Some Fixed-Point Results via Mix-Type Contractive Condition

We consider a ﬁ xed-point problem for mappings involving a rational type and almost type contraction on complete metric spaces. To do this, we are using F -contraction and ð H , φ Þ -contraction. We also present an example to illustrate our result.


Introduction
The beginning of metrical fixed point theory is related to Banach's Contraction Principle, presented in 1922 [1], which says that any contraction self-map on M has a unique fixed point whenever ðM, dÞ is complete. Afterwards, the crucial role of the principle in existence and uniqueness problems arising in mathematics has been realized which fact directed the researchers to extend and generalize the principle in many ways (see [2][3][4][5][6][7]).
In the studies of generalizations and modifications of contractions, an interesting generalization was given by Wardowski [8] using a new concept F-contraction. Then, many authors gave some results using this concept in different type metric spaces. One of them is given by Jleli et al. [9] by introducing a family H of functions H : ½0,∞Þ 3 → 0,∞Þ with the certain assumption. Also, you can find this type generalizations in [10][11][12].
In this paper, we consider a fixed-point problem for mappings involving a rational type contraction and almost contraction. Firstly, we recall some basic on the notions of F -contraction and ðH, ϕÞ-contraction.

Preliminaries
Let F be the family of all functions F : ℝ + = ½0,∞Þ → ℝ satisfying the following conditions: (F1) F is nondecreasing; (F2) for every sequence fα n g of positive numbers lim n→+∞ α n = 0 if and only if lim n→+∞ Fðα n Þ = −∞; Definition 1. (see [8]). Let ðM, dÞ be a metric space and Y : M → M be a mapping. Given F ∈ F, we say that Y is F -contraction, if there exists τ > 0 such that Taking in (1) different functions F ∈ F, one gets a variety of F-contractions, and some of them being already known in the literature. You can see this contractions in [8]. In addition, Wardowski concluded that every F-contraction Y is a contractive mapping, i.e., Thus, every F-contraction is a continuous mapping.
Definition 3. (see [9]). Let ðM, dÞ be a metric space, ϕ : M → 0, +∞ be a given function, and H ∈ H . Then, Y : M → M is called a ðH, ϕÞ-contraction with respect to the metric d if and only if for some constant k ∈ 0, 1½.

Now, we set
Furthermore, we say that Y is a ϕ-Picard operator if and only if the following condition holds Theorem 4. (see [9]). Let ðM, dÞ be a C:M:S, ϕ : M → 0, +∞ be a given function and H ∈ H . Suppose that the following conditions hold (A1) ϕ is lower semicontinuous (l.s.c.); (A2) Y : M → M is a ðH, ϕÞ-contraction with respect to the metric d. Then, (i) Y is a ϕ-Picard operator (ii) For all μ ∈ M and for all n ∈ ℕ, we have where Recently, Vetro ([13]) generalized Theorem 4 by using F -H-contraction.

Main Results
We first introduce the rational type F-H-contraction.
Using (13) with μ = ς and γ = w, we obtain which is a contradiction. So, we have w = ς, and the fixed point is unique. Now, we can show the existence of a fixed point. Take a point μ 0 ∈ M and create the fμ n g sequence starting at μ 0 . We emphasize that if μ k−1 = μ k for some k ∈ ℕ, then ς = μ k−1 = μ k = Yμ k−1 = Yς; that is, ς is a fixed point of Y such that ϕðςÞ = 0. In fact, by Lemma 10, Hðdðμ k−1 , μ k Þ, ϕðμ k−1 Þ, ϕðμ k ÞÞ = 0 and by the property (H 1 ) of the function H, we have ϕðςÞ = 0. So, we can suppose that μ n−1 ≠ μ n for every n ∈ ℕ.
In this step, we show that fμ n g is a Cauchy. By Lemma 10, we say that There exists k ∈ 0, 1½ such that h k n Fðh n Þ → 0 as n → +∞ by he property (F 3 ) of F: Using (13) with μ = μ n−1 and γ = μ n , we get for all n ∈ ℕ; that is, we deduce that This provides that ∑ +∞ n=1 h n is convergent. By the property (H 1 ) of the function H, also, the series ∑ +∞ n=1 dðμ n , μ n+1 Þ is convergent and hence fμ n g is a Cauchy sequence. Now, since ðM, dÞ is complete, there exists ς ∈ M such that By (13), taking into account that ϕ is a l.s.c. function, we have that is, ϕðςÞ = 0. Now, show that ς is a fixed point. If there exists a subsequence fμ n k g of fμ n g such that μ n k = ς or Y μ n k = Yς, for all k ∈ ℕ, then ς is a fixed point. Otherwise, we can assume that μ n ≠ ς and Yμ n ≠ Yς for all n ∈ ℕ. So, using (13) with μ = μ n and γ = ς, we deduce that
Imposing that F is a continuous function and relaxing the hypothesis ðF 3 Þ, we can give t Theorem 12.
Proof. Following the similar arguments as in the proof of Theorem 11, we obtain easily the uniqueness of the fixed point. The existence of a fixed point, we take a point μ 0 ∈ M and create the fμ n g sequence starting at μ 0 . Clearly, if μ k−1 = μ k for some k ∈ ℕ, then ς = μ k−1 = μ k = Yμ k−1 = Yς; that is, ς is a fixed point of Y such that ϕðςÞ = 0 (see the proof of Theorem 11), and so we have already done. So, we can suppose that μ n−1 ≠ μ n for every n ∈ ℕ. Now, showing that fμ n g is a Cauchy. Let us admit the opposite. Then, there exists a positive real number ε and two sequences fm k g and fn k g such that By Lemma 10, we say that dðμ n−1 , μ n Þ → 0, ϕðμ n Þ → 0, as n → +∞. This implies Now, the hypothesis that dðμ m k , μ n k Þ > ε ensures that Using the continuity of H, we have Using again (29), with μ = μ m k −1 and γ = μ n k −1 , we get for all k ∈ ℕ. Letting k → +∞ in the previous inequality, since the function F is continuous, we get which leads to contradiction. It follows that fμ n g is a Cauchy sequence. Now, since ðM, dÞ is complete, there exists some ς ∈ M such that lim n→+∞ μ n = ς: By (29), using lower semicontinuity of ϕ, we get that is, ϕðςÞ = 0. Now, show that ς is a fixed point of Y.
Clearly, ς is a fixed point of Y if there exists a subsequence fμ n k g of fμ n g such that μ n k = ς or Yμ n k = Yς, for all k ∈ ℕ. Otherwise, we can assume that μ n ≠ ς and Yμ n ≠ Yς for all n ∈ ℕ. Then, the property (H 1 ) of the function H ensures that HðdðYμ n , YςÞ, ϕðYμ n Þ, ϕðYςÞÞ > 0 for all n ∈ ℕ. So, using (29) with μ = μ n and γ = ς, we deduce that