The purpose of this paper is to obtain Opial-type inequalities that are useful to study various qualitative properties of certain differential equations involving impulses. After we obtain some Opial-type inequalities, we apply our results to certain differential equations involving impulses. Opial-type inequalities are very useful to study various qualitative properties of differential equations. For a good reference of the work on such inequalities together with various applications, we recommend the monograph [1] . In this paper we obtain some Opial-type inequalities that involve Stieltjes derivatives which are applicable to differential equations with impulses. Differential equations involving impulses arise in various real world phenomena, we refer to the monograph [8] . To obtain our results in this paper we need some preliminaries. Let R be the set of all real numbers. Assume that [ a , b ] ⊂ R is a bounded interval. A function f : [ a , b ] → R is called regulated on [ a , b ] if both exist for every point s ∈ [ a , b ), t ∈ ( a , b ], respectively. Let G ([ a , b ]) be the set of all regulated functions on [ a , b ]. For f ∈ G ([ a , b ]) we define f ( a −) = f ( a ); f ( b +) = f ( b ). For convenience we define Remark 2.1. Let f ∈ G ([ a , b ]). Since both f ( s +) and f ( s −) exist for every s ∈ [ a , b ] it is obvious that f is bounded on [ a , b ], and since f is the uniform limit of step functions, f is Borel measurable (see [3, Theorem 3.1.]). For a closed interval I = [ c , d ], we define f ( I ) = f ( d ) − f ( c ). A tagged interval ( τ , [ c , d ]) in [ a , b ] consists of an interval [ c , d ] ⊂ [ a , b ] and a point τ ⊂ [ c , d ]. Let I_i = [ c_i , d_i ] ⊂ [ a , b ], i = 1, …, m . We say that the intervals I_i are pairwise non-overlapping if for i ≠ j where int ( I ) denotes the interior of an interval I . A finite collection {( τ_i , I_i ) : i = 1, 2, …, m } of pairwise non-overlapping tagged intervals is called a tagged partition of [ a , b ] if . A positive function δ on [ a , b ] is called a gauge on [ a , b ]. From now on we use notation . Definition 2.2 ( [6 , 9] ) . Let δ be a gauge on [ a , b ]. A tagged partition of [ a , b ] is said to be δ − fine if for every we have Moreover if a δ −fine partition P satisfies the implications then it is called a δ *−fine partition of [ a , b ]. The following lemma implies that for a gauge δ on [ a , b ] there exists a δ *−fine partition of [ a , b ]. This also implies the existence of a δ − partition of [ a , b ]. Lemma 2.3 ( [6] ) . Let δ be a gauge on [ a , b ] and a dense subset ­ Ω⊂ ( a , b ) be given . Then there exists a δ*−fine partition of [ a , b ] such that t_i ∈ Ω for We now give a formal definition of two types of the Kurzweil integrals. Definition 2.4 ( [6 , 9] ) . Assume that f , g : [ a , b ] → R are given. We say that f d g is Kurzweil integrable (or shortly, K- integrable ) on [ a , b ] and v ∈ R is its integral if for every ε > 0 there exists a gauge δ on [ a , b ] such that provided is a δ −fine tagged partition of [ a , b ]. In this case we define (or, shortly, ). If, in the above definition, δ −fine is replaced by δ *−fine, then we say that f d g is Kurzweil* integrable (or, shortly, K*- integrable ) on [ a , b ] and we define . Remark 2.5. By the above definition it is obvious that K-integrability implies K*-integrability. The following results are needed in this paper. For other properties of the K-integrals, see, e.g., [2 , 7 , 9 , 10] . In this paper BV ([ a , b ]) denotes the set of all functions that are of bounded variation on [ a , b ]. Theorem 2.6 ([ 11 , 2.15. Theorem]) . Assume that f ∈ G ([ a , b ]) and g ∈ BV ([ a , b ]). Then both f d g and g d f are K-integrable on [ a , b ] and Remark 2.7. In the above theorem, the sum Σ _t∈[a,b] [Δ ⁻ f ( t )Δ ⁻ g ( t )−Δ ⁺ f ( t )Δ + g ( t )] is actually a countable sum because every regulated function has only countable dis-continuities. Theorem 2.8 ([ 10 , p. 40, 4.25. Theorem]) . Let h ∈ BV ([ a , b ]), g : [ a , b ] → R and f : [ a , b ] → R . If the integral g dh exists and f is bounded on [ a , b ], then the integral exists if and only if the integral exists and in this case we have Theorem 2.9 ([ 10 , p. 34, 4.13. Corollary]) . Assume that f ∈ G ([ a , b ]) and g ∈ BV ([ a , b ]). Then we have for every t ∈ [ a , b ] The following result is the Hölder’s inequality for K-integral. In this paper we frequently use this inequality. Theorem 2.10. (Hölder’s inequality) Assume that f , g ∈ G ([ a , b ]) and h is a nondecreasing function defined on [ a , b ]. Let . Then we have Proof . The proof of this theorem is very similar to the proof of the classical Hölder’s inequality. So we omit the proof.   ☐ In this section we state the results in [4 , 5] that are essential to obtain our main results. Throughout this section, we assume that f ∈ G ([ a , b ]) and g is a nondecreasing function on [ a , b ]. We say that the function g is not locally constant at t ∈ ( a , b ) if there exists η > 0 such that g is not constant on ( t − ε , t + ε ) for every 0 < ε < η . We also say that the function g is not locally constant at a and b , respectively if there exist η , η * > 0 such that g is not constant on [ a , a + ε ), ( b − ε *, b ] respectively, for every ε ∈ (0, η ), ε *∈ (0, η* ). Definition 3.1 ( [4] ) . If g is not locally constant at t ∈ ( a , b ), we define provided that the limit exists. If g is not locally constant at t = a and t = b respectively, we define respectively, provided that the limits exist. Frequently we use instead of . If both f and g are constant on some neighborhood of t , then we define . Remark 3.2. It is obvious that if g is not continuous at t then exists. Thus if does not exist then g is continuous at t . is called a Stieltjes derivative of f with respect to g . Theorem 3.3 ( [4] ) . Assume that if g is not locally constant at t ∈ [ a , b ]. If f is continuous at t or g is not continuous at t; then we have K*-integrals recover Stieltjes derivatives. Theorem 3.4 ( [4] ) . Assume that if g is constant on some neighborhood of t then there is a neighborhood of t where both f and g are constant. Suppose that exists at every t ∈ [ a , b ] − { c ₁ , c ₂ , …}, where f is continuous at every t ∈ { c ₁ , c ₂ , …}. Then we have Lemma 3.5 ( [4] ) . Assume that if g is constant on some neighborhood of t then there is a neighborhood of t such that both f ₁ and f ₂ are constant there. If both and exist and f ₁ , f ₂ ∈ G ([ a , b ]), t hen we have Similarly to the Riemann integral we have the following integration by parts formula. Theorem 3.6. (Integration by Parts) Assume that functions f, g, h ∈ G ([ a , b ]) are all left-continuous and h is nondecreasing. Suppose that both and exist for every t ∈ [ a , b ] and . Then we have Proof . By Theorem 2.8 and Theorem 3.4 we have So by Theorem 2.6 we get This completes the proof.   ☐ Let The Heaviside function H_τ : R → {0, 1} is defined by Using the Heaviside function H_τ , we define function ϕ : [ a , b ] → R by Remark 3.7. It is obvious that the function ϕ is strictly increasing and of bounded variation on [ a , b ], and left-continuous on [ a , b ]. Lemma 3.8 ( [5] ) . Assume that f ∈ G ([ a , b ]) and f' ( t ) exists for t ≠ t_k , Then we have In this section we obtain some Opial-type integral inequalities involving Stieltjes derivatives. The Opial-type inequalities have many interesting applications in the theory of differential equations(see, e.g., [1] ). Throughout this paper we always assume that and that a function α : [ a , b ] → R is strictly increasing on [ a , b ], and continuous at t ≠ t_k , and Δ α ( t_k ) ≠ 0; for every . Remark 4.1. Note that strictly increasing implies nondecreasing, and a nondecreasing function is regulated. Let PC ([ a , b ]) = { u ∈ G ([ a , b ]) : u is continuous at every t ≠ t_k , }. From now on we always assume that u , , and we define The following result is an Opial-type inequality with Stieltjes derivatives. Theorem 4.2. Assume that u ( a ) = u ( b ) = 0. If both u and α are left-continuous on [ a , b ], then we have where K_α = inf _{h∈[a, b]} max{ α ( h ) − α ( a ), α ( b ) − α ( h )}. Proof . Let for t ∈[ a , b ], By Theorem 2.9, the functions, y and z are left-continuous on [ a , b ]. Also by Theorem 3.3, we have and we have by Theorem 3.4 and u ( a ) = u ( b ) = 0 for t ∈ [ a , b ]. So by Theorem 3.4, Lemma 3.5, and using Hölder’s inequality, we get and similarly we obtain So we have The proof is complete.   ☐ A slightly more general result is as follows. Theorem 4.3. Assume that u ( b ) = 0. If both u and α are left-continuous on [ a , b ], then we have Proof . From (4.2) we have So we get This gives (4.3). The proof is complete.   ☐ More generally we have the following result. Theorem 4.4. Let p ≥ 0, q ≥ 1, r ≥ 0, m ≥ 1 be real numbers and let f ∈ PC ([ a , b ]) be a positive function on [ a , b ] with inf _{s∈[a, b]} f ( s ) > 0. Assume that both functions u and α are left-continuous on [ a , b ]. If u ( b ) = 0, then we have where , for m ≠ 1, and for m = 1. Proof . Let for t ∈ [ a , b ], Then by Theorem 3.4, | u ( t )| ≤ z ( t ) and by Theorem 2.9, z is left-continuous, and non-increasing on [ a , b ]. If t ≠ t_k , , Then by Theorem 3.3, exists, and by Theorem 2.9, z is continuous at t . Using the Mean Value Theorem and by the definition of the Stieltjes derivatives, if z is not locally constant at t , then we have, If z is constant on some neighborhood of t , then since , the above equality is also true. If t = t_k , , since z is non-increasing on [ a , b ], and and by the Mean Value Theorem, and by the definition of the Stieltjes derivatives, we have, where z ( t_k +) ≤ ω ≤ z ( t_k ) = z( t_k −): Thus we have Let . Then by hypotheses, β is strictly increasing on [ a , b ]. Since we have by Theorem 3.4 and (4.5), since z ( b ) = 0 and , Using Hölder’s inequality with indices m , , we have Integrating (4.6) on [ a , b ] and using Hölder’s inequality with indices q , , and considering by Theorem 2.8, we get If , then is obviously true, otherwise, dividing both sides of (4.7) by and then taking the q th power on both sides of the resulting inequality we get also (4.8). Using the Hölder’s inequality with indice , , we have, by (4.8), Using Hölder’s inequality with indices , , we get by (4.9) If , then the inequality is obviously true, otherwise, dividing both sides of (4.10) by and then taking the power on both sides of the resulting inequality we get also (4.11). Since | u | ≤ z and we have This gives (4.4). The proof is complete.   ☐ In this section we always assume that both functions u and u' are left- continuous on [ a , b ], and that α = ϕ (see (3.2)). Consider the following impulsive differential equation: for , where q ₁ ∈ PC ([ a , b ]): Now we define Since by Lemma 3.8 for the equation (5.1) implies the following equation: where We need the following result. Lemma 5.1. If the function u satisfies the equation (5.1) and c ∈ [ a , b ], then we have Proof . In the proof, we frequently use Lemma 3.8, . , and , . This gives (5.4). And This gives (5.5). And This gives (5.6). Also This gives (5.7). The proof is complete.   ☐ Theorem 5.2. Assume that u satisfies the equation (5.1) and u' ( a ) = 0, u ( a ) ≠ 0. If we have where , then u ( t ) ≠ 0 for every t ∈ [ a , b ]. Proof . Assume that there is a number c ∈ ( a , b ] with u ( c ) = 0. Then multiplying both sides of (5.2) by u and integrating we have Using Theorem 3.3, Lemma 3.5 and Theorem 3.6, and u ( c ) = Q ( a ) = 0, we get, since, by Theorem 2.9 and Remark 3.7, Q is left-continuous on [ a , b ]; and Δ α ( t_k ) = Δ ⁺ α ( t_k ) = 1, q ( t_k ) = 0, , Since both u and u' are left-continuous By Lemma 3.8 and Lemma 5.1, we get, since u ( c ) = u' ( a ) = 0, By (5.9), (5.10) and (5.11), we have Hence by Theorem 4.3 and Lemma 5.1, we get If then, since , and u' _α ( t_k ) = u ( t_k +) − u ( t_k ) = 0. This implies that u is a constant on [ a , c ]. So u ( c ) = u ( a ) ≠0. But this is a contradiction to u ( c ) = 0. Hence we conclude that . In (5.12), canceling , we get a contradiction to (5.8). This completes the proof.   ☐ In the following result we apply Theorem 4.4. Theorem 5.3. Let q ∈ PC ([ a , b ]) and let α = ϕ (see (3.2)). If u ∈ PC ([ a , b ]) is left-continuous and a nontrivial solution of the following equation: then we have Proof . Substituting f ≡ 1, p = 0, q = 1, r = 0 into Theorem 4.4, then we have So we have Canceling , we get (5.13).   ☐