Function bounds for solutions of Volterra integro dynamic equations on time scales

Introducing shift operators on time scales we construct the integro-dynamic equa- tion corresponding to the convolution type Volterra differential and difference equations in particular cases T = R and T = Z. Extending the scope of time scale variant of Gronwall's inequality we determine function bounds for the solutions of the integro dynamic equation.


Introduction
In this paper, we are concerned with the investigation of function bounds for the solutions of integro dynamic equation of type which includes the following Volterra equations in particular cases: • Volterra integro differential equation of convolution type: For T = R with δ − (s, t) = t − s and t 0 = 0 x ′ (t) = −a(t)x(t) + t 0 b(t − s)x(s)ds, t ∈ [0, ∞). (1.2) • Volterra integral equation with nonconvolutional kernel: For T = R with δ − (s, t) = t/s and t 0 = 1 x ′ (t) = −a(t)x(t) • Volterra integro difference equation of convolution type: For T = Z with δ − (s, t) = t − s + λ and t 0 = λ ∆x(t) = −a(t)x(t) where ∆ is the forward difference operator.• Volterra integro q−difference equation: For T = q Z with δ − (s, t) = t/s and t 0 = 1 where ∆ q is the q-difference operator given by ∆ q x(t) = x(qt)−x(t) (q−1)t .
Many papers have appeared in the literature on Volterra equations on particular time scales such as R, Z and q N .An early contribution to integro q-difference equations was made by Trjitzinsky [19].In [15] and [16] Elaydi dealt with stability analysis of convolution type Volterra integro difference equations of the form (1.4).In [4], Becker derived an extension of Gronwall's inequality to find function bounds for the solutions of Eq. (1.2), where a, b : [0, ∞) → R are continuous functions and b is nonnegative.Since the time scale theory provides a wide perspective for the unification of discrete and continuous analyses, Volterra integro dynamic equations on general time scales became topic of several research papers.For instance, boundedness of the solutions of nonlinear Volterra integro-dynamic equations on time scales has been investigated in [3] by means of nonnegative definite Lyapunov functionals on time scales.Furthermore, in [2], existence of periodic solutions of nonlinear system of Volterra type integro-dynamic equations has been shown using the topological degree method and Schaefer's fixed point theorem.However, to the best of our knowledge, function bounds for the solutions of Volterra integral equations of the form (1.3) has not been treated elsewhere before.
Motivated by the results of [4], we bring the integro dynamic equation (1.1) under investigation to obtain more general results which are not known even for the above mentioned particular cases.Some applications are also given to illustrate the usefulness of our results.
The remaining part of this paper is organized as follows: In the second section, we propose an extension of Gronwall's inequality ( [7, Corollary 6.7, p.257]) on time scales.In the third section, we introduce the shift operators δ ± to construct the kernel of integro dynamic equation (1.1).In the last section, we give several theorems and corollaries regarding the function bounds for the solutions of (1.1).Hence, it turns out that the results in Sections III and Section IV are valid only for the time scales containing an initial point t 0 so that there exist shift operators δ ± (s, t) on [t 0 , ∞) T .
For the sake of brevity, we assume familiarity with time scale calculus.For a comprehensive review on fundamental aspects of the theory we refer the reader to [7] and [8].
Throughtout the paper, we denote by σ and ρ the forward and backward jump operators, σ : T → T and ρ : T → T, defined by σ(t) := inf {s ∈ T : s > t} and ρ(t) = sup {s ∈ T : s < t}, respectively.A point t ∈ T is said to be right dense (right scattered) if σ(t) = t (σ(t) > t).We say t ∈ T is left dense (left scattered) if ρ(t) = t (ρ(t) < t).The graininness (step size) function it is continuous at right dense points and its left sided limits exists (finite) at left dense points.We use the notation C rd to indicate the set of rd-continuous function on T. Hereafter, we shall denote by [u, v) T the time scale interval [u, v) ∩ T. The intervals (u, v) T , [u, v] T , and (u, v] T are defined similarly.
We list the following theorems which will be needed at several occasions throughout this study.
Theorem 1 (First Mean Value Theorem).[8,Theorem 5.41. p. 142] Let f and g be bounded and ∆ integrable functions on [u, v] T , and let g be nonnegative (or nonpositive) on Then there exists a real number Λ satisfying the inequalities m ≤ Λ ≤ M such that

Gronwall's inequality
There is no doubt that Gronwall's inequality [14, p.293] plays a substantial role in the investigation of stability and convergence properties of solutions of Volterra integral equations.The purpose of this section is to extend the scope of time scale analogue of Gronwall's inequality, which will be used to obtain function bounds for the solutions of Volterra integro-dynamic equations on time scales.A variant of Gronwall's inequality on time scales is given as follows: Theorem 5. [7, Corollary 6.7, p.257] Let y ∈ C rd and ω ∈ R + , ω ≥ 0, and α ∈ R. Then For more on Gronwall's inequalities on time scales we refer to [7, p.256], [18], and [20].
One may easily see by setting and y(t) = 1 for t ∈ [0, 1] and y(t) = 0 for t ∈ [2, ∞) that nonnegativity condition on the function ω in Theorem 5 cannot be omitted.However, in the next theorem, we keep positive regressivity condition ω ∈ R + and rule out nonnegativity condition on ω by making more stringent assumption than (2.1).Therefore, we obtain important relaxations for the particular cases.For instance, if T = R, then µ(t) = 0, i.e., all functions ω : R → R are positively regressive, if T = Z, the functions ω : Z → R satisfying ω(t) > −1 for all t ∈ Z are positively regressive.That is, the following result is valid for all functions ω : R → R and for all functions ω : Z → R satisfying ω(t) > −1.

Theorem 6 (An extension of Gronwall's inequality). Let f and γ be continuous functions on
Hereafter, we present some results which are essential for the proof of Theorem 6.
One can similarly prove the next result by reversing the directions of the inequalities (2.8-2.10).

Corollary 1. Let f, γ, and ξ be rd-continuous functions on
Now, we are ready to prove Theorem 6.
Case II.Now, suppose γ changes sign.Hereafter, we will use continuity of the function γ on [t 0 , T ) T to show that the interval [t 0 , T ) T can be partitioned into disjoint subintervals of the form [t n−1 , t n ) T so that γ is strictly negative, strictly positive, or identically zero on each of the open intervals (t n−1 , t n ) T .Let us define the set S ⊂ [t 0 , T ) T as follows It is obvious that the set S consists only of right scattered points of [t 0 , T ) T and can be expressed as follows Let us separate this set from [t 0 , T ) T and define Denote the set of zeros of γ in K by A, i.e., Since the single point set {0} is closed in R, we get by continuity of γ that the set is closed in K (here, we are considering R with its standard topology and the subset K with the subspace topology inherited from the topology on R).Thus, the complement that either γ(c) = 0 or γ(c)γ(σ(c)) < 0. This is not possible since A ∩ (A ∪ S) = ∅.So, there is an increasing sequence (t n ) n∈J of distinct points such that and the values of γ on (t n−1 , t n ) are always positive or always negative or always zero.Quantitative properties of the index set J depends on the function γ and the interval [t 0 , T ) T , i.e., the set J can be either a finite set {1, 2, ..., N } or the set of natural numbers N.
From Case I we know that the inequality To see that (2.15) holds in the case when t n−1 is left scattered, assume that t n−2 ∈ [t 0 , t n−1 ) T is a point such that σ(t n−2 ) = t n−1 and define the function By (2.2) we have and therefore, Hence, (2.15) holds in any case.Thus, by (2.13) and (2.15) we get that In this section, we introduce the shift operators δ ± : [t 0 , ∞) T → R to construct the integrodynamic equation An arbitrary time scale (e.g., T =q N ) does not have to include t − s and 0. Therefore, different than the kernel b(t − s) and the lower limit 0 of the integral in (1.2), we use b(δ − (s, t)) and t 0 in (3.1), respectively.An intuition for the determination of the shift operator δ − can be developed by understanding the idea behind the use of b(t − s) in (1.2).Informally, the expression b(t − s) in (1.2) can be regarded as a shift (or delay) of the function b.However, since t − s / ∈ q N for the time scale T = q N , the expression b(t − s) cannot be used as the shift of b.On the other hand, t/s ∈ q N for all t, s ∈ q N satisfying t ≥ s ≥ 1. Inspired by these examples and common properties of the operations t − s and t/s, we can construct backward shift operator δ − on time scales.Similarly, we can describe properties of the forward shift operator δ + considering the properties of the operations t + s and ts.
Definition 2. Suppose we are given an initial point t 0 ∈ T so that there exists operators δ ± : 2 Given a fixed element T 0 ∈ [t 0 , ∞) T , the functions δ± are strictly increasing with respect to their second arguments, i.e., T 0 ≤ t < u implies δ ± (T 0 , t) < δ ± (T 0 , u), Then the operators δ − and δ + associated with the initial point t 0 are called backward and forward shift operators on [t 0 , ∞) T , respectively.
Generalized shifts and the associated geometry on a general time scale were first dealt with in [11].Also, generalized convolution on time scales was treated by [13].Afterwards, in [10, Definition 2.1] shift operators was defined to propose convolution on time scales.Note that the shift operators δ ± defined here are different than the ones in the above mentioned literature.
This example shows that we can define different type shift operators on the same time scale.For instance, on T = R, we have the shift operators δ ± (s, t) = t ± s and δ ± (s, t) = ts ±1 with the initial points 0 and 1, respectively.
i.For a fixed T ∈ (t 0 , ∞) T we have Proof.(i) follows from P.2 and P.3.To show (ii), use P.4-5 and P.7 to obtain For the proof of (iii) we use (ii) to get This, along with P. 4 and P.6, yields (iv) is direct implication of (iii) since u = δ − (s, t) implies s = δ − (u, t) and (v) is obtained by assuming u = δ − (s, t), using (ii), i.e., t = δ + (u, s), and To verify (vi) take u = δ − (s, t) use (ii) to get t = δ + (s, u) and (vii) is proven by substituting s = v = t 0 in P.6.The proof is complete.
Notice that shift operators δ ± are defined once the initial point t 0 ∈ T is known.For instance, we choose the initial point t 0 = 0 to define shift operators δ ± (s, t) = t ± s on T = R.However, if we take the initial point λ ∈ (0, ∞) then we can define new shift operators by δ ± (s, t) = t ∓ λ ± s and in terms of δ ± as Example 2. In the following table, we give several particular time scales to show the change in the formula of shift operators as the initial points change.In general, let T be a time scale with shift operators δ ± associated with initial point t 0 .Choosing a new initial point λ ∈ [t 0 , ∞) T , we can define the new shift operators δ ± associated with λ by Using P.1-7, one may easily verify that the new shift operators δ ± satisfy the following properties: the functions δ± are strictly increasing with respect to their second arguments, i.e., Moreover, the properties given in Lemma 4 are also valid for the operators δ ± .

Function Bounds for Solutions of Volterra Equations
Hereafter, we suppose that T is a time scale including an initial point t 0 so that there exists shift operators δ ± satisfying properties P.1-7 in Definition 2. Let a, b : [t 0 , ∞) T → R be two continuous functions with b(t) ≥ 0 for all t ∈ [t 0 , ∞) T and −a ∈ R + .Hereafter, we denote by x(t) := x(t, t 0 , x 0 ) the unique differentiable solution of satisfying x(t 0 ) = x 0 .For the existence and boundedness of such a solution we refer the reader to [1], [3], and [12].For brevity, we shall use the notation x(t) instead of x(t, t 0 , x 0 ).
To see nonnegativity of x(t) on the interval (T, ∞) T it suffices to prove that the set M − given by M − := {t ∈ (T, ∞) T : x(t) < 0} EJQTDE, 2010 No. 7, p. 12 is empty.Suppose contrary that M − = ∅ and denote by t 1 the real number Henceforth, we show that t 1 ∈ M − , i.e., x(t 1 ) < 0. It follows from continuity of x that x(t 1 ) ≤ 0, in which the case x(t 1 ) = 0 leads to a contradiction in the sign of x ∆ (t 1 ).To see this, let x(t 1 ) = 0. Thus, t 1 is right dense, and hence, M − includes a continuous interval (t 1 , a) on which x is nonincreasing, i.e., x ∆ (t 1 ) ≤ 0. On the other hand, from (4.1) and Theorem 1 we arrive at where Λ 1 is a real number satisfying in which m 1 and M 1 are given by On the other hand, similar to (4.8) we get that t 0 ≤ s ≤ T implies This shows that b(δ − (s, t 1 )) is not equally zero on the interval [t 0 , T ) T .From (4.11-4.12)we find Therefore, we have t 1 ∈ M − , i.e., x(t 1 ) < 0. Since x(T )x(t 1 ) < 0, Theorem 2 guarantees the existence of a c ∈ [T, t 1 ) such that x(c) = 0 or x(c)x(σ(c)) < 0, where x(c)x(σ(c)) < 0 is not possible.To see this, we show that the set given by This along with −a ∈ R + , i.e., 1 − µ(t)a(t) > 0, implies where we also used (4.1) and (4.10).This leads to a contradiction.Hence, we have x(c) = 0 for a c ∈ (T, t 1 ) T .This shows that the set is non-empty.Let η = sup M 0 .It follows from continuity of x that η ∈ M 0 .Since D = ∅, as a consequence of Theorem 2, there cannot be any element t ∈ (η, t 1 ) T such that x(t) > 0. Thus, (η, t 1 ) T = ∅ and then x is strictly decreasing on [η, t 1 ) T , i.e., x ∆ (η) < 0 by Lemma 1.However, we get from Theorem 1 that where Λ is a real number satisfying 0 ≤ Λ ≤ sup {x(t) : t ∈ (t 0 , η) T }.We obtain this contradiction by assuming that M = ∅.Consequently, M − = ∅, i.e., x(t) ≥ 0 for all t ∈ [t 0 , T ) T .Taking the integral in (4.Consequently, the result (4.4) follows from inequalities (4.15) and (4.18).
To see that (4.5) holds for x 0 = x(t 0 ) < 0, it is enough to employ (4.4) by taking into account that −x(t) is the unique solution of Eq. (4.1) satisfying the initial condition −x(t 0 ) = −x 0 > 0.
The proof is complete.
In the next corollary, we shall provide lower and upper bounds for X(t) by using the results of Theorem 7. To be able to employ Theorem 7 in the analysis, first of all the kernel of the integral term in (4.19) should include the shift operator associated with λ.As we have mentioned in Section 3 we may move the initial point t 0 to λ, and define the new shift operators δ ± associated with λ as in (3.2) and (3.
the set A is composed of disjoint open intervals in K, each of which have one of the following forms: (a, b) ∩ A or [t 0 , b) ∩ A, where a, b ∈ [t 0 , T ] T and a < b.We conclude from Theorem 2 that on each of these open intervals, the function γ is either strictly positive or strictly negative.Because, if there exist two points t 1 , t 2 ∈ (a, b) ∩ A such that γ(t 1 )γ(t 2 ) < 0 then, Theorem 2 implies the existence of a point c ∈ [t 1 , t 2 ) ∩ A such EJQTDE, 2010 No. 7, p. 7 13)is satisfied, where [t n−1 , t n ) T is any subinterval of the above mentioned partition of [t 0 , T ) T .The rest of the proof proceeds by induction.Suppose that (2.3) holds on [t 0 , t n−1 ) T = ∪ n−1 k=1 [t k−1 , t k ), i.e., f (t) ≤ f (t 0 )e −γ (t, t 0 ) for t ∈ [t 0 , t n−1 ) T .(2.14)If t n−1 is a left dense point, then continuity of both sides of (2.14) implies This shows that (2.3) holds on the interval [t 0 , t n ) T .By induction we conclude that (2.3) holds on the entire interval [t 0 , T ) T .For the proof of second statement of theorem we reverse the directions of inequalities (2.2-2.3) and invoke Corollary 1 to modify the proof of the first statement accordingly.The proof is complete.EJQTDE, 2010 No. 7, p. 8 3. Shift operators on time scales