0, F : (0, ∞) → R 1 is an increasing function and that T : X → X is a mapping such that for any (x, y) ∈ E(G) for which x ̸ = y, (T(x), T(y)) ∈ E(G) and F(ρ(T(x), T(y))) + τ ≤ F(ρ(x, y)). (56) Equation ( 56) implies the following proposition.
Proposition 4 implies the following result.
Proposition 5 implies the following result.
Assume that there exists m 0 ∈ N such that the following property holds: (P) for any pair of nonnegative integers i < j, there is p ∈ {j, . . ., j + m 0 } for which Then, the sequence {T n (x)} ∞ n=0 converges and its limit is a fixed point of T if the graph of T is closed.
Since the collection of all points at which F is not continuous is countable, we may assume that the function F is continuous at ϵ. Choose δ ∈ (0, ϵ/4) and set In view of (59), there exists a natural number n 0 for which Assume that j > i ≥ n 0 are integers.We show that ρ(T i (x), T j (x)) ≤ ϵ.

Conclusions
We consider three fixed point problems, and for each of them establish the existence of a fixed point.In the first and the third cases, we consider single-valued Wardowski type contraction, while in the second case we study Cho-type set-valued contractions.In the second case, we also study approximate fixed points.
Funding: This research received no external funding.