Browsing by Author "Özbekler, Abdullah"
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Article Citation Count: 15Disconjugacy via Lyapunov and Vallee-Poussin type inequalities for forced differential equations(Elsevier Science inc, 2015) Agarwal, Ravi P.; Ozbekler, Abdullah; MathematicsIn the case of oscillatory potentials, we present some new Lyapunov and Vallee-Poussin type inequalities for second order forced differential equations. No sign restriction is imposed on the forcing term. The obtained inequalities generalize and compliment the existing results in the literature. (C) 2015 Elsevier Inc. All rights reserved.Conference Object Citation Count: 3Forced Oscillation of Second-Order Impulsive Differential Equations with Mixed Nonlinearities(Springer, 2013) Ozbekler, A.; Zafer, A.; MathematicsIn 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.Article Citation Count: 10Forced oscillation of second-order nonlinear differential equations with positive and negative coefficients(Pergamon-elsevier Science Ltd, 2011) Ozbekler, A.; Wong, J. S. W.; Zafer, A.; MathematicsIn this paper we give new oscillation criteria for forced super- and sub-linear differential equations by means of nonprincipal solutions. (c) 2011 Elsevier Ltd. All rights reserved.Article Citation Count: 1Forced oscillation of sublinear impulsive differential equations via nonprincipal solution(Wiley, 2018) Mostepha, Naceri; Ozbekler, Abdullah; MathematicsIn this paper, we give new oscillation criteria for forced sublinear impulsive differential equations of the form (r(t)x')' + q(t)vertical bar x vertical bar(gamma-1) x = f(t), t not equal theta(i); Delta r(t)x' + q(i)vertical bar x vertical bar(gamma-1) x = f(i), t = theta(i), where gamma is an element of(0, 1), under the assumption that associated homogenous linear equation (r(t)z')' + q(t)z = 0, t not equal theta(i); Delta r(t)z' + q(i)z = 0, t = theta(i). is nonoscillatory.Article Citation Count: 16Forced oscillation of super-half-linear impulsive differential equations(Pergamon-elsevier Science Ltd, 2007) Oezbekler, A.; Zafer, A.; MathematicsBy using a Picone type formula in comparison with oscillatory unforced half-linear equations, we derive new oscillation criteria for second order forced super-half-linear impulsive differential equations having fixed moments of impulse actions. In the superlinear case, the effect of a damping term is also considered. (c) 2007 Elsevier Ltd. All rights reserved.Article Citation Count: 19Interval criteria for the forced oscillation of super-half-linear differential equations under impulse effects(Pergamon-elsevier Science Ltd, 2009) Ozbekler, A.; Zafer, A.; MathematicsIn this paper, we derive new interval oscillation criteria for a forced super-half-linear impulsive differential equation having fixed moments of impulse actions. The results are extended to a more general class of nonlinear impulsive differential equations. Examples are also given to illustrate the relevance of the results. (C) 2009 Elsevier Ltd. All rights reserved.Article Citation Count: 0DE LA VALLEE POUSSIN INEQUALITY FOR IMPULSIVE DIFFERENTIAL EQUATIONS(Walter de Gruyter Gmbh, 2021) Akgol, Sibel Dogru; Ozbekler, Abdullah; MathematicsThe de la Vallee Poussin inequality is a handy tool for the investigation of disconjugacy, and hence, for the oscillation/nonoscillation of differential equations. The results in this paper are extensions of former those of Hartman and Wintner [Quart. Appl. Math. 13 (1955), 330-332] to the impulsive differential equations. Although the inequality first appeared in such an early date for ordinary differential equations, its improved version for differential equations under impulse effect never has been occurred in the literature. In the present study, first, we state and prove a de la Vallee Poussin inequality for impulsive differential equations, then we give some corollaries on disconjugacy. We also mention some open problems and finally, present some examples that support our findings. (C) 2021 Mathematical Institute Slovak Academy of SciencesBook Part Citation Count: 0De La Vallée Poussin-type inequality for impulsive dynamic equations on time scales(De Gruyter, 2023) Akgöl,S.D.; Özbekler,A.; MathematicsWe derive a de La Vallée Poussin-type inequality for impulsive dynamic equations on time scales. This inequality is often used in conjunction with disconjugacy and/or (non)oscillation. Hence, it appears to be a very useful tool for the qualitative study of dynamic equations. In this work, generalizing the classical de La Vallée Poussin inequality for impulsive dynamic equations on arbitrary time scales, we obtain a dis-conjugacy criterion and some results on nonoscillation. We also present illustrative examples that support our findings. © 2023 Walter de Gruyter GmbH, Berlin/Bostonl. All rights reserved.Article Citation Count: 1LYAPUNOV AND HARTMAN-TYPE INEQUALITIES FOR HIGHER-ORDER DISCRETE FRACTIONAL BOUNDARY VALUE PROBLEMS(Univ Miskolc inst Math, 2023) Oguz, Arzu Denk; Alzabut, Jehad; Ozbekler, Abdullah; Jonnalagadda, Jagan Mohan; MathematicsBy employing Green's function, we obtain new Lyapunov and Hartman-type inequalities for higher-order discrete fractional boundary value problems. Reported results essentially generalize some theorems existing in the literature. As an application, we discuss the corresponding eigenvalue problems.Book Citation Count: 19Lyapunov Inequalities and Applications(Springer International Publishing, 2021) Agarwal,R.P.; Bohner,M.; Özbekler,A.; MathematicsThis book provides an extensive survey on Lyapunov-type inequalities. It summarizes and puts order into a vast literature available on the subject, and sketches recent developments in this topic. In an elegant and didactic way, this work presents the concepts underlying Lyapunov-type inequalities, covering how they developed and what kind of problems they address. This survey starts by introducing basic applications of Lyapunov's inequalities. It then advances towards even-order, odd-order, and higher-order boundary value problems; Lyapunov and Hartman-type inequalities; systems of linear, nonlinear, and quasi-linear differential equations; recent developments in Lyapunov-type inequalities; partial differential equations; linear difference equations; and Lyapunov-type inequalities for linear, half-linear, and nonlinear dynamic equations on time scales, as well as linear Hamiltonian dynamic systems. Senior undergraduate students and graduate students of mathematics, engineering, and science will benefit most from this book, as well as researchers in the areas of ordinary differential equations, partial differential equations, difference equations, and dynamic equations. Some background in calculus, ordinary and partial differential equations, and difference equations is recommended for full enjoyment of the content. © Springer Nature Switzerland AG 2021. All rights reserved.Article Citation Count: 15Lyapunov type inequalities for even order differential equations with mixed nonlinearities(Springeropen, 2015) Agarwal, Ravi P.; Ozbekler, Abdullah; MathematicsIn 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 Count: 17Lyapunov type inequalities for mixed nonlinear Riemann-Liouville fractional differential equations with a forcing term(Elsevier, 2017) Agarwal, Ravi P.; Ozbekler, Abdullah; MathematicsIn 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 Citation Count: 8LYAPUNOV TYPE INEQUALITIES FOR nTH ORDER FORCED DIFFERENTIAL EQUATIONS WITH MIXED NONLINEARITIES(Amer inst Mathematical Sciences-aims, 2016) Agarwal, Ravi P.; Ozbekler, Abdullah; MathematicsIn the case of oscillatory potentials, we present Lyapunov type inequalities for nth order forced differential equations of the form x((n))(t) + Sigma(m)(j=1) qj (t)vertical bar x(t)vertical bar(alpha j-1)x(t)= f(t) satisfying the boundary conditions x(a(i)) = x(1)(a(i)) = x(11)(ai) = center dot center dot center dot = x((ki))(ai) = 0; i = 1, 2,..., r, where a(1) < a(2) < ... < a(r), 0 <= k(i) and Sigma(r)(j=1) k(j) + r = n: r >= 2. No sign restriction is imposed on the forcing term and the nonlinearities satisfy 0 < alpha(l) < ... < alpha a(j) < 1 < alpha a(j+1) < ... < alpha(m) < 2. The obtained inequalities generalize and compliment the existing results in the literature.Review Citation Count: 3Lyapunov type inequalities for second order forced mixed nonlinear impulsive differential equations(Elsevier Science inc, 2016) Agarwal, Ravi P.; Ozbekler, Abdullah; MathematicsIn 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 Count: 6LYAPUNOV TYPE INEQUALITIES FOR SECOND ORDER SUB AND SUPER-HALF-LINEAR DIFFERENTIAL EQUATIONS(Dynamic Publishers, inc, 2015) Agarwal, Ravi P.; Ozbekler, Abdullah; MathematicsIn the case of oscillatory potential, we present a Lyapunov type inequality for second order differential equations of the form (r(t)Phi(beta)(x'(t)))' + q(t)Phi(gamma)(x(t)) = 0, in the sub-half-linear (0 < gamma < beta) and the super-half-linear (0 < beta < gamma < 2 beta) cases where Phi(*)(s) = vertical bar s vertical bar*(-1)s.Article Citation Count: 5Lyapunov type inequalities for second order sub and super-half-linear differential equations(Dynamic Publishers, 2015) Agarwal,R.P.; Özbekler,A.; MathematicsIn the case of oscillatory potential, we present a Lyapunov type inequality for second order differential equations of the form (Equation Presented) in the sub-half-linear (0 < γ < β) and the super-half-linear (0 < β < γ < 2β) cases where Φ∗ (s) =|s|∗-1s.Article Citation Count: 1Lyapunov type inequalities for second-order differential equations with mixed nonlinearities(Walter de Gruyter GmbH, 2016) Agarwal,R.P.; Özbekler,A.; MathematicsIn this paper,we present some new Lyapunov and Hartman type inequalities for second-order equations with mixed nonlinearities: x''(t) + p(t)|x(t)|β?1x(t) + q(t)|x(t)|y?1x(t) = 0, where p(t), q(t) are real-valued functions and 0 < γ < 1 < β < 2. No sign restrictions are imposed on the potential functions p(t) and q(t). The inequalities obtained generalize the existing results for the special cases of this equation in the literature. © 2016 by De Gruyter.Article Citation Count: 2Lyapunov 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, Abdullah; MathematicsIn 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 Count: 1Lyapunov-type inequalities for Lidstone boundary value problems on time scales(Springer-verlag Italia Srl, 2020) Agarwal, Ravi P.; Oguz, Arzu Denk; Ozbekler, Abdullah; MathematicsIn 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 Count: 33Lyapunov-type inequalities for mixed non-linear forced differential equations within conformable derivatives(Springer, 2018) Abdeljawad, Thabet; Agarwal, Ravi P.; Alzabut, Jehad; Jarad, Fahd; Ozbekler, Abdullah; MathematicsWe 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.
