FUNCTIONAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH STATE-DEPENDENT DELAY
|
|
- Vivian Cook
- 5 years ago
- Views:
Transcription
1 Dynamic Systems and Applications 8 (29) FUNCTIONAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH STATE-DEPENDENT DELAY MOHAMED ABDALLA DARWISH AND SOTIRIS K. NTOUYAS Department of Mathematics, Faculty of Science Alexandria University at Damanhour, 225 Damanhour, Egypt Department of Mathematics, University of Ioannina, 45 Ioannina, Greece ABSTRACT. In this paper we study the existence of solutions for the initial value problem for functional differential equations, as well as, for neutral functional differential equations of fractional order with state-dependent delay. The nonlinear alternative of Leray-Schauder type is the main tool in our analysis. AMS (MOS) Subject Classification. 26A33, 26A42, 34K3. INTRODUCTION The purpose of this paper is to study the existence of solutions for initial value problems (IVP for short) for both functional differential equations of fractional order with state-dependent delay (.) D β y(t) = f(t, y ρ(t,yt)), for each t J = [, b], < β <, (.2) y(t) = ϕ(t), t (, ] as well as for neutral functional differential equations of fractional order with statedependent delay (.3) D β [y(t) g(t, y ρ(t,yt))] = f(t, y ρ(t,yt)), for each t J, (.4) y(t) = ϕ(t), t (, ], where D β is the standard Riemman-Liouville fractional derivative. Here, f : J B R, g : J B R and ρ : J B (, b] are appropriate given functions, ϕ B, ϕ() =, g(, ϕ) = and B is called a phase space that will be defined later (see Section 2). For any function y defined on (, b] and any t J, we denote by y t the element of B defined by y t (θ) = y(t + θ), θ (, ]. Received July 5, $5. c Dynamic Publishers, Inc.
2 54 M. A. DARWISH AND S. K. NTOUYAS The notion of the phase space B plays an important role in the study of both qualitative and quantitative theory for functional differential equations. A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato [2] (see also Kappel and Schappacher [9] and Schumacher [3]). For a detailed discussion on this topic we refer the reader to the book by Hino et al [8]. Functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equation has received a significant amount of attention in the last years, we refer to [2, 3, 5, 7, 3, 4, 5] and the references therein. On the other hand, the first serious attempt to give a logical definition of a fractional derivative is due to Liouville. Now, the fractional calculus topic is enjoying growing interest among scientists and engineers, see [9, 7, 2, 22, 24, 25, 26, 27] and references therein. Differential equations of fractional order play a very important role in describing some real world problems. For example some problems in physics, mechanics and other fields can be described with the help of fractional differential equations, see [8,, 7, 23, 27, 28, 29] and references therein. The theory of differential equations of fractional order has recently received a lot of attention and now constitutes a significant branch of nonlinear analysis. Numerous research papers and monographs have appeared devoted to fractional differential equations, for example see [, 2, 2, 26, 3]. Our approach is based on the nonlinear alternative of Leray-Schauder type []. These results can be considered as a contribution to this emerging field. 2. PRELIMINARIES In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let R + = {x R : x > }, and let C (R + ) be the space of all continuous functions on R +. Consider also the space C (R + ) of all continuous real functions on R + = {x R : x }, which later identify by abuse of notation, with the class of all f C (R + ) such that lim f(t) = f(+ ) R. t + By C(J, R) we denote the Banach space of all continuous functions from J into R with the norm y := sup{ y(t) : t J}, where denotes a suitable complete norm on R. Now, we recall some definitions and facts about fractional derivatives and fractional integrals of arbitrary orders, see [2, 25, 26, 27].
3 FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS 54 Definition 2.. The fractional primitive of order β > of a function h : R + R of order β R + is defined by I β h(t) = (t s) β h(s)ds, provided the right hand side exists pointwise on R +. Γ is the gamma function. For instance, I β h exists for all β >, when h C (R + ) L loc (R+ ); note also that when h C (R + ) then I β h C (R + ) and moreover I β h() =. Definition 2.2. The fractional derivative of order β > of a continuous function h : R + R is given by d β h(t) dt β = d Γ( β) dt = d dt I β a h(t). a (t s) β h(s)ds For our existence results we will need the following lemma. Lemma 2.3. [6] Let < β < and let h : (, b] R be continuous and lim h(t) = t + h( + ) R. Then y is a solution of the fractional integral equation y(t) = (t s) β h(y(s))ds, if and only if, y is a solution of the initial value problem for the fractional differential equation D β y(t) = h(y(t)), t (, b], y() =. In this paper, we will employ an axiomatic definition for the phase space B which is similar to those introduced in [8]. More precisely, B will be a linear space of all functions from (, ] to R endowed with a seminorm B satisfying the following axioms: (A) If y : (, b] R, b >, is continuous on J and y B, then for every t J the following conditions hold: (i) y t B, (ii) y t B K(t) sup{ y(s) : s t} + M(t) y B, (iii) y(t) H y t B, where H > is a constant, K : [, ) [, ) is continuous, M : [, ) [, ) is locally bounded and H, K, M are independent of y( ). (A-) For the function y( ) in (A), y t is a B-valued continuous function on [, b]. (A-2) The space B is complete. The next lemma is a consequence of the phase space axioms and is proved in [3].
4 542 M. A. DARWISH AND S. K. NTOUYAS Lemma 2.4. Let ϕ B and I = (γ, ] be such that ϕ t B for every t I. Assume that there exists a locally bounded function J ϕ : I [, ) such that ϕ t B J ϕ (t) ϕ B for every t I. If y : (, b] R is continuous on J and y = ϕ, then y t B (M b +J ϕ (max{γ, s }) ϕ B +K b sup{ y(θ) : θ [, max{, s}]}, s (γ, b], where, we denoted K b = sup t J K(t) and M b = sup t J M(t). Finally, we state the following generalization of Gronwall s lemma for singular kernels, whose proof can be found in [6], Lemma 7.., will be essential for our main results. Lemma 2.5. Let v : [, b] [, ) be a real function and w( ) is a nonnegative, locally integrable function on [, b] and there are constants a > and < β < such that v(s) v(t) w(t) + a (t s) ds, β then, there exists a constant K = K(β) such that for every t [, b]. v(t) w(t) + Ka w(s) (t s) β ds, 3. FDEs OF FRACTIONAL ORDER In this section, the nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions of problem (.) (.2). Let us start by defining what we mean by a solution of problem (.) (.2). Definition 3.. A function y : (, b] R is said to be a solution of (.) (.2) if y = ϕ, y ρ(s,ys) B for every s J and y(t) = f(s, y ρ(s,ys)) (t s) β ds, t J. In what follows we assume that ρ : J B (, b] is continuous and ϕ B and f satisfies the following hypotheses: (H) f is a continuous function; (H2) There exist p, q C(J, R + ) such that f(t, u) p(t) + q(t) u B for t J and each u B, and I β p < + ;
5 FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS 543 (H3) The function t ϕ t is well defined and continuous from the set R(ρ ) = {ρ(s, ψ) : (s, ψ) J B, ρ(s, ψ) } into B. Moreover, there exists a continuous and bounded function J ϕ : R(ρ ) (, ) such that ϕ t B J ϕ (t) ϕ B for every t R(ρ ). Remark 3.2. The hypothesis (H3) is adapted from [3], where we refer for remarks concerning this hypothesis. Theorem 3.3. Assume that the hypotheses (H) (H3) hold. If ρ(t, ψ) t for every (t, ψ) J B, then the IVP (.) (.2) has at least one solution on (, b]. Proof. Let Y = {u C(J, R) : u() = ϕ() = } endowed with the uniform convergence topology and N : Y Y be the operator defined by Ny(t) = f(s, ȳ ρ(s,ȳs)) ds, t J, (t s) β where ȳ : (, b] R is such that ȳ = ϕ and ȳ = y on J. From axiom (A) and our assumption on ϕ, we infer that Ny( ) is well defined and continuous. Let ϕ : (, b] R be the extension of ϕ to (, b] such that ϕ(θ) = ϕ() = on J and J ϕ = sup{j ϕ : s R(ρ )}. We will prove that N( ) is completely continuous from B r ( ϕ J, Y ) into B r ( ϕ J, Y ). Step : N is continuous on B r ( ϕ J, Y ). This was proved in [3, p. 55, Step 3]. Step 2: The set N(B r ( ϕ J, Y ))(t) = {Ny(t) : y B r ( ϕ J, Y )} is relatively compact in R for every t J. The case t = is obvious. Let < ǫ < t b. If y B r ( ϕ J, Y ), from Lemma 2.4 follows that and so ȳ ρ(t,ȳt) B r = (M b + J ϕ ) ϕ B + K b r (Ny)(t) = t f(s, ȳ ρ(s,ȳs)) (t s) β ds p(s) + q(s) ȳ ρ(s,ȳs) B (t s) β ds = bβ p Γ(β + ) + bβ q Γ(β + ) r := l Step 3: N maps bounded sets into equicontinuous sets of Y.
6 544 M. A. DARWISH AND S. K. NTOUYAS Let t, t 2 [, b], t < t 2 and let B r as in Step 2. Let y B r. Then for each t [, b], we have (Ny)(t 2 ) (Ny)(t ) = [(t 2 s) β (t s) β ]f(s, ȳ ρ(s,ȳs)) ds + 2 f(s, ȳ ρ(s,ȳs)) t (t 2 s) ds β p + r q t [(t s) β (t 2 s) β ]ds + p + r q t2 ds t (t 2 s) β p + r q [(t 2 t ) β + t β Γ(β + ) tβ 2 ] + p + r q (t 2 t ) β Γ(β + ) 2( p + r q ) (t 2 t ) β. Γ(β + ) As t t 2 the right-hand side of the above inequality tends to zero. The equicontinuity for the cases t < t 2 and t t 2 is obvious. As a consequence of Steps to 3, together with the Arzelá-Ascoli theorem, we can conclude that N is continuous and completely continuous. Step 4: (A priori bounds). with y λn(y) for λ (, ) and y U. We now show there exists an open set U Y Let y Y and y = λn(y) for some < λ <. Then for each t [, b] we have This implies by (H2) [ t f(s, ȳ ] ρ(s,ȳs)) y(t) = λ ds. (t s) β y(t) = t f(s, ȳ ρ(s,ȳs)) (t s) β ds bβ p Γ(β + ) + q p(s) + q(s)[(m b + J ϕ ) ϕ B + K b sup{ ȳ(s) : s [, t]}] ds (t s) β (M b + J ϕ ) ϕ B + K b sup{ ȳ(s) : s [, t]} (t s) β ds, since ρ(s, ȳ s ) s for every s J. Here J φ = sup{j φ (s) : s R(ρ )}. If µ(t) = (M b + J ϕ ) ϕ B +K b sup{ ȳ(s) : s [, max{, ρ(s, ȳ t )}] then we obtain µ(t) bβ p Γ(β + ) + q (t s) β µ(s)ds.
7 FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS 545 Then from Lemma 2.5, there exists K = K(β) such that we have Then Set µ(t) bβ p Γ(β + ) + q = bβ p Γ(β + ) bβ p Γ(β + ) K(β) (t s) β bβ p Γ(β + ) ds { + q } K(β) (t s) β ds { + q } Γ(β + ) K(β)bβ. { µ bβ p + q } Γ(β + ) Γ(β + ) K(β)bβ := M. U = {y Y : y < M + }. N : U Y is continuous and completely continuous. From the choice of U, there is no y U such that y = λn(y), for λ (, ). As a consequence of the nonlinear alternative of Leray-Schauder type [], we deduce that N has a fixed point y in U. 4. NFDEs OF FRACTIONAL ORDER In this section we give existence results for the IVP (.3) (.4). Definition 4.. A function y : (, b] R is said to be a solution of (.3) (.4) if y = ϕ, y ρ(s,ys) B for every s J and y(t) = g(s, y ρ(s,ys)) + (t s) β f(s, y ρ(s,ys))ds, t J. Theorem 4.2. Assume (H) (H2) and the following condition: (H4) the function g is continuous and completely continuous, and for any bounded set Q in B C([, b], R), the set {t g(t, y t ) : y Q} is equicontinuous in C([, b], R), and there exist constants K b d <, d 2 such that g(t, u) d u B + d 2, t [, b], u B. If ρ(t, ψ) t for every (t, ψ) J B, then the IVP (.3) (.4) has at least one solution on (, b]. Proof. Consider the operator N : C((, b], R) C((, b], R) defined by, ϕ(t), if t (, ], N (y)(t) = ϕ() g(, ϕ) + g(t, y ρ(t,yt)) + (t s) β f(s, y ρ(s,ys))ds, if t [, b].
8 546 M. A. DARWISH AND S. K. NTOUYAS In analogy to Theorem 3.3, we consider the operator N : Y Y defined by, t, (N y)(t) = g(t, ȳ ρ(s,ȳs)) + (t s) β f(s, ȳ ρ(s,ȳs))ds, t [, b]. We shall show that the operator N is continuous and completely continuous. Using (H4) it suffices to show that the operator N 2 : Y Y defined by, N 2 (y)(t) = (t s) β f(s, ȳ ρ(s,ȳs)) ds, t [, b], is continuous and completely continuous. This was proved in Theorem 3.3. y U. and We now show there exists an open set U Y with y λn (y) for λ (, ) and Let y Y and y = λn (y) for some < λ <. Then [ y(t) = λ g(s, ȳ ρ(s,ȳs)) + ] (t s) β f(s, ȳ ρ(s,ȳs)) ds, t [, b], y(t) d ((M b + J ϕ ) ϕ B + K b sup{ y(s) : s [, t]}) + d 2 + (t s) β [p(s) + q(s)((m b + J ϕ ) ϕ B + K b sup{ y(s) : s [, t]})ds d ((M b + J ϕ ) ϕ B + K b sup{ y(s) : s [, t]}) + d 2 + bβ p Γ(β + ) + q for t (, b]. If µ(t) = sup{ y(s) : s [, t]} then (t s) β ((M b + J ϕ ) ϕ B + K b sup{ y(s) : s [, t]})ds, µ(t) d (M b + J ϕ ) ϕ B + d K b µ(t) + d 2 + bβ p Γ(β + ) + bβ q Γ(β + ) + q K b (t s) β µ(s) ds d (M b + J ϕ ) ϕ B + d K b µ(t) + d 2 (t s) β (M b + J ϕ ) ϕ B + bβ p Γ(β + ) + bβ q Γ(β + ) bβ (M b + J ϕ ) ϕ B +K b q (t s) β µ(s) ds, t (, b],
9 FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS 547 or µ(t) K b d + K b q K b d Consequently µ K b d + K b q K b d [ d (M b + J ϕ ) ϕ B + d 2 + bβ p (t s) β µ(s) ds, t (, b]. [ d (M b + J ϕ ) ϕ B + d 2 + bβ p (t s) β µ(s) ds and by Lemma 2.5, there exists K = K(β) such that where µ Λ + Λ 2 K(β) Λ = K b d Set ] Γ(β + ) + bβ q Γ(β + ) (M b + J ϕ ) ϕ B ] Γ(β + ) + bβ q Γ(β + ) (M b + J ϕ ) ϕ B (t s) β Λ ds Λ + Λ 2 K(β)Λ b β := L, [ ] d (M b + J ϕ ) ϕ B + d 2 + bβ p Γ(β + ) + bβ q Γ(β + ) (M b + J ϕ ) ϕ B, Λ = K b K b d q. U = {y Y : y < L + }. From the choice of U there is no y U such that y = λn (y) for λ (, ). As a consequence of the nonlinear alternative of Leray-Schauder type [], we deduce that N has a fixed point y in U. Then N has a fixed point, which is a solution of the IVP (.3) (.4). Acknowledgement. This work was completed when the first author was visiting the Department of Mathematics, Massachusetts Institute of Technology, USA. It is a pleasure for him to express gratitude for the warm hospitality. Also, the authors would like to thank the referee for his corrections and valuable remarks. REFERENCES [] O.P. Agrawal, Analytical schemes for a new class of fractional differential equations, J. Phys. A 4 (2) (27), [2] W. Aiello, H.I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM Journal Applied Mathematics, 52(3) (992),
10 548 M. A. DARWISH AND S. K. NTOUYAS [3] M. Bartha, Periodic solutions for differential equations with state-dependent delay and positive feedback, Nonlinear Analysis TMA 53(6) (23), [4] M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (28), [5] Y. Cao, J. Fan and T.C. Gard, The effects of state-dependent time delay on a stage-structured population growth model, Nonlinear Analysis TMA, 9(2) (992), [6] D. Delboso and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 24 (996), [7] A. Domoshnitsky, M. Drakhlin and E. Litsyn, On equations with delay depending on solution, Nonlinear Analysis TMA, 49(5) (22), [8] M. Garh, A. Rao and S.L. Kalla, Fractional generalization of temperature fields problems in oil strata, Mat. Bilten 3 (26), [9] L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (99), [] W. G. Glockle and T. F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (995), [] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 23. [2] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 2 (978), 4. [3] E. Hernández, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. Real World Appl. 7(4) (26), [4] E. Hernández, M. Pierri and G. Goncalves, Existence results for an impulsive abstract partial differential equation with state-dependent delay, Comput. Math. Appl. 52(3-4) (26), [5] E. Hernández, R. Sakthivel, Rathinasamy and A. Tanaka, Existence results for impulsive evolution differential equations with state-dependent delay, Electron. J. Differential Equations 28 (28),. [6] D. Henry, Geometric Theory of Semilinear Parabolic Partial Differential Equations, Springer- Verlag, Berlin/New York, 989. [7] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2. [8] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, 473, Springer-Verlag, Berlin, 99. [9] F. Kappel and W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations 37 (98), [2] A. A. Kilbas, H. M. Srivastava and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 24. Elsevier Science B.V., Amsterdam, 26. [2] V. Lakshmikantham and J.V. Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math. () (28), [22] V. Lakshmikantham, L. Wen and B. Zhang, Theory of Differential Equations with Unbounded Delay, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 994. [23] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanis, in Fractals and Fractional Calculus in Continuum Mechanics (A. Carpinteri and F. Mainardi, Eds), pp , Springer-Verlag, Wien, 997.
11 FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS 549 [24] F. Metzler, W. Schick, H. G. Kilian and T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 3 (995), [25] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 993. [26] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 999. [27] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 993. [28] R. K. Saxena and S. L. Kalla, On a fractional generalization of free electron laser equation, Appl. Math. Comput. 43 (23), [29] R. K. Saxena, A. M. Mathai and H. L. Haubold, On generalized fractional kinetic equations, Phys. A 344 (24), [3] K. Schumacher, Existence and continuous dependence for differential equations with unbounded delay, Arch. Rational Mech. Anal. 64 (978), [3] C. Yu and G. Gao, Existence of fractional differential equations, J. Math. Anal. Appl. 3 (25),
Existence results for fractional order functional differential equations with infinite delay
J. Math. Anal. Appl. 338 (28) 134 135 www.elsevier.com/locate/jmaa Existence results for fractional order functional differential equations with infinite delay M. Benchohra a, J. Henderson b, S.K. Ntouyas
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 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 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 informationFractional Order Riemann-Liouville Integral Equations with Multiple Time Delays
Applied Mathematics E-Notes, 12(212), 79-87 c ISSN 167-251 Available free at mirror sites of http://www.math.nthu.edu.tw/amen/ Fractional Order Riemann-Liouville Integral Equations with Multiple Time Delays
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 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 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 informationApproximate controllability of impulsive neutral functional differential equations with state-dependent delay via fractional operators
International Journal of Computer Applications (975 8887) Volume 69 - No. 2, May 23 Approximate controllability of impulsive neutral functional differential equations with state-dependent delay via fractional
More informationON QUADRATIC INTEGRAL EQUATIONS OF URYSOHN TYPE IN FRÉCHET SPACES. 1. Introduction
ON QUADRATIC INTEGRAL EQUATIONS OF URYSOHN TYPE IN FRÉCHET SPACES M. BENCHOHRA and M. A. DARWISH Abstract. In this paper, we investigate the existence of a unique solution on a semiinfinite interval for
More informationFractional Differential Inclusions with Impulses at Variable Times
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 7, Number 1, pp. 1 15 (212) http://campus.mst.edu/adsa Fractional Differential Inclusions with Impulses at Variable Times Mouffak Benchohra
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 informationPartial Hadamard Fractional Integral Equations
Advances in Dynamical Syems and Applications ISSN 973-532, Volume, Number 2, pp. 97 7 (25) http://campus.m.edu/adsa Partial Hadamard Fractional Integral Equations Saïd Abbas University of Saïda Laboratory
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 informationINITIAL VALUE PROBLEMS OF FRACTIONAL ORDER HADAMARD-TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equation, Vol. 205 205), No. 77, pp. 9. ISSN: 072-669. URL: http://ejde.math.txtate.edu or http://ejde.math.unt.edu ftp ejde.math.txtate.edu INITIAL VALUE PROBLEMS OF
More informationON SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES
Applied Mathematics and Stochastic Analysis 15:1 (2002) 45-52. ON SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES M. BENCHOHRA Université de Sidi Bel Abbés Département de Mathématiques
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS TO FUNCTIONAL INTEGRO-DIFFERENTIAL FRACTIONAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 212 212, No. 13, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationTomasz Człapiński. Communicated by Bolesław Kacewicz
Opuscula Math. 34, no. 2 (214), 327 338 http://dx.doi.org/1.7494/opmath.214.34.2.327 Opuscula Mathematica Dedicated to the Memory of Professor Zdzisław Kamont GLOBAL CONVERGENCE OF SUCCESSIVE APPROXIMATIONS
More informationExistence Results for Multivalued Semilinear Functional Differential Equations
E extracta mathematicae Vol. 18, Núm. 1, 1 12 (23) Existence Results for Multivalued Semilinear Functional Differential Equations M. Benchohra, S.K. Ntouyas Department of Mathematics, University of Sidi
More informationNonlocal Fractional Semilinear Delay Differential Equations in Separable Banach spaces
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn:2319-765x. Volume 10, Issue 1 Ver. IV. (Feb. 2014), PP 49-55 Nonlocal Fractional Semilinear Delay Differential Equations in Separable Banach
More informationExistence Results on Nonlinear Fractional Differential Inclusions
Bol. Soc. Paran. Mat. (3s.) v. ():????. c SPM ISSN-2175-1188 on line ISSN-378712 in press SPM: www.spm.uem.br/bspm doi:1.5269/bspm.4171 Existence Results on Nonlinear Fractional Differential Inclusions
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 informationINITIAL AND BOUNDARY VALUE PROBLEMS FOR NONCONVEX VALUED MULTIVALUED FUNCTIONAL DIFFERENTIAL EQUATIONS
Applied Mathematics and Stochastic Analysis, 6:2 23, 9-2. Printed in the USA c 23 by North Atlantic Science Publishing Company INITIAL AND BOUNDARY VALUE PROBLEMS FOR NONCONVEX VALUED MULTIVALUED FUNCTIONAL
More informationSecond order Volterra-Fredholm functional integrodifferential equations
Malaya Journal of Matematik )22) 7 Second order Volterra-Fredholm functional integrodifferential equations M. B. Dhakne a and Kishor D. Kucche b, a Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada
More informationOn a perturbed functional integral equation of Urysohn type. Mohamed Abdalla Darwish
On a perturbed functional integral equation of Urysohn type Mohamed Abdalla Darwish Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia Department of
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 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 informationApproximating solutions of nonlinear second order ordinary differential equations via Dhage iteration principle
Malaya J. Mat. 4(1)(216) 8-18 Approximating solutions of nonlinear second order ordinary differential equations via Dhage iteration principle B. C. Dhage a,, S. B. Dhage a and S. K. Ntouyas b,c, a Kasubai,
More informationEXISTENCE RESULTS FOR NONLINEAR FUNCTIONAL INTEGRAL EQUATIONS VIA NONLINEAR ALTERNATIVE OF LERAY-SCHAUDER TYPE
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA IAŞI Tomul LII, s.i, Matematică, 26, f.1 EXISTENCE RESULTS FOR NONLINEAR FUNCTIONAL INTEGRAL EQUATIONS VIA NONLINEAR ALTERNATIVE OF LERAY-SCHAUDER TYPE
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 informationEXISTENCE OF SOLUTIONS FOR A SECOND ORDER ABSTRACT FUNCTIONAL DIFFERENTIAL EQUATION WITH STATE-DEPENDENT DELAY
Electronic Journal of Differential Equations, Vol. 27(27), No. 21, pp. 1 1. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationIMPULSIVE NEUTRAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH STATE DEPENDENT DELAYS AND INTEGRAL CONDITION
Electronic Journal of Differential Equations, Vol. 213 (213), No. 273, pp. 1 13. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu IMPULSIVE NEUTRAL
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 informationExistence And Uniqueness Of Mild Solutions Of Second Order Volterra Integrodifferential Equations With Nonlocal Conditions
Applied Mathematics E-Notes, 9(29), 11-18 c ISSN 167-251 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Existence And Uniqueness Of Mild Solutions Of Second Order Volterra Integrodifferential
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 informationEXISTENCE OF SOLUTIONS TO FRACTIONAL ORDER ORDINARY AND DELAY DIFFERENTIAL EQUATIONS AND APPLICATIONS
Electronic Journal of Differential Equations, Vol. 211 (211), No. 9, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF SOLUTIONS
More informationNONLOCAL INITIAL VALUE PROBLEMS FOR FIRST ORDER FRACTIONAL DIFFERENTIAL EQUATIONS
Dynamic Systems and pplications 2 (2) 247-26 NONLOCL INITIL VLUE PROBLEMS FOR FIRST ORDER FRCTIONL DIFFERENTIL EQUTIONS BDELKDER BOUCHERIF ND SOTIRIS K. NTOUYS Department of Mathematical Sciences, King
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 informationSolvability of Neumann boundary value problem for fractional p-laplacian equation
Zhang Advances in Difference Equations 215) 215:76 DOI 1.1186/s13662-14-334-1 R E S E A R C H Open Access Solvability of Neumann boundary value problem for fractional p-laplacian equation Bo Zhang * *
More informationExistence results for rst order boundary value problems for fractional dierential equations with four-point integral boundary conditions
Miskolc Mathematical Notes HU e-issn 787-243 Vol. 5 (24), No, pp. 5-6 DOI:.854/MMN.24.5 Existence results for rst order boundary value problems for fractional dierential equations with four-point integral
More informationOn the existence of solution to a boundary value problem of fractional differential equation on the infinite interval
Shen et al. Boundary Value Problems 5 5:4 DOI.86/s366-5-59-z R E S E A R C H Open Access On the existence of solution to a boundary value problem of fractional differential equation on the infinite interval
More informationUpper and lower solutions method and a fractional differential equation boundary value problem.
Electronic Journal of Qualitative Theory of Differential Equations 9, No. 3, -3; http://www.math.u-szeged.hu/ejqtde/ Upper and lower solutions method and a fractional differential equation boundary value
More informationLOCAL FIXED POINT THEORY INVOLVING THREE OPERATORS IN BANACH ALGEBRAS. B. C. Dhage. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 24, 24, 377 386 LOCAL FIXED POINT THEORY INVOLVING THREE OPERATORS IN BANACH ALGEBRAS B. C. Dhage Abstract. The present
More informationPOSITIVE SOLUTIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH DERIVATIVE TERMS
Electronic Journal of Differential Equations, Vol. 212 (212), No. 215, pp. 1 27. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu POSITIVE SOLUTIONS
More informationResearch Article A New Fractional Integral Inequality with Singularity and Its Application
Abstract and Applied Analysis Volume 212, Article ID 93798, 12 pages doi:1.1155/212/93798 Research Article A New Fractional Integral Inequality with Singularity and Its Application Qiong-Xiang Kong 1 and
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 informationCorrespondence should be addressed to Yagub A. Sharifov,
Abstract and Applied Analysis Volume 212, Article ID 59482, 14 pages doi:1.1155/212/59482 Research Article Existence and Uniqueness of Solutions for the System of Nonlinear Fractional Differential Equations
More informationExistence of solutions for a coupled system of. fractional differential equations
Existence of solutions for a coupled system of fractional differential equations Zhigang Hu, Wenbin Liu, Wenjuan Rui Department of Mathematics, China University of Mining and Technology, Xuzhou 228, PR
More informationEXISTENCE OF SOLUTIONS FOR SECOND ORDER SEMILINEAR FUNCTIONAL INTEGRODIFFERENTIAL INCLUSIONS IN BANACH SPACES
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVII, s.i a, Matematică, 21, f.1. EXISTENCE OF SOLUTIONS FOR SECOND ORDER SEMILINEAR FUNCTIONAL INTEGRODIFFERENTIAL INCLUSIONS IN BANACH SPACES
More informationBOUNDARY VALUE PROBLEMS FOR FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS INVOLVING GRONWALL INEQUALITY IN BANACH SPACE
J. Appl. Math. & Informatics Vol. 34(216, No. 3-4, pp. 193-26 http://dx.doi.org/1.14317/jami.216.193 BOUNDARY VALUE PROBLEMS FOR FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS INVOLVING GRONWALL INEQUALITY IN
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 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 informationExistence of Solutions for Nonlocal Boundary Value Problems of Nonlinear Fractional Differential Equations
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 7, Number 1, pp. 31 4 (212) http://campus.mst.edu/adsa Existence of Solutions for Nonlocal Boundary Value Problems of Nonlinear Fractional
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 informationON PERIODIC BOUNDARY VALUE PROBLEMS OF FIRST-ORDER PERTURBED IMPULSIVE DIFFERENTIAL INCLUSIONS
Electronic Journal of Differential Equations, Vol. 24(24), No. 84, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ON PERIODIC
More informationEXISTENCE RESULTS FOR BOUNDARY-VALUE PROBLEMS WITH NONLINEAR FRACTIONAL DIFFERENTIAL INCLUSIONS AND INTEGRAL CONDITIONS
Electronic Journal of Differential Equations, Vol. 21(21), No. 2, pp. 1 16. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE RESULTS FOR
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 informationMild Solution for Nonlocal Fractional Functional Differential Equation with not Instantaneous Impulse. 1 Introduction
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.21(216) No.3, pp.151-16 Mild Solution for Nonlocal Fractional Functional Differential Equation with not Instantaneous
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 informationEXISTENCE THEOREMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXVIII, 2(29), pp. 287 32 287 EXISTENCE THEOREMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES A. SGHIR Abstract. This paper concernes with the study of existence
More informationPositive solutions for nonlocal boundary value problems of fractional differential equation
Positive solutions for nonlocal boundary value problems of fractional differential equation YITAO YANG Tianjin University of Technology Department of Applied Mathematics No. 39 BinShuiWest Road, Xiqing
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR CAPUTO-HADAMARD SEQUENTIAL FRACTIONAL ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equation, Vol. 207 207), No. 36, pp.. ISSN: 072-669. URL: http://ejde.math.txtate.edu or http://ejde.math.unt.edu EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR CAPUTO-HADAMARD
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 informationx(t)), 0 < t < 1, 1 < q 2, 0 < p < 1 x(1). x(0) = 0, x (1) = αi p 0
Journal of Fractional Calculus and Applications, Vol. 3. July 22, No. 9, pp. 4. ISSN: 29-5858. http://www.fcaj.webs.com/ EXISTENCE RESULTS FOR FIRST ORDER BOUNDARY VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL
More informationExistence and stability of fractional implicit differential equations with complex order
Res. Fixed Point Theory Appl. Volume 218, Article ID 21827, 1 pages eissn 2581-647 Results in Fixed Point Theory and Applications RESEARCH ARTICLE Existence and stability of fractional implicit differential
More informationA generalized Gronwall inequality and its application to a fractional differential equation
J. Math. Anal. Appl. 328 27) 75 8 www.elsevier.com/locate/jmaa A generalized Gronwall inequality and its application to a fractional differential equation Haiping Ye a,, Jianming Gao a, Yongsheng Ding
More informationFUZZY SOLUTIONS FOR BOUNDARY VALUE PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXV 1(26 pp. 119 126 119 FUZZY SOLUTIONS FOR BOUNDARY VALUE PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS A. ARARA and M. BENCHOHRA Abstract. The Banach fixed point theorem
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 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 informationResearch Article Solvability for a Coupled System of Fractional Integrodifferential Equations with m-point Boundary Conditions on the Half-Line
Abstract and Applied Analysis Volume 24, Article ID 29734, 7 pages http://dx.doi.org/.55/24/29734 Research Article Solvability for a Coupled System of Fractional Integrodifferential Equations with m-point
More informationFRACTIONAL FOURIER TRANSFORM AND FRACTIONAL DIFFUSION-WAVE EQUATIONS
FRACTIONAL FOURIER TRANSFORM AND FRACTIONAL DIFFUSION-WAVE EQUATIONS L. Boyadjiev*, B. Al-Saqabi** Department of Mathematics, Faculty of Science, Kuwait University *E-mail: boyadjievl@yahoo.com **E-mail:
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 informationFractional order Pettis integral equations with multiple time delay in Banach spaces
An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S. Tomul LXIII, 27, f. Fractional order Pettis integral equations with multiple time delay in Banach spaces Mouffak Benchohra Fatima-Zohra Mostefai Received:
More informationON THE BEHAVIOR OF SOLUTIONS OF LINEAR NEUTRAL INTEGRODIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY
Georgian Mathematical Journal Volume 11 (24), Number 2, 337 348 ON THE BEHAVIOR OF SOLUTIONS OF LINEAR NEUTRAL INTEGRODIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY I.-G. E. KORDONIS, CH. G. PHILOS, I. K.
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR THE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES
Electronic Journal of Differential Equations, Vol. 29(29), No. 129, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
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 informationOn four-point nonlocal boundary value problems of nonlinear impulsive equations of fractional order
On four-point nonlocal boundary value problems of nonlinear impulsive equations of fractional order Dehong Ji Tianjin University of Technology Department of Applied Mathematics Hongqi Nanlu Extension,
More informationApplied Mathematics Letters. A reproducing kernel method for solving nonlocal fractional boundary value problems
Applied Mathematics Letters 25 (2012) 818 823 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml A reproducing kernel method for
More informationOn three and four point boundary value problems for second order dierential inclusions. Mouak Benchohra and Sotiris K. Ntouyas
Miskolc Mathematical Notes HU e-issn 1787-2413 Vol. 2 (21), No 2, pp. 93-11 DOI: 1.18514/MMN.21.4 On three and four point boundary value problems for second order dierential inclusions Mouak Benchohra
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 informationResearch Article Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic Semigroup
Abstract and Applied Analysis Volume 212, Article ID 93518, 15 pages doi:1.1155/212/93518 Research Article Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic
More informationExistence results for nonlinear fractional differential equation with nonlocal integral boundary conditions
Malaya J. Mat. 4326 392 43 Existence results for nonlinear fractional differential equation with nonlocal integral boundary conditions Renu Chaudhary a and Dwijendra N Pandey a, a Department of Mathematics,
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 informationPositive solutions of the three-point boundary value problem for fractional-order differential equations with an advanced argument
RESEARCH Open Access Positive solutions of the three-point boundary value problem for fractional-order differential equations with an advanced argument Guotao Wang *, SK Ntouyas 2 and Lihong Zhang * Correspondence:
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR FOURTH-ORDER BOUNDARY-VALUE PROBLEMS IN BANACH SPACES
Electronic Journal of Differential Equations, Vol. 9(9), No. 33, pp. 1 8. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationON A TWO-VARIABLES FRACTIONAL PARTIAL DIFFERENTIAL INCLUSION VIA RIEMANN-LIOUVILLE DERIVATIVE
Novi Sad J. Math. Vol. 46, No. 2, 26, 45-53 ON A TWO-VARIABLES FRACTIONAL PARTIAL DIFFERENTIAL INCLUSION VIA RIEMANN-LIOUVILLE DERIVATIVE S. Etemad and Sh. Rezapour 23 Abstract. We investigate the existence
More informationDynamic Systems and Applications xx (200x) xx-xx ON TWO POINT BOUNDARY VALUE PROBLEMS FOR SECOND ORDER DIFFERENTIAL INCLUSIONS
Dynamic Systems and Applications xx (2x) xx-xx ON WO POIN BOUNDARY VALUE PROBLEMS FOR SECOND ORDER DIFFERENIAL INCLUSIONS LYNN ERBE 1, RUYUN MA 2, AND CHRISOPHER C. ISDELL 3 1 Department of Mathematics,
More informationDELAY INTEGRO DIFFERENTIAL EQUATIONS OF MIXED TYPE IN BANACH SPACES. Tadeusz Jankowski Technical University of Gdańsk, Poland
GLASNIK MATEMATIČKI Vol. 37(57)(22), 32 33 DELAY INTEGRO DIFFERENTIAL EQUATIONS OF MIXED TYPE IN BANACH SPACES Tadeusz Jankowski Technical University of Gdańsk, Poland Abstract. This paper contains sufficient
More informationALMOST PERIODIC SOLUTIONS OF NONLINEAR DISCRETE VOLTERRA EQUATIONS WITH UNBOUNDED DELAY. 1. Almost periodic sequences and difference equations
Trends in Mathematics - New Series Information Center for Mathematical Sciences Volume 10, Number 2, 2008, pages 27 32 2008 International Workshop on Dynamical Systems and Related Topics c 2008 ICMS in
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 informationFunctional Differential Equations with Causal Operators
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.11(211) No.4,pp.499-55 Functional Differential Equations with Causal Operators Vasile Lupulescu Constantin Brancusi
More informationAdvances in Difference Equations 2012, 2012:7
Advances in Difference Equations This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Extremal
More informationA fixed point theorem for multivalued mappings
Electronic Journal of Qualitative Theory of Differential Equations 24, No. 17, 1-1; http://www.math.u-szeged.hu/ejqtde/ A fixed point theorem for multivalued mappings Cezar AVRAMESCU Abstract A generalization
More informationDifferentiability with Respect to Lag Function for Nonlinear Pantograph Equations
Differentiability with Respect to Lag Function for Nonlinear Pantograph Equations Alexandru Tămăşan Differentiability with respect to λ (, 1) is provided for the nonlinear pantograph equations y (t) =
More informationFinite Difference Method for the Time-Fractional Thermistor Problem
International Journal of Difference Equations ISSN 0973-6069, Volume 8, Number, pp. 77 97 203) http://campus.mst.edu/ijde Finite Difference Method for the Time-Fractional Thermistor Problem M. R. Sidi
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER NONLINEAR HYPERBOLIC SYSTEM
Electronic Journal of Differential Equations, Vol. 211 (211), No. 78, 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 informationOn Some Stochastic Fractional Integro-Differential Equations
Advances in Dynamical Systems and Applications. ISSN 973-5321 Volume 1 Number 1 (26), pp. 49 57 c Research India Publications http://www.ripublication.com/adsa.htm On Some Stochastic Fractional Integro-Differential
More informationAN EXTENSION OF THE LAX-MILGRAM THEOREM AND ITS APPLICATION TO FRACTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 215 (215), No. 95, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu AN EXTENSION OF THE
More informationOn the fixed point theorem of Krasnoselskii and Sobolev
Electronic Journal of Qualitative Theory of Differential Equations 2011, No. 5, 1-6; http://www.math.u-szeged.hu/ejqtde/ On the fixed point theorem of Krasnoselskii and Sobolev Cristina G. Fuentes and
More informationExistence of Positive Periodic Solutions of Mutualism Systems with Several Delays 1
Advances in Dynamical Systems and Applications. ISSN 973-5321 Volume 1 Number 2 (26), pp. 29 217 c Research India Publications http://www.ripublication.com/adsa.htm Existence of Positive Periodic Solutions
More information