Weak Wardowski contractive multivalued mappings and solvability of generalized ϕ -Caputo fractional snap boundary inclusions

. In this paper, we introduce the notion of weak Wardowski contractive multivalued mappings and investigate the solvability of generalized ϕ -Caputo snap boundary fractional differential inclusions. Our results utilize some existing results regarding snap boundary fractional differential inclusions. An example is given to illustrate the applicability of our main results


Introduction
Fractional integro-differential operators have many applications to investigate the mathematical modeling of physical phenomenon in many technological fields, namely, mechanics and physics. To see several published papers in this regard, the reader is referred to [2,7,8,13]. Among these works, the Riemann-Liouville and Caputo integro-differential operators are the most fractional operators, which have been used. Recently, a new fractional integro-differential operator, namely, ϕ-Caputo fractional derivative, which means that the fractional derivative is defined with respect to another strictly increasing differentiable function, was introduced in [14] and used in [6]. Then some researchers used this operator in different subjects (see, for example, [1,3,5,11,15,23]).
In [9], da C. Sousa and de Oliveira introduced a fractional derivative with respect to another function, the so-called ψ-Hilfer fractional derivative, and discussed some properties and important results of the fractional calculus. In this sense, they presented some results involving uniformly convergent sequence of function, uniformly continuous function, and examples including the Mittag-Leffler function with one parameter. Finally, they presented a wide class of integrals and fractional derivatives by means of the fractional integral with respect to another function and the ψ-Hilfer fractional derivative.
In [10], da C. Sousa and de Oliveira studied a Leibniz-type rule for the ψ-Hilfer fractional differential operator in two forms. They also presented some specific cases of Leibniz-type rule for this operator. In [19], these authors also presented a differential operator of arbitrary order defined by means of a Caputo-type modification of the generalized fractional derivative. As an application, they proved the fundamental theorem of fractional calculus associated with this differential operator.
Recently, Samei et al. [22] investigated the following ϕ-Caputo fractional differential inclusion: c D k;ϕ a + c D r;ϕ a + c D p;ϕ a + c D q;ϕ a + z(t) ∈ f t, z(t), c D q;ϕ a + z(t), c D p;ϕ where c D η;ϕ a + is the ϕ-Caputo fractional-order derivative introduced by Jarad et al. [14], f : [a, b] × R 4 → P(R) is a multivalued function, η belong to {q, p, r, k} such that 0 < q, p, r, k 1, the increasing function The authors investigated the solvability of the above-mentioned problem by using the α-ψ-contractive multivalued mappings defined in [17].
In this paper, we pursue two goals: In Section 3, we introduce a new multivalued contraction named the weak Wardowski multivalued contraction and prove the existence of a fixed point for such mappings. In Section 4, we use our new contraction to show that the above ϕ-Caputo fractional differential inclusion (1) is solvable when the right-hand function f : [a, b]×R 4 → P(R) does not always involve α-ψ-contraction for multivalued mappings.

Preliminaries and auxiliary notions
Let (X, d) is a metric space. Following [18], denote by P cb (X) the class of all nonempty closed bounded subsets of X. Let H be the Hausdorff-Pompeiu metric on P cb (X) induced by the metric d given as An element θ ∈ X is said to be a fixed point of a multivalued mapping T : Recently, Parvaneh and Farajzadeh [20] introduced and obtained the weak Wardowski contractions and obtained some fixed point theorems for this contractions via the notion of measure of noncompactness.
Denote by Ξ the set of all functions F : [0, ∞] → [−∞, ∞] so that: (δ 1 ) F is increasing and continuous; As examples of elements of Ξ: Denote by Θ the collection of all functions ϑ : R → (0, ∞) such that ϑ is continuous. As examples of elements of Θ : Now, let us recall some introductive definitions of fractional differential equations [16,21]. For a continuous function f : [a, b] → R, the Riemann-Liouville integral of fractional order α is defined by The Caputo derivative of fractional order α is defined by for n − 1 < α < n, n = [α] + 1. Here the Riemann-Liouville fractional derivative of order α is defined by for n − 1 < α < n, n = [α] + 1.
provided that the right-hand side of equality is finite-valued.

Definition 2. (See
provided that the right-hand side of equality is finite-valued. In the similar manner, if ϕ(t) = t, then it is obvious that the ϕ-Riemann-Liouville derivative (6) reduces to the standard Riemann-Liouville derivative (4). Inspired by these operators, Almeida presented a new ϕ-version of the Caputo derivative in the following formulation. (7) provided that the right-hand side of equality possesses values finitely. https://www.journals.vu.lt/nonlinear-analysis It should be noted that if ϕ(s) = s, then it is obvious that the ϕ-Caputo derivative of order r in formula (7) reduces to the standard Caputo derivative of order r in (3). In the following, some useful specifications of the ϕ-Caputo and ϕ-Riemann-Liouville integro-derivative operators can be seen. Let AC([a, b], R) stand for the set of absolutely Lemma 1. (See [14].) Let n = [r] + 1. For a real mapping f ∈ AC n ([a, b], R), For a real mapping f ∈ C([a, b], R), we have:

Main results
In 2005, Echenique [12] started combining fixed point theory and graph theory. Consider a directed graph G on a metric space (X, d) such that the set of its vertices V (G) coincides with X (i.e., V (G) = X), and the set of its edges Let us also assume that G has no parallel edges. We can identify G with the pair (V (G), E(G)). The graph G is called a (C)-graph if for any sequence {x n } in X such that x n → x and (x n , x n+1 ) ∈ E(G) for all n ∈ N, there exists a subsequence {x n k } such that (x n k , x) ∈ E(G) for all k ∈ N. Now, we are ready to state and prove the main results of this study.
Let T : X → P cb (X) be a multivalued mapping. We say that T is a weak Wardowski multivalued contraction if there exist F ∈ Ξ and ϑ ∈ Θ such that Theorem 1. Let (X, d) be a complete metric space, and let G be a directed graph on X. Assume that T : X → P cb (X) is a weak Wardowski multivalued contraction satisfying comparable approximate valued property. If G is a c-graph, then T has a fixed point.
Taking limit in both sides of (11), Thus, {ς n } is a Cauchy sequence in the complete metric space (X, d), hence there is z ∈ X so that lim n→∞ ς n = z.
We claim that d(z, T z) = 0. Suppose to the contrary d(z, T z) = 0.
Since G is a (C)-graph, there exists a subsequence {x n k } such that (x n k , x) ∈ E(G) for all k ∈ N.
We have Also, Passing to the limit through (12), we obtain F(d(z, T z)) < F(d(z, T z)), which is a contradiction. Thus, d(z, T z) = 0. Now, since G has comparable approximate valued property, there exists u ∈ X such that u ∈ T z, (z, u) ∈ E(G), and d(z, u) = d(z, T z). Consequently, d(z, u) = 0 and so z = u ∈ T z. The proof is completed.
Denote by P cp (X) the family of all nonempty compact subsets of X.
Corollary 1. Let (X, d) be a complete metric space, and let G be a directed graph on X. Assume that T : X → P cp (X) is a weak Wardowski multivalued contraction. Moreover, assume that Graph(T ) = {(x, y): y ∈ T x} ⊆ E(G). If G is a c-graph, then T has a fixed point.
Theorem 2. Let f : [a, b] × R 4 → P cp (R) be a multivalued mapping. Suppose that the following conditions are satisfied: Then the inclusion problem (1) has at least one solution.
Proof. We shall show that the multivalued mapping U defined in (13) has a fixed point. Let z, z ∈ C and * 1 ∈ U(z ) and choose z 1 ∈ S f,z such that * (14) we have for all z, z ∈ C. Thus, there exists Υ ∈f z such that Now, define a multivalued mapping N : [a, b] → P(C) as As z 1 and ζ(t) are measurable, so there is N(·) ∩f z (·). Now, let z 2 ∈f z (t) be such that Now, we define * 2 ∈ U(z) as * Therefore, * (ϕ(b) − ϕ(a)) q+p+r+k Γ(q + p + r + k + 1) + (ϕ(b) − ϕ(a)) p+r+k Γ(p + r + k + 1) Thus, Now, taking a graph G on C such that E(G) = C × C, all the conditions of Corollary 1 are satisfied. Thus, U has a fixed point, and so the problem (1) has a solution.

Conclusion
In this paper, we first introduce a new multivalued contraction called weak Wardowski multivalued contraction and show that such mappings have fixed points. Second, we use our new contraction to show that the ϕ-Caputo fractional differential inclusion (1) is solvable when the right-hand function f : [a, b] × R 4 → P(R) does not require the α-ψcontractive condition for multivalued mappings. An example is given to show the usability of our new results. We intend to develop a coupled fixed point theorem for two variable multivalued mappings satisfying a weak Wardowski-type multivalued contraction in the future. Then we propose to investigate the solvability of the ϕ-Caputo fractional differential systems of inclusions (1) when the right-hand functions satisfy a weak Wardowski multivalued contraction.