• Title, Summary, Keyword: fractional boundary value problems

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MULTIPLE POSITIVE SOLUTIONS OF INTEGRAL BOUNDARY VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS

  • Liu, Xiping;Jin, Jingfu;Jia, Mei
    • Journal of applied mathematics & informatics
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    • v.30 no.1_2
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    • pp.305-320
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    • 2012
  • In this paper, we study a class of integral boundary value problems for fractional differential equations. By using some fixed point theorems, the results of existence of at least three positive solutions for the boundary value problems are obtained.

BOUNDARY VALUE PROBLEMS FOR FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS INVOLVING GRONWALL INEQUALITY IN BANACH SPACE

  • KARTHIKEYAN, K.;CHANDRAN, C.;TRUJILLO, J.J.
    • Journal of applied mathematics & informatics
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    • v.34 no.3_4
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    • pp.193-206
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    • 2016
  • In this paper, we study boundary value problems for fractional integrodifferential equations involving Caputo derivative in Banach spaces. A generalized singular type Gronwall inequality is given to obtain an important priori bounds. Some sufficient conditions for the existence solutions are established by virtue of fractional calculus and fixed point method under some mild conditions.

THREE-POINT BOUNDARY VALUE PROBLEMS FOR A COUPLED SYSTEM OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

  • Yang, Wengui
    • Journal of applied mathematics & informatics
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    • v.30 no.5_6
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    • pp.773-785
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    • 2012
  • In this paper, we establish sufficient conditions for the existence and uniqueness of solutions to a general class of three-point boundary value problems for a coupled system of nonlinear fractional differential equations. The differential operator is taken in the Caputo fractional derivatives. By using Green's function, we transform the derivative systems into equivalent integral systems. The existence is based on Schauder fixed point theorem and contraction mapping principle. Finally, some examples are given to show the applicability of our results.

EXISTENCE OF EVEN NUMBER OF POSITIVE SOLUTIONS TO SYSTEM OF FRACTIONAL ORDER BOUNDARY VALUE PROBLEMS

  • Krushna, B.M.B.;Prasad, K.R.
    • Journal of the Chungcheong Mathematical Society
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    • v.31 no.2
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    • pp.255-268
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    • 2018
  • We establish the existence and multiplicity of positive solutions to a coupled system of fractional order differential equations satisfying three-point boundary conditions by utilizing Avery-Henderson functional fixed point theorems and under suitable conditions.

THREE-POINT BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

  • Khan, Rahmat Ali
    • Journal of applied mathematics & informatics
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    • v.31 no.1_2
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    • pp.221-228
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    • 2013
  • The method of upper and lower solutions and the generalized quasilinearization technique is developed for the existence and approximation of solutions to boundary value problems for higher order fractional differential equations of the type $^c\mathcal{D}^qu(t)+f(t,u(t))=0$, $t{\in}(0,1),q{\in}(n-1,n],n{\geq}2$ $u^{\prime}(0)=0,u^{\prime\prime}(0)=0,{\ldots},u^{n-1}(0)=0,u(1)={\xi}u({\eta})$, where ${\xi},{\eta}{\in}(0,1)$, the nonlinear function f is assumed to be continuous and $^c\mathcal{D}^q$ is the fractional derivative in the sense of Caputo. Existence of solution is established via the upper and lower solutions method and approximation of solutions uses the generalized quasilinearization technique.

HIGHER ORDER NONLOCAL NONLINEAR BOUNDARY VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS

  • Khan, Rahmat Ali
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.329-338
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    • 2014
  • In this paper, we study the method of upper and lower solutions and develop the generalized quasilinearization technique for the existence and approximation of solutions to some three-point nonlocal boundary value problems associated with higher order fractional differential equations of the type $$^c{\mathcal{D}}^q_{0+}u(t)+f(t,u(t))=0,\;t{\in}(0,1)$$ $$u^{\prime}(0)={\gamma}u^{\prime}({\eta}),\;u^{\prime\prime}(0)=0,\;u^{\prime\prime\prime}(0)=0,{\ldots},u^{(n-1)}(0)=0,\;u(1)={\delta}u({\eta})$$, where, n-1 < q < n, $n({\geq}3){\in}\mathbb{N}$, 0 < ${\eta},{\gamma},{\delta}$ < 1 and $^c\mathcal{D}^q_{0+}$ is the Caputo fractional derivative of order q. The nonlinear function f is assumed to be continuous.

NONLOCAL BOUNDARY VALUE PROBLEMS FOR HILFER FRACTIONAL DIFFERENTIAL EQUATIONS

  • Asawasamrit, Suphawat;Kijjathanakorn, Atthapol;Ntouyas, Sotiris K.;Tariboon, Jessada
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1639-1657
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    • 2018
  • In this paper, we initiate the study of boundary value problems involving Hilfer fractional derivatives. Several new existence and uniqueness results are obtained by using a variety of fixed point theorems. Examples illustrating our results are also presented.

POSITIVE SOLUTIONS OF MULTI-POINT BOUNDARY VALUE PROBLEMS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION AT RESONANCE

  • Yang, Aijun;Ge, Weigao
    • The Pure and Applied Mathematics
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    • v.16 no.2
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    • pp.213-225
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    • 2009
  • This paper deals with the existence of positive solutions for a kind of multi-point nonlinear fractional differential boundary value problem at resonance. Our main approach is different from the ones existed and our main ingredient is the Leggett-Williams norm-type theorem for coincidences due to O'Regan and Zima. The most interesting point is the acquisition of positive solutions for fractional differential boundary value problem at resonance. And an example is constructed to show that our result here is valid.

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NEHARI MANIFOLD AND MULTIPLICITY RESULTS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEMS WITH p-LAPLACIAN

  • Ghanmi, Abdeljabbar;Zhang, Ziheng
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1297-1314
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    • 2019
  • In this work, we investigate the following fractional boundary value problems $$\{_tD^{\alpha}_T({\mid}_0D^{\alpha}_t(u(t)){\mid}^{p-2}_0D^{\alpha}_tu(t))\\={\nabla}W(t,u(t))+{\lambda}g(t){\mid}u(t){\mid}^{q-2}u(t),\;t{\in}(0,T),\\u(0)=u(T)=0,$$ where ${\nabla}W(t,u)$ is the gradient of W(t, u) at u and $W{\in}C([0,T]{\times}{\mathbb{R}}^n,{\mathbb{R}})$ is homogeneous of degree r, ${\lambda}$ is a positive parameter, $g{\in}C([0,T])$, 1 < r < p < q and ${\frac{1}{p}}<{\alpha}<1$. Using the Fibering map and Nehari manifold, for some positive constant ${\lambda}_0$ such that $0<{\lambda}<{\lambda}_0$, we prove the existence of at least two non-trivial solutions

EXISTENCE OF SOLUTIONS OF A CLASS OF IMPULSIVE PERIODIC TYPE BVPS FOR SINGULAR FRACTIONAL DIFFERENTIAL SYSTEMS

  • Liu, Yuji
    • Korean Journal of Mathematics
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    • v.23 no.1
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    • pp.205-230
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    • 2015
  • A class of periodic type boundary value problems of coupled impulsive fractional differential equations are proposed. Sufficient conditions are given for the existence of solutions of these problems. We allow the nonlinearities p(t)f(t, x, y) and q(t)g(t, x, y) in fractional differential equations to be singular at t = 0, 1 and be involved a sup-multiplicative-like function. So both f and g may be super-linear and sub-linear. The analysis relies on a well known fixed point theorem. An example is given to illustrate the efficiency of the theorems.