Oscillatory Solutions of Nonlinear Fractional Difference Equations
|
|
- Rodney Evans
- 5 years ago
- Views:
Transcription
1 International Journal of Difference Equations ISSN , Volume 3, Number, pp Oscillaty Solutions of Nonlinear Fractional Difference Equations G. E. Chatzarakis School of Pedagogical and Technological Education ASPETE Department of Electrical and Electronic Engineering Educats N. Heraklio, Athens, 42, Greece. geaxatz@otenet.gr and gea.xatz@aspete.gr P. Gokulraj, T. Kalaimani and V. Sadhasivam Thiruvalluvar Government Arts College PG and Research Department of Mathematics Rasipuram, Namakkal , Tamil Nadu, India. gokulxlr8@gmail.com, kalaimaths4@gmail.com, ovsadha@gmail.com Abstract In this paper, we study the oscillaty behavi of the fractional difference equation of the fm α xt γ + qtfxt = 0, t N t0 + α, where α denotes the Riemann left fractional difference operat of der α, 0 < α and γ > 0 is a quotient of odd positive integers. We establish some oscillaty criteria f the above equation, using the Riccati transfmation and Hardy type inequalities. Examples are provided to illustrate our theetical results. AMS Subject Classifications: 26A33, 39A2. Keywds: Oscillation, difference equations, fractional sum. Introduction Fractional difference equations have received considerable attention during the recent years. Fractional calculus finds significant application in the fields of viscoelasticity, Received July 4, 207; Accepted December 9, 207 Communicated by Agnieszka Malinowska
2 20 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam capacit they, electrical circuits, electro-analytical chemistry, tum growth models, neurology, control they, statistics and a review on this direction, see [6, 8, 20, 22, 23, 25, 26]. Significant progress has been made in the study of fractional differential equations, see [6, 7, 0, 2, 4, 27, 3]. In contrast, very little progress has been made in they of fractional difference equations, see [ 5, 8, 9,, 5, 7, 2]. In particular, we observe that the oscillation of fractional difference equations has been studied by many auths in recent researches [3,9,24,28,29]. This is one of reasons to study difference equations with fractional der. Strong interest in the fractional difference equation. is motivated by the fact that it represents a discrete analogue of the following fractional differential equation D α a xt + qtfxt = 0 f 0 α <, t [a, + ], a > 0, where D α a denotes the Riemann Liouville differential operat of der α and the above problem was investigated by Wang et al [30]. The objective is to study the oscillaty behavi of the solutions of fractional difference equations of the fm α xt γ + qtfxt = 0 f 0 < α, t N t0 + α.. Here α denotes the Riemann left fractional difference operat and γ > 0 is a quotient of odd positive integers. In the paper, we assume the conditions H qt is a positive sequence and f : R R is a continuous function such that fx k f a positive constant k, n is a natural number f all x 0 and [ c qt H2 x n ] γ N f t t0 where c < 0 and N > 0. xt α xt + M, xt α xt M, t t 0 f all α xt + 0, α xt 0 and f some positive constants M, M xt 2 xtxt + J, 2 xt T f some positive constants J and T. A solution xt of. is said to be oscillaty if it has no last zero, i.e., if xt = 0, then there exists a t 2 > t such that xt 2 = 0. Equation. itself is said to be oscillaty if every solution of. is oscillaty. A solution xt which is not oscillaty is called nonoscillaty. 2 Preliminaries In this section, we present some preliminary results from discrete fractional calculus. We will make use of these results, throughout the paper.
3 Oscillaty Solutions of Nonlinear Fractional Difference Equations 2 Definition 2. See [23]. Let ν > 0. The νth fractional sum f is defined by ν ft = Γν t ν t s ν fs, s=a where f is defined f s a mod, ν ft is defined f t a + ν mod and t ν Γt + = Γt ν +. The fractional sum ν f maps functions defined in N a to functions defined in N a+ν. Definition 2.2 See [23]. Let µ > 0 and m < µ < m, where m denotes a positive integer, m = µ. Set ν = m µ. The µth der Riemann left fractional difference is defined as µ ft = m ν ft = m ν ft, where ν ft is νth fractional sum. Lemma 2.3 See [29]. If Gt = t s α xs, t +α then Gt = Γ α α xt. Lemma 2.4 See [7]. If X and Y are nonnegative, then mxy m X m m Y m f m >. 3 Main Results Theem 3.. Suppose that H and H2 hold and q γ s =. Furtherme, assume that there exists a positive sequence rt such that krsqs r +s 2 =, 3. 4rs + where r + s = max{ rs, 0}. Then every solution of. is oscillaty.
4 22 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam Proof. Suppose to the contrary that xt is a nonoscillaty solution of.. Without loss of generality, we can assume that xt is an eventually positive solution of.. Then there exists t > t 0 such that xt > 0, Gt > 0 and fxt > 0 f t t, where Gt is defined as in Lemma 2.3. From. we have α xt γ = qtfxt < 0 f t t. Thus α xt γ is an eventually non increasing sequence. Next we show that α xt γ is eventually positive. Suppose there exists an integer t > t 0 such that α xt γ = c < 0 f t t, so that α xt γ α xt γ = c < 0, α xt γ c, α xt c γ. Applying Lemma 2.3, we get that Gt Γ α c qt γ γ,, qt γ Thus i.e., Gt Γ α c γ qt γ qt γ. c γ Gt Γ α qt γ, qt Gt Γ α Nqt γ. Summing both sides of the last inequality from t to t, we get Gt Gt + Γ α Nqt γ as t, which contradicts the fact that Gt > 0. Hence α xt γ is eventually positive. Define the function wt by the Riccati substitution wt = rt α xt γ. x γ t
5 Oscillaty Solutions of Nonlinear Fractional Difference Equations 23 Since rt > 0, xt > 0 and α xt γ > 0, we have wt > 0. Now [ ] rt α xt γ wt = x γ t [ ] α xt γ = rt + α xt + γ rt x γ t x γ t + [ ] x γ t α xt γ α xt γ x γ t = rt x γ tx γ t + r [ ] +t qtfxt wt + + rt rt rt + x γ t + r +t wt + rtqtk rt + [ ] α xt γ x γ t w 2 t + rt + α xt γ α xt + γ r [ +t rt + wt + rtqtk rt + r +t wt + rtqtk M rt + wt + + rt rt + [ α xt γ x γ t x γ tx γ t + xt α xt + ] γ w 2 t + γ rt + w2 t r + t Let X = wt + and Y = rt + 2. Using Lemma 2.4 and setting rt + M γ m = 2, we obtain M 2 γ r + t wt + rt + 2 rt + M γ rt + w2 t + r +t 2 4rt + which implies that r + t wt + rt + rt + w2 t + r +t 2 4rt + M, γ wt krtqt + r +t 2 4rt +. Summing the above inequality from t to t, we get ws krsqs + r +s 2 4rs + wt wt krsqs + r +s 2 4rs +,, ]
6 24 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam i.e., krsqs r +s 2 wt 4rs + wt wt < f t t. Letting t, we have krsqs r +s 2 wt 4rs + <, which contradicts 3.. The proof is complete. Theem 3.2. Suppose that H and H2 hold and q γ s =. Furtherme, assume that there exists a positive sequence rt, and a double positive sequence Ht, s such that Ht, t = 0 f t t 0, Ht, s > 0 f t > s t 0, s Ht, s = Ht, s + Ht, s 0 f t > s t 0. If Ht, t 0 rsqsht, s k h2 +t, srs + =, 3.3 4Ht, s where h + t, s = 2 Ht, s + Ht, s r +s rs + and r +s = max{ rs, 0}, then every solution of equation. is oscillaty. Proof. Suppose to the contrary that xt is a non-oscillaty solution of.. Without loss of generality, we can assume that xt is an eventually positive solution of.. Proceeding as in Theem 3., we arrive at equation 3.2. Multiplying 3.2 by Ht, s and summing from t to t, we get Ht, s ws r+ s ws + Ht, s rs + rsqskht, s Ht, s rs + w2 s +,
7 Oscillaty Solutions of Nonlinear Fractional Difference Equations 25 rsqskht, s Ht, s ws r+ s + ws + Ht, s Ht, s rs + rs + w2 s +. Using the summation by parts fmula, we have that Ht, s ws = [Ht, sws] t + which implies that k rsqsht, s = Ht, t wt + ws + 2 Ht, s ws + 2 Ht, s, Ht, t wt + ws + 2 Ht, s r+ s + ws + Ht, s Ht, s rs + rs + w2 s + Ht, t wt + 2 Ht, s + r +s Ht, s ws + rs + Ht, s rs + w2 s + Ht, t wt + h + t, sws + Ht, s rs + w2 s +, where h + t, s = 2 Ht, s + r +s Ht, s rs + Ht, t wt + h + t, sws + Ht, s rs + w2 s Set X = Ht, s rs + ws + and Y = h + t, s. 2 Ht, s rs +
8 26 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam Using Lemma 2.4 with m = 2, we have that 2 Ht, s rs + ws + h + t, s 2 Ht, s rs + Ht, s rs + w2 s + h2 +t, srs +, 4Ht, s h + t, sws + Ht, s rs + w2 s + h2 +t, srs +. 4Ht, s From equation 3.4, we have 2 Ht, s 0 f t > s t 0, 0 < Ht, t Ht, t 0 f t > s t 0, rsqsht, s k Ht, t wt + k h2 +t, srs +, 4Ht, s rsqsht, s k h2 +t, srs + k Ht, t 4Ht, s wt k Ht, t 0 wt. Since 0 < Ht, s Ht, t 0 f t > s t 0 then we have 0 < t > s t 0. Hence it follows that + Ht, t 0 Ht, t 0 Ht, t 0 t rsqsht, s k h2 +t, srs + 4Ht, s rsqsht, s k h2 +t, srs + 4Ht, s rsqsht, s k h2 +t, srs + 4Ht, s t rsqsht, s + k wt Ht, t 0 t Letting t, we have rsqs + k wt. Ht, t 0 rsqsht, s k h2 +t, srs + 4Ht, s Ht, s Ht, t 0 f
9 Oscillaty Solutions of Nonlinear Fractional Difference Equations 27 t which contradicts 3.3. The proof is complete. rsqs + k wt <, Theem 3.3. Suppose that H and H2 hold. Furtherme assume that there exists a positive sequence rt such that [ [ ] γ J qsk + T ] M M r +s =, 3.5 where r + s = max{ rs, 0}. Then every solution of. is oscillaty. Proof. Suppose to the contrary that xt is a nonoscillaty solution of.. Without loss of generality, we can assume that xt is an eventually positive solution of.. We proceed as in Theem 3. to get that α xt γ is positive. Now define the following function, using Riccati substitution wt = α xt γ x γ t + α xt + rt. Thus [ α xt γ wt = x γ t [ ] α xt γ wt = x γ t ] + α xt + rt, + [ α xt] + rt. Hence, [ ] wt xγ t [ α xt γ ] α xt γ x γ t xt + + rt x γ tx γ t + M = qtfxt [ ] α xt γ x γ t xt x γ t + x γ t x γ t rt M [ ] α γ [ ] γ xt xt qtk + xt xt + M 2 xt + rt [ ] γ [ ] γ xt xt qtk + T M xt xt + M + rt = qtk [ ] xt 2 γ + T M γ xtxt + M + rt qtk J γ M + T γ M + r +t [ ] γ J = qtk + T M M + r +t.
10 28 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam Summing the above inequality from t to t, we get wt wt [ [ ] γ J qsk + T ] M M + r +s, [ [ ] γ J qsk + T ] M M r +s wt wt wt <. Taking t and sup, we get [ [ ] γ J qsk + T ] M M r +s wt <, which contradicts 3.5. The proof is complete. 4 Examples Example 4.. Consider the nonlinear fractional difference equation 0.5 xt γ + tfxt = 0 f t N t0 +0.5, 4. where α = 0.5, qt = t and γ > 0 is a quotient of odd positive integers. We apply Theem 3.3 with rt = t γ+, qt = t, T =, M = and J =. It is easy to see that H and H2 hold. Then we have [ [ ] γ J qsk + T ] M M r +s = = s = s =, [ s s γ+ ] [s 2 s γ ] [ ] s γ+2 that is condition 3.5 of Theem 3.3 is satisfied. Therefe, all solutions of 4. are oscillaty. Example 4.2. Consider the nonlinear fractional difference equation 0.5 xt γ + tfxt = 0 f t N t0 +0.5, 4.2 s γ
11 Oscillaty Solutions of Nonlinear Fractional Difference Equations 29 where qt = t, α = 0.5, and γ > 0 is a quotient of odd positive integers. Clearly t = and conditions H and H2 hold. We apply Theem 3. with rt =, k = and M =, we obtain t2 krsqs r +s 2 4rs + = = =, s s s + 2 s2 4s 4 s + 2 4s 3 that is, condition 3. of Theem 3. is satisfied. Therefe all solutions of 4.2 are oscillaty. Acknowledgement The auths would like to thank the referee f the constructive remarks which greatly improved the paper. References [] G. A. Anastassiou, Discrete fractional calculus and inequalities, /abs/ v. [2] F. M. Atici and P. W. Eloe, A transfm method in discrete fractional calculus, Int. J. Difference Equ., [3] F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operat, Electron. J. Qual. They Differ. Equ., [4] F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Am. Math. Soc [5] F. Chen, Fixed points and asymptotic stability of nonlinear fractional difference equations, Electron. J. Qual. They Differ. Equ., [6] D. Chen, Oscillaty behavi of a class of Fractional differential equations with damping, U. Politeh. Buch. Ser. A, [7] D. X. Chen, Oscillation criteria of fractional differential equations, Adv. Differ. Equ.,
12 30 G. E. Chatzarakis, P. Gokulraj, T. Kalaimani and V. Sadhasivam [8] F. Chen, Z. Liu, Asymptotic Stability Results f Nonlinear Fractional Difference Equations, J. Appl. Math., doi:0.55/202/ [9] F. Chen, X. Luo, Y. Zhou, Existence Results f Nonlinear Fractional Difference Equation, Adv. Differ. Equ., doi:0.55/202/ [0] D. Chen, P. Qu, Y. Lan, Fced oscillation of certain fractional differential equations, Adv. Differ. Equ., [] J. B. Diaz and T. J. Olser, Differences of Fractional Order, Math. Comput., [2] K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, 200. [3] A. Gege Maria Selvam, M. Reni Sagayaraj and M. Paul Loganathan, Oscillaty behavi of a class of fractional difference equations with damping, Int. J. Appl. Math. Res., [4] S. R. Grace, R. P. Agarwal, Patricia J. Y. Wong, A. Zafer, On the Oscillation of Fractional Differential Equations, Fract. Calc. Appl. Anal., [5] H. Gray and N. Zhang, On a new definition of the fractional difference, Math. Comput., [6] Z. Han, Y. Zhao, Y. Sun, C. Zhang, Oscillation f a class of fractional differential equation, Discrete Dyn. Nat. Soc [7] G. H. Hardy, J. E. Littlewood, G.Polya, Inequalities, Cambridge University Press, Cambridge 959. [8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, They and Applications of Fractional Differential Equations, Nth-Holland Math. Studies 204, Elsevier, Amsterdam, [9] W. N. Li, Oscillation results f certain fced fractional difference equations with damping term, Adv. Differ. Equ., [20] R. Matusu, Application of fractional der calculus to control they, Int. J. Math. Models Methods Appl. Sci., [2] H. T. Michael, The They of Discrete Fractional Calculus: Development and Application Spring 4-20, Dissertations, Theses, and Student Research Papers in Mathematics, Paper 27.
13 Oscillaty Solutions of Nonlinear Fractional Difference Equations 3 [22] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New Yk, USA, 993. [23] K. S. Miller and B. Ross, Fractional Difference Calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and their Applications, Nihon University, Japan, 988. [24] M. Paul Loganathan, M. Reni Sagayaraj, A. Gege Maria Selvam, On the oscillation of non-linear fractional difference equations, Math. Aeterna, [25] I. Petras, Control of Fractional-Order Chuas System, arxiv:nlin/ v. [26] I. Podlubny, Fractional Differential Equations, Academic Press, USA, 999. [27] C. Qi and J. Cheng, Interval oscillation criteria f a class of fractional differential equations with damping term, Math. Probl. Eng., [28] M. Reni Sagayaraj, A. Gege Maria Selvam and M. Paul Loganathan, Oscillation criteria f a class of discrete nonlinear fractional equations, Bull. Soc. Math. Serv. Stand., [29] A. Secer and H. Adiguzel, Oscillation of solutions f a class of nonlinear fractional difference equations, J. Nonlinear Sci. Appl., [30] Y. Wang, Z. Han, P. Zhao and S. Sun, On the oscillation and asymptotic behavi f a kind of fractional differential equations, Adv. Differ. Equ., [3] S. Xiang, Z. Han, P. Zhao and Y. Sun, Oscillaty behavi f a class of differential equations with fractional der derivatives, Abstr. Appl. Anal.,
Oscillation theorems for nonlinear fractional difference equations
Adiguzel Boundary Value Problems (2018) 2018:178 https://doi.org/10.1186/s13661-018-1098-4 R E S E A R C H Open Access Oscillation theorems for nonlinear fractional difference equations Hakan Adiguzel
More informationOscillation results for certain forced fractional difference equations with damping term
Li Advances in Difference Equations 06) 06:70 DOI 0.86/s66-06-0798- R E S E A R C H Open Access Oscillation results for certain forced fractional difference equations with damping term Wei Nian Li * *
More informationMulti-Term Linear Fractional Nabla Difference Equations with Constant Coefficients
International Journal of Difference Equations ISSN 0973-6069, Volume 0, Number, pp. 9 06 205 http://campus.mst.edu/ijde Multi-Term Linear Fractional Nabla Difference Equations with Constant Coefficients
More informationOSCILLATORY PROPERTIES OF A CLASS OF CONFORMABLE FRACTIONAL GENERALIZED LIENARD EQUATIONS
IMPACT: International Journal of Research in Humanities, Arts and Literature (IMPACT: IJRHAL) ISSN (P): 2347-4564; ISSN (E): 2321-8878 Vol 6, Issue 11, Nov 2018, 201-214 Impact Journals OSCILLATORY PROPERTIES
More informationPositive solutions for discrete fractional intiail value problem
Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 4, No. 4, 2016, pp. 285-297 Positive solutions for discrete fractional intiail value problem Tahereh Haghi Sahand University
More informationFractional differential equations with integral boundary conditions
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 8 (215), 39 314 Research Article Fractional differential equations with integral boundary conditions Xuhuan Wang a,, Liping Wang a, Qinghong Zeng
More informationResearch Article Convergence and Divergence of the Solutions of a Neutral Difference Equation
Journal of Applied Mathematics Volume 2011, Article ID 262316, 18 pages doi:10.1155/2011/262316 Research Article Convergence and Divergence of the Solutions of a Neutral Difference Equation G. E. Chatzarakis
More informationOn Two-Point Riemann Liouville Type Nabla Fractional Boundary Value Problems
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 13, Number 2, pp. 141 166 (2018) http://campus.mst.edu/adsa On Two-Point Riemann Liouville Type Nabla Fractional Boundary Value Problems
More informationBoundary value problems for fractional differential equations with three-point fractional integral boundary conditions
Sudsutad and Tariboon Advances in Difference Equations 212, 212:93 http://www.advancesindifferenceequations.com/content/212/1/93 R E S E A R C H Open Access Boundary value problems for fractional differential
More informationMonotone Iterative Method for a Class of Nonlinear Fractional Differential Equations on Unbounded Domains in Banach Spaces
Filomat 31:5 (217), 1331 1338 DOI 1.2298/FIL175331Z Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Monotone Iterative Method for
More informationON THE OSCILLATION OF THE SOLUTIONS TO LINEAR DIFFERENCE EQUATIONS WITH VARIABLE DELAY
Electronic Journal of Differential Equations, Vol. 008(008, No. 50, pp. 1 15. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp ON THE
More informationExistence and Uniqueness Results for Nonlinear Implicit Fractional Differential Equations with Boundary Conditions
Existence and Uniqueness Results for Nonlinear Implicit Fractional Differential Equations with Boundary Conditions Mouffak Benchohra a,b 1 and Jamal E. Lazreg a, a Laboratory of Mathematics, University
More informationOscillation results for difference equations with oscillating coefficients
Chatzarakis et al. Advances in Difference Equations 2015 2015:53 DOI 10.1186/s13662-015-0391-0 R E S E A R C H Open Access Oscillation results f difference equations with oscillating coefficients Gege
More informationOscillation criteria for second-order half-linear dynamic equations on time scales
P a g e 46 Vol.10 Issue 5(Ver 1.0)September 2010 Global Journal of Science Frontier Research Oscillation criteria for second-order half-linear dynamic equations on time scales Zhenlai Han a,b, Tongxing
More informationOscillation of second-order differential equations with a sublinear neutral term
CARPATHIAN J. ATH. 30 2014), No. 1, 1-6 Online version available at http://carpathian.ubm.ro Print Edition: ISSN 1584-2851 Online Edition: ISSN 1843-4401 Oscillation of second-order differential equations
More informationOn boundary value problems for fractional integro-differential equations in Banach spaces
Malaya J. Mat. 3425 54 553 On boundary value problems for fractional integro-differential equations in Banach spaces Sabri T. M. Thabet a, and Machindra B. Dhakne b a,b Department of Mathematics, Dr. Babasaheb
More informationExistence of triple positive solutions for boundary value problem of nonlinear fractional differential equations
Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 5, No. 2, 217, pp. 158-169 Existence of triple positive solutions for boundary value problem of nonlinear fractional differential
More informationCritical exponents for a nonlinear reaction-diffusion system with fractional derivatives
Global Journal of Pure Applied Mathematics. ISSN 0973-768 Volume Number 6 (06 pp. 5343 535 Research India Publications http://www.ripublication.com/gjpam.htm Critical exponents f a nonlinear reaction-diffusion
More informationNontrivial solutions for fractional q-difference boundary value problems
Electronic Journal of Qualitative Theory of Differential Equations 21, No. 7, 1-1; http://www.math.u-szeged.hu/ejqtde/ Nontrivial solutions for fractional q-difference boundary value problems Rui A. C.
More informationEXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO HIGHER-ORDER NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH INTEGRAL BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 212 (212), No. 234, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationSOME RESULTS FOR BOUNDARY VALUE PROBLEM OF AN INTEGRO DIFFERENTIAL EQUATIONS WITH FRACTIONAL ORDER
Dynamic Systems and Applications 2 (2) 7-24 SOME RESULTS FOR BOUNDARY VALUE PROBLEM OF AN INTEGRO DIFFERENTIAL EQUATIONS WITH FRACTIONAL ORDER P. KARTHIKEYAN Department of Mathematics, KSR College of Arts
More informationExistence of solutions for multi-point boundary value problem of fractional q-difference equation
Electronic Journal of Qualitative Theory of Differential Euations 211, No. 92, 1-1; http://www.math.u-szeged.hu/ejtde/ Existence of solutions for multi-point boundary value problem of fractional -difference
More informationOscillatory Behavior of Third-order Difference Equations with Asynchronous Nonlinearities
International Journal of Difference Equations ISSN 0973-6069, Volume 9, Number 2, pp 223 231 2014 http://campusmstedu/ijde Oscillatory Behavior of Third-order Difference Equations with Asynchronous Nonlinearities
More informationANALYSIS OF NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS
International Journal of Analysis and Applications ISSN 229-8639 Volume 7, Number (205), 79-95 http://www.etamaths.com ANALYSIS OF NONLINEAR FRACTIONAL NABLA DIFFERENCE EQUATIONS JAGAN MOHAN JONNALAGADDA
More informationANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS INVOLVING A NEW FRACTIONAL DERIVATIVE WITHOUT SINGULAR KERNEL
Electronic Journal of Differential Equations, Vol. 217 (217), No. 289, pp. 1 6. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ANALYSIS AND APPLICATION OF DIFFUSION EQUATIONS
More informationA study on nabla discrete fractional operator in mass - spring - damper system
NTMSCI 4, No. 4, 137-144 (2016) 137 New Trends in Mathematical Sciences http://dx.doi.org/10.20852/ntmsci.2016422559 A study on nabla discrete fractional operator in mass - spring - damper system Okkes
More informationExistence of solutions of fractional boundary value problems with p-laplacian operator
Mahmudov and Unul Boundary Value Problems 25 25:99 OI.86/s366-5-358-9 R E S E A R C H Open Access Existence of solutions of fractional boundary value problems with p-laplacian operator Nazim I Mahmudov
More informationOscillation of second-order nonlinear difference equations with sublinear neutral term
Mathematica Moravica Vol. 23, No. (209), 0 Oscillation of second-order nonlinear difference equations with sublinear neutral term Martin Bohner, Hassan A. El-Morshedy, Said R. Grace and Ilgin Sağer Abstract.
More informationIMPROVEMENTS OF COMPOSITION RULE FOR THE CANAVATI FRACTIONAL DERIVATIVES AND APPLICATIONS TO OPIAL-TYPE INEQUALITIES
Dynamic Systems and Applications ( 383-394 IMPROVEMENTS OF COMPOSITION RULE FOR THE CANAVATI FRACTIONAL DERIVATIVES AND APPLICATIONS TO OPIAL-TYPE INEQUALITIES M ANDRIĆ, J PEČARIĆ, AND I PERIĆ Faculty
More informationFRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE. Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi
Opuscula Math. 37, no. 2 27), 265 28 http://dx.doi.org/.7494/opmath.27.37.2.265 Opuscula Mathematica FRACTIONAL BOUNDARY VALUE PROBLEMS ON THE HALF LINE Assia Frioui, Assia Guezane-Lakoud, and Rabah Khaldi
More informationThe main objective of this work is to establish necessary and sufficient conditions for oscillations of (1.1), under the assumptions
Journal of Applied Mathematics and Computation (JAMC), 2018, 2(3), 100-106 http://www.hillpublisher.org/journal/jamc ISSN Online:2576-0645 ISSN Print:2576-0653 Necessary and Sufficient Conditions for Oscillation
More informationNo. 5 Discrete variational principle the first integrals of the In view of the face that only the momentum integrals can be obtained by the abo
Vol 14 No 5, May 005 cfl 005 Chin. Phys. Soc. 1009-1963/005/14(05)/888-05 Chinese Physics IOP Publishing Ltd Discrete variational principle the first integrals of the conservative holonomic systems in
More informationOscillation Criteria for Certain nth Order Differential Equations with Deviating Arguments
Journal of Mathematical Analysis Applications 6, 601 6 001) doi:10.1006/jmaa.001.7571, available online at http://www.idealibrary.com on Oscillation Criteria for Certain nth Order Differential Equations
More informationEXISTENCE THEOREM FOR A FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM
Fixed Point Theory, 5(, No., 3-58 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html EXISTENCE THEOREM FOR A FRACTIONAL MULTI-POINT BOUNDARY VALUE PROBLEM FULAI CHEN AND YONG ZHOU Department of Mathematics,
More informationLOCAL EXTREMA OF POSITIVE SOLUTIONS OF NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 158, pp. 1 11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LOCAL EXTREMA OF POSITIVE SOLUTIONS OF
More informationNonlocal problems for the generalized Bagley-Torvik fractional differential equation
Nonlocal problems for the generalized Bagley-Torvik fractional differential equation Svatoslav Staněk Workshop on differential equations Malá Morávka, 28. 5. 212 () s 1 / 32 Overview 1) Introduction 2)
More informationA. Then p P( ) if and only if there exists w Ω such that p(z)= (z U). (1.4)
American International Journal of Research in Science, Technology, Engineering & Mathematics Available online at http://www.iasir.net ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 2328-3629
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1. Yong Zhou. Abstract
EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1 Yong Zhou Abstract In this paper, the initial value problem is discussed for a system of fractional differential
More informationSome New Inequalities Involving Generalized Erdélyi-Kober Fractional q-integral Operator
Applied Mathematical Sciences, Vol. 9, 5, no. 7, 3577-359 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/ams.5.539 Some New Inequalities Involving Generalized Erdélyi-Kober Fractional q-integral Operator
More informationExistence of Minimizers for Fractional Variational Problems Containing Caputo Derivatives
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 1, pp. 3 12 (2013) http://campus.mst.edu/adsa Existence of Minimizers for Fractional Variational Problems Containing Caputo
More informationExistence of Ulam Stability for Iterative Fractional Differential Equations Based on Fractional Entropy
Entropy 215, 17, 3172-3181; doi:1.339/e1753172 OPEN ACCESS entropy ISSN 199-43 www.mdpi.com/journal/entropy Article Existence of Ulam Stability for Iterative Fractional Differential Equations Based on
More informationSolution of fractional oxygen diffusion problem having without singular kernel
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 1 (17), 99 37 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Solution of fractional oxygen diffusion
More informationPositive solutions for a class of fractional boundary value problems
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 1, 1 17 ISSN 1392-5113 http://dx.doi.org/1.15388/na.216.1.1 Positive solutions for a class of fractional boundary value problems Jiafa Xu a, Zhongli
More informationFUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM
Electronic Journal of Differential Equations, Vol. 28(28), No. 22, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) FUNCTIONAL
More informationARCHIVUM MATHEMATICUM (BRNO) Tomus 32 (1996), 13 { 27. ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR DIFFERENCE EQUATION
ARCHIVUM MATHEMATICUM (BRNO) Tomus 32 (996), 3 { 27 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR DIFFERENCE EQUATION P. J. Y. Wong and R. P. Agarwal Abstract. We oer sucient conditions for the
More informationarxiv: v1 [math.na] 8 Jan 2019
arxiv:190102503v1 [mathna] 8 Jan 2019 A Numerical Approach for Solving of Fractional Emden-Fowler Type Equations Josef Rebenda Zdeněk Šmarda c 2018 AIP Publishing This article may be downloaded for personal
More informationAbdulmalik Al Twaty and Paul W. Eloe
Opuscula Math. 33, no. 4 (23, 63 63 http://dx.doi.org/.7494/opmath.23.33.4.63 Opuscula Mathematica CONCAVITY OF SOLUTIONS OF A 2n-TH ORDER PROBLEM WITH SYMMETRY Abdulmalik Al Twaty and Paul W. Eloe Communicated
More informationA Numerical Scheme for Generalized Fractional Optimal Control Problems
Available at http://pvamuedu/aam Appl Appl Math ISSN: 1932-9466 Vol 11, Issue 2 (December 216), pp 798 814 Applications and Applied Mathematics: An International Journal (AAM) A Numerical Scheme for Generalized
More informationIterative scheme to a coupled system of highly nonlinear fractional order differential equations
Computational Methods for Differential Equations http://cmde.tabrizu.ac.ir Vol. 3, No. 3, 215, pp. 163-176 Iterative scheme to a coupled system of highly nonlinear fractional order differential equations
More informationSolution and stability of a reciprocal type functional equation in several variables
Available online at www.tjnsa.co J. Nonlinear Sci. Appl. 7 04, 8 7 Research Article Solution and stability of a reciprocal type functional equation in several variables K. Ravi a,, E. Thandapani b, B.V.
More informationBOUNDARY VALUE PROBLEMS OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION
U.P.B. Sci. Bull., Series A, Vol. 79, Iss. 4, 2017 ISSN 1223-7027 BOUNDARY VALUE PROBLEMS OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION Lianwu Yang 1 We study a higher order nonlinear difference equation.
More informationLeighton Coles Wintner Type Oscillation Criteria for Half-Linear Impulsive Differential Equations
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 5, Number 2, pp. 25 214 (21) http://campus.mst.edu/adsa Leighton Coles Wintner Type Oscillation Criteria for Half-Linear Impulsive Differential
More informationNecessary and Sufficient Condition for Oscillation Solution of Nonlinear Second Order Difference Equations
Necessary and Sufficient Condition for Oscillation Solution of Nonlinear Second Order Difference Equations C. Jayakumar 1 and A. Merlin Vinola 2 1 (Assistant Professor, Department of Mathematics, Mahendra
More informationPOSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF SINGULAR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION
International Journal of Pure and Applied Mathematics Volume 92 No. 2 24, 69-79 ISSN: 3-88 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/.2732/ijpam.v92i2.3
More informationMONOTONE POSITIVE SOLUTION OF NONLINEAR THIRD-ORDER TWO-POINT BOUNDARY VALUE PROBLEM
Miskolc Mathematical Notes HU e-issn 177-2413 Vol. 15 (214), No. 2, pp. 743 752 MONOTONE POSITIVE SOLUTION OF NONLINEAR THIRD-ORDER TWO-POINT BOUNDARY VALUE PROBLEM YONGPING SUN, MIN ZHAO, AND SHUHONG
More informationFRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX MAPPINGS AND APPLICATIONS TO SPECIAL MEANS AND A MIDPOINT FORMULA
Journal of Applied Mathematics, Statistics and Informatics (JAMSI), 8 (), No. FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX MAPPINGS AND APPLICATIONS TO SPECIAL MEANS AND A MIDPOINT FORMULA
More informationApplied Mathematics Letters
Applied Mathematics Letters 24 (211) 219 223 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Laplace transform and fractional differential
More informationMIXED TYPE OF ADDITIVE AND QUINTIC FUNCTIONAL EQUATIONS. Abasalt Bodaghi, Pasupathi Narasimman, Krishnan Ravi, Behrouz Shojaee
Annales Mathematicae Silesianae 29 (205, 35 50 Prace Naukowe Uniwersytetu Śląskiego nr 3332, Katowice DOI: 0.55/amsil-205-0004 MIXED TYPE OF ADDITIVE AND QUINTIC FUNCTIONAL EQUATIONS Abasalt Bodaghi, Pasupathi
More informationON THE FRACTAL HEAT TRANSFER PROBLEMS WITH LOCAL FRACTIONAL CALCULUS
THERMAL SCIENCE, Year 2015, Vol. 19, No. 5, pp. 1867-1871 1867 ON THE FRACTAL HEAT TRANSFER PROBLEMS WITH LOCAL FRACTIONAL CALCULUS by Duan ZHAO a,b, Xiao-Jun YANG c, and Hari M. SRIVASTAVA d* a IOT Perception
More informationExistence, Uniqueness and Stability of Hilfer Type Neutral Pantograph Differential Equations with Nonlocal Conditions
International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 6, Issue 8, 2018, PP 42-53 ISSN No. (Print) 2347-307X & ISSN No. (Online) 2347-3142 DOI: http://dx.doi.org/10.20431/2347-3142.0608004
More informationDETERMINATION OF AN UNKNOWN SOURCE TERM IN A SPACE-TIME FRACTIONAL DIFFUSION EQUATION
Journal of Fractional Calculus and Applications, Vol. 6(1) Jan. 2015, pp. 83-90. ISSN: 2090-5858. http://fcag-egypt.com/journals/jfca/ DETERMINATION OF AN UNKNOWN SOURCE TERM IN A SPACE-TIME FRACTIONAL
More informationHERMITE HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS
HERMITE HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRALS MARIAN MATŁOKA Abstract: In the present note, we have established an integral identity some Hermite-Hadamard type integral ineualities for the
More informationExistence and Uniqueness of Anti-Periodic Solutions for Nonlinear Higher-Order Differential Equations with Two Deviating Arguments
Advances in Dynamical Systems and Applications ISSN 973-531, Volume 7, Number 1, pp. 19 143 (1) http://campus.mst.edu/adsa Existence and Uniqueness of Anti-Periodic Solutions for Nonlinear Higher-Order
More informationCollege, Nashik-Road, Dist. - Nashik (MS), India,
Approximate Solution of Space Fractional Partial Differential Equations and Its Applications [1] Kalyanrao Takale, [2] Manisha Datar, [3] Sharvari Kulkarni [1] Department of Mathematics, Gokhale Education
More informationOscillation Theorems for Second-Order Nonlinear Dynamic Equation on Time Scales
Appl. Math. Inf. Sci. 7, No. 6, 289-293 (203) 289 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/0.2785/amis/070608 Oscillation heorems for Second-Order Nonlinear
More informationRESOLVENT OF LINEAR VOLTERRA EQUATIONS
Tohoku Math. J. 47 (1995), 263-269 STABILITY PROPERTIES AND INTEGRABILITY OF THE RESOLVENT OF LINEAR VOLTERRA EQUATIONS PAUL ELOE AND MUHAMMAD ISLAM* (Received January 5, 1994, revised April 22, 1994)
More informationA General Boundary Value Problem For Impulsive Fractional Differential Equations
Palestine Journal of Mathematics Vol. 5) 26), 65 78 Palestine Polytechnic University-PPU 26 A General Boundary Value Problem For Impulsive Fractional Differential Equations Hilmi Ergoren and Cemil unc
More informationBLOW-UP OF SOLUTIONS FOR A NONLINEAR WAVE EQUATION WITH NONNEGATIVE INITIAL ENERGY
Electronic Journal of Differential Equations, Vol. 213 (213, No. 115, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu BLOW-UP OF SOLUTIONS
More informationOscillation Criteria for Delay and Advanced Difference Equations with General Arguments
Advances in Dynamical Systems Applications ISSN 0973-531, Volume 8, Number, pp. 349 364 (013) http://campus.mst.edu/adsa Oscillation Criteria for Delay Advanced Difference Equations with General Arguments
More informationSOLUTION OF THE ULAM STABILITY PROBLEM FOR CUBIC MAPPINGS. John Michael Rassias National and Capodistrian University of Athens, Greece
GLASNIK MATEMATIČKI Vol. 36(56)(2001), 63 72 SOLUTION OF THE ULAM STABILITY PROBLEM FOR CUBIC MAPPINGS John Michael Rassias National and Capodistrian University of Athens, Greece Abstract. In 1968 S. M.
More informationAsymptotic Behavior of a Higher-Order Recursive Sequence
International Journal of Difference Equations ISSN 0973-6069, Volume 7, Number 2, pp. 75 80 (202) http://campus.mst.edu/ijde Asymptotic Behavior of a Higher-Order Recursive Sequence Özkan Öcalan Afyon
More informationPositive solutions for integral boundary value problem of two-term fractional differential equations
Xu and Han Boundary Value Problems (28) 28: https://doi.org/.86/s366-8-2-z R E S E A R C H Open Access Positive solutions for integral boundary value problem of two-term fractional differential equations
More informationPOSITIVE SOLUTIONS TO SINGULAR HIGHER ORDER BOUNDARY VALUE PROBLEMS ON PURELY DISCRETE TIME SCALES
Communications in Applied Analysis 19 2015, 553 564 POSITIVE SOLUTIONS TO SINGULAR HIGHER ORDER BOUNDARY VALUE PROBLEMS ON PURELY DISCRETE TIME SCALES CURTIS KUNKEL AND ASHLEY MARTIN 1 Department of Mathematics
More informationMEASURE OF NONCOMPACTNESS AND FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES
Communications in Applied Analysis 2 (28), no. 4, 49 428 MEASURE OF NONCOMPACTNESS AND FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES MOUFFAK BENCHOHRA, JOHNNY HENDERSON, AND DJAMILA SEBA Laboratoire
More informationOscillation by Impulses for a Second-Order Delay Differential Equation
PERGAMON Computers and Mathematics with Applications 0 (2006 0 www.elsevier.com/locate/camwa Oscillation by Impulses for a Second-Order Delay Differential Equation L. P. Gimenes and M. Federson Departamento
More informationOne point compactification for generalized quotient spaces
@ Applied General Topology c Universidad Politécnica de Valencia Volume 11, No. 1, 2010 pp. 21-27 One point compactification for generalized quotient spaces V. Karunakaran and C. Ganesan Abstract. The
More informationDisconjugate operators and related differential equations
Disconjugate operators and related differential equations Mariella Cecchi, Zuzana Došlá and Mauro Marini Dedicated to J. Vosmanský on occasion of his 65 th birthday Abstract: There are studied asymptotic
More informationDiscrete Population Models with Asymptotically Constant or Periodic Solutions
International Journal of Difference Equations ISSN 0973-6069, Volume 6, Number 2, pp. 143 152 (2011) http://campus.mst.edu/ijde Discrete Population Models with Asymptotically Constant or Periodic Solutions
More informationTRIPLE POSITIVE SOLUTIONS FOR A CLASS OF TWO-POINT BOUNDARY-VALUE PROBLEMS
Electronic Journal of Differential Equations, Vol. 24(24), No. 6, pp. 8. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) TRIPLE POSITIVE
More informationNUMERICAL SOLUTION OF FRACTIONAL ORDER DIFFERENTIAL EQUATIONS USING HAAR WAVELET OPERATIONAL MATRIX
Palestine Journal of Mathematics Vol. 6(2) (217), 515 523 Palestine Polytechnic University-PPU 217 NUMERICAL SOLUTION OF FRACTIONAL ORDER DIFFERENTIAL EQUATIONS USING HAAR WAVELET OPERATIONAL MATRIX Raghvendra
More informationA Comparison Result for the Fractional Difference Operator
International Journal of Difference Equations ISSN 0973-6069, Volume 6, Number 1, pp. 17 37 (2011) http://campus.mst.edu/ijde A Comparison Result for the Fractional Difference Operator Christopher S. Goodrich
More informationSolutions of the Diophantine Equation p x + (p+6) y = z 2 when p, (p + 6) are Primes and x + y = 2, 3, 4
Annals of Pure and Applied Mathematics Vol. 17, No. 1, 2018, 101-106 ISSN: 2279-087X (P), 2279-0888(online) Published on 3 May 2018 www.researchmathsci.g DOI: http://dx.doi.g/10.22457/apam.v17n1a11 Annals
More information1 Introduction ON NABLA DISCRETE FRACTIONAL CALCULUS OPERATOR FOR A MODIFIED BESSEL EQUATION. Resat YILMAZER a,, and Okkes OZTURK b
ON NABLA DISCRETE FRACTIONAL CALCULUS OPERATOR FOR A MODIFIED BESSEL EQUATION by Resat YILMAZER a,, and Okkes OZTURK b a Department of Mathematics, Firat University, 23119, Elazig, Turkey b Department
More informationNecessary and Sufficient Conditions for Oscillation of Certain Higher Order Partial Difference Equations
International Journal of Difference Equations ISSN 0973-6069, Volume 4, Number 2, pp 211 218 (2009 http://campusmstedu/ijde Necessary and Sufficient Conditions for Oscillation of Certain Higher Order Partial
More informationBritish Journal of Applied Science & Technology 10(2): 1-11, 2015, Article no.bjast ISSN:
British Journal of Applied Science & Technology 10(2): 1-11, 2015, Article no.bjast.18590 ISSN: 2231-0843 SCIENCEDOMAIN international www.sciencedomain.org Solutions of Sequential Conformable Fractional
More informationOn the fractional-order logistic equation
Applied Mathematics Letters 20 (2007) 817 823 www.elsevier.com/locate/aml On the fractional-order logistic equation A.M.A. El-Sayed a, A.E.M. El-Mesiry b, H.A.A. El-Saka b, a Faculty of Science, Alexandria
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics ON A HYBRID FAMILY OF SUMMATION INTEGRAL TYPE OPERATORS VIJAY GUPTA AND ESRA ERKUŞ School of Applied Sciences Netaji Subhas Institute of Technology
More informationMahmoud M. El-Borai a, Abou-Zaid H. El-Banna b, Walid H. Ahmed c a Department of Mathematics, faculty of science, Alexandria university, Alexandria.
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 52 On Some Fractional-Integro Partial Differential Equations Mahmoud M. El-Borai a, Abou-Zaid H. El-Banna b, Walid H. Ahmed c
More informationAsymptotic stability of solutions of a class of neutral differential equations with multiple deviating arguments
Bull. Math. Soc. Sci. Math. Roumanie Tome 57(15) No. 1, 14, 11 13 Asymptotic stability of solutions of a class of neutral differential equations with multiple deviating arguments by Cemil Tunç Abstract
More informationJUNXIA MENG. 2. Preliminaries. 1/k. x = max x(t), t [0,T ] x (t), x k = x(t) dt) k
Electronic Journal of Differential Equations, Vol. 29(29), No. 39, pp. 1 7. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu POSIIVE PERIODIC SOLUIONS
More informationNON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXVI, 2 (2017), pp. 287 297 287 NON-MONOTONICITY HEIGHT OF PM FUNCTIONS ON INTERVAL PINGPING ZHANG Abstract. Using the piecewise monotone property, we give a full description
More informationA computationally effective predictor-corrector method for simulating fractional order dynamical control system
ANZIAM J. 47 (EMA25) pp.168 184, 26 168 A computationally effective predictor-corrector method for simulating fractional order dynamical control system. Yang F. Liu (Received 14 October 25; revised 24
More informationDIfferential equations of fractional order have been the
Multistage Telescoping Decomposition Method for Solving Fractional Differential Equations Abdelkader Bouhassoun Abstract The application of telescoping decomposition method, developed for ordinary differential
More informationEXACT TRAVELING WAVE SOLUTIONS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING THE IMPROVED (G /G) EXPANSION METHOD
Jan 4. Vol. 4 No. 7-4 EAAS & ARF. All rights reserved ISSN5-869 EXACT TRAVELIN WAVE SOLUTIONS FOR NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USIN THE IMPROVED ( /) EXPANSION METHOD Elsayed M.
More informationSMOOTHNESS PROPERTIES OF SOLUTIONS OF CAPUTO- TYPE FRACTIONAL DIFFERENTIAL EQUATIONS. Kai Diethelm. Abstract
SMOOTHNESS PROPERTIES OF SOLUTIONS OF CAPUTO- TYPE FRACTIONAL DIFFERENTIAL EQUATIONS Kai Diethelm Abstract Dedicated to Prof. Michele Caputo on the occasion of his 8th birthday We consider ordinary fractional
More informationMULTIPLICITY OF CONCAVE AND MONOTONE POSITIVE SOLUTIONS FOR NONLINEAR FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS
Fixed Point Theory, 4(23), No. 2, 345-358 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html MULTIPLICITY OF CONCAVE AND MONOTONE POSITIVE SOLUTIONS FOR NONLINEAR FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS
More informationTrigonometric Recurrence Relations and Tridiagonal Trigonometric Matrices
International Journal of Difference Equations. ISSN 0973-6069 Volume 1 Number 1 2006 pp. 19 29 c Research India Publications http://www.ripublication.com/ijde.htm Trigonometric Recurrence Relations and
More informationBull. Math. Soc. Sci. Math. Roumanie Tome 60 (108) No. 1, 2017, 3 18
Bull. Math. Soc. Sci. Math. Roumanie Tome 6 8 No., 27, 3 8 On a coupled system of sequential fractional differential equations with variable coefficients and coupled integral boundary conditions by Bashir
More informationNontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation
Advances in Dynamical Systems and Applications ISSN 973-532, Volume 6, Number 2, pp. 24 254 (2 http://campus.mst.edu/adsa Nontrivial Solutions for Boundary Value Problems of Nonlinear Differential Equation
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics OPTIMAL OSCILLATION CRITERIA FOR FIRST ORDER DIFFERENCE EQUATIONS WITH DELAY ARGUMENT GEORGE E. CHATZARAKIS, ROMAN KOPLATADZE AND IOANNIS P. STAVROULAKIS Volume 235 No. 1
More information