tailieunhanh - New inequalities of Opial type for conformable fractional integrals

In this paper, some Opial-type inequalities for conformable fractional integrals are obtained using the remainder function of Taylor’s theorem for conformable integrals. | Turk J Math (2017) 41: 1164 – 1173 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article New inequalities of Opial type for conformable fractional integrals Mehmet Zeki SARIKAYA, H¨ useyin BUDAK∗ Department of Mathematics, Faculty of Science and Arts, D¨ uzce University, D¨ uzce, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, some Opial-type inequalities for conformable fractional integrals are obtained using the remainder function of Taylor’s theorem for conformable integrals. Key words: Opial inequality, H¨ older’s inequality, conformable fractional integrals 1. Introduction In 1960, Opial established the following interesting integral inequality [10]: Theorem 1 Let x(t) ∈ C (1) [0, h] be such that x(0) = x(h) = 0, and x(t) > 0 in (0, h) . Then the following inequality holds: ∫h 0 h |x(t)x (t)| dt ≤ 4 ′ ∫h (x′ (t)) dt 2 () 0 The constant h/4 is the best possible. Opial’s inequality and its generalizations, extensions, and discretizations play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations. Over the last 20 years a large number of papers have appeared in the literature that deals with the simple proofs, various generalizations, and discrete analogues of Opial’s inequality and its generalizations; see [2,4,5,11–14,16,17]. The purpose of this paper is to establish some Opial-type inequalities for conformable integrals. The structure of this paper is as follows. In Section 2, we give the definitions of conformable derivatives and conformable integrals and introduce several useful notations for conformable integrals used in our main results. In Section 3, the main result is presented. Using the remainder function of Taylor’s theorem for conformable integrals, we establish .