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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.Review Citation - WoS: 3Citation - Scopus: 3Lyapunov Type Inequalities for Second Order Forced Mixed Nonlinear Impulsive Differential Equations(Elsevier Science inc, 2016) Agarwal, Ravi P.; Ozbekler, AbdullahIn this paper, we present some new Lyapunov and Hartman type inequalities for second order forced impulsive differential equations with mixed nonlinearities: x ''(t) + p(t)vertical bar x(t)vertical bar(beta-1)x(t) + q(t)vertical bar x(t)vertical bar(gamma-1)x(t) = f(t), t not equal theta(i); Delta x'(t) + p(i)vertical bar x(t)vertical bar(beta-1)x(t) + q(i)vertical bar x(t)vertical bar(gamma-1) x(t) = f(i), t = theta(i), where p, q, f are real-valued functions, {p(i)}, {q(i)}, {f(i)} are real sequences and 0 < gamma < 1 < beta < 2. No sign restrictions are imposed on the potential functions p, q and the forcing term f and the sequences {p(i)}, {q(i)}, {f(i)}. The inequalities obtained generalize and complement the existing results for the special cases of this equation in the literature. (C) 2016 Elsevier Inc. All rights reserved.Article Citation - WoS: 27Citation - Scopus: 29Oscillation of Solutions of Second Order Mixed Nonlinear Differential Equations Under Impulsive Perturbations(Pergamon-elsevier Science Ltd, 2011) Ozbekler, A.; Zafer, A.New oscillation criteria are obtained for second order forced mixed nonlinear impulsive differential equations of the form (r(t)Phi(alpha)(x'))' + q(t)(Phi)(x) + Sigma(n)(k=1)q(k)(t)Phi beta(k)(x ) = e(t), t not equal theta(I) x(theta(+)(i)) = ajx(theta(+)(i)) = b(i)x'(theta(i)) where Phi(gamma):= ,s vertical bar(gamma-1)s and beta(1) > beta(2) > ... > beta(m) > alpha > beta(m+1)> ... > beta(n) > beta(n) > 0. If alpha = 1 and the impulses are dropped, then the results obtained by Sun and Wong [Y.G. Sun, J.S.W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007) 549-560] are recovered. Examples are given to illustrate the results. (C) 2011 Elsevier Ltd. All rights reserved.Article Citation - WoS: 5Citation - Scopus: 5Sturmian theory for second order differential equations with mixed nonlinearities(Elsevier Science inc, 2015) Ozbekler, A.In the paper, Sturmian comparison theory is developed for the pair of second order differential equations; first of which is the nonlinear differential equations (m(t)y')' + s(t)y' + Sigma(n)(i=1)q(i)(t)vertical bar y vertical bar(proportional to j-1)y = 0, with mixed nonlinearities alpha(1) > ... > alpha(m) > 1 > alpha(m+1) > ... > alpha(n), and the second is the non-selfadjoint differential equations (k(t)x')' + r(t)x' + p(t)x = 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. (C) 2015 Elsevier Inc. All rights reserved.Conference Object Citation - WoS: 3Citation - Scopus: 3Forced Oscillation of Second-Order Impulsive Differential Equations With Mixed Nonlinearities(Springer, 2013) Ozbekler, A.; Zafer, A.In this paper we give new oscillation criteria for a class of second-order mixed nonlinear impulsive differential equations having fixed moments of impulse actions. The method is based on the existence of a nonprincipal solution of a related second-order linear homogeneous equation.
