Browsing by Author "Ozbekler, 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.; Özbekler, Abdullah; 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.Article Citation Count: 0Forced Oscillation of Delay Difference Equations Via Nonprincipal Solution(Wiley, 2018) Ozbekler, Abdullah; Özbekler, Abdullah; MathematicsIn 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 Count: 1Forced Oscillation of Sublinear Impulsive Differential Equations Via Nonprincipal Solution(Wiley, 2018) Mostepha, Naceri; Özbekler, Abdullah; 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: 0De La Vallee Poussin Inequality for Impulsive Differential Equations(Walter de Gruyter Gmbh, 2021) Akgol, Sibel Dogru; Akgöl, Sibel Doğru; Ozbekler, Abdullah; Özbekler, 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 SciencesArticle Citation Count: 1Lyapunov and Hartman-Type Inequalities for Higher-Order Discrete Fractional Boundary Value Problems(Univ Miskolc inst Math, 2023) Oguz, Arzu Denk; Özbekler, Abdullah; 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.Article Citation Count: 15Lyapunov Type Inequalities for Even Order Differential Equations With Mixed Nonlinearities(Springeropen, 2015) Agarwal, Ravi P.; Özbekler, Abdullah; 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.; Özbekler, Abdullah; 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.; Özbekler, Abdullah; 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.; Özbekler, Abdullah; 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 Differential Equations(Dynamic Publishers, inc, 2015) Agarwal, Ravi P.; Özbekler, Abdullah; 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: 2Lyapunov type inequalities for second-order forced dynamic equations with mixed nonlinearities on time scales(Springer-verlag Italia Srl, 2017) Özbekler, Abdullah; 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.; Özbekler, Abdullah; 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; Özbekler, Abdullah; 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.Article Citation Count: 4New Criteria on Oscillatory and Asymptotic Behavior of Third-Order Nonlinear Dynamic Equations With Nonlinear Neutral Terms(Mdpi, 2021) Grace, Said R.; Özbekler, Abdullah; Alzabut, Jehad; Ozbekler, Abdullah; MathematicsIn the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a new approach as far as the nonlinear neutral terms are concerned. The main results are illustrated by some particular examples.Article Citation Count: 0On the Oscillation of Discrete Volterra Equations With Positive and Negative Nonlinearities(Rocky Mt Math Consortium, 2018) Ozbekler, Abdullah; Özbekler, Abdullah; MathematicsIn this paper, we give new oscillation criteria for discrete Volterra equations having different nonlinearities such as super-linear and sub-linear cases. We also present some new sufficient conditions for oscillation under the effect of the oscillatory forcing term.Article Citation Count: 4On the Oscillation of Even-Order Nonlinear Differential Equations With Mixed Neutral Terms(Hindawi Ltd, 2021) Kaabar, Mohammed K. A.; Özbekler, Abdullah; Grace, Said R.; Alzabut, Jehad; Ozbekler, Abdullah; Siri, Zailan; Özbekler, Abdullah; Özbekler, Abdullah; Mathematics; MathematicsThe oscillation of even-order nonlinear differential equations (NLDiffEqs) with mixed nonlinear neutral terms (MNLNTs) is investigated in this work. New oscillation criteria are obtained which improve, extend, and simplify the existing ones in other previous works. Some examples are also given to illustrate the validity and potentiality of our results.Article Citation Count: 9On the Oscillation of Non-Linear Fractional Difference Equations With Damping(Mdpi, 2019) Alzabut, Jehad; Özbekler, Abdullah; Muthulakshmi, Velu; Ozbekler, Abdullah; Adigilzel, Hakan; MathematicsIn studying the Riccati transformation technique, some mathematical inequalities and comparison results, we establish new oscillation criteria for a non-linear fractional difference equation with damping term. Preliminary details including notations, definitions and essential lemmas on discrete fractional calculus are furnished before proceeding to the main results. The consistency of the proposed results is demonstrated by presenting some numerical examples. We end the paper with a concluding remark.Article Citation Count: 0On the Oscillation of Volterra Integral Equations With Positive and Negative Nonlinearities(Wiley-blackwell, 2016) Ozbekler, Abdullah; Özbekler, Abdullah; MathematicsIn the paper, we give new oscillation criteria for Volterra integral equations having different nonlinearities such as superlinearity and sublinearity. We also present some new sufficient conditions for oscillation under the effect of oscillatory forcing term. Copyright (C) 2015 JohnWiley & Sons, Ltd.Article Citation Count: 0Oscillation Criteria for Non-Canonical Second-Order Nonlinear Delay Difference Equations With a Superlinear Neutral Term(Texas State Univ, 2023) Vidhyaa, Kumar S.; Özbekler, Abdullah; Thandapani, Ethiraju; Alzabut, Jehad; Ozbekler, Abdullah; MathematicsWe obtain oscillation conditions for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term. To cope with non-canonical types of equations, we propose new oscillation criteria for the main equation when the neutral coefficient does not satisfy any of the conditions that call it to either converge to 0 or & INFIN;. Our approach differs from others in that we first turn into the non-canonical equation to a canonical form and as a result, we only require one condition to weed out non-oscillatory solutions in order to induce oscillation. The conclusions made here are new and have been condensed significantly from those found in the literature. For the sake of confirmation, we provide examples that cannot be included in earlier works.Article Citation Count: 1Oscillation Results for a Class of Nonlinear Fractional Order Difference Equations with Damping Term(Hindawi Ltd, 2020) Özbekler, Abdullah; Alzabut, Jehad; Jacintha, Mary; Ozbekler, Abdullah; MathematicsThe paper studies the oscillation of a class of nonlinear fractional order difference equations with damping term of the form Delta[psi(lambda)z(eta) (lambda)] + p(lambda)z(eta) (lambda) + q(lambda)F(Sigma(lambda-1+mu)(s=lambda 0) (lambda - s - 1)((-mu)) y(s)) = , where z(lambda) = a(lambda) + b(lambda)Delta(mu) y(lambda), Delta(mu) stands for the fractional difference operator in Riemann-Liouville settings and of order mu, 0 < mu <= 1, and eta >= 1 is a quotient of odd positive integers and lambda is an element of N lambda 0+1-mu. New oscillation results are established by the help of certain inequalities, features of fractional operators, and the generalized Riccati technique. We verify the theoretical outcomes by presenting two numerical examples.
