Browsing by Author "Mustafa, Octavian G."

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Citation - WoS: 12
Citation - Scopus: 12
A Kamenev-Type Oscillation Result For a Linear (1+Alpha)-Order Fractional Differential Equation
(Elsevier Science inc, 2015) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; O'Regan, Donal; 56389; Matematik
We investigate the eventual sign changing for the solutions of the linear equation (x((alpha)))' + q(t)x = t >= 0, when the functional coefficient q satisfies the Kamenev-type restriction lim sup 1/t epsilon integral(t)(to) (t - s)epsilon q(s)ds = +infinity for some epsilon > 2; t(0) > 0. The operator x((alpha)) is the Caputo differential operator and alpha is an element of (0, 1). (C) 2015 Elsevier Inc. All rights reserved.
A Kamenev-type oscillation result for a linear (1+alpha)-order fractional differential equation
(Elsevier Science Inc., 2015) Baleanu, Dumitru; Mustafa, Octavian G.; O'Regan, Donal; Matematik
We investigate the eventual sign changing for the solutions of the linear equation (x((alpha)))' + q(t)x = t >= 0, when the functional coefficient q satisfies the Kamenev-type restriction lim sup 1/t epsilon integral(t)(to) (t - s)epsilon q(s)ds = +infinity for some epsilon > 2; t(0) > 0. The operator x((alpha)) is the Caputo differential operator and alpha is an element of (0, 1)
Citation - WoS: 18
Citation - Scopus: 22
A Nagumo-like uniqueness theorem for fractional differential equations
(Iop Publishing Ltd, 2011) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; O'Regan, Donal; Matematik
We extend to fractional differential equations a recent generalization of the Nagumo uniqueness theorem for ordinary differential equations of first order.
Citation - WoS: 9
Citation - Scopus: 8
A uniqueness criterion for fractional differential equations with Caputo derivative
(Springer, 2013) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; O'Regan, Donal; Matematik
We investigate the uniqueness of solutions to an initial value problem associated with a nonlinear fractional differential equation of order alpha a(0,1). The differential operator is of Caputo type whereas the nonlinearity cannot be expressed as a Lipschitz function. Instead, the Riemann-Liouville derivative of this nonlinearity verifies a special inequality.
Citation - WoS: 102
Citation - Scopus: 104
An existence result for a superlinear fractional differential equation
(Pergamon-elsevier Science Ltd, 2010) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.; Matematik
We establish the existence and uniqueness of solution for the boundary value problem D-0(t)alpha(x') + a(t)x(lambda) = 0, t > 0, x' (0) = 0, lim(t ->+infinity) x(t) = 1, where D-0(t)alpha designates the Riemann-Liouville derivative of order alpha epsilon (0, 1) and lambda > 1. Our result might be useful for establishing a non-integer variant of the Atkinson classical theorem on the oscillation of Emden-Fowler equations. (C) 2010 Elsevier Ltd. All rights reserved.
Asymptotic Integration of (1 + Α) -Order Fractional Differential Equations
(2011) Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.; 56389; Matematik
We establish the long-time asymptotic formula of solutions to the (1+α)-order fractional differential equation 0iOt1+αx+a(t)x=0, t>0, under some simple restrictions on the functional coefficient a(t), where 0iOt1+α is one of the fractional differential operators 0Dtα(x′), (0Dtαx)′= 0Dt1+αx and 0Dtα(tx′-x). Here, 0Dtα designates the Riemann-Liouville derivative of order α∈(0,1). The asymptotic formula reads as [b+O(1)] ·xsmall+c·xlarge as t→+∞ for given b, c∈R, where xsmall and xlarge represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation 0iOt1+αx=0, t>0
Asymptotic integration of (1+alpha)-order fractional differential equations
(Pergamon-Elsevier Science Ltd, 2011) Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.; Matematik
We establish the long-time asymptotic formula of solutions to the (1 + alpha)-order fractional differential equation (i)(0)O(t)(1+alpha)x + a (t)x = 0, t > 0, under some simple restrictions on the functional coefficient a(t), where (i)(0)O(t)(1+alpha)x is one of the fractional differential operators D-0(t)alpha(x'), ((0)D(t)(alpha)x)' = D-0(t)1+alpha x and D-0(t)alpha(tx' - x). Here, D-0(t)alpha designates the Riemann-Liouville derivative of order a E (0, 1). The asymptotic formula reads as [b + O(1)] . x(small) + c . x(large) as t -> +infinity for given b, c E is an element of R, where x(small) and x(large) represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation (i)(0)O(t)(1+alpha)x = 0, t > 0.
Asymptotic Integration of (1+Alpha)-Order Fractional Differential Equations
(Pergamon-Elsevier Science LTD, 2011) Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.; 56389; Matematik
We establish the long-time asymptotic formula of solutions to the (1 + alpha)-order fractional differential equation (i)(0)O(t)(1+alpha)x + a (t)x = 0, t > 0, under some simple restrictions on the functional coefficient a(t), where (i)(0)O(t)(1+alpha)x is one of the fractional differential operators D-0(t)alpha(x'), ((0)D(t)(alpha)x)' = D-0(t)1+alpha x and D-0(t)alpha(tx' - x). Here, D-0(t)alpha designates the Riemann-Liouville derivative of order a E (0, 1). The asymptotic formula reads as [b + O(1)] . x(small) + c . x(large) as t -> +infinity for given b, c E is an element of R, where x(small) and x(large) represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation (i)(0)O(t)(1+alpha)x = 0, t > 0. (C) 2011 Elsevier Ltd. All rights reserved.
Citation - WoS: 28
Citation - Scopus: 30
Asymptotic integration of (1+alpha)-order fractional differential equations
(Pergamon-elsevier Science Ltd, 2011) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.; 56389; Matematik
We establish the long-time asymptotic formula of solutions to the (1 + alpha)-order fractional differential equation (i)(0)O(t)(1+alpha)x + a (t)x = 0, t > 0, under some simple restrictions on the functional coefficient a(t), where (i)(0)O(t)(1+alpha)x is one of the fractional differential operators D-0(t)alpha(x'), ((0)D(t)(alpha)x)' = D-0(t)1+alpha x and D-0(t)alpha(tx' - x). Here, D-0(t)alpha designates the Riemann-Liouville derivative of order a E (0, 1). The asymptotic formula reads as [b + O(1)] . x(small) + c . x(large) as t -> +infinity for given b, c E is an element of R, where x(small) and x(large) represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation (i)(0)O(t)(1+alpha)x = 0, t > 0. (C) 2011 Elsevier Ltd. All rights reserved.
Citation - WoS: 24
Citation - Scopus: 32
Asymptotic integration of some nonlinear differential equations with fractional time derivative
(Iop Publishing Ltd, 2011) Baleanu, Dumitru; Baleanu, Dumitru; Agarwal, Ravi P.; Mustafa, Octavian G.; Cosulschi, Mirel; Matematik
We establish that, under some simple integral conditions regarding the nonlinearity, the (1 + alpha)-order fractional differential equation D-0(t)alpha (x') + f (t, x) = 0, t > 0, has a solution x is an element of C([0, +infinity), R) boolean AND C-1((0, +infinity), R), with lim(t SE arrow 0) [t(1-alpha)x'(t)] is an element of R, which can be expanded asymptotically as a+bt(alpha)+O(t(alpha-1)) when t ->+infinity for given real numbers a, b. Our arguments are based on fixed point theory. Here, D-0(t)alpha designates the Riemann-Liouville derivative of order alpha is an element of (0, 1).
Citation - WoS: 15
Citation - Scopus: 28
Asymptotically linear solutions for some linear fractional differential equations
(Hindawi Publishing Corporation, 2010) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.; Matematik
We establish here that under some simple restrictions on the functional coefficient a(t) the fractional differential equation 0D(t)(alpha)[tx' - x + x(0)] + a(t)x = 0, t > 0, has a solution expressible as ct + d + o(1) for t -> +infinity, where D-0(t)alpha designates the Riemann-Liouville derivative of order alpha is an element of (0, 1) and c, d is an element of R.
Citation - WoS: 5
On A Fractional Differential Equation With İnfinitely Many Solutions
(Springer international Publishing Ag, 2012) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; O'Regan, Donal; 56389; Matematik
We present a set of restrictions on the fractional differential equation , , where and , that leads to the existence of an infinity of solutions (a continuum of solutions) starting from . The operator is the Caputo differential operator.
Citation - WoS: 3
Citation - Scopus: 3
On an inverse scattering algorithm for the Camassa-Holm equation
(Taylor & Francis Ltd, 2008) Mustafa, Octavian G.; O'Regan, Donal
We present a clarification of a recent inverse scattering algorithm developed for the Camassa-Holm equation.
Citation - WoS: 66
Citation - Scopus: 72
On L-p-solutions for a class of sequential fractional differential equations
(Elsevier Science inc, 2011) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.; Matematik
Under some simple conditions on the coefficient a( t), we establish that the initial value problem ((0)D(t)(alpha)x)' + a(t)x = 0; t > 0; lim(t SE arrow 0)[t(1-alpha)x(t)] = 0 has no solution in L-p((1, +infinity), R), where p-1/p > alpha > 1/p and D-0(t)alpha designates the Riemann-Liouville derivative of order alpha Our result might be useful for developing a non-integer variant of H. Weyl's limit-circle/limit-point classification of differential equations. (C) 2011 Elsevier Inc. All rights reserved.
Citation - WoS: 2
On oscillatory solutions of certain forced Emden–Fowler like equations
(Academic Press inc Elsevier Science, 2008) Mustafa, Octavian G.
We give a constructive proof of existence to oscillatory solutions for the differential equations x ''(t) + a(t)vertical bar x(t)vertical bar lambda sign[x(t)] = e(t), where t >= t(0) >= 1 and lambda > 1, that decay to 0 when t -> infinity as 0(t(-mu)) for mu > 0 as close as desired to the "critical quantity" mu* = 2/lambda-1 For this class of equations, we have lim(t ->+infinity) E(t) = 0, where E(t) 0 and E ''(t) e(t) throughout [t(0) + infinity). We also establish that for any mu > mu* and any negative-valued E(t) = 0(t(-mu)) as t ->+infinity the differential equation has a negative-valued solution decaying to 0 at +infinity as o(t(-mu)). In this way, we are not in the reach of any of the developments from the recent paper [C.H Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732]. (C) 2008 Elsevier Inc. All rights reserved.
Citation - WoS: 16
Citation - Scopus: 18
On the asymptotic integration of a class of sublinear fractional differential equations
(Aip Publishing, 2009) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; Matematik
We estimate the growth in time of the solutions to a class of nonlinear fractional differential equations D-0+(alpha)(x-x(0))=f(t,x) which includes D-0+(alpha)(x-x(0))=H(t)x(lambda) with lambda is an element of(0,1) for the case of slowly decaying coefficients H. The proof is based on the triple interpolation inequality on the real line and the growth estimate reads as x(t)=o(t(a alpha)) when t ->+infinity for 1>alpha>1-a>lambda>0. Our result can be thought of as a noninteger counterpart of the classical Bihari asymptotic integration result for nonlinear ordinary differential equations. By a carefully designed example we show that in some circumstances such an estimate is optimal.
Citation - WoS: 9
Citation - Scopus: 8
On the existence interval for the initial value problem of a fractional differential equation
(Hacettepe Univ, Fac Sci, 2011) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; Matematik
We compute via a comparison function technique, a new bound for the existence interval of the initial value problem for a fractional differential equation given by means of Caputo derivatives. We improve in this way the estimate of the existence interval obtained very recently in the literature.
Citation - WoS: 151
Citation - Scopus: 181
On the global existence of solutions to a class of fractional differential equations
(Pergamon-elsevier Science Ltd, 2010) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; Matematik
We present two global existence results for an initial value problem associated to a large class of fractional differential equations. Our approach differs substantially from the techniques employed in the recent literature. By introducing an easily verifiable hypothesis, we allow for immediate applications of a general comparison type result from [V. Lakshmikantham, AS. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA 69 (2008), 2677-2682]. (C) 2009 Elsevier Ltd. All rights reserved.
Citation - WoS: 73
Citation - Scopus: 85
On the solution set for a class of sequential fractional differential equations
(Iop Publishing Ltd, 2010) Baleanu, Dumitru; Baleanu, Dumitru; Mustafa, Octavian G.; Agarwal, Ravi P.; Matematik
We establish here that under some simple restrictions on the functional coefficient a(t) the solution set of the fractional differential equation ((0)D(t)(alpha)x)' + a(t) x = 0 splits between eventually small and eventually large solutions as t -> +infinity, where D-0(t)alpha designates the Riemann-Liouville derivative of the order alpha is an element of (0, 1).
Citation - WoS: 0
Citation - Scopus: 1
Positive solutions of some elliptic differential equations with oscillating nonlinearity
(Taylor & Francis Ltd, 2012) Jarad, Fahd; Jarad, Fahd; Mustafa, Octavian G.; O'Regan, Donal; Matematik
We discuss the occurrence of positive solutions which decay to 0 as vertical bar xj vertical bar ->+infinity the differential equation Delta u+f(x, u)+g(vertical bar x vertical bar)x . del u=0, vertical bar xj vertical bar>R>0, x is an element of R-n, where n >= 3, g is nonnegative valued and f has alternating sign, by means of the comparison method. Our results complement several recent contributions from Ehrnstrom and Mustafa [ M. Ehrnstrom, O.G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. TMA 67 (2007), pp. 1147-1154].