Common and Coincidence Fixed-Point Theorems for (cid:61) -Contractions with Existence Results for Nonlinear Fractional Differential Equations

: In this paper, we derive the coincidence ﬁxed-point and common ﬁxed-point results for (cid:61) -type mappings satisfying certain contractive conditions and containing fewer conditions imposed on function (cid:61) with regard to generalized metric spaces (in terms of Jleli Samet ). Finally, a fractional boundary value problem is reduced to an equivalent Volterra integral equation, and the existence results of common solutions are obtained with the use of proved ﬁxed-point results.


Introduction
One of the most significant basic fixed-point results is the well-known Banach's fixedpoint theorem (abbreviated BFPT) [1].Due to the numerous uses of this principle in other disciplines of mathematics, numerous writers have expanded, generalized, and enhanced it in numerous ways by taking into account alternative mappings or space types.Wardowski [2] provided a striking and significant generalization of this nature.He provided this to introduce the idea of -contraction as Definition 1.Let (V, d) be a metric space.A mapping Ω : V → V is said to be an -contraction, if there exist ∈ ∆( ) and λ > 0 such that for all µ, ∈ V λ + (d(Ωµ, Ω )) ≤ (d(µ, )), (1) where ∆( ) is the family of all mappings : (0, +∞) → (−∞, ∞) meeting the criteria listed below.
Wardowski's result is given as follows: Theorem 1 ([2]).Let (V, d) be a complete metric space and Ω : V → V be an -contraction.Then, µ * ∈ V is a unique fixed point of Ω and for every µ 0 ∈ V, a sequence {Ω p µ 0 } p∈N is convergent to µ * .
However, the concept of standard metric space is generalized in a number of ways (see [17][18][19][20][21][22][23][24]).Jleli and Samet provided one of the most common generalizations of metric spaces in [25], which recapitulates a broad class of topological spaces, including b-metric spaces, standard metric spaces, dislocated metric spaces, and modular spaces.They expanded BFPT, Cirić's fixed-point theorem and a fixed-point result attributed to Ran and Reurings, among other fixed-point theorems.Additionally, Altun et al. obtained a fixedpoint theorem of the Feng-Liu type with regard to generalized metric spaces in [26], while Karapinar et al. gained fixed-point theorems within fairly broad contractive conditions in generalized metric spaces in [27].In the framework of generalized metric spaces, Saleem et al. [28] recently demonstrated a few novel fixed-point theorems, coincidence point theorems, and a common fixed-point theorem for multivalued -contraction involving a binary relation that is not always a partial order.
Henceforth, let V be a non-empty set and Ł : V × V → [0, +∞] be a given mapping.Following Jleli and Samet [25], for every µ ∈ V, define the set Definition 2 ([25]).Let V be a non-empty set and Ł : V × V → [0, +∞] be a function which fulfils the following criteria for all µ, ∈ V: Then Ł is called a generalized metric and the pair (V, Ł) is called a generalized metric space.We renamed it as κ-generalized metric space (abbreviated, a κ-GMS).
In the sequel, N = {1, 2, 3, . . .}, N 0 = N ∪ {0} and R indicate the set of all positive integers, the set of all non-negative integers and the set of all real numbers, respectively, and R indicates the set of all real numbers.Let ζ be self-mapping on a non-empty set V, P(V) be the collection of all non-empty subsets of V, C(V) be the collection of all non-empty closed subsets of V, and Ω : V → P(V) be a set-valued mapping.We denoted by Coi(V, ζ, Ω) the set of all coincidence points of ζ & Ω in V and by Com(V, ζ, Ω) the set of all common fixed points of ζ & Ω in V.A non-empty subset ∼ of the Cartesian product V × V is a binary relation on V.For simplicity, we denote µ ∼ if (µ, ) ∈∼. [29] contains the concepts of preorder, partial order, transitivity, reflexivity, and antisymmetry.Definition 3 ([27]).Let a binary relation on the κ-GMS (V, Ł) be defined as ∼.If a sequence µ p ⊆ V µ p ∼ µ p+1 for all p ∈ N, then the sequence is ∼-non-decreasing.Definition 4 ([27]).If each ∼-non-decreasing and Ł-Cauchy sequence is Ł-convergent in V, then a κ-GMS (V, Ł) is ∼-non-decreasing complete.
Remark 2. Keep in mind that every κ-GMS that is complete also happens to be ∼-non-decreasing complete, while the opposite is false, as evidenced by the case below.

Definition 6 ([28]
).Let (V, Ł) be a κ-GMS furnished with a preorder ∼, ζ : V → V and By obtaining inspiration from the work of Derouiche and Ramoul [16] and by following the direction of Saleem et al. [28], in this paper, we prove the coincidence point theorem and common fixed-point theorem in generalized metric spaces for mappings satisfying certain contractive conditions and containing fewer conditions imposed on function .
The paper is organized as follows: We renamed the generalized metric space (in the sense of Jleli and Samet) as κ-generalized metric space and consider the κ-generalized metric space for κ ∈ (0, 1].Then, we derive the common fixed-point and coincidence fixed-point results in the setting of this space.Lastly, by using these results, we proved the existence results of common solutions of fractional boundary value problems.

Coincidence Point Theorems
In this section, we prove the coincidence point theorems.
From the closeness of Ωτ, we have ζτ ∈ Ωτ.

Case: II When
So, in this case, inequality (11) holds true for (t) = ln(t + 1) and λ(t) = ln(t) for all t ∈ (0, ∞).Hence, all the conditions of Theorem 2 are fulfilled and {0, 1} is the set of coincidence points of ζ and Ω.
Remark 3. Note that in Example 3, the function : (0, +∞) → R defined by (t) = ln(t + 1) belongs to hc .But does not satisfy ( 2 ).Indeed, for any sequence π p ∈ (0, +∞) such that lim p→∞ π p = 0, we have Next, from Theorem 2 we obtain the following by using the fact that a partial order is a preorder ∼.
There exists a sequence {µ p : Ω 1 and Ω 2 owns a common fixed point in V.
Hence, all the conditions of Theorem 3 are fulfilled and 0 is the common fixed point of Ω 1 and Ω 2 .
By defining C(V) = V in Theorem 3, we obtain the following: for all ē, f ∈ V.Then, 1.
There exists a sequence {µ p : Ω 1 and Ω 2 owns a common fixed-point in V.
There exists a sequence {µ p : Ω 1 owns a fixed-point in V.
Ω 1 and Ω 2 owns a common fixed-point in V.

Existence of Common Solution of Nonlinear Fractional Differential Equations with Nonlocal Boundary Conditions
In this section, we present the application of our results to prove the existence of the common solutions for the following boundary value problems involving Caputo fractional derivative.
Definition 7.For a continuous function u : [0, ∞) → R, the Caputo derivative of fractional order α is defined as where [α] denotes the integer part of the real number α.
Definition 8.The Riemann-Liouville fractional integral of order α is defined as provided the integral exists.
Lemma 3 ([31]).For α > 0, the general solution of the fractional differential equation c D α x( ) = 0 is given by In view of Lemma 3, it follows that for some In the following, we obtain the Volterra integral equation of the fractional differential equation boundary value problem.
Proof.From Lemma 3, the general solution for the problem (69) is where b i ∈ R. By using the boundary conditions ℘(0 Let V = C(I, R) be the space of all continuous real valued functions on I, where I = [0, 1].Then, V is a complete metric space with respect to metric Ł(x, y) = sup ∈I |x( ) − y( )|.Since every metric space is κ-GMS for κ = 1; henceforth, we assume that (V, Ł) is complete is κ-GMS.Define the operators A, L : V → V as follows: and Note that a common fixed point of operators (71) and ( 72) is the common solutions of (62) and (63).We consider the following set of assumptions in the following: Theorem 4. Suppose that hypothesis (H1)-(H3) hold.Then, the boundary value problems (62) and (63) have a common solution in V.

Common Solution to Integral Inclusions
In this section, we present the existence of common solutions to the integral inclusions.For this, let V = C(J, R) be the space of all continuous real valued functions on J, where J = [a, b].Then, V is a complete metric space with respect to metric Ł(x, y) = sup t∈J |x(t) − y(t)|.Since every metric space is GMS(JS), throughout this section we assume that (V, Ł) is complete and is GMS(JS).Consider the following integral inclusions: k(t, s)L(s, π(s))ds (76) and ξ(t) ∈ q(t) k(t, s)M(s, ξ(s))ds (77) for t ∈ J, where α, β : J → J, q : J → V, k : I × J → R are continuous and L, M : J × V → P(R), P(R) denotes the collection of all nonempty, compact, and convex subsets of R. For each x ∈ V, the operators L(., x) and M(., y) are lower semi-continuous.Define the multivalued operators Ω, Ω 1 : V → C(V) as follows: k(t, s)L(s, π(s))ds, t ∈ J (78) and α k(t, s)M(s, ξ(s))ds, t ∈ J (79) Note that a common fixed point of multivalued operators (78) and ( 79) is the common solution of integral inclusions (76) and (77).We consider the following set of assumptions in the following.Proof.Let x, y ∈ V. Denote L x = L x (s, x(s)) and M y = M y (s, y(s)).Now for L x : J → P(R) and M y : J → P(R), by Micheal's selection theorem, there exists continuous operators l x , m y : J × J → R with l x (s) ∈ L x (s) and m y (s) ∈ M y (s) for s ∈ J. So, we have u = β(t) α(t) k(t, s)l x (s)ds + q(t) ∈ Ω ē(t) and v = β(t) α(t) k(t, s)m y (s)ds + q(t) ∈ Ω 1 f (t).Thus, the operators Ω ē and Ω 1 f is nonempty and closed (see [33]).By hypothesis (H4)-(H6) and by using Cauchy-Schwartz inequality, we obtain Ł(u, v) = sup   Hence, (34) is satisfied for (℘) = ln(℘) and λ(℘) = θ > 0 for all ℘ ∈ (0, ∞).Thus, all hypotheses of Theorem 3 are satisfied, and therefore Ω and Ω 1 have a common fixed point.It further implies that integral inclusions (76) and (77) have a common solution in I.

Lastly, we present an open problem for future work as follows: Open Problem
Let (V, Ł) be a κ-GMS for any κ > 0l then, can Theorems 2 and 3 still be proved?

Conclusions
We have proved the coincidence fixed-point and common fixed-point theorems in the setting of generalized metric spaces (in the terms of Jleli and Samet) for -type mappings