Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle

Following the idea of T.A. Burton, of progressive contractions, presented in some examples (T.A. Burton, A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions, Fixed Point Theory, 20 (2019), No. 1, 107-113) and the forward step method (I.A. Rus, Abstract models of step method which imply the convergence of successive approximations, Fixed Point Theory, 9 (2008), No. 1, 293-307), in this paper we give some variants of contraction principle in the case of operators with Volterra property. The basic ingredient in the theory of step by step contraction is G-contraction (I.A. Rus, Cyclic representations and fixed points, Ann. T. Popoviciu Seminar of Functional Eq. Approxim. Convexity, 3 (2005), 171-178). The relevance of step by step contraction principle is illustrated by applications in the theory of differential and integral equations.


Introduction
Following an idea of T.A. Burton ([7], [8], [9], . . . ) of progressive contractions, and the forward step method ( [21]), in this paper we give some variants of contraction principle in the case of operators with Volterra property. The basic ingredient in our variant, step by step contraction principle, is G-contraction ( [20]). Some applications to differential and integral equations are also given. In connection with our abstract results, a conjecture is formulated.

G-contractions
Let (X, d) be a metric space and G ⊂ X × X be a nonempty subset. An operator f : X → X is a G-contraction if there exists l ∈]0, 1[ such that, d(f (x), f (y)) ≤ ld(x, y), ∀(x, y) ∈ G.
(2) Let A i ⊂ X, i = 1, p, be nonempty closed subsets such that: , a G-contraction is a cyclic contraction of Kirk-Srinivasan-Veeramani (see the references in [20]). We suppose that there exists L H > 0 such that If, L H (b − c) < 1 and if we take For other examples of G-contractions see [20] and [24], pp. 282-284.

Weakly Picard operators
). An operator f : X → X is weakly Picard operator (W P O) if the sequence, (f n (x)) n∈N , converges for all x ∈ X and the limit (which generally depend on x) is a fixed point of f .
If an operator f is W P O and the fixed point set of f , F f = {x * }, then by definition f is Picard operator (P O).
For a W P O, f : X → X, we define the operator f ∞ : We remark that, f ∞ (X) = F f , i.e., f ∞ is a set retraction of X on F f . For the case of ordered L-spaces, we have some properties of W P O and P O.
Then, the operator f : If f 0 is a P O, then f is a P O.

Operators with Volterra property with respect to a subinterval
Let (B, +, R, |·|) be a Banach space, a, b, c ∈ R, a < c < b. In what follows, we consider on C([a, b], B), C([a, c], B) norms of uniform convergence (max-norm, · , Bielecki norm, · τ ). In, we consider a subset defined by, , has the Volterra property with respect to the subinterval, [a, c], if the following implication holds, , has the Volterra property if it has the Volterra property with respect to each subinterval, [a, t], for a < t < b.
This operator has the Volterra property with respect to the subinterval [a, c], but V has not the Volterra property.
, is an operator with Volterra property with respect to [a, c], then the operator V induces an operator, V 1 , on C([a, c], B), defined by The first abstract result of our paper is the following.
Theorem 3.4. In terms of the above notations, we suppose that: (1) V has the Volterra property with respect to [a, c]; (2) V 1 is a contraction; (3) V is a G-contraction. Then: , is a contraction, i.e., it has a unique fixed point, x * , and x * [a,c] = x * 1 . From these we have (i), (ii) and (iii). (3.1) We are looking for the solution of this equation in C[a, b]. In addition, we suppose that: (2 ) there exists L K > 0 such that: In this case: V (x)(t) = the second part of (3.1);   y(θ)))ds.
We remark that, f 0 is a P O, and f 1 (x, ·) : Then, x n → x * [a,c] as n → ∞, y n → x * [c,b] as n → ∞. We denote, Then, u n ∈ C[a, b], for n ∈ N * , and, u n+1 = V (u n ) with u n → x * as n → ∞, i.e., V is a P O. This result is very important because we can apply for V , the Abstract Gronwall Lemma. So we have:

Also, from the Abstract Comparison Lemma we have a comparison result for equation (3.1).
Remark 3.9. For the functional integral equations with maxima, see [1], [11], [16], [22], [13], . . . We are looking for solutions of these equations in C ([a, b], B). To do this, in addition, we suppose that: (2 ) there exists L K > 0 such that, (3 ) there exists L H > 0 such that, In the case of equation (3.2) we have: V (x)(t) = the second part of (3.2); First, we remark that V has the Volterra property with respect to the subinterval [a, c]. Remark 3.11. In a similar way, as in the case of Example 3.6, the Conjecture 3.5 is a theorem for the operator V in Example 3.10.
If B := R m , then we can work with vectorial max-norms and with vectorial Bielecki norms.

Operators with Volterra property
. We also consider the following sets, For, x k ∈ C([t 0 , t k ], B), k = 1, m − 1, we denote, The second basic result of this paper is the following. Theorem 4.1 (Theorem of step by step contraction). We suppose that: (1) V has the Volterra property; (2) V 1 is a contraction; Then: , .
Proof. It follows from successive (step by step !) application of Theorem 3.4, for the pairs, (V k+1 , V k ), k = 1, m − 1, with V k+1 as V and V k as V 1 .  x(θ))ds, t ∈ [a, b] (4.1) By step by step contraction principle we shall prove that, if there exists L K > 0 such that, then the equation First, we remark that V has the Volterra property. In this case: . So, V 1 is a contraction with respect to max-norm. In a similar way, V 2 is a G 1 -contraction, V k is a G k−1 -contraction and V is G m−1 -contraction. So, we are in the conditions of Theorem 4.1. From this theorem we have that: The equation (4.1) has in C[a, b] a unique solution, x * . Moreover, x n (θ))ds; , and x n+1 (t) = x n (θ))ds, n ∈ N; . . .

Step by step generalized contraction principles
There is a large class of generalized contraction principle (see, for example, [24], [2], [17]). As an example in what follows, we consider the case of ϕ-contractions.
Let (X, d) be a metric space, G ⊂ X × X a nonempty subset and f : X → X be an operator.
In the terms of notations in section 4, in a similar way as in the case of Theorem 4.1, we have: Theorem 5.2 (Theorem of step by step ϕ-contraction). We suppose that: (1) V has the Volterra property; (2) V 1 is a ϕ-contraction; (3) V k is a (G k−1 , ϕ)-contraction, for k = 2, m. Then: , .
. Problem 5.3. For which generalized contractions we have step by step corresponding result ? If such generalized contractions are found, then the problem is to give relevant applications of such result.