We firstly establish some new theorems on time scales, and then, by employing them together with a new comparison result and the monotone iterative technique, we show the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem: , u(0) = u(ρ(a)), where is a time scale.

1. Introduction

In this paper, we are concerned with the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem:

()

where

is a time scale and f, g satisfy

(S₁) .

By proving a new comparison result and developing the monotone iterative technique, we show the extremal solutions of the periodic boundary value problem of nabla integrodifferential equations of Volterra type on time scales.

The study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which may be an arbitrary closed subset of the reals. Many results on this issue have been well documented in the monographs [1, 2] written by Bohner and Peterson. Moreover, an integrodifferential equation on time scales (including time scale ) finds many applications in various mathematical problems [3]. And this leads to the extensive study of the existence of extremal solutions to such kind of equations; see Agarwal et al. [4], Franco [5], Guo [6], Z. He and X. He [7], Nieto and Rodríguez-López [8], Song [9], Xu and Nieto [10], Xing et al. [11], and the references therein. However, to the best of the authors′ knowledge, most of them are of ordinary integrodifferential equations and delta integrodifferential equations on time scales, while the nabla integrodifferential equations on time scales have rarely been considered up to now; the main reason is that the theory on nabla derivatives on time scales is not complete. So, in order to study PBVP (1) to fill the gap, we need firstly to establish some new theorems on time scales, including the Induction Principle and Mean Value Theorem, which are very important for getting our main results, and this will be shown in Section 2.2.

In addition, monotone iterative technique coupled with the method of upper and lower solutions has been widely used in the treatment of existence results of initial and boundary value problems for nonlinear differential equations in recent years. The basic idea is that using the upper and lower solutions as an initial iteration one can construct monotone sequences from a corresponding linear problem, and these sequences converge monotonically to the minimal and maximal solutions of the nonlinear problem. When the method is applied to nabla differential equations on time scales, it needs a suitable nabla differential inequality as a comparison principle; this will be shown in Section 3.

For some other work on time scales, we refer the readers to Aderson [12], Agarwal et al. [13], Tisdell et al. [14, 15], and the references therein.

We will assume the following throughout: by we mean that , where 0 < a. And we denote by J.

2. Preliminary

2.1. Some Definitions and Lemmas

For convenience, in this subsection, we give some definitions and lemmas on time scales, which can be found in book [1, 2].

Definition 1 (see [1], page 1.)Let be a time scale. For , one defines the forward jump operator by

()

while one defines the backward jump operator

()

Definition 2 (see [2], page 47.)If has a right-scattered minimum m, define ; otherwise, set . The backward graininess function is defined by

()

Definition 3 (see [1], page 47.)For and , define the nabla derivative of f at t, denoted by f^∇(t), to be the number (provided it exists) with the property that, given any ɛ > 0, there is a neighborhood U of t such that

()

for all s ∈ U.

Definition 4 (see [1], page 48.)The function p is ν-regressive if

()

Define the ν-regressive class of functions on

to be

()

, then we define circle plus addition by

()

Definition 5 (see [1], page 48.)For , define circle minus p by

()

Definition 6 (see [1], page 49.)For h > 0, let

()

Define the ν-cylinder transformation

()

where Log is the principle logarithm function. For h = 0, we define

for all

Definition 7 (see [1], page 49.)If , then we define the nabla exponential function by

()

where the ν-cylinder transformation

is as in Definition 6.

Remark 8. From Definitions 6 and 7 we know that 0 ≤ ν(t)p(t) < 1 implies .

Lemma 9 (see [1], page 51.)Let and . Then one has the following:

(i)
and ;
(ii)
;
(iii)
;
(iv)
;
(v)
;
(vi)
;
(vii)
;
(viii)
.

Lemma 10 (see [1], page 48.)Assume that are nabla differentiable at . Then

(i)
the sum is nabla differentiable at t with
()
(ii)
the product is nabla differentiable at t, and the product rules
()
(iii)
if g(t)g^ρ(t) ≠ 0, then f/g is nabla differentiable at t, and we get the quotient rule
()
(iv)
if f and f^∇ are continuous, then
()

2.2. Some New Results on Time Scales

In order to get Theorem 16 which plays an important role in getting our main results, in this subsection, we need firstly to establish Lemmas 11, 14, and 15. The counterpart about delta derivatives can be found in book [1, 2].

Lemma 11 (left-forward induction principle). Let and assume that

()

is a family of statements satisfying the following.

(i)
The statement S(t₀) is true.
(ii)
If t ∈ (−∞, t₀] is left-scattered and S(t) is true, then S(ρ(t)) is also true.
(iii)
If t ∈ (−∞, t₀] is left-dense and S(t) is true, then there is a neighborhood U of t such that S(s) is true for all s ∈ U∩(−∞, t).
(iv)
If t ∈ (−∞, t₀) is right-dense and S(s) is true for all s ∈ (t, t₀], then S(t) is true.

Then S(t) is true for all t ∈ (−∞, t₀].

Proof. Let

()

We want to show S^* = ∅. To achieve a contradiction we assume S^* ≠ ∅. But since S^* is closed and nonempty, we have

()

We claim that S(t^*) is true. If t^* = t₀, then S(t^*) is true from (I). If t^* ≠ t₀ and σ(t^*) = t^*, then S(t^*) is true from (IV). Finally, if σ(t^*) > t^*, then S(t^*) is true from (II). Hence, in any case,

()

Thus, t^* cannot be left-scattered (as t^* = sup⁡S^*), and

(or S^* = ∅). Hence t^* is left-dense. But now (III) leads to a contradiction. The proof is complete.

Next, for convenience, we give a definition.

Definition 12. A continuous function is called pre-nabla-differentiable with (region of differentiation) D, provided is countable and contains no left-scattered elements of , and f is nabla differentiable at each t ∈ D.

Remark 13. This is an example; let and let be defined by

()

Then f is pre-nabla-differentiable with

()

Lemma 14 (Mean Value Theorem). Let f and g be real-valued functions defined on and both pre-nabla-differentiable with D; then

()

implies

()

Proof. Let , with r < s and denote . Let ɛ > 0; we now show by induction that

()

holds for all t ∈ [r, s]. Note that, once we have shown this, the claim of the Mean Value Theorem follows. We now check the conditions given in Lemma 11 as follows.

(I) The statement S(s) is trivially satisfied.

(II) Let t be left-scattered and assume that S(t) holds. Then t ∈ D and

()

Therefore S(ρ(t)) is true.

(III) Suppose S(t) is true and t ≠ s is left-dense; that is, ρ(t) = t. We consider two cases; namely, t ∈ D and t ∉ D. First of all, suppose t ∈ D. Then f and g are differentiable at t and hence there exists a neighborhood of t with

()

Thus

()

That is,

()

Hence we have for all τ ∈ U∩(−∞, t)

()

Thus S(τ) is true for all τ ∈ U∩(−∞, t).

For the second case, suppose t ∉ D. Then t = t_m for . Since f and g are pre-nabla-differentiable, they are continuous and hence there exists a neighborhood U of t with

()

Therefore

()

That is,

()

and hence

()

Thus again S(τ) follows for all τ ∈ U∩(−∞, t).

(IV) Now let t be right-dense and suppose that S(τ) is true for τ > t. Then

()

implies that S(t) is true as both f and g are continuous at t.

Lemma 15. Suppose f is pre-nabla-differentiable with D and U is a compact interval with endpoints ; then

()

Proof. Suppose f is pre-nabla-differentiable with D and with r ≤ s. Defining

()

then

()

By Lemma 14, we get

()

so that

()

This completes the proof.

Theorem 16. Suppose is pre-nabla-differentiable with D for each . Assume that for each there exists a compact interval neighborhood U(t) such that the sequence

()

Then the limit mapping f = lim⁡_n→∞f_n is predifferentiable with D and one has

()

Proof. Let t ∈ D. Without loss of generality we can assume that ρ(t) ∈ U(t). Letting , there exists such that

()

Also by Lemma 15

()

holds for all r ∈ U(t), and m, n > N. Since

converges uniformly on U(t)∩D, there exists

such that

()

Hence,

()

for all r ∈ U(t), and m, n > N so that, by letting m → ∞,

()

for all r ∈ U(t), and

. Let

()

Then there exists

such that

()

and since

is nabla differentiable at t, there also exists a neighborhood W of t with

()

Altogether we have now, for all r ∈ U(t)∩W,

()

which implies that f is nabla differentiable at t with f^∇(t) = g(t); that is,

; the proof is complete.

3. Some Important Lemmas

In this section, we will give some lemmas which are important for the main results.

Lemma 17 (comparison result). Suppose that there is a function and G = max⁡{g(t, s) : t, s ∈ J × J} > 0. Assume that there exist a positive function and a nonnegative function on J, such that α = sup⁡_t∈J{ν(t)p(t)} < 1 and

()

Then u(t) ≤ 0 for all t ∈ J provided that QGMρ(a)/p₀ ≤ 1, where Q = max⁡_t∈J{q(t)}, p₀ = min⁡_t∈J{p(t)},

Proof. Denote . By Definition 5 and (52), we know that ⊖_ν(−p) = p(t)/(1 + pν); thus, by Lemma 9, we have

()

Next, we try to show that m(t) ≤ 0 for all t ∈ J. Otherwise, we have one of the following two cases:

(i)
m(t) ≥ 0 for all t ∈ J and sup⁡_t∈J{m(t)} > 0;
(ii)
there exists t₁, t₂ ∈ J such that p(t₁) > 0 and p(t₂) < 0.

In case (i), since ν(t)p(t) ≥ 0 and α = sup⁡{ν(t)p(t)} < 1, we have ⊖_ν(−p) = p(t)/(1 + ν(t)p(t)) ≥ 0; also 0 < 1 − ν(t) ⊖_ν(−p) = 1 − ν(p/(1 + ν(t)p(t))) = (1/(1 + ν(t)p(t))) < 1, so, by Remark 8, we have . Together with

()

it follows that

On the other hand, from (53), we have

()

which implies that m(t) is nonincreasing and hence m(0) ≥ m(ρ(a)). Thus, we have m(t) = c = const.>0. Therefore, from

, we have

for all t ∈ J. Since

is positive and decreasing on J, u(0) > u(ρ(a)) holds. This is a contradiction.

In case (ii), we have two subcases:

()

When (ii₁) holds, suppose that m(t₂) = −λ = :min⁡_t∈Jm(t) with λ > 0, and we claim that there exists t₃ ∈ [t₂, ρ(a)) such that m^∇(t₃) ≥ λ/ρ(a). Otherwise, m^∇(t) < λ/ρ(a) for all t ∈ [t₂, ρ(a)); it follows that

()

which is a contradiction. Further,

()

Thus, m^∇(t₃) < QGMλ/p₀. Then it follows, together with m^∇(t₃) ≥ λ/ρ(a), that QGMρ(a)/p₀ > 1, provided that QGMρ(a)/p₀ ≤ 1. It is a contradiction and therefore (ii₁) cannot occur.

When (ii₂) holds, we have . Thus

()

Let t₄ = inf⁡{t : m(t) > 0, t ∈ J}, and choose t₅ such that m(t₅) = min⁡{m(t), t ∈ [0, t₄]} = :λ with λ > 0. As in (ii₁), we can similarly prove that there exists a t₆ ∈ [t₅, t₄) such that m^∇(t₆) ≥ λ/ρ(a). On the other hand, deducing as before, we have

()

Thus it follows that QGMρ(a)/p₀ > 1, which contradicts the condition QGMρ(a)/p₀ ≤ 1.

To sum up, we have m(t) ≤ 0 for all t ∈ J. Thus, u(t) ≤ 0 for all t ∈ J. The proof is completed.

Lemma 18 (comparison result). Suppose that g(t, s), p(t), and q(t) satisfy all the conditions in Lemma 17 and u satisfy (52); then u(t) ≤ 0 for all t ∈ J provided that ρ(a)[P + ρ(a)QG] ≤ 1, where Q, G are defined as in Lemma 17, and P = max⁡_t∈J{p(t)}.

Proof. If the conclusion is not true, we have one of the following two cases:

(i)
u(t) ≥ 0 for all t ∈ J and sup⁡_t∈J{u(t)} > 0;
(ii)
there exist t₁, t₂ ∈ J such that u(t₁) > 0 and u(t₂) < 0.

In case (i), we have

()

which implies that u(t) is nonincreasing and so u(0) ≥ u(ρ(a)). Then it follows from u(0) ≤ u(ρ(a)) that u(t) = c, where c is a constant. Hence u^∇(t) ≡ 0. On the other hand,

()

where p₀ = min⁡_t∈J{p(t)}. This is a contradiction.

In case (ii), we have two subcases:

()

When (ii₁) holds, suppose that u(t₃) = −λ = :min⁡_t∈J{u(t)} with λ > 0, and we claim that there exists a t₄ ∈ [t₃, ρ(a)) such that u^∇(t₄) ≥ λ/ρ(a). Otherwise u^∇(t) < λ/ρ(a) for all t ∈ [t₃, ρ(a)); it therefore follows that

()

which is a contradiction. On the other hand,

()

Then, together with u^∇(t₄) ≥ λ/ρ(a), it follows that

()

It is a contradiction and therefore (ii₁) does not hold. Similarly, we can prove that case (ii₂) is also wrong. Hence the proof is complete.

Lemma 19 (existence result). Assume that all the assumptions on g(t, s), q(t), and p(t) in Lemma 17 are satisfied. Then, for any , the periodic boundary value problem

()

has a unique solution u_h(t) provided ρ(a)QG < p₀, where

()

Proof. We will prove the conclusion by Banach contraction Principle. First, define a Banach Space as follows:

()

We define an operator on E as

()

For any two functions u₁, u₂ ∈ E, there holds

()

where the integral

()

Thus we have

()

This implies by condition ρ(a)QG < p₀ that T is a contraction operator on E and therefore by Banach Contraction Principle there exists exactly one u(t) ∈ E such that u = Tu; that is,

()

Next, we will show that u(t) is a solution of (67). In fact,

()

Moreover, by Lemma 9(iii), there holds

()

Thus, the proof is complete.

Lemma 20 (compact result). Assume that {f_n} _n∈N is a function sequence on J satisfying the following conditions:

(i)
{f_n} _n∈N is bounded on J;
(ii)
is bounded on J.

Then there is a subsequence of {f_n} _n∈N that converges uniformly on J.

Proof. From the assumption, there exists a positive number R such that for all t ∈ J and . Since J is bounded, we can choose such that , where δ_ɛ = ɛ/3R and V(r_i, δ_ɛ) = {t ∈ J, |r_i − t | < δ_ɛ}.

If there exists some , which means 0 < r_j < a and , then we can find

()

Since J is a closed subset of

, it is clear in this case that

. Then we can define

()

We claim that

()

In fact, there are two cases to consider.

Case I (r_j ∉ J). If r_j < t < r_j+1, then t ≥ β_j and by (78) we have

()

If r_j−1 < t < r_j, then t ≤ α_j and by (78) we have

()

Case II (r_j ∈ J). We have in this case

()

Next we will show that there exists a subsequence

of {f_n} convergent on Ω. In fact, since {f_n(r₁)} is bounded, it has a convergent subsequence

. Similarly,

is bounded and therefore we can choose a convergent subsequence

. If we repeat this process, we get

()

Then

is convergent on Ω if we choose {n^′} = {n_p}. From the above argument, for any r_k, there exists a constant N_ɛ(r_k) such that

for

. For any fixed t ∈ J, there is some V(r_j, δ_ɛ) such that t ∈ V(r_i, δ_ɛ). Thus if we set

()

then when

, there holds

()

It therefore follows that

is convergent uniformly on J. Thus the proof is complete.

4. Main Results

Denote

()

In this section, we will make use of iterative technique to prove the main theorem.

Definition 21. Functions are said to be an upper and a lower solution of (1), respectively, if

()

Theorem 22. Assume that v₀(t) ≤ u₀(t) are a lower and an upper solution of (1), respectively. Further, suppose that there exist two positive functions such that one of the following conditions holds:

(H) α = sup⁡_t∈J{ν(t)p(t)} < 1, ρ(a)GQ/p₀ < min⁡{1, 1/M};
(H^′) α = sup⁡_t∈J{ν(t)p(t)} < 1, ρ(a)GQ/p₀ < 1 and ρ(a)(P + ρ(a)QG) ≤ 1.

And

()

where G, M, and p₀ are defined as in Lemma 17; P is as defined in Lemma 18. Then PBVP (1) has a maximal solution u^*(t) and a minimal solution v^*(t) in

; moreover, there exist monotone iterative sequences {u_n(t)} and {v_n(t)} such that u_n(t) → u^*(t), v_n(t) → v^*(t) (n → ∞) uniformly for t ∈ J.

Proof. Without loss of generality, we suppose that (H) holds. For any , we consider the following nonhomogeneous linear integrodifferential equations on time scales:

()

where (Fh)(t) = f(t, h(t), (Gh)(t)) + p(t)h(t) + q(t)(Gh)(t). It follows by Lemma 11 that (89) has a unique solution

, and

()

We define an operator

and will prove that (a) v₀ ≤ Av₀(t), Au₀ ≤ u₀ and (b) A is increasing in [v₀, u₀]. To prove (a), we set v₁ = Av₀; we have

()

Denoting k(t) = v₀ − v₁, then, by Definition 21 and (53), we have

()

Thus by Lemma 17 we know that k(t) ≤ 0 for all t ∈ J, which implies that v₀ ≤ v₁; that is, v₀ ≤ Av₀. Essentially, with the same method, we can show that u₀ ≥ Au₀.

To prove (b), we firstly show that F is increasing. Letting , then by (88) we have

()

Thus, F(D) = {F(u(t)), u(t) ∈ D}⊂[Fv₀, Fu₀] is bounded. Next, we show that A is increasing in [v₀, u₀]. Setting

, then

()

Denote

. By (88) and (93), we have

()

It follows by Lemma 17 that −l(t) ≤ 0, ∀ t ∈ J, which implies that

; that is, A is increasing; thus (b) has been proved.

Let u_n = Au_n−1 and v_n = Av_n−1 (n = 1,2, ….). By (a) and (b), we have

()

Next we will show that both {u_n} and {v_n} have convergent subsequences. Let

. It is obvious that U is a bounded set; in fact,

()

From the above discussion, we have

()

In view of the properties of p, q, G, and F, we know that

is bounded on J; by Lemma 20, we know that there exists a subsequence of {u_n} which converges uniformly on J to some

. Since

is nonincreasing, we see that

itself converges uniformly on J to

. Then, by the continuity of p, q, G and F, we have

()

By the definition of F, we know that

converges uniformly on J to f(t, u^*(t), (Gu^*)(t)). Thus, we have

()

which implies that u^* is solution to PBVP (1). Essentially, with the same method, we can prove that {v_n} converges uniformly on J to some

, and v^* is also a solution to PBVP (1).

Finally, we try to show that u^* and v^* are maximal solution and minimal solution to PBVP (1), respectively, on [v₀, u₀].

Suppose that is any solution to PBVP (1) in [v₀, u₀]; then there exists k ∈ N such that v_k(t) ≤ u(t) ≤ u_k(t). Denoting r(t) = u − u_k+1, then by (88) we have

()

Thus, by Lemma 17, we obtain that r(t) ≤ 0 for all t ∈ J, which implies that u(t) ≤ u_k+1(t). Similarly, we can show that v_k+1(t) ≤ u(t) for all t ∈ J. Consequently, by induction, we have v_n(t) ≤ u(t) ≤ u_n(t) for all t ∈ J, n = 1,2, …. Then, by taking limits, we get v^*(t) ≤ u(t) ≤ u^*(t) all t ∈ J, which implies that u^* and v^* are maximal solution and minimal solution to PBVP (1), respectively. The proof is complete.

As an application, we consider the second order PBVP on time scales:

()

where J = [0, a],

. We make the following assumptions.

H₁ There exist satisfying and
()
H₂ There exist two positive functions p(t), q(t) with α = sup⁡{ν(t)q(t)} < 1 such that
()

Theorem 23. Suppose that conditions (H₁) and (H₂) are satisfied. Then there exist monotone sequences , such that

()

and

uniformly on J. Letting

()

then v^*(t), u^*(t) are the minimal and maximal solutions of (102) satisfying

Proof. Let y^∇(t) = u(t). Then , and therefore BVP (102) reduces to the following PBVP:

()

where

. Obviously, this is a PBVP of type (1) with g(t, s) = 1. Hence, the conclusion of Theorem 23 follows immediately from Theorem 22. The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The study was supported by the Fundamental Research Funds for the Central Universities (no. 2652012141) and Beijing Higher Education Young Elite Teacher Project.

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Extremal Solutions to Periodic Boundary Value Problem of Nabla Integrodifferential Equation of Volterra Type on Time Scales

Abstract

1. Introduction

2. Preliminary

2.1. Some Definitions and Lemmas

2.2. Some New Results on Time Scales

3. Some Important Lemmas

4. Main Results

Conflict of Interests

Acknowledgments

References

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Extremal Solutions to Periodic Boundary Value Problem of Nabla Integrodifferential Equation of Volterra Type on Time Scales

Abstract

1. Introduction

2. Preliminary

2.1. Some Definitions and Lemmas

2.2. Some New Results on Time Scales

3. Some Important Lemmas

4. Main Results

Conflict of Interests

Acknowledgments

References

References

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