Infinite Geraghty type extensions and their applications on integral equations

In this article, we discuss about a series of infinite dimensional extensions of some theorems given in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018), (Fisher in Math. Mag. 48(4):223–225, 1975), and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). We also prove a similar Geraghty type construction for Fisher (Math. Mag. 48(4):223–225, 1975) in an infinite dimension using similar techniques as in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018) and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). As an application, we ensure the existence of solutions for infinite dimensional Fredholm integral equation and Uryshon type integral equation.

Next, we discuss some of the preliminaries which will be needed later for proving our main theorems. The following result is due to Geraghty [24]. Then T admits a unique fixed point u ∈ X and {T n x} converges to u for each x ∈ X.
We denote by G the set of all functions β given in Theorem 1.1. We denote by N (resp. N 0 ) the set of positive (nonnegative) integers.
Further, let us recall some results and definitions useful in our main results. for all x, y ∈ X, where c ∈ (0, 1/2), then T has a unique fixed point u ∈ X. Theorem 1.3 [2] Let (X, d) be a complete metric space and T : X → X be a mapping. If T verifies d(Tx, Ty) ≤ c d(Tx, y) + d (Ty, x) for all x, y ∈ X, where c ∈ (0, 1/2), then T possesses a unique fixed point u ∈ X. for all x, y ∈ X, where β ∈ G. Theorem 1.4 (see [3]) Let (X, d) be a complete metric space and T : X → X be a mapping. If T is a Kannan-Geraghty contraction on (X, d), then T possesses a unique fixed point u ∈ X, and for any x 0 ∈ X, {T n x 0 } converges to u. Then there exists a point u ∈ X such that T(u, u, . . . , u) = u.
The aim of the next section is to generalize and extend Theorem 1.1, Theorem 1.4, and Theorem 1.5 as well as the k-dimensional extension of the result given in [24] to an infinite dimension. We denote the infinite tuples of points (x 1 ,

Main results
From now on, let us consider that (X, d) is a complete metric space and k ∈ N.
Definition 2.2 (H k contraction) An operator T : ∞ l=1 X → X is called an H k contraction if and only if it satisfies the following inequality: where k ∈ N is any fixed number and c ∈ (0, 1). Definition 2.3 (Kannan-Geraghty H k contraction) An operator T : k l=1 X → X is called a Kannan-Geraghty contraction of dimension k if and only if it satisfies the following inequality: Definition 2.4 (Extended Kannan-Geraghty H k contraction) An operator T : ∞ l=1 X → X is called an extended Kannan-Geraghty H k contraction if and only if it satisfies the following: Definition 2.5 (Fisher-Geraghty H k contraction) An operator T : k l=1 X → X is called a Fisher-Geraghty H k contraction of dimension k if and only if it satisfies the following: Definition 2.6 (Extended Fisher-Geraghty H k contraction) An operator T : ∞ l=1 X → X is called an extended Kannan-Geraghty H k contraction if and only if it satisfies the following: Example 2.3 As an example for Definition 2.3 and Definition 2.5, we can consider, for The theory of Picard operators was developed by Rus and is used a lot in proving the existence and uniqueness of a solution of different types of integral or differential equations. For more details, see [39,40]. Definition 2.7 (Picard operator) Let (X, d) be a metric space. An operator T : X → X is a Picard operator if there exists x * ∈ X such that FixT = {x * } and the sequence (T n (x 0 )) n∈N converges to x * for all x 0 ∈ X.
Inspired by the Picard operator definition, we extend this notion for the k-dimensional and infinite dimensional cases as follows. Definition 2.8 (k-Picard sequence with respect to the operator T) Let T : k i=1 X → X be any operator, and let us choose x 1 , x 2 , . . . , x k ∈ X. The k-Picard sequence with respect to the operator T based on the base point set {x 1 , x 2 , . . . , x k } is defined as Example 2.5 If we fix k = 1, then the base point set is singleton and the 1-Picard sequence with respect to T based on {x 0 } is basically the Picard sequence of T based on the base point {x 0 } defined by x n := T(x n-1 ) for all n ≥ 1 and some x 0 ∈ X. Now, we define the following notions. ∞ i=1 X → X be any operator, and let us choose x 1 , x 2 , . . . , x k , . . . ∈ X. The k-Picard sequence with respect to the operator T based on the base point set {x 1 , x 2 , . . . , x k , . . .} is defined as Example 2.6 If we fix k = 1, then the base point set is singleton and the infinite 1-Picard sequence with respect to T based on the base point {x 0 } is basically the sequence defined by Let us give our first main fixed point result, which is a generalization of the Banach contraction principle with respect to the infinite-dimensional notion introduced in our paper.
Theorem 2.1 Let (X, d) be a complete metric space. T : ∞ l=1 X → X is an H k contraction for some k ∈ N. Then there exists u ∈ X such that T((u) ∞ i=1 ) = u and the infinite k-Picard sequence for T converges to u.
Proof Let x 1 , x 2 . . . , x k ∈ X. Then an infinite k-Picard sequence is defined as follows: ∀n ∈ N we define We claim that We have That is, Without loss of generality, assume that d((x n+k+1 , x n+k+2 ) > 0 for each k.
Then we get a contradiction. Hence, for each k, We claim lim n→∞ d(x n+k , x n+k+1 ) = 0. Suppose on the contrary that r > 0. We have We now claim that {x n+k } n∈N is Cauchy, and we prove it by contradiction. Suppose on the contrary that there is ε > 0 such that we can find some subsequences Moreover, corresponding to each n(q), we can choose least of such m(q) satisfying (8). Then From (7), (9) and using the triangle inequality, we get and Letting q → ∞ in (10) and using (11), we get On the other hand, if Using (7) and (12) we get By (1) and (17) we get If we suppose Letting q → ∞ and using (1), (7), (17), and (18), we get Since β ∈ G, we get Then We get a contradiction to (17). Hence, {x n+k } n∈N is a Cauchy sequence.
We claim that d( Letting n → ∞ and using (7), (19), we get Then we have Letting n → ∞ and using (19), we get lim n→∞ β(d((u, T((u) ∞ i=1 ))) ≥ 1. Since β ∈ G, one writes lim n→∞ β(d((u, T((u) ∞ i=1 ))) = 1. Then we have lim n→∞ d((u, Remark 2.1 Theorem 2.1 is a proper generalization of Theorem 1.5 since in the case of the simplest operator on ∞ i=1 X → X the contraction condition of Theorem 1.5 is not applicable; on the other hand, the H k contraction (see Definition 2.2) is easily applicable for the infinite case. Also, if we restrict the operator to any finite k dimension through an easy calculation, it is obvious that it is an equivalent statement of Theorem 1.5. Theorem 2.2 . Let (X, d) be a complete metric space and T : ∞ l=1 X → X be an extended Kannan-Geraghty H k contraction for some k ∈ N. Then there is u ∈ X such that T((u) ∞ i=1 ) = u, and for any x 1 , . . . , x k ∈ X, the infinite k-Picard sequence converges to u.
Proof Let x 1 , . . . , x k ∈ X. For all n ∈ N, we define the infinite k-Picard sequence as follows: We claim that lim n→∞ d(x n+k , x n+k+1 ) = 0. Then we have Thus, d(x n+k+1 , x n+k+2 ) < d(x n+k , x n+k+1 ). This sequence is decreasing, so there is r ≥ 0 so that We claim that r = 0. If we consider on the contrary and suppose r > 0, we have Therefore, From (21) we get 2r 2r ≤ β(d(x n+k , x n+k+1 )), and so lim n→∞ β(d(x n+k , x n+k+1 )) ≥ 1. Since β ∈ G, we have lim n→∞ β(d(x n+k , x n+k+1 )) ≤ 1. Using the well-known "sandwich theorem", we obtain Further, we have For large enough n, m ∈ N, one has d(x n+k , x m+k ) < ε for fixed ε > 0. Then {x n+k } n∈N is a Cauchy sequence. Since (X, d) is a complete metric space, there exists u ∈ X such that We claim that T((u) ∞ i=1 ) = u. If we suppose on the contrary that d(u, T((u) ∞ i=1 )) > 0. Then, by (21) and (23), for arbitrary ε > 0 and sufficiently large n, we get ≤ ε. This is a contradiction.
Next, we will provide a new result for a multivalued proper extension of Theorem 1.4, which is also a generalization of Kannan (Theorem 1.3) as a result of Theorem 2.2.

Corollary 2.1 Let (X, d) be a complete metric space and T be a Kannan-Geraghty H k contraction. Then T has a fixed point and every k-Picard sequence for T converges to u.
Proof Let us choose x 0 , . . . , x k ∈ X. We define a k-Picard sequence by If we follow the same steps as in the proof of Theorem 2.2, we get the required result.
Remark 2.2 For k = 1, we get Theorem 1.4 in [3] which proves that Corollary 2.1 is a proper generalization of Theorem 1.4.

Theorem 2.3 Let (X, d) be a complete metric space and T :
∞ l=1 X → X be an extended Fisher-Geraghty H k contraction for some k ∈ N. Then there exists u ∈ X such that T((u) ∞ i=1 ) = u, and for any x 1 , x 2 , . . . x k , the infinite k-Picard sequence converges to u.
At the limit, we have Then lim n→∞ d(x n+k , x n+k+1 ) = 0. We claim that {x n+k } n∈N is Cauchy, and we want to prove this by contradiction. Then, using the contrary, there exists ε > 0 such that we can find subsequences {x m(q)+k } p∈N , {x n(q)+k } p∈N with m(q) > n(q) > q such that, for every q, we have Moreover, corresponding to each n(q) we can choose m(q) satisfying (28) so that Using (28), (29) and the triangle inequality, we get and Letting q → ∞ in (30) and (27), we get Then from (31) and (27) we get a contradiction. Then Since (X, d) is complete, there exists u ∈ X such that We claim that T((u) ∞ i=1 ) = u. If we suppose on the contrary that d(u, T((u) ∞ i=1 )) > 0, then, by (26) and (33), for arbitrary ε > 0 and sufficiently large n, we get

Corollary 2.2 Let (X, d) be a complete metric space and T be a Fisher-Geraghty H k contraction. Then T has a fixed point.
Proof Choose any x 0 , . . . , x k ∈ X. Define Using the same steps as in the proof of Theorem 2.3, we get the conclusion.
Remark 2.3 For k = 1, we get a new type of extension of Theorem 1.3 proved in [2], and for any k, Corollary 2.2 also gives the multidimensional extension of the same theorem stated in [2].   The Fredholm integral equations play an important role in modeling of physics phenomena described by two or three dimensions. In the same way, they have applications in astrophysics models thinking of the four dimensions of a neutron star or a black hole.

Applications to integral equations
Thinking of these aspects, if we extend to infinity the dimension "n" of the previous multi-dimensional integral equation, we introduce a new notion, the infinite dimensional Fredholm integral equation, as follows: is an unknown function. Further, let us give our first application of the main results of this paper by proving the existence of a solution of infinite dimensional Fredholm integral equation (35).
Assume that the following hold: We have the following estimation: . . ds n . . .
Taking supremum on both sides, we get Further, since 1 γ < β(z), for z = d(u k , u k+1 ) and δ ∈ (0, 1), we obtain In conclusion, all the hypotheses of Theorem 2.2 are accomplished. Then the operator T has a fixed point, which means the infinite dimensional Fredholm integral equation (35) has a solution.
The following application involves another type of integral equations: Urysohn type integral equations. We extend the known cases of this type of integral equations to infinite dimensional Urysohn integral equations: is an unknown function. For this new type of Urysohn integral equation, let us give the following result.
Then the infinite dimensional Urysohn integral equation (38) has a solution.
Proof It is easy to check that the space X = (C[0, 1], R) endowed with the metric d defined by relation (39) is a complete metric space.
To prove the existence of a solution of infinite dimensional Urysohn integral equation, we shall show that the operator T defined by (40) has a fixed point.
We have the following estimation: Applying maximum on both sides, we get max t∈ [a,b] Tu Then we have Since 1 < β(z) and for z = M k ((u