Existence Theory of Second Order Random Differential Equations

Size: px
Start display at page:

Download "Existence Theory of Second Order Random Differential Equations"

Transcription

1 Global Journal of Mahemaical Sciences: Theory and Pracical. ISSN Volume 4, Number 3 (22), pp Inernaional Research Publicaion House hp:// Exisence Theory of Second Order Random Differenial Equaions D.S. Palimkar Deparmen of Mahemaics, Vasanrao Naik College, Nanded- 4363, M.S., India palimkar@rediffmail.com Absrac In his paper, an exisence resul for nonlinear second order ordinary random differenial equaions is proved under a Carah eodory condiion. My invesigaions is placed in he Banach space of coninuous real valued funcions on closed and bounded inervals of he real line ogeher wih an applicaion of he random version of he Leray-Schauder principle. Keywor: Nonlinear random differenial equaion, muli-valued funcion, local araciviy, global araciviy, boundary value problem. Mahemaics Subjec Classificaions: 6H25, 47H4, 47N2. Saemen of he Problem Le R denoe he real line and le J = [, T ] be a closed and bounded inerval in R. le C ( J, R) denoe he class of realvalued funcions defined and coninuously differeniable on J. Given a measurable space ( Ω, A) and for a given measurable funcion x : Ω C ( J, R), consider he boundary value problem of second order ordinary random differenial equaions (RDE), x"(, ω) = f(, x(, ω), ω) a. e. J, (.) x(, ω) = q ( ω), x'(, ω) = q ( ω), for all ω Ω, where f : J R Ω R, q, q: Ω R. By a random soluion of he RDE. (.), I mean a measurable funcion x : Ω AC ( J, R) ha saisfies he equaions in (.) where AC ( J, R) is he space of

2 34 D.S. Palimkar real valued funcions defined and absoluely coninuously differeniable on J. The RDE (.) is no new o he heory random differenial equaions. When he random parameer ω is absen, he RDE (.) reduces o he classical RDE of second order ordinary differenial equaions (ODE), x"( ) = f (, x( )) a. e. J, x() = x, x'() = x, (.2) where f : J R R. The classical ODE (.2) has been sudied in he lieraure by several auhors for differen aspecs of he soluions. See for example, Heikkilä and Lakshikanham [9] and he references herein. In his paper, I discuss he RDE (.) for exisence of random soluions, under suiable condiions of he nonlineariy f which hereby generalize several exisence resuls of he RDE (.2) proved in he above papers. My analyses on he random of he nonlinear alernaive of Leray-Schauder ype(dhage[4, 5] and an algebraic random fixed poin heorem of Dhage[4]. Auxiliary Resuls Theorem 2. (Dhage [4, 5] ) Le E be a separable Banach space and le Q: Ω E E be a compleely coninuous random operaor. Then, eiher. he random equaion Q( ω ) x= xhas a random soluion, i.e. here is a measurable funcion ξ : Ω E such ha Q( ω) ξω ( ) = ξω ( ) for all ω Ω, or 2. he se ε = { x : Ω E is measurable / λω ( ) Q( ω) x = x} is unbounded for some measurable λ : Ω R wih < λ( ω) < on Ω. An immediae corollary o above heorem in applicable form is Corollary 2. Le E be a separable Banach space and le Q: Ω E E be a compleely coninuous random operaor. Then, eiher. he random equaion Q( ω ) x= x has a random soluion, i.e. here is a measurable funcion ξ : Ω E such ha Q( ω) ξω ( ) = ξω ( ) for all ω Ω, or 2. hese ε = { x : Ω E is measurable / λq( ω) x = x} is unbounded for some ω Ω saisfying < λ <. The following heorem is used in he sudy of nonlinear disconinuous random differenial equaions. Theorem 2.2 (Carah eodory) Le Q: Ω E E be a mapping such ha Q(., x ) is measurable for all x E and Q( ω,.) is coninuous for allω Ω. Then he map ( ω, x ) Q( ω, x ) is joinly measurable. The following lemma is useful in he sudy of second order iniial value problems of ordinary random differenial equaions via fixed poin echniques.

3 Exisence Theory of Second Order Random Differenial Equaions 35 Lemma2.. For any funcion h: J L( J, R), a funcion x : J C ( J, R) is a soluion o he differenial equaion, () () ( ) ( ) x" = h a. e. J, x = q, x = q, (2.) if and only if i is a soluion of he inegral equaion. () = + + () x q q ( s) h s. (2.2) Exisence Resuls Some basic definiions are needed summarise as follows Definiion 3. A Funcion f : J R Ω Ris called random Carah eodory if he following condiions are saisfied:-. he map(, ω) (, x, ω) is joinly measurable for all x R, and 2. he map x f(, x, ω) is coninuous for all J and ω Ω. Definiion 3.2 A Carah eodory funcion f : J R Ω Ris called random L - Caraheodory if for each real number r > here is a measurable and bounded funcion h : Ω L( J, R) such ha r f (, x, ω) h (, ω) a.e. J r for all ω Ω and x R wih x r. Similarly, a Carah eodory funcion f is called random L R -Carah eodory if here is a measurable and bounded funcion h: Ω L( J, R) such ha f (, x, ω) h(, ω) a.e. J for all ω Ω and x R. I consider he following se of hypoheses in wha follows: (H ) The funcions q, q : Ω Rare measurable and bounded wih Q ( ) = Sup q ω Ω ω and Q ( ) = Sup q ω Ω ω. (H ) The funcions f is random Carah eodory on J R Ω. (H 2 ) There exiss a measurable and bounded funcion γ : Ω L( J, R) and a coninuous and nondecreasing funcion : R (, ) such ha γ +

4 36 D.S. Palimkar f (, x, ω) γ(, ω) ψ( x) a. e. J dr for all ω Ω and x R.Moreover, assume ha = for all C. ψ () r C Main Exisence Resul Theorem 3. Assume ha he hypoheses (H ) (H 2 ) hold. Suppose ha C dr > γ ( ω) L ψ ( r) (3.) for all ω Ω, where C = Q + QT. Then he RDE (.) has a random soluion defined on J. Proof. Se E = CJR (, ) and define a mapping Q: Ω E E by Q( ω) x( ) = q( ω) + q( ω) + ( s) f ( s, x( s, ω), ω) (3.2) for all J and ω Ω. Now he map q( ω) + q( ω) is coninuous for all ω Ω. Again, as he indefinie inegral is coninuous on J, Q( ω) defines a mapping Q: Ω E E. we show ha Q saisfies all he condiions of Corollary 2. on E. Firs, I show ha Q is random operaor on E. Since f (, xω, ) is random Carah eodory, he map ω f(, x, ω) is measurable in view of Theorem 2.2 Similarly, he produc ( s) f( s, x( s, ω), ω) of a coninuous and a measurable funcion is again measurable. Furher, he inegral is a limi of a finie sum of measurable funcions, herefore, he map ω q ( ω) + q ( ω) + ( s) f ( s, x( s, ω), ω) = Q( ω) x( ) is measurable. As a resul, Q is a random operaor on Ω E ino E. Le B be a bounded subse of E. hen, here is real number r > such ha x r for all x B. Nex, I show ha he random operaor Q( ω ) is coninuous on B. le { x n } be a sequence of poins in B converging o he poin x B. Then i is enough o prove ha Q( ) x () = Q( ) x() for all J and ω Ω. By he dominaed lim ω ω n n convergence heorem, hen,

5 Exisence Theory of Second Order Random Differenial Equaions 37 lim Q( ω ) x ( ) n n = lim q ( ω) + q ( ω) + ( s) f ( s, x ( s, ω), ω) n lim [ n ] = q ( ω) + q ( ω) + ( s) f ( s, x ( s, ω), ω) n [ ] = q ( ω) + q ( ω) + ( s) f ( s, x( s, ω), ω) = Q( ω) x( ) for all J and ω Ω. This shows ha Q( ω ) is a coninuous random operaor on E. Now, I show ha Q( ω ) is a oally bounded random operaor on E. I prove ha Q( ω)( B) is a oally bounded subse of E for each bounded subse B of E. To finish, i is enough o prove ha Q( ω)( B) is a uniformly bounded and equi-coninuous se in E for each ω Ω. Since he map ω γ(, ω) is bounded, by hypohesis (H ), here is a consan c such ha γω ( ) L c for all ω Ω. Leω Ω be fixed. Then for any x : Ω B, one has, Q( ω) x( ) q ( ω) + q ( ω) + ( s) f ( s, x( s, ω), ω) q ( ω ) + q ( ω ) + ( s) γ ( s, ω ) ψ ( x( s, ω ) ) Q + Q + ( s) γ ( s, ω ) ψ ( x( ω ) ) T ( Q + Q T ) + T γ ( ω) ψ ( r ) K, L n for all J, where K = ( Q + Q T ) + ctψ ( r ). This shows ha Q( ω )( B) is a uniformly bounded subse of E for each ω Ω. Nex, I show ha Q( ω)( B) is an equi-coninuous se in E. Le x B be arbirary. Then, for any, 2 J, one has Q( ω) x( ) Q( ω) x( ) q ( ω) q ( ω) ( s) f ( s, x( s, ω), ω) ( s) f ( s, x( s, ω), ω) 2

6 38 D.S. Palimkar q ( ω ) 2 + ( s) f ( s, x( s, ω), ω) ( s) f ( s, x( s, ω), ω) ( s) f ( s, x( s, ω), ω) ( s) f ( s, x( s, ω), ω) 2 2 q ( ω ) 2 T + ( ) f ( s, x( s, ω), ω) + ( s) f ( s, x( s, ω), ω) Q + ( ) f ( s, x( s, ω), ω) + T f ( s, x( s, ω), ω) T Q + f ( s, x( s, ω), ω) T T f ( s, x( s, ω), ω) Q + γ ( s, ω) ψ ( x( s, ω) T Tγ ( s, ω) ψ ( x( s, ω) Q + γ ( ω) ψ ( r) p(, ω) p(, ω) 2 L [ Q + cψ ( r)] + p(, ω) p(, ω) (3.3) for all ω Ω, where 2 2 p(, ω) = Tγ( s, ωψ ) ( r). 2 Hence, for all, J 2, Q( ω) x( ) Q( ω) x( 2) as 2, uniformly for all x B and ω Ω. Therefore, Q( ω)( B) is an equi-coninuous se in E. As Q( ω )( B) is uniformly bounded and equi-coninuous, i is compac by he Arzela-Ascolli heorem for each ω Ω. Consequenly, Q( ω) is a compleely coninuous random operaor on B. Finally, I prove ha he se ε given in conclusion (ii) of corollary 2. does no hold. Le u ε be arbirary and le ω Ω be fixed. Then u (, ω) = λq( ω) u ( ) for all J and ω Ω, where < λ <. Then, one has u (, ω) λ Q( ω) u ()

7 Exisence Theory of Second Order Random Differenial Equaions 39 q ( ω ) + q ( ω ) + ( s) f ( s, u( s, ω ), ω ) Q + Q T + ( s) γ( s, ω) ψ ( u( s, ω) ) C + T + γ ( s, ω) ψ ( u( s, ω) ) for all J andω Ω, wherec = Q + QT. m Le mω s [, ] * (, ω) u (, ω) (, ) = sup us (, ω). Then, here is a =. Then from he inequaliy (3.3) i follows ha * [, ] such ha m * (, ω) = u (, ω) * = + + Q Q T T γ ( s, ω) ψ ( u( s, ω) ) C + T + T γ ( s, ω) ψ ( m( s, ω )). ω γ ω ψ ω Pu, w(, ) = C + T ( s, ) ( m( s, )) for J. Now differeniaing his wih respec o, we obain w (, ω) = Tγ(, ω) ψ ( m(, ω) w(, ω) = C, for J. Fom he above inequaliy, I obain w (, ω) Tγ(, ω) ψ ( w(, ω), w(, ω) = C, or w (, ω ) ψ ( w (, ω)) w(, ω ) = C. Tγ (, ω ), Inegraing from o, we ge w ( s, ω ) T γ( s, ω ) ψ ( ws (, ω )). By change of variable,

8 3 D.S. Palimkar w(, ω ) C dr ψ dr T γ ( ω ) < = L ( r) ψ ( r) C. Now an applicaion of he mean value heorem for inegral calculus, here exiss a consan M > such ha u (, ω) m (, ω) w (, ω) M for all J andω Ω. Hence he condiion (ii) of Corollary 2. does no hold. As a resul, he condiion(i) hol and he operaor equaion Q( ω ) x = x has a random soluion. This furher implies ha he random differenial equaion(.)has a random soluion defined on Ω J. This complees he proof. Acknowledgemen This aricle is an some oupu resul of Minor Research Projec funded by UGC, New Delhi, India. References [] T. Bharucha-Reid, On he heory of random equaions, Proc. Symp.Appl.6 h, (963), 4-69, Ame.Soc., Providence, Rhode, Island, (964). [2] A.T. Bharucha-Reid, Random Inegral Equaions, Academic Press, New York, 972. [3] K. Deimling, Muli-valued Differenial Equaions, De Gruyer, Berlin, 998. [4] B. C. Dhage, Some algebraic and opological random fixed poin heorems wih applicaions o nonlinear random inegral equaions, Tamkang J. Mah. 35(24), [5] B. C. Dhage, A random version of a Schaefer ype fixed poin heorem wih applicaions o funcional random inegral equaions, Nonlinear Func. Anal. Appl 9 (24), [6] B. C. Dhage, Monoone ieraive echnique for Carah eodory heory of nonlinear funcional random inegral equaions, Tamkang J. Mah. 35 (24), [7] K. P. Gupa, Measure Theory, Springer-Verlag, New York, Berlin, 97. [8] P.Hans, Random Fixed Poin Theorem, Transacions of Firs Prague Conference on Informaion Theory, Saisical Decision Funcions, Pure and Applied Mahemaics, Marcel Dekker, New York, 994. [9] S. Heikkila and V. Lakshmikanham, Monoone ieraive echnique for disconinuous onlinear differenial equaions, Pure and Applied Mahs., Marcel Dekker, New York, 994. [] J. Heimmelberg, Measurable relaions, Fund. Mah. 87 (975), [] S. Ioh, Random fixed poin heorems wih applicaions o random differenial equaions in Banach spaces, J. Mah. Anal. Appl. 67 (979),

9 Exisence Theory of Second Order Random Differenial Equaions 3 [2] G. S. Ladde and V. Lakshmikanham, Random Di erenial Inequaliies, Academic Press, New York, 98. [3] V. Lakshmikanham and S. Leela, Remarks on firs and second order periodic boundary valueproblem, Nonlinear Anal. 8 (984), [4] E.Zeidler, Nonlinear Funcional analysis and Applicaions:Par-I, SpringerVerlag, 985

10

Approximating positive solutions of nonlinear first order ordinary quadratic differential equations

Approximating positive solutions of nonlinear first order ordinary quadratic differential equations Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/233835.25.2367 APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Approximaing posiive soluions of nonlinear firs order ordinary quadraic

More information

The Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales

The Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions

More information

Positive continuous solution of a quadratic integral equation of fractional orders

Positive continuous solution of a quadratic integral equation of fractional orders Mah. Sci. Le., No., 9-7 (3) 9 Mahemaical Sciences Leers An Inernaional Journal @ 3 NSP Naural Sciences Publishing Cor. Posiive coninuous soluion of a quadraic inegral equaion of fracional orders A. M.

More information

On Oscillation of a Generalized Logistic Equation with Several Delays

On Oscillation of a Generalized Logistic Equation with Several Delays Journal of Mahemaical Analysis and Applicaions 253, 389 45 (21) doi:1.16/jmaa.2.714, available online a hp://www.idealibrary.com on On Oscillaion of a Generalized Logisic Equaion wih Several Delays Leonid

More information

Ann. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:

Ann. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL: Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS

More information

CONTRIBUTION TO IMPULSIVE EQUATIONS

CONTRIBUTION TO IMPULSIVE EQUATIONS European Scienific Journal Sepember 214 /SPECIAL/ ediion Vol.3 ISSN: 1857 7881 (Prin) e - ISSN 1857-7431 CONTRIBUTION TO IMPULSIVE EQUATIONS Berrabah Faima Zohra, MA Universiy of sidi bel abbes/ Algeria

More information

ON FRACTIONAL RANDOM DIFFERENTIAL EQUATIONS WITH DELAY. Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong

ON FRACTIONAL RANDOM DIFFERENTIAL EQUATIONS WITH DELAY. Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong Opuscula Mah. 36, no. 4 (26), 54 556 hp://dx.doi.org/.7494/opmah.26.36.4.54 Opuscula Mahemaica ON FRACTIONAL RANDOM DIFFERENTIAL EQUATIONS WITH DELAY Ho Vu, Nguyen Ngoc Phung, and Nguyen Phuong Communicaed

More information

Monotonic Solutions of a Class of Quadratic Singular Integral Equations of Volterra type

Monotonic Solutions of a Class of Quadratic Singular Integral Equations of Volterra type In. J. Conemp. Mah. Sci., Vol. 2, 27, no. 2, 89-2 Monoonic Soluions of a Class of Quadraic Singular Inegral Equaions of Volerra ype Mahmoud M. El Borai Deparmen of Mahemaics, Faculy of Science, Alexandria

More information

Existence of positive solution for a third-order three-point BVP with sign-changing Green s function

Existence of positive solution for a third-order three-point BVP with sign-changing Green s function Elecronic Journal of Qualiaive Theory of Differenial Equaions 13, No. 3, 1-11; hp://www.mah.u-szeged.hu/ejqde/ Exisence of posiive soluion for a hird-order hree-poin BVP wih sign-changing Green s funcion

More information

Properties Of Solutions To A Generalized Liénard Equation With Forcing Term

Properties Of Solutions To A Generalized Liénard Equation With Forcing Term Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick

More information

Existence of positive solutions for second order m-point boundary value problems

Existence of positive solutions for second order m-point boundary value problems ANNALES POLONICI MATHEMATICI LXXIX.3 (22 Exisence of posiive soluions for second order m-poin boundary value problems by Ruyun Ma (Lanzhou Absrac. Le α, β, γ, δ and ϱ := γβ + αγ + αδ >. Le ψ( = β + α,

More information

EXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN DIFFERENCE-DIFFERENTIAL EQUATIONS

EXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN DIFFERENCE-DIFFERENTIAL EQUATIONS Elecronic Journal of Differenial Equaions, Vol. 29(29), No. 49, pp. 2. ISSN: 72-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu EXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN

More information

Some New Uniqueness Results of Solutions to Nonlinear Fractional Integro-Differential Equations

Some New Uniqueness Results of Solutions to Nonlinear Fractional Integro-Differential Equations Annals of Pure and Applied Mahemaics Vol. 6, No. 2, 28, 345-352 ISSN: 2279-87X (P), 2279-888(online) Published on 22 February 28 www.researchmahsci.org DOI: hp://dx.doi.org/.22457/apam.v6n2a Annals of

More information

Asymptotic instability of nonlinear differential equations

Asymptotic instability of nonlinear differential equations Elecronic Journal of Differenial Equaions, Vol. 1997(1997), No. 16, pp. 1 7. ISSN: 172-6691. URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp (login: fp) 147.26.13.11 or 129.12.3.113 Asympoic insabiliy

More information

EXISTENCE OF NON-OSCILLATORY SOLUTIONS TO FIRST-ORDER NEUTRAL DIFFERENTIAL EQUATIONS

EXISTENCE OF NON-OSCILLATORY SOLUTIONS TO FIRST-ORDER NEUTRAL DIFFERENTIAL EQUATIONS Elecronic Journal of Differenial Equaions, Vol. 206 (206, No. 39, pp.. ISSN: 072-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu EXISTENCE OF NON-OSCILLATORY SOLUTIONS TO

More information

Convergence of the Neumann series in higher norms

Convergence of the Neumann series in higher norms Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann

More information

Research Article Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations

Research Article Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations Hindawi Publishing Corporaion Boundary Value Problems Volume 11, Aricle ID 19156, 11 pages doi:1.1155/11/19156 Research Aricle Exisence and Uniqueness of Periodic Soluion for Nonlinear Second-Order Ordinary

More information

Research Article Existence of the Solution for System of Coupled Hybrid Differential Equations with Fractional Order and Nonlocal Conditions

Research Article Existence of the Solution for System of Coupled Hybrid Differential Equations with Fractional Order and Nonlocal Conditions Inernaional Differenial Equaions Volume 26, Aricle ID 4726526, 9 pages hp://dx.doi.org/.55/26/4726526 Research Aricle Exisence of he Soluion for Sysem of Coupled Hybrid Differenial Equaions wih Fracional

More information

TO our knowledge, most exciting results on the existence

TO our knowledge, most exciting results on the existence IAENG Inernaional Journal of Applied Mahemaics, 42:, IJAM_42 2 Exisence and Uniqueness of a Periodic Soluion for hird-order Delay Differenial Equaion wih wo Deviaing Argumens A. M. A. Abou-El-Ela, A. I.

More information

EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar

EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 ISSN 225-353 EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D S Palikar Depare of Maheaics, Vasarao Naik College, Naded

More information

A Necessary and Sufficient Condition for the Solutions of a Functional Differential Equation to Be Oscillatory or Tend to Zero

A Necessary and Sufficient Condition for the Solutions of a Functional Differential Equation to Be Oscillatory or Tend to Zero JOURNAL OF MAEMAICAL ANALYSIS AND APPLICAIONS 24, 7887 1997 ARICLE NO. AY965143 A Necessary and Sufficien Condiion for he Soluions of a Funcional Differenial Equaion o Be Oscillaory or end o Zero Piambar

More information

Solution of Integro-Differential Equations by Using ELzaki Transform

Solution of Integro-Differential Equations by Using ELzaki Transform Global Journal of Mahemaical Sciences: Theory and Pracical. Volume, Number (), pp. - Inernaional Research Publicaion House hp://www.irphouse.com Soluion of Inegro-Differenial Equaions by Using ELzaki Transform

More information

Research Article Existence and Uniqueness of Positive and Nondecreasing Solutions for a Class of Singular Fractional Boundary Value Problems

Research Article Existence and Uniqueness of Positive and Nondecreasing Solutions for a Class of Singular Fractional Boundary Value Problems Hindawi Publishing Corporaion Boundary Value Problems Volume 29, Aricle ID 42131, 1 pages doi:1.1155/29/42131 Research Aricle Exisence and Uniqueness of Posiive and Nondecreasing Soluions for a Class of

More information

Example on p. 157

Example on p. 157 Example 2.5.3. Le where BV [, 1] = Example 2.5.3. on p. 157 { g : [, 1] C g() =, g() = g( + ) [, 1), var (g) = sup g( j+1 ) g( j ) he supremum is aken over all he pariions of [, 1] (1) : = < 1 < < n =

More information

Sobolev-type Inequality for Spaces L p(x) (R N )

Sobolev-type Inequality for Spaces L p(x) (R N ) In. J. Conemp. Mah. Sciences, Vol. 2, 27, no. 9, 423-429 Sobolev-ype Inequaliy for Spaces L p(x ( R. Mashiyev and B. Çekiç Universiy of Dicle, Faculy of Sciences and Ars Deparmen of Mahemaics, 228-Diyarbakir,

More information

L 1 -Solutions for Implicit Fractional Order Differential Equations with Nonlocal Conditions

L 1 -Solutions for Implicit Fractional Order Differential Equations with Nonlocal Conditions Filoma 3:6 (26), 485 492 DOI.2298/FIL66485B Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma L -Soluions for Implici Fracional Order Differenial

More information

Oscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson

Oscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,

More information

On Gronwall s Type Integral Inequalities with Singular Kernels

On Gronwall s Type Integral Inequalities with Singular Kernels Filoma 31:4 (217), 141 149 DOI 1.2298/FIL17441A Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma On Gronwall s Type Inegral Inequaliies

More information

MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE

MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE Topics MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES 2-6 3. FUNCTION OF A RANDOM VARIABLE 3.2 PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE 3.3 EXPECTATION AND MOMENTS

More information

CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS

CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS SARAJEVO JOURNAL OF MATHEMATICS Vol.10 (22 (2014, 67 76 DOI: 10.5644/SJM.10.1.09 CERTAIN CLASSES OF SOLUTIONS OF LAGERSTROM EQUATIONS ALMA OMERSPAHIĆ AND VAHIDIN HADŽIABDIĆ Absrac. This paper presens sufficien

More information

Undetermined coefficients for local fractional differential equations

Undetermined coefficients for local fractional differential equations Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani

More information

On Two Integrability Methods of Improper Integrals

On Two Integrability Methods of Improper Integrals Inernaional Journal of Mahemaics and Compuer Science, 13(218), no. 1, 45 5 M CS On Two Inegrabiliy Mehods of Improper Inegrals H. N. ÖZGEN Mahemaics Deparmen Faculy of Educaion Mersin Universiy, TR-33169

More information

On two general nonlocal differential equations problems of fractional orders

On two general nonlocal differential equations problems of fractional orders Malaya Journal of Maemaik, Vol. 6, No. 3, 478-482, 28 ps://doi.org/.26637/mjm63/3 On wo general nonlocal differenial equaions problems of fracional orders Abd El-Salam S. A. * and Gaafar F. M.2 Absrac

More information

On a Fractional Stochastic Landau-Ginzburg Equation

On a Fractional Stochastic Landau-Ginzburg Equation Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn

More information

Variational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations

Variational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 1, Issue 6 Ver. II (Nov - Dec. 214), PP 48-54 Variaional Ieraion Mehod for Solving Sysem of Fracional Order Ordinary Differenial

More information

Existence of multiple positive periodic solutions for functional differential equations

Existence of multiple positive periodic solutions for functional differential equations J. Mah. Anal. Appl. 325 (27) 1378 1389 www.elsevier.com/locae/jmaa Exisence of muliple posiive periodic soluions for funcional differenial equaions Zhijun Zeng a,b,,libi a, Meng Fan a a School of Mahemaics

More information

EXISTENCE AND ITERATION OF MONOTONE POSITIVE POLUTIONS FOR MULTI-POINT BVPS OF DIFFERENTIAL EQUATIONS

EXISTENCE AND ITERATION OF MONOTONE POSITIVE POLUTIONS FOR MULTI-POINT BVPS OF DIFFERENTIAL EQUATIONS U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 2 ISSN 223-727 EXISTENCE AND ITERATION OF MONOTONE POSITIVE POLUTIONS FOR MULTI-POINT BVPS OF DIFFERENTIAL EQUATIONS Yuji Liu By applying monoone ieraive meho,

More information

arxiv:math/ v1 [math.nt] 3 Nov 2005

arxiv:math/ v1 [math.nt] 3 Nov 2005 arxiv:mah/0511092v1 [mah.nt] 3 Nov 2005 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON AND S. M. GONEK Absrac. Le πs denoe he argumen of he Riemann zea-funcion a he poin 1 + i. Assuming

More information

STABILITY OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS

STABILITY OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS Elecronic Journal of Differenial Equaions, Vol. 217 217, No. 118, pp. 1 14. ISSN: 172-6691. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu STABILITY OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

More information

arxiv: v1 [math.gm] 4 Nov 2018

arxiv: v1 [math.gm] 4 Nov 2018 Unpredicable Soluions of Linear Differenial Equaions Mara Akhme 1,, Mehme Onur Fen 2, Madina Tleubergenova 3,4, Akylbek Zhamanshin 3,4 1 Deparmen of Mahemaics, Middle Eas Technical Universiy, 06800, Ankara,

More information

EXERCISES FOR SECTION 1.5

EXERCISES FOR SECTION 1.5 1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler

More information

EIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON TIME SCALES

EIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON TIME SCALES Elecronic Journal of Differenial Equaions, Vol. 27 (27, No. 37, pp. 3. ISSN: 72-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu EIGENVALUE PROBLEMS FOR SINGULAR MULTI-POINT DYNAMIC EQUATIONS ON

More information

Olaru Ion Marian. In 1968, Vasilios A. Staikos [6] studied the equation:

Olaru Ion Marian. In 1968, Vasilios A. Staikos [6] studied the equation: ACTA UNIVERSITATIS APULENSIS No 11/2006 Proceedings of he Inernaional Conference on Theory and Applicaion of Mahemaics and Informaics ICTAMI 2005 - Alba Iulia, Romania THE ASYMPTOTIC EQUIVALENCE OF THE

More information

POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER

POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER POSITIVE PERIODIC SOLUTIONS OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER GUANG ZHANG AND SUI SUN CHENG Received 5 November 21 This aricle invesigaes he exisence of posiive

More information

CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR

CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR Annales Academiæ Scieniarum Fennicæ Mahemaica Volumen 31, 2006, 39 46 CHARACTERIZATION OF REARRANGEMENT INVARIANT SPACES WITH FIXED POINTS FOR THE HARDY LITTLEWOOD MAXIMAL OPERATOR Joaquim Marín and Javier

More information

POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION

POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION Novi Sad J. Mah. Vol. 32, No. 2, 2002, 95-108 95 POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION Hajnalka Péics 1, János Karsai 2 Absrac. We consider he scalar nonauonomous neural delay differenial

More information

An Introduction to Malliavin calculus and its applications

An Introduction to Malliavin calculus and its applications An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214

More information

An Invariance for (2+1)-Extension of Burgers Equation and Formulae to Obtain Solutions of KP Equation

An Invariance for (2+1)-Extension of Burgers Equation and Formulae to Obtain Solutions of KP Equation Commun Theor Phys Beijing, China 43 2005 pp 591 596 c Inernaional Academic Publishers Vol 43, No 4, April 15, 2005 An Invariance for 2+1-Eension of Burgers Equaion Formulae o Obain Soluions of KP Equaion

More information

Engineering Letter, 16:4, EL_16_4_03

Engineering Letter, 16:4, EL_16_4_03 3 Exisence In his secion we reduce he problem (5)-(8) o an equivalen problem of solving a linear inegral equaion of Volerra ype for C(s). For his purpose we firs consider following free boundary problem:

More information

On the Oscillation of Nonlinear Fractional Differential Systems

On the Oscillation of Nonlinear Fractional Differential Systems On he Oscillaion of Nonlinear Fracional Differenial Sysems Vadivel Sadhasivam, Muhusamy Deepa, Nagamanickam Nagajohi Pos Graduae and Research Deparmen of Mahemaics,Thiruvalluvar Governmen Ars College (Affli.

More information

Boundedness and Exponential Asymptotic Stability in Dynamical Systems with Applications to Nonlinear Differential Equations with Unbounded Terms

Boundedness and Exponential Asymptotic Stability in Dynamical Systems with Applications to Nonlinear Differential Equations with Unbounded Terms Advances in Dynamical Sysems and Applicaions. ISSN 0973-531 Volume Number 1 007, pp. 107 11 Research India Publicaions hp://www.ripublicaion.com/adsa.hm Boundedness and Exponenial Asympoic Sabiliy in Dynamical

More information

Existence of non-oscillatory solutions of a kind of first-order neutral differential equation

Existence of non-oscillatory solutions of a kind of first-order neutral differential equation MATHEMATICA COMMUNICATIONS 151 Mah. Commun. 22(2017), 151 164 Exisence of non-oscillaory soluions of a kind of firs-order neural differenial equaion Fanchao Kong Deparmen of Mahemaics, Hunan Normal Universiy,

More information

Essential Maps and Coincidence Principles for General Classes of Maps

Essential Maps and Coincidence Principles for General Classes of Maps Filoma 31:11 (2017), 3553 3558 hps://doi.org/10.2298/fil1711553o Published by Faculy of Sciences Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma Essenial Maps Coincidence

More information

Nonlinear Fuzzy Stability of a Functional Equation Related to a Characterization of Inner Product Spaces via Fixed Point Technique

Nonlinear Fuzzy Stability of a Functional Equation Related to a Characterization of Inner Product Spaces via Fixed Point Technique Filoma 29:5 (2015), 1067 1080 DOI 10.2298/FI1505067W Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma Nonlinear Fuzzy Sabiliy of a Funcional

More information

The Existence, Uniqueness and Stability of Almost Periodic Solutions for Riccati Differential Equation

The Existence, Uniqueness and Stability of Almost Periodic Solutions for Riccati Differential Equation ISSN 1749-3889 (prin), 1749-3897 (online) Inernaional Journal of Nonlinear Science Vol.5(2008) No.1,pp.58-64 The Exisence, Uniqueness and Sailiy of Almos Periodic Soluions for Riccai Differenial Equaion

More information

Chapter 2. First Order Scalar Equations

Chapter 2. First Order Scalar Equations Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.

More information

SOLUTIONS APPROACHING POLYNOMIALS AT INFINITY TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

SOLUTIONS APPROACHING POLYNOMIALS AT INFINITY TO NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS Elecronic Journal of Differenial Equaions, Vol. 2005(2005, No. 79, pp. 1 25. ISSN: 1072-6691. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu (login: fp SOLUIONS APPROACHING POLYNOMIALS

More information

Omega-limit sets and bounded solutions

Omega-limit sets and bounded solutions arxiv:3.369v [mah.gm] 3 May 6 Omega-limi ses and bounded soluions Dang Vu Giang Hanoi Insiue of Mahemaics Vienam Academy of Science and Technology 8 Hoang Quoc Vie, 37 Hanoi, Vienam e-mail: dangvugiang@yahoo.com

More information

arxiv: v1 [math.fa] 9 Dec 2018

arxiv: v1 [math.fa] 9 Dec 2018 AN INVERSE FUNCTION THEOREM CONVERSE arxiv:1812.03561v1 [mah.fa] 9 Dec 2018 JIMMIE LAWSON Absrac. We esablish he following converse of he well-known inverse funcion heorem. Le g : U V and f : V U be inverse

More information

On the probabilistic stability of the monomial functional equation

On the probabilistic stability of the monomial functional equation Available online a www.jnsa.com J. Nonlinear Sci. Appl. 6 (013), 51 59 Research Aricle On he probabilisic sabiliy of he monomial funcional equaion Claudia Zaharia Wes Universiy of Timişoara, Deparmen of

More information

International Journal of Pure and Applied Mathematics Volume 56 No ,

International Journal of Pure and Applied Mathematics Volume 56 No , Inernaional Journal of Pure and Applied Mahemaics Volume 56 No. 2 2009, 165-172 THE GENERALIZED SOLUTIONS OF THE FUZZY DIFFERENTIAL INCLUSIONS Andrej V. Plonikov 1, Naalia V. Skripnik 2 1 Deparmen of Numerical

More information

arxiv: v1 [math.ca] 15 Nov 2016

arxiv: v1 [math.ca] 15 Nov 2016 arxiv:6.599v [mah.ca] 5 Nov 26 Counerexamples on Jumarie s hree basic fracional calculus formulae for non-differeniable coninuous funcions Cheng-shi Liu Deparmen of Mahemaics Norheas Peroleum Universiy

More information

Dynamical systems method for solving linear ill-posed problems

Dynamical systems method for solving linear ill-posed problems ANNALES POLONICI MATHEMATICI * (2*) Dynamical sysems mehod for solving linear ill-posed problems by A. G. Ramm (Manhaan, KS) Absrac. Various versions of he Dynamical Sysems Mehod (DSM) are proposed for

More information

On Fixed Point Theorem in Fuzzy2- Metric Spaces for Integral type Inequality

On Fixed Point Theorem in Fuzzy2- Metric Spaces for Integral type Inequality On Fixed Poin Theorem in Fuzzy- Meric Spaces for Inegral ype Inequaliy Rasik M. Pael, Ramakan Bhardwaj Research Scholar, CMJ Universiy, Shillong, Meghalaya, India Truba Insiue of Engineering & Informaion

More information

arxiv: v1 [math.pr] 19 Feb 2011

arxiv: v1 [math.pr] 19 Feb 2011 A NOTE ON FELLER SEMIGROUPS AND RESOLVENTS VADIM KOSTRYKIN, JÜRGEN POTTHOFF, AND ROBERT SCHRADER ABSTRACT. Various equivalen condiions for a semigroup or a resolven generaed by a Markov process o be of

More information

A remark on the H -calculus

A remark on the H -calculus A remark on he H -calculus Nigel J. Kalon Absrac If A, B are secorial operaors on a Hilber space wih he same domain range, if Ax Bx A 1 x B 1 x, hen i is a resul of Auscher, McInosh Nahmod ha if A has

More information

A New Perturbative Approach in Nonlinear Singularity Analysis

A New Perturbative Approach in Nonlinear Singularity Analysis Journal of Mahemaics and Saisics 7 (: 49-54, ISSN 549-644 Science Publicaions A New Perurbaive Approach in Nonlinear Singulariy Analysis Ta-Leung Yee Deparmen of Mahemaics and Informaion Technology, The

More information

Cash Flow Valuation Mode Lin Discrete Time

Cash Flow Valuation Mode Lin Discrete Time IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics

More information

LIMIT AND INTEGRAL PROPERTIES OF PRINCIPAL SOLUTIONS FOR HALF-LINEAR DIFFERENTIAL EQUATIONS. 1. Introduction

LIMIT AND INTEGRAL PROPERTIES OF PRINCIPAL SOLUTIONS FOR HALF-LINEAR DIFFERENTIAL EQUATIONS. 1. Introduction ARCHIVUM MATHEMATICUM (BRNO) Tomus 43 (2007), 75 86 LIMIT AND INTEGRAL PROPERTIES OF PRINCIPAL SOLUTIONS FOR HALF-LINEAR DIFFERENTIAL EQUATIONS Mariella Cecchi, Zuzana Došlá and Mauro Marini Absrac. Some

More information

4 Sequences of measurable functions

4 Sequences of measurable functions 4 Sequences of measurable funcions 1. Le (Ω, A, µ) be a measure space (complee, afer a possible applicaion of he compleion heorem). In his chaper we invesigae relaions beween various (nonequivalen) convergences

More information

Generalized Snell envelope and BSDE With Two general Reflecting Barriers

Generalized Snell envelope and BSDE With Two general Reflecting Barriers 1/22 Generalized Snell envelope and BSDE Wih Two general Reflecing Barriers EL HASSAN ESSAKY Cadi ayyad Universiy Poly-disciplinary Faculy Safi Work in progress wih : M. Hassani and Y. Ouknine Iasi, July

More information

Efficient Solution of Fractional Initial Value Problems Using Expanding Perturbation Approach

Efficient Solution of Fractional Initial Value Problems Using Expanding Perturbation Approach Journal of mahemaics and compuer Science 8 (214) 359-366 Efficien Soluion of Fracional Iniial Value Problems Using Expanding Perurbaion Approach Khosro Sayevand Deparmen of Mahemaics, Faculy of Science,

More information

Stability of General Cubic Mapping in Fuzzy Normed Spaces

Stability of General Cubic Mapping in Fuzzy Normed Spaces An. Ş. Univ. Ovidius Consanţa Vol. 20, 202, 29 50 Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces S. Javadi, J. M. Rassias Absrac We esablish some sabiliy resuls concerning he general cubic funcional

More information

SOME PROPERTIES OF GENERALIZED STRONGLY HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM NOOR, KHALIDA INAYAT NOOR, SABAH IFTIKHAR AND FARHAT SAFDAR

SOME PROPERTIES OF GENERALIZED STRONGLY HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM NOOR, KHALIDA INAYAT NOOR, SABAH IFTIKHAR AND FARHAT SAFDAR Inernaional Journal o Analysis and Applicaions Volume 16, Number 3 2018, 427-436 URL: hps://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-427 SOME PROPERTIES OF GENERALIZED STRONGLY HARMONIC

More information

Homotopy Perturbation Method for Solving Some Initial Boundary Value Problems with Non Local Conditions

Homotopy Perturbation Method for Solving Some Initial Boundary Value Problems with Non Local Conditions Proceedings of he World Congress on Engineering and Compuer Science 23 Vol I WCECS 23, 23-25 Ocober, 23, San Francisco, USA Homoopy Perurbaion Mehod for Solving Some Iniial Boundary Value Problems wih

More information

Correspondence should be addressed to Nguyen Buong,

Correspondence should be addressed to Nguyen Buong, Hindawi Publishing Corporaion Fixed Poin Theory and Applicaions Volume 011, Aricle ID 76859, 10 pages doi:101155/011/76859 Research Aricle An Implici Ieraion Mehod for Variaional Inequaliies over he Se

More information

2. Nonlinear Conservation Law Equations

2. Nonlinear Conservation Law Equations . Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear

More information

Coincidence Points of a Sequence of Multivalued Mappings in Metric Space with a Graph

Coincidence Points of a Sequence of Multivalued Mappings in Metric Space with a Graph mahemaics Aricle Coincidence Poins of a Sequence of Mulivalued Mappings in Meric Space wih a Graph Muhammad Nouman Aslam Khan,2,, Akbar Azam 2, and Nayyar Mehmood 3, *, School of Chemical and Maerials

More information

Fractional Method of Characteristics for Fractional Partial Differential Equations

Fractional Method of Characteristics for Fractional Partial Differential Equations Fracional Mehod of Characerisics for Fracional Parial Differenial Equaions Guo-cheng Wu* Modern Teile Insiue, Donghua Universiy, 188 Yan-an ilu Road, Shanghai 51, PR China Absrac The mehod of characerisics

More information

BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NON-DOUBLING MANIFOLDS WITH ENDS

BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NON-DOUBLING MANIFOLDS WITH ENDS BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NON-DOUBLING MANIFOLDS WITH ENDS XUAN THINH DUONG, JI LI, AND ADAM SIKORA Absrac Le M be a manifold wih ends consruced in [2] and be he Laplace-Belrami operaor on M

More information

Stability and Bifurcation in a Neural Network Model with Two Delays

Stability and Bifurcation in a Neural Network Model with Two Delays Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, 175-1731 Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy

More information

EXISTENCE OF S 2 -ALMOST PERIODIC SOLUTIONS TO A CLASS OF NONAUTONOMOUS STOCHASTIC EVOLUTION EQUATIONS

EXISTENCE OF S 2 -ALMOST PERIODIC SOLUTIONS TO A CLASS OF NONAUTONOMOUS STOCHASTIC EVOLUTION EQUATIONS Elecronic Journal of Qualiaive Theory of Differenial Equaions 8, No. 35, 1-19; hp://www.mah.u-szeged.hu/ejqde/ EXISTENCE OF S -ALMOST PERIODIC SOLUTIONS TO A CLASS OF NONAUTONOMOUS STOCHASTIC EVOLUTION

More information

Mapping Properties Of The General Integral Operator On The Classes R k (ρ, b) And V k (ρ, b)

Mapping Properties Of The General Integral Operator On The Classes R k (ρ, b) And V k (ρ, b) Applied Mahemaics E-Noes, 15(215), 14-21 c ISSN 167-251 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Mapping Properies Of The General Inegral Operaor On The Classes R k (ρ, b) And V k

More information

STABILITY OF PEXIDERIZED QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FUZZY NORMED SPASES

STABILITY OF PEXIDERIZED QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FUZZY NORMED SPASES Novi Sad J. Mah. Vol. 46, No. 1, 2016, 15-25 STABILITY OF PEXIDERIZED QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FUZZY NORMED SPASES N. Eghbali 1 Absrac. We deermine some sabiliy resuls concerning

More information

DISCRETE GRONWALL LEMMA AND APPLICATIONS

DISCRETE GRONWALL LEMMA AND APPLICATIONS DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE MAA NORTH CENTRAL SECTION MEETING AT UND 24 OCTOBER 29 Gronwall s lemma saes an inequaliy ha is useful in he heory of differenial equaions. Here is

More information

Stochastic Model for Cancer Cell Growth through Single Forward Mutation

Stochastic Model for Cancer Cell Growth through Single Forward Mutation Journal of Modern Applied Saisical Mehods Volume 16 Issue 1 Aricle 31 5-1-2017 Sochasic Model for Cancer Cell Growh hrough Single Forward Muaion Jayabharahiraj Jayabalan Pondicherry Universiy, jayabharahi8@gmail.com

More information

A NEW APPROACH FOR STUDYING FUZZY FUNCTIONAL EQUATIONS

A NEW APPROACH FOR STUDYING FUZZY FUNCTIONAL EQUATIONS IJMMS 28:12 2001) 733 741 PII. S0161171201006639 hp://ijmms.hindawi.com Hindawi Publishing Corp. A NEW APPROACH FOR STUDYING FUZZY FUNCTIONAL EQUATIONS ELIAS DEEBA and ANDRE DE KORVIN Received 29 January

More information

A Sharp Existence and Uniqueness Theorem for Linear Fuchsian Partial Differential Equations

A Sharp Existence and Uniqueness Theorem for Linear Fuchsian Partial Differential Equations A Sharp Exisence and Uniqueness Theorem for Linear Fuchsian Parial Differenial Equaions Jose Ernie C. LOPE Absrac This paper considers he equaion Pu = f, where P is he linear Fuchsian parial differenial

More information

On the Fourier Transform for Heat Equation

On the Fourier Transform for Heat Equation Applied Mahemaical Sciences, Vol. 8, 24, no. 82, 463-467 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/.2988/ams.24.45355 On he Fourier Transform for Hea Equaion P. Haarsa and S. Poha 2 Deparmen of Mahemaics,

More information

Variation of Parameters for Causal Operator Differential Equations

Variation of Parameters for Causal Operator Differential Equations Applied Mahemaics, 217, 8, 1883-192 hp://wwwscirporg/journal/am ISSN Online: 2152-7393 ISSN Prin: 2152-7385 Variaion of Parameers for Causal Operaor Differenial Equaions Reza R Ahangar Mahemaics Deparmen,

More information

SOME INEQUALITIES AND ABSOLUTE MONOTONICITY FOR MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND

SOME INEQUALITIES AND ABSOLUTE MONOTONICITY FOR MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND Commun. Korean Mah. Soc. 3 (6), No., pp. 355 363 hp://dx.doi.org/.434/ckms.6.3..355 SOME INEQUALITIES AND ABSOLUTE MONOTONICITY FOR MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND Bai-Ni Guo Feng Qi Absrac.

More information

Method For Solving Fuzzy Integro-Differential Equation By Using Fuzzy Laplace Transformation

Method For Solving Fuzzy Integro-Differential Equation By Using Fuzzy Laplace Transformation INERNAIONAL JOURNAL OF SCIENIFIC & ECHNOLOGY RESEARCH VOLUME 3 ISSUE 5 May 4 ISSN 77-866 Meod For Solving Fuzzy Inegro-Differenial Equaion By Using Fuzzy Laplace ransformaion Manmoan Das Danji alukdar

More information

L p -L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity

L p -L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity ANNALES POLONICI MATHEMATICI LIV.2 99) L p -L q -Time decay esimae for soluion of he Cauchy problem for hyperbolic parial differenial equaions of linear hermoelasiciy by Jerzy Gawinecki Warszawa) Absrac.

More information

POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE

POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE Urainian Mahemaical Journal, Vol. 55, No. 2, 2003 POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE A. G. Mazo UDC 517.983.27 We invesigae properies of posiive and monoone differenial sysems wih

More information

A Note on Superlinear Ambrosetti-Prodi Type Problem in a Ball

A Note on Superlinear Ambrosetti-Prodi Type Problem in a Ball A Noe on Superlinear Ambrosei-Prodi Type Problem in a Ball by P. N. Srikanh 1, Sanjiban Sanra 2 Absrac Using a careful analysis of he Morse Indices of he soluions obained by using he Mounain Pass Theorem

More information

Ordinary Differential Equations

Ordinary Differential Equations Lecure 22 Ordinary Differenial Equaions Course Coordinaor: Dr. Suresh A. Karha, Associae Professor, Deparmen of Civil Engineering, IIT Guwahai. In naure, mos of he phenomena ha can be mahemaically described

More information

Improved Approximate Solutions for Nonlinear Evolutions Equations in Mathematical Physics Using the Reduced Differential Transform Method

Improved Approximate Solutions for Nonlinear Evolutions Equations in Mathematical Physics Using the Reduced Differential Transform Method Journal of Applied Mahemaics & Bioinformaics, vol., no., 01, 1-14 ISSN: 179-660 (prin), 179-699 (online) Scienpress Ld, 01 Improved Approimae Soluions for Nonlinear Evoluions Equaions in Mahemaical Physics

More information

OSCILLATION CONSTANT FOR MODIFIED EULER TYPE HALF-LINEAR EQUATIONS

OSCILLATION CONSTANT FOR MODIFIED EULER TYPE HALF-LINEAR EQUATIONS Elecronic Journal of Differenial Equaions, Vol. 205 (205), No. 220, pp. 4. ISSN: 072-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu OSCILLATION CONSTANT FOR MODIFIED EULER

More information

MATH 5720: Gradient Methods Hung Phan, UMass Lowell October 4, 2018

MATH 5720: Gradient Methods Hung Phan, UMass Lowell October 4, 2018 MATH 5720: Gradien Mehods Hung Phan, UMass Lowell Ocober 4, 208 Descen Direcion Mehods Consider he problem min { f(x) x R n}. The general descen direcions mehod is x k+ = x k + k d k where x k is he curren

More information

Dynamic Systems and Applications 12 (2003) A SECOND-ORDER SELF-ADJOINT EQUATION ON A TIME SCALE

Dynamic Systems and Applications 12 (2003) A SECOND-ORDER SELF-ADJOINT EQUATION ON A TIME SCALE Dynamic Sysems and Applicaions 2 (2003) 20-25 A SECOND-ORDER SELF-ADJOINT EQUATION ON A TIME SCALE KIRSTEN R. MESSER Universiy of Nebraska, Deparmen of Mahemaics and Saisics, Lincoln NE, 68588, USA. E-mail:

More information