Fixed Point, Data Dependence, and Well-Posed Problems for Multivalued Nonlinear Contractions

. The aim of the paper is to discuss data dependence, existence of ﬁ xed points, strict ﬁ xed points, and well posedness of some multivalued generalized contractions in the setting of complete metric spaces. Using auxiliary functions, we introduce Wardowski type multivalued nonlinear operators that satisfy a novel class of contractive requirements. Furthermore, the existence and data dependence ﬁ ndings for these multivalued operators are obtained. A nontrivial example is also provided to support the results. The results generalize, improve, and extend existing results in the literature.


Introduction and Preliminaries
Let ðZ, dÞ be a metric space (in short MS).The set of all nonempty subsets of Z is denoted by PðZÞ, the set of all nonempty closed subsets of Z is denoted by CLðZÞ, the set of all nonempty closed and bounded subsets of Z is denoted by CBðZÞ, and the set of all nonempty compact subsets of X is denoted by KðZÞ.It is obvious that CBðZÞ includes K ðZÞ.For U, V ∈ CBðZÞ, define H : CBðZÞ × CBðZÞ ⟶ ½0, ∞Þ by where Dðu, VÞ = inf fdðu, vÞ: v ∈ Vg.Such a function H is called the Pompei-Hausdorff metric induced by d, for more details, see, e.g., [1].
Lemma 1 [2].Let ðZ, dÞ be a MS and A, B ∈ CLðZÞ with H ðA, BÞ > 0.Then, for each h > 1 and for each a ∈ A, there exists b = bðaÞ ∈ B such that dða, bÞ < hHðA, BÞ: If Ω : Z ⟶ PðZÞ is a multivalued operator, then an element ϖ ∈ Z is called a fixed point for Ω if x ∈ Ωϖ.The sym-bol fix Ω = fϖ ∈ Z : x ∈ Ωϖg denotes the fixed point set of Ω .On the other hand, a strict fixed point for Ω is an element ϖ ∈ Z with the property fxg = Ωϖ.The set of all strict fixed points of Ω is denoted by SFix Ω.
Banach's contraction principle [3] is the most fundamental result in metric fixed point theory.Since then, many authors have extended and generalized Banach's contraction principle in many ways.Extensions of Banach's contraction principle have spawned a wealth of literature.(see [13,29]).One of an attractive and important generalization is given by Wardowski in [10].He introduced a new type of contraction called F-contraction and proved a new fixed point theorem concerning F-contraction.

Very recently, Iqbal and Rizwan
Definition 8 [26].Let ðZ, dÞ be a complete MS and Ω : Z ⟶ CBðZÞ be a mapping.Then, Ω is a multivalued F -contraction, if there exists τ > 0 and F ∈ ΔðFÞ such that for all ϖ, ω ∈ Z, Theorem 9 [26].Let ðZ, dÞ be a complete MS and Ω : Z ⟶ KðZÞ be a multivalued F -contraction, and then Ω has a fixed point in Z.
Afterwards, Olgun et al. [27] proved the nonlinear case of Theorem 9 as follows.
Lemma 11.If ρ ∈ P and u, v ∈ ½0,∞Þ are such that Proof.Without loss of generality, we can suppose that u Then, F satisfies ðϜ 1Þ, ðϜ 2 ′ Þ, and F that is continuous but does not satisfies ðϜ 3Þ.
Examples 4-6 clearly show that there exist some functions F : ð0,∞Þ ⟶ ℝ which does not satisfy the condition of continuity, ðϜ 1Þ, ðϜ 2Þ, and ðϜ 3Þ at a time.By getting inspiration from this, in this paper, we prove fixed point results for contractive conditions involving functions F, not necessarily continuous and belongs to ΔðϜ Þ by taking support of a continuous function from P .Our results generalize many results appearing recently in the literature including Altun et al. [32], Olgun et al., [27] Sgroi and Vetro [33], Vetro [34], Wardowski [24], and Wardowski and Dung [35].
Proof.Let ϖ 0 ∈ Z be an arbitrary point and ϖ 1 ∈ Ωϖ 0 .Assume that ϖ 1 ∈Ωϖ 1 ; otherwise, ϖ 1 is a fixed point of Ω, and the proof is complete.Then, Dðϖ 1 , Ωϖ 1 Þ > 0 and consequently HðΩϖ 0 , From (N1) and (N2), we get By using Lemma 11,(20) implies 3 Journal of Function Spaces Next, arguing as previous, we get Also, by using Lemma 11, from (N1) and (N2), we obtain Continuing in the same manner, we get a sequence for all n ∈ ℕ. (23) implies that fdðϖ n , ϖ n+1 Þg n∈ℕ is a decreasing sequence of positive real numbers.Hence, from (N1) and (N2), we get Thus, for all n ∈ ℕ, Since χ ∈ Φ, there exists h > 0 and n 0 ∈ ℕ such that χðd ðϖ n , ϖ n+1 ÞÞ > h, for all n ≥ n 0 .From (25), we obtain which further implies that Now from ðϜ 3Þ, there exists k ∈ ð0, 1Þ such that Then, from (26), for all n ∈ ℕ, we have Taking limit n ⟶ ∞, in (30) and using ( 27) and ( 29), we have Observe that from (31), there exist sn 1 ∈ ℕ such that n ðHðΩϖ n , Ωϖ n+1 ÞÞ k ≤ 1 for all n ≥ n 1 .Thus, for all n ≥ n 1 , we have which further implies that Now, in order to show that fϖ n g n∈ℕ is Cauchy sequence, consider m, n ∈ ℕ such that m > n > n 1 .From (33), we get As a result of the above and the series' convergence, ∑ ∞ i=n ð1/i 1/k Þ, we receive that fϖ n g n∈ℕ is Cauchy sequence.Since Z is a complete space, so there exists ϖ * ∈ Z such that Now, Since F 1 is nondecreasing function, we obtain for all ϖ 4 Journal of Function Spaces , ω ∈ Z.
We claim that ϖ * is fixed point of Z. On contrary, suppose that Dðϖ * , Ωϖ * Þ > 0 and by equation ( 37), we have Passing to limit as n ⟶ ∞ in the above inequality, we obtain which implies by Lemma 1 that and F 2 ðuÞ = ln u + u for all u ∈ ð0,∞Þ, respectively.Then F 1 is nondecreasing, F 2 satisfy the conditions ðϜ 2′Þ and ð Ϝ 3Þ and F 1 ðuÞ ≤ F 2 ðuÞ for all u > 0.
Corollary 16.Let ðZ, dÞ be a complete MS and Ω : Z ⟶ KðZÞ be a multivalued mapping.Assume that there exist χ ∈ Φ, a non decreasing real valued function F 1 on ð0, ∞Þ and a real valued function F 2 on ð0, ∞Þ satisfying condition ðϜ 2 ′ Þ and ðϜ 3Þ such that (N1) and the following condition holds: HðΩϖ, ΩωÞÞ > 0 implies χðdðϖ, ωÞÞ + F 2 ðHðΩϖ, ΩωÞÞ ≤ F 1 ðNðϖ, ωÞÞ for all ϖ, ω ∈ Z, where a, b, c, e ≥ 0 and a where a, b, c, e > 0 and a + b + c + 2e < 1.Then ρ ∈ P and result follows from Theorem 12. ☐ Now, in the next Theorem, we replace the condition ðϜ 3Þ of F 2 by continuity of F 1 in hypothesis of Theorem 12 and obtain another fixed point result.
Proof.Let ϖ 0 ∈ Z be an arbitrary point and ϖ 1 ∈ Ωϖ 0 .Then, as in proof of Theorem 12, we get a sequence Taking n ⟶ ∞ in (52), we get F 2 ðHðΩϖ n , Ωϖ n+1 ÞÞ ⟶ −∞ and by ðϜ 2 ′ Þ, we have which further implies that Next, we claim that If (55) is not true, then there exists δ > 0 such that for all r ≥ 0, there exists Also, there exists r 0 ∈ ℕ such that Consider two subsequences fϖ n k g and fϖ m k g of fϖ n g satisfying Observe that where m k is chosen as minimal index for which (59) is 6 Journal of Function Spaces satisfied.Also, note that because of (58) and (59), the case n k+1 ≤ n k is impossible.Thus, n k+2 ≤ m k for all k.It implies Using triangle inequality and by ( 58) and (59), we have Letting limit k ⟶ ∞ in (61) and using (53), we get Now, by using ( 53) and (62), we obtain Then, from ðN1Þ, ðN2Þ, and monotonicity of F 1 , we get Since F 1 is continuous, so by passing the limit k ⟶ ∞, using equations (62) and (63), we have Now, since ρ ∈ ℙ, we have ρð1, 0, 0, 1, 1Þ ∈ ð0, 1; so, (65) implies which is a contradiction to (17).Hence, (55) holds, which implies that fϖ n g is a Cauchy sequence.Completeness of Z ensures the existence of ϖ * ∈ Z such that By following the same steps as in the proof of Theorem 12, we get ϖ * ∈ Ωϖ * .This completes the proof.☐ Corollary 18.Let ðZ, dÞ be a complete MS and Ω : Z ⟶ KðZÞ be a multivalued mapping.Assume that there exists χ ∈ Φ, a continuous, nondecreasing real-valued function F 1 on ð0, ∞Þ and a real valued function F 2 on ð0, ∞Þ satisfying condition ðϜ 2 ′ Þ such that (N1) and the following condition holds: where Then, fix Ω is nonempty.
Next, we consider Ωϖ that are closed subsets of Z instead of compact subsets for all Z and obtain the following theorems.
Corollary 21.Let ðZ, dÞ be a complete MS and Ω : Z ⟶ CðZÞ be a multivalued mapping.Assume that there exists χ ∈ Φ, F ∈ ΔðϜ * Þ and a real-valued function L on ð0, ∞Þ such that (G1) and the following condition holds: where ℘ i ≥ 0 and Then, fix Ω is nonempty.
Proof.Let ϖ 0 ∈ Z be an arbitrary point and ϖ 1 ∈ Ωϖ 0 .Then, as in proof of Theorem 20, we get a sequence Next, we claim that If (102) is not true, then there exists δ > 0 such that for all r ≥ 0, there exists Also, there exists r 0 ∈ ℕ such that Consider two subsequences fϖ n k g and fϖ m k g of fx n g; then, as is proof of Theorem 17, we get Then, from (G1), (G2), and monotonicity of F, we get Since F is continuous, so by passing the limit k ⟶ ∞, using equations ( 105) and (106), we have Now, since ρ ∈ ℙ, we have ρð1, 0, 0, 1, 1Þ ∈ ð0, 1; so, (107) implies which is a contradiction to (17).Hence, (102) holds, which implies that fϖ n g is Cauchy sequence.Completeness of Z ensures the existence of ϖ * ∈ Z such that By following the same steps as in the proof of Theorem 20, we get ϖ * ∈ Ωϖ * .This completes the proof.☐ Corollary 25.Let ðZ, dÞ be a complete MS and Ω : Z ⟶ CðZÞ be a multivalued mapping.Assume that there exists χ ∈ Φ, a nondecreasing and continuous real-valued function F : ð0,∞Þ ⟶ ℝ satisfying condition ðϜ 2 ′ Þ and a realvalued function L on ð0, ∞Þ such that (G1) and the following condition hold: where Then, fix Ω is nonempty.

Strict Fixed Points and Well Posedness
Firstly, we define the notions of well posedness of a fixed point problem.

Conclusion
In the theory of set-valued dynamic systems, fixed points and strict fixed points of multivalued operators are essential notions.A rest point of the dynamic system can be read as a fixed point for the multivalued map Ω, whereas a strict fixed point for Ω can be viewed as the system's endpoint.We have made a contribution in this approach by establishing some basic problems in multivalued fixed point and strict fixed point theory.We have proved several existence and data dependence results for multivalued nonlinear mappings satisfying a new class of contractive conditions via auxiliary functions.The obtained outcomes are backed up by a nontrivial example.The findings add to and expand on some of the most recent results in the literature.