Optimal and Nonoptimal Gronwall Lemmas

: In this paper, we study some optimal inequalities of the Riccati type and of the Bihari type. We also consider nonoptimal inequalities of the Wendorf type. At the same time, we get a partial answer to Problems 5 and 9, formulated by I. A. Rus. This paper is also motivated by the fact that, in many inequalities, the upper bound is not an optimal one.


Introduction
Generally, inequalities have an important role in the development of all branches of mathematics-for instance, in differential and integral equations theory, partial differential equations theory, in the qualitative theory of differential equations, and stability theory. Many inequalities appear in modern science, in economical theory, in engineering, etc.
In what follows, we will consider some concrete optimal and nonoptimal Gronwall lemmas for differential equations and integral equations.
We will study inequalities of the form where A : X → X is a Picard operator and (X, →, ≤) is an ordered L-space. We first recall some notions and notations used in the paper. Let F A = {x ∈ X | A (x) = x} be the fixed points set of A. Let A 0 := 1 X , A 1 := A, ..., A n+1 := A • A n , n ∈ N. Definition 1 ([1]). Let X be a nonempty set. Let s(X) := {(x n ) n∈N |x n ∈ X, n ∈ N}. Let c(X) be a subset of s(X) and Lim : c(X) → X be an operator. The triple (X, c(X), Lim) is called an L-space (denoted by (X, →)) if the following conditions are satisfied: (i) if x n = x for all n ∈ N, then (x n ) n∈N ∈ c(X) and Lim(x n ) n∈N = x; (ii) if (x n ) n∈N ∈ c(X) and Lim(x n ) n∈N = x, then, for all subsequences (x n i ) i∈N of (x n ) n∈N , we have that (x n i ) i∈N ∈ c(X) and Lim(x n i ) i∈N = x.
If we denote by x * A the unique fixed point of A, then we have: Lemma 2. (Abstract Gronwall Comparison Lemma [7]). Let (X, →, ≤) be an ordered L-space and A, B : X → X two operators. We suppose that: (i) A and B are Picard operators; (ii) A is an increasing operator; (iii) A ≤ B.
If we denote by x * A the unique fixed point of A and by x * B the unique fixed point of B, then We can conclude, from Lemma 1, that x * A is the upper optimal bound for the solutions of the inequality (1).
So, if we can determine the explicit form for x * A , then we have a bound for the solutions of the inequality (1) and x * A is the optimal bound. If the optimal bound x * A cannot be derived explicitly or can not be found, then we apply Lemma 2 and the bound x * B is not optimal but it is useful for applications. One of the most interesting nonoptimal inequalities is the Wendorff inequality (see Example 3).
In the paper [8], I. A. Rus proposed ten open problems regarding the theory of Gronwall lemmas. In this paper, we present some partial responses to Problems 5 and 9. We recall these problems bellow.
Problem 5. In which Gronwall lemmas are the upper bounds fixed points of the operator A? Problem 9. Give new concrete and abstract Gronwall lemmas. These problems were also studied by Craciun and Lungu in [4], Lungu and Rus [7], and Lungu and Ciplea [9].

Optimal Gronwall Lemmas
In this section, we consider some optimal lemmas, which are consequences of the Abstract Gronwall Lemma (AGL).

Optimal Riccati Type Inequality
In what follows, we present an upper bound of the solutions of the Riccati type inequality, which is the fixed point of the corresponding operator. Hence, we have an optimal Gronwall lemma.
In C ([a, b] , R) we consider the Bielecki norm: Lemma 3. We assume that: Then: (a) there exists a unique solution u * ∈ B (c, R) of the equation (Riccati type equation): and y 1 (x) is a particular solution of the Riccati type equation

Proof. (a) Let
A : X → X be the operator defined by: The operator is increasing, by the assumption of (ii). Using conditions (i)-(iii), it follows that τ . Hence We chose τ > 0 such that thus, the operator A is a contraction, which implies that A is a Picard operator. Hence, there exists a unique solution u * ∈ B (c, R) to Equation (3). (b) We determine the fixed point of the operator A, which is a solution of the equation . This equation is equivalent to the following Cauchy problem: If y 1 is a solution of the Riccati Equation (3), then the solution of the Cauchy problem (6) u * A (x) is given in (5), and it is the optimal solution of inequality (4). Therefore, Lemma 3 is an AGL consequence and it is an optimal Gronwall lemma. Example 1. We consider u ∈ C ([0, 0.5] , R + ). If u ∈ B (0, 1) satisfies the inequality is the solution of the Cauchy problem For example, u 1 (x) = x 2 /2 and u 2 (x) = x 2 /4 satisfy inequality (7) and, geometrically, if we represent the functions u 1 , u 2 and u * A , we have that the graphs of u 1 , u 2 are below u * A (see Figure 1).

Optimal Bihari Type Inequality
In [4], an upper bound of the solutions of the Bihari inequality has been given, which is the fixed point of the corresponding operator.
Here, we will show that Lemma 2.2 from [4] is an optimal one. If we consider the inequality where p ∈ C ([a, b] , R + ) , V is continuous, positive, increasing and the Lipschitz function, then x a p (s) ds and F −1 is the inverse of F.
where · τ is the Bielecki norm and A : X → X, , the fixed point of the operator A is Hence, the Bihari inequality is an optimal one. Example 2. We consider y ∈ C π 6 , π 3 , R + . If y satisfies the inequality satisfy inequality (8) and, geometrically, we have that the graphs of y 1 , y 2 are below y * (see Figure 2). (blue color), together on π 6 , π 3 .

Nonoptimal Gronwall Lemmas
In some concrete Gronwall lemmas, only the following implication holds: In this part we consider consequences of Lemma 2. We consider (X, →, ≤) := C (D) , and · τ is the Bielecki norm on C (D) : Lemma 4. (Wendorff type inequality) ( [10][11][12][13][14]). We assume that We consider the operator A : X → X, This operator is an increasing Picard operator, but the function u (x, y) = c exp x 0 y 0 v (s, t) dsdt is not a fixed point of operator A. Remark 1. The right side of (9) is not a fixed point of operator A, so the concrete Lemma 4 is not a consequence of the abstract Gronwall Lemma 1.
In this case, the corresponding operator A : X → X is A (u) (x, y) := 1 + hence, here, Lemma 2 is applied. We have where u * B (x, y) = exp x 2 y and u * A (x, y) is the fixed point of operator A. Geometrically, the surface u * B (x, y) = exp x 2 y , represented in Figure 3, is above the surface corresponding to the optimal solution u * A , which cannot be explicitly derived.

Example 4.
We consider the inequality (see [15]) where u is a continuous and positive function for all x ≥ a and K is of C 1 class after x and continuous after s, K (x, s) ≥ 0 for x ≥ s ≥ a and c > 0. Then, In this case, operator A is We remark that the function is not the fixed point of operator A; therefore, inequality (11) is nonoptimal.

Conclusions
In this paper, we studied concrete optimal Gronwall lemmas corresponding to Riccati and Bihari type inequalities and nonoptimal Gronwall lemmas corresponding to Wendorf type inequalities. Moreover, we obtained a partial response to Problems 5 and 9, formulated by I. A. Rus in [8]. Some geometrical meanings were also given. In the case containing the functions of one variable, where optimal Gronwall lemmas were applied, the curve corresponding to every solution of the given inequality was below the curve corresponding to the optimal solution. In the case containing the functions of two variables, the surface corresponding to each solution of the given inequality was below the surface of the corresponding optimal solution.