Abstract
In this manuscript, we propose some sufficient conditions for the existence of solution for the multivalued orthogonal ℱ -contraction mappings in the framework of orthogonal metric spaces. As a consequence of results, we obtain some interesting results. Also as application of the results obtained, we investigate Ulam’s stability of fixed point problem and present a solution for the Caputo-type nonlinear fractional integro-differential equation. An example is also provided to illustrate the usability of the obtained results.
Highlights
Introduction andPreliminaries e theory of multivalued mappings has an important role in mathematics and allied sciences because of its many applications, for instance, in real and complex analysis as well as in optimal control problems
We establish some results on the existence of fixed point for weak orthogonal multivalued contraction mappings using conditions of Wardowski [3–5]
Let A and B be two nonempty subsets of an orthogonal set (X, ⊥). e set A is orthogonal to set B is denoted by ⊥1 and defined as follows: A⊥1B, if for every a ∈ A and b ∈ B, a⊥b
Summary
We establish some results on the existence of fixed point for weak orthogonal multivalued contraction mappings using conditions of Wardowski [3–5]. (iii) If xn is an orthogonal sequence in X such that xn ⟶ x∗ ∈ X, xn⊥x∗ or x∗⊥xn for all n ∈ N (iv) If F ∈ F, there exists τ > 0 such that for all x, y ∈ X with x⊥y satisfying the following: H(Tx, Ty) > 0, τ + F(H(Tx, Ty)) ≤ F(d(x, y)). Continuing this process, we construct an orthogonal sequence xn in X such that xn+1 ∈ Txn for all n ∈ N ∪ {0}. From (19), we can get a sequence xn in X such that there exists xn+1 ∈ Txn and F(d(xn, xn+1)) < F(d(xn−1, xn)).
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