Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces

Feng, Meiqiang; Ji, Dehong; Ge, Weigao

doi:10.1016/j.cam.2007.11.003

Cited by 74 publications

(49 citation statements)

References 25 publications

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“…Remark 3.1 When q(t) ≡ 0, l = 1, the problem (1.1) reduces to the problem studied in [1], and so our results generalize and include some results in [1].…”

Section: H(t S)dssupporting

confidence: 61%

“…The existence of positive solutions for second-order boundary value problems has been studied by many authors using various methods (see [1][2][3][4][5][6]). Recently, the integral boundary value problems have been studied extensively.…”

Section: Introductionmentioning

confidence: 99%

“…Feng et al [9] considered the boundary value problems with one-dimensional (1D) p-Laplacian and impulse effects subject to the integral boundary condition. This study in this article is motivated by Feng and Ge [1], who applied a fixed point theorem [10] in cone to the second-order differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…First, the author discusses the existence results in the case q C[0, 1], not q(t) = 0 as in [1]. Second, comparing with [1], let us consider the existence results in the case l > 0, not l = 1 as in [1].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Existence of positive solutions for nonlocal second-order boundary value problem with variable parameter in Banach spaces

Zhang

2011

Fixed Point Theory Appl

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“…Remark 3.1 When q(t) ≡ 0, l = 1, the problem (1.1) reduces to the problem studied in [1], and so our results generalize and include some results in [1].…”

Section: H(t S)dssupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Existence of positive solutions for nonlocal second-order boundary value problem with variable parameter in Banach spaces

Zhang

2011

Fixed Point Theory Appl

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“…Especially, the existence of positive solutions of nonlinear BVP with integral boundary conditions has been extensively studied by many authors (see [9][10][11][12][13][14][15][16][17][18] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Multiple positive solutions for a fourth-order integral boundary value problem on time scales

Dong

2011

Bound Value Probl

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In this article, we investigate the multiplicity of positive solutions for a fourth-order system of integral boundary value problem on time scales. The existence of multiple positive solutions for the system is obtained by using the fixed point theorem of cone expansion and compression type due to Krasnosel'skill. To demonstrate the applications of our results, an example is also given in the article.

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Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

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Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments -distributed and variable fractional order differential operators.Fractional order derivatives interpolate integer order derivatives to real (not necessarily fractional) or complex order derivatives. There are different types of fractional derivatives not always equivalent. The first attempt to develop the fractional calculus systematically was taken by Liouville (1832) and Riemann (1847) in the first half of the nineteenth century, even though discussions on non-integer order derivatives had been started long ago. 1 In the 1870s Letnikov and Grünwald independently used an approach for the definition of the fractional order derivative and integral different from that of Riemann and Liouville. The Cauchy problem for fractional order differential equations with the Riemann-Liouville derivative is not well posed (Section 3.3), that is the Cauchy problem in this case is unphysical. In the 1960s Caputo and Djrbashian introduced independently, so-called, a regularization of the Riemann-Liouville fractional derivative, which was later named a fractional derivative in the sense of Caputo-Djrbashian (Section 3.5). The usefulness of the Caputo-Djrbashian derivative is that the Cauchy problem for fractional order differential equations with the Caputo-Djrbashian derivative is well posed. In the operators language one can write the latter in the form DJ = I, where I is the identity operator, which means that the operator D is a left inverse to the operator J. One can easily check that D is not a right inverse to J, since, according to the same fundamental theorem of calculus, for any differentiable function f the equality JD f (t) = f (t) − f (0) holds. The similar relations are true by induction for operators D n and J n , where D n = d n dt n , "n-th derivative," and J n is the n-fold integration operator. Namely,andThus D n is the left inverse to J n , and is the right inverse to J n in the class of functions satisfying additional conditions: f (k) (0) = 0, k = 0,..., n − 1. These relations between "differentiation" and "integration" operators valid for n = 1, 2, . . . , form the basis for the definitions of fractional derivatives in the sense of RiemannLiouville and Caputo-Djrbashian, as soon as the fractional order integration operator is defined. Fractional order integration operator 123 Fractional order integration operatorIn this section we introduce the fractional order integration operator of order α > 0 (ℜ(α) > 0). One can verify (by changing order of integration) that the n-fold integration operatorTaking into account the relationship Γ (n) = (n − 1)!, whereis the Euler's ...

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Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces

Cited by 74 publications

References 25 publications

Existence of positive solutions for nonlocal second-order boundary value problem with variable parameter in Banach spaces

Existence of positive solutions for nonlocal second-order boundary value problem with variable parameter in Banach spaces

Multiple positive solutions for a fourth-order integral boundary value problem on time scales

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

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