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Article Citation - WoS: 2Citation - Scopus: 3Lyapunov-Type Inequalities for Lidstone Boundary Value Problems on Time Scales(Springer-verlag Italia Srl, 2020) Agarwal, Ravi P.; Oguz, Arzu Denk; Ozbekler, AbdullahIn this paper, we establish new Hartman and Lyapunov-type inequalities for even-order dynamic equations x.2n (t) + (-1)n-1q(t) xs (t) = 0 on time scales T satisfying the Lidstone boundary conditions x.2i (t1) = x.2i (t2) = 0; t1, t2. [t0,8) T for i = 0, 1,..., n - 1. The inequalities obtained generalize and complement the existing results in the literature.Article Citation - WoS: 35Citation - Scopus: 43Lyapunov-Type Inequalities for Mixed Non-Linear Forced Differential Equations Within Conformable Derivatives(Springer, 2018) Abdeljawad, Thabet; Agarwal, Ravi P.; Alzabut, Jehad; Jarad, Fahd; Ozbekler, AbdullahWe state and prove new generalized Lyapunov-type and Hartman-type inequalities fora conformable boundary value problem of order alpha is an element of (1,2] with mixed non-linearities of the form ((T alpha X)-X-a)(t) + r(1)(t)vertical bar X(t)vertical bar(eta-1) X(t) + r(2)(t)vertical bar x(t)vertical bar(delta-1) X(t) = g(t), t is an element of (a, b), satisfying the Dirichlet boundary conditions x(a) = x(b) = 0, where r(1), r(2), and g are real-valued integrable functions, and the non-linearities satisfy the conditions 0 < eta < 1 < delta < 2. Moreover, Lyapunov-type and Hartman-type inequalities are obtained when the conformable derivative T-alpha(a) is replaced by a sequential conformable derivative T-alpha(a) circle T-alpha(a), alpha is an element of (1/2,1]. The potential functions r(1), r(2) as well as the forcing term g require no sign restrictions. The obtained inequalities generalize some existing results in the literature.Article Citation - WoS: 1Citation - Scopus: 4Second Order Oscillation of Mixed Nonlinear Dynamic Equations With Several Positive and Negative Coefficients(Amer inst Mathematical Sciences-aims, 2011) Ozbekler, Abdullah; Zafer, Agacik; MathematicsNew oscillation criteria are obtained for superlinear and sublinear forced dynamic equations having positive and negative coefficients by means of nonprincipal solutions.Article Citation - WoS: 16Citation - Scopus: 16Lyapunov Type Inequalities for Even Order Differential Equations With Mixed Nonlinearities(Springeropen, 2015) Agarwal, Ravi P.; Ozbekler, AbdullahIn the case of oscillatory potentials, we present Lyapunov and Hartman type inequalities for even order differential equations with mixed nonlinearities: x((2n))(t) + (-1)(n-1) Sigma(m)(i=1) q(i)(t)vertical bar x(t)vertical bar(alpha i-1) x(t) = 0, where n,m epsilon N and the nonlinearities satisfy 0 < alpha(1) < center dot center dot center dot < alpha(j) < 1 < alpha(j+1) < center dot center dot center dot < alpha(m) < 2.Article Citation - WoS: 2Citation - Scopus: 4A Sturm Comparison Criterion for Impulsive Hyperbolic Equations(Springer-verlag Italia Srl, 2020) Ozbekler, Abdullah; Isler, Kubra UsluIn this paper, we investigate the Sturmian comparison theory for hyperbolic equations with fixed moments of effects. The results obtained extend the results of those existing in the literature for Sturmian comparison theory on ordinary and impulsive differential equations to impulsive hyperbolic equations.Article Citation - WoS: 1Citation - Scopus: 1Sturmian Comparison Theory for Half-Linear and Nonlinear Differential Equations Via Picone Identity(Wiley, 2017) Ozbekler, AbdullahIn this paper, Sturmian comparison theory is developed for the pair of second-order differential equations; first of which is the nonlinear differential equations of the form (m(t) Phi(beta)(y'))' + Sigma(n)(i=1) q(i)(t) Phi(alpha i)(y) = 0 and the second is the half-linear differential equations (k(t)Phi(beta)(x'))' + p(t)Phi(beta)(x) = 0 where Phi(alpha)(s) = vertical bar s vertical bar(alpha-1)s and alpha(1) > ... > alpha(m) > beta > alpha(m+1) > ... > alpha(n) > 0. Under the assumption that the solution of has two consecutive zeros, we obtain Sturm-Picone type and Leighton type comparison theorems for by employing the new nonlinear version of Picone formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for (1). Examples are given to illustrate the relevance of the results. Copyright (c) 2016 John Wiley & Sons, Ltd.Article Forced Oscillation of Delay Difference Equations Via Nonprincipal Solution(Wiley, 2018) Ozbekler, AbdullahIn this paper, we obtain a new oscillation result for delay difference equations of the form Delta(r(n)Delta x(n)) + a(n)x(tau n) = b(n); n is an element of N under the assumption that corresponding homogenous equation Delta(r(n)Delta z(n)) + a(n)z(n+1) = 0; n is an element of N is nonoscillatory, where tau(n) <= n + 1. It is observed that the oscillation behaviormay be altered due to presence of the delay. Extensions to forced Emden-Fowler-type delay difference equations Delta(r(n)Delta x(n)) + a(n)vertical bar x(tau n)vertical bar(alpha-1)x(tau n) = b(n); n is an element of N in the sublinear (0 < alpha < 1) and the superlinear (1 < alpha) cases are also discussed.Article Citation - WoS: 3Citation - Scopus: 3Lyapunov type inequalities for second-order forced dynamic equations with mixed nonlinearities on time scales(Springer-verlag Italia Srl, 2017) Agarwal, Ravi P.; Cetin, Erbil; Ozbekler, AbdullahIn this paper, we present some newHartman and Lyapunov inequalities for second-order forced dynamic equations on time scales T with mixed nonlinearities: x(Delta Delta)(t) + Sigma(n)(k=1) qk (t)vertical bar x(sigma) (t)vertical bar (alpha k-1) x(sigma) (t) = f (t); t is an element of [t(0), infinity)(T), where the nonlinearities satisfy 0 < alpha(1) < ... < alpha(m) < 1 < alpha(m+1) < ... < alpha(n) < 2. No sign restrictions are imposed on the potentials qk, k = 1, 2, ... , n, and the forcing term f. The inequalities obtained generalize and compliment the existing results for the special cases of this equation in the literature.Article Citation - WoS: 19Citation - Scopus: 21Lyapunov Type Inequalities for Mixed Nonlinear Riemann-Liouville Fractional Differential Equations With a Forcing Term(Elsevier, 2017) Agarwal, Ravi P.; Ozbekler, AbdullahIn this paper, we present some new Lyapunov and Hartman type inequalities for Riemann-Liouville fractional differential equations of the form ((a)D(alpha)x)(t) + p(t) vertical bar x(t) vertical bar(mu-1) x(t) + q(t) vertical bar x(t) vertical bar(gamma-1) x(t) = f(t), where p, q, f are real-valued functions and 0 < gamma < 1 < mu < 2. No sign restrictions are imposed on the potential functions p, q and the forcing term f. The inequalities obtained generalize and compliment the existing results for the special cases of this equation in the literature. (C) 2016 Elsevier B.V. All rights reserved.Article Sturmian Comparison Theory for Half-Linear and Nonlinear Differential Equations Via Picone Identity(Univ Nis, Fac Sci Math, 2017) Ozbekler, AbdullahIn this paper, Sturmian comparison theory is developed for the pair of second order differential equations; first of which is the nonlinear differential equations of the form (m(t)Phi(beta) (y'))' + Sigma (n) (i = 1) qi(t)Phi(alpha i) (y) = 0 and the second is the half-linear differential equations (k(t)Phi(beta) (x'))' + p(t) Phi(beta) (x) = 0 where Phi(*)(s) = \ s \(*-1)s and alpha(1) > . . .> alpha(m) > beta > alpha(m + 1) > . . . > alpha(n) > 0. Under the assumption that the solution of Eq. (2) has two consecutive zeros, we obtain Sturm-Picone type and Leighton type comparison theorems for Eq. (1) by employing the new nonlinear version of Picone's formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for Eq. (1). Examples are given to illustrate the relevance of the results.
