Basic theory of differential equations with mixed perturbations of the second type on time scales

In this paper, we develop the theory of differential equations with mixed perturbations of the second type on time scales. We give an existence theorem for differential equations with mixed perturbations of the second type on time scales under Lipschitz condition. We also present some fundamental differential inequalities on time scales, which are utilized to investigate the existence of extremal solutions. We establish the comparison principle for differential equations with mixed perturbations of the second type on time scales. Our results in this paper extend and improve some well-known results.

Let T be a time scale, and let J = [t 0 , t 0 + a] T = [t 0 , t 0 + a] ∩ T be a bounded interval in T for some t 0 ∈ R and a > 0. We denote by C rd (J × R, R) the class of rd-continuous functions g : J × R → R. For basic definitions and useful lemmas from the theory of calculus on time scales, we refer to [1].
By a solution of the DETS (1) we mean a -differentiable function u such that (i) the function t → u-k (t,u) f (t,u) is -differentiable for each u ∈ R, and (ii) u satisfies equations (1).
As far as we know, there are no results for the DETS (1). From the works mentioned we consider the theory of DETS (1). We give an existence theorem for the DETS (1) under Lipschitz conditions. We also present some fundamental differential inequalities on time scales (DITS), which are utilized to investigate the existence of extremal solutions. We establish the comparison principle for the DETS (1). Our results in this paper extend and improve some well-known results.
The paper is organized as follows. In Sect. 2, we give an existence theorem for the DETS (1) under Lipschitz conditions by the fixed point theorem in Banach algebra due to Dhage. In Sect. 3, we establish some fundamental DITS to strict inequalities for the DETS (1). In Sect. 4, we present existence results of maximal and minimal solutions for HDTS. In Sect. 5, we prove the comparison principle for the DETS (1), which is followed by the conclusion in Sect. 6.

Existence result
In this section, we discuss the existence results for the DETS (1). We place the DETS (1) in the space C rd (J, R) of rd-continuous functions defined on J with the supremum norm · defined as u = sup t∈J u(t) and the multiplication "·" in C rd (J, R) defined as for u, v ∈ C rd (J, R). Clearly, C rd (J, R) is a Banach algebra with respect to these norm and multiplication. By L 1 (J, R) we denote the space of Lebesgue -integrable functions on J equipped with the norm · L 1 defined as The following fixed point theorem in a Banach algebra due to Dhage [16] is useful in the proofs of our main results. We present the following hypotheses.
There exist constants L 1 > 0 and L 2 > 0 such that for all u ∈ R. Lemma 2.2 Suppose that (A 0 ) holds. Then for any v ∈ L 1 (J, R), the -differentiable function u is a solution of the DETS and if and only if u satisfies the integral equation Proof Let u be a solution of problem (2)-(3). Applying the -integral to (2) from t 0 to t, we obtain that is, Thus (4) holds.
Conversely, suppose that u satisfies equation (4). By direct differentiation, applying the -derivative to both sides of (4), we get that (2) is satisfied. Thus, substitute t = t 0 in (4) implies Now we will give the following existence theorem for the DETS (1).
then the DETS (1) has a solution defined on J.
Proof Set U = C rd (J, R) and define the subset S of U by Clearly, S is a closed, convex, and bounded subset of the Banach space U. By Lemma 2.2 the DETS (1) is equivalent to the nonlinear integral equation Define three operators A, C : U → U and B : S → U by and Then equation (6) is transformed into the operator equation as Next, we prove that the operators A, B, and C satisfy all the conditions of Lemma 2.1. First, we prove that A is a Lipschitz operator on U with Lipschitz constant L 1 . Let u, v ∈ U. Then by (A 1 ) Taking the supremum over t, then we have Similarly, we can get that C is also a Lipschitz operator on U with Lipschitz constant L 2 .
Next, we prove that B is a compact continuous operator from S into U. First, we prove that B is continuous on S. Let {u n } be a sequence in S converging to a point u ∈ S. Then by the Lebesgue dominated convergence theorem adapted to time scale we have for all t ∈ J. This shows that B is a continuous operator on S. Next, we prove that B is a compact operator on S. It suffices to show that B(S) is a uniformly bounded and equicontinuous set in U. Take arbitrary u ∈ S. Then by (A 2 ) for all t ∈ J. Taking the supremum over t, we have for all u ∈ S. This shows that B is uniformly bounded on S.
On the other hand, let t 1 , t 2 ∈ J. Then for any u ∈ S, we get Since the function p is continuous on compact J, it is uniformly continuous. Hence, for ε > 0, there exists δ > 0 such that for all t 1 , t 2 ∈ J and u ∈ S. This shows that B(S) is an equicontinuous set in U. Now the set B(S) is uniformly bounded and equicontinuous set in U, so it is compact by Arzelà-Ascoli theorem. Thus B is a compact operator on S. Next, we show that (c) of Lemma 2.1 is satisfied. Let u ∈ U and v ∈ S be such that u = AuBv + Cu. Then by assumption (A 1 ) we have Thus we get Taking the supremum over t, we have This shows that (c) of Lemma 2.1 is satisfied.
Finally, we obtain and so Thus all the conditions of Lemma 2.1 are satisfied, and hence the operator equation AuBu + Cu = u has a solution in S. Therefore the DETS (1) has a solution defined on J.

Differential inequalities on time scales
In this section, we establish DITS for the DETS (1).

Theorem 3.1 Suppose that (A 0 ) holds. Assume that there exist -differentiable functions
and one of the inequalities being strict. Then v(t 0 ) < w(t 0 ) implies v(t) < w(t) (12) for all t ∈ J.
Proof Assume that inequality (11) is strict. Suppose that the claim is false. Then there and for all t ∈ J. Then we obtain V (t 1 ) = W (t 1 ), and by (A 0 ) we have V (t) < W (t) for all t < t 1 .
Since V (t 1 ) = W (t 1 ), we get for sufficiently small h < 0. This inequality implies that because of (A 0 ). Then we obtain This is a contradiction with v(t 1 ) = w(t 1 ). Hence inequality (12) is valid.
The next result is concerned with nonstrict DITS, which needs a Lipshitz condition. (10) and (11). Suppose that there exists a real number K > 0 such that
Then from (13) we have for all t ∈ J, and thus that is, for all t ∈ J. Also, we get w ε (t 0 ) > w(t 0 ) > v(t 0 ). Hence Theorem 3.1 with w = w ε implies that v(t) < w ε (t) for all t ∈ J. By the arbitrariness of ε > 0, taking the limits as ε → 0, we have v(t) ≤ w(t) for all t ∈ J.

Existence of maximal and minimal solutions
In this section, we prove the existence of maximal and minimal solutions for the DETS (1) on J = [t 0 , t 0 + a] T . We discuss the case of maximal solution only. Similarly, the case of minimal solution can be obtained with the same arguments with appropriate modifications. Given an arbitrary small number ε > 0, we discuss the following initial value problem of DETS: where f ∈ C rd (J × R, R \ {0}) and k, g ∈ C rd (J × R, R).
An existence theorem for the DETS (14) can be stated as follows. Proof By hypothesis, since there exists ε 0 > 0 such that for all 0 < ε ≤ ε 0 . The rest of the proof is similar to that of Theorem 2.1, and we omit it.
Our main existence theorem for maximal solution for the DETS (1) is the following: Proof Let {ε n } ∞ 0 be a decreasing sequence of positive numbers such that lim n→∞ ε n = 0, where ε 0 is a positive number satisfying the inequality Such a number ε 0 exists in view of inequality (5). Then for any solution x of the DETS (1), by Theorem 4.1 we get

Comparison theorems on time scales
The main problem of the DITS is to estimate a bound for the solution set for the DITS related to the DETS (1). In this section, we present the maximal and minimal solutions serving as bounds for the solutions of the related DITS to the DETS (1) on J = [t 0 , t 0 + a] T .
Theorem 5.1 Suppose that (A 0 )-(A 2 ) and inequality (5) hold. Assume that there exists a -differentiable function u such that for all t ∈ J, where r is a maximal solution of the the DETS (1) on J.
Then ρ(t) ≤ y(t) for all t ∈ J, where ρ is a minimal solution of the DETS (1) on J.

Conclusion
In this paper, we have developed the theory of the DETS (1). By the fixed point theorem in Banach algebra due to Dhage we have presented an existence theorem for the DETS (1) under Lipschitz conditions. We have also established some DITS for the DETS (1), which are used to investigate the existence of extremal solutions, and the comparison principle for the DETS (1). Our results in this paper extend and improve some well-known results.