Second-order ordinary differential systems with nonlocal Neumann conditions at resonance

Mawhin, Jean; Szymańska‐Dȩbowska, Katarzyna

doi:10.1007/s10231-015-0532-9

Cited by 8 publications

(8 citation statements)

References 8 publications

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“…More precisely, we focus on the case the condition (1.5) is satisfied. We point out that, in contrast to the papers [16] and [23], both the boundary conditions are nonlocal. To provide the main results of the paper, we apply the celebrated Mawhin continuation theorem (see, for example, [8,15]).…”

Section: Introductionmentioning

confidence: 63%

“…Their tool was the fixed point index for compact operators in cones. Other types of nonlocal Neumann problems than those discussed in this paper can be found, for instance, in [16,21,23].…”

Section: Introductionmentioning

confidence: 75%

“…Such conditions were introduced in 1971 by Nirenberg [17] in the context of elliptic boundary value problems, generalizing the one introduced in 1970 by Landesman and Lazer [12] for semilinear elliptic equations with resonant linear part. For a detailed exposition and applications, see for example [1,16,21,23].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A topological degree approach to a nonlocal Neumann problem for a system at resonance

Szymańska‐Dȩbowska

Zima

2019

J. Fixed Point Theory Appl.

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In this paper, we investigate the existence of solutions for a system of second-order differential equations with two nonlocal Neumann boundary conditions at resonance. The existence results are obtained by applying the Mawhin continuation theorem, where the required a priori estimates are derived using some conditions of the Nirenberg type.Mathematics Subject Classification. Primary 34B10; Secondary 34B15, 47H10, 54H25.where γ(t) = t and δ(t) = 1. This means that if (1.4) holds, then the homogeneous linear problem corresponding to (1.1), that is,has nontrivial solutions. In particular, (1.4) is satisfied if(1.5)(1.6)Vol. 21 (2019) Nonlocal Neumann problem at resonance

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Section: Introductionmentioning

confidence: 63%

“…Their tool was the fixed point index for compact operators in cones. Other types of nonlocal Neumann problems than those discussed in this paper can be found, for instance, in [16,21,23].…”

Section: Introductionmentioning

confidence: 75%

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A topological degree approach to a nonlocal Neumann problem for a system at resonance

Szymańska‐Dȩbowska

Zima

2019

J. Fixed Point Theory Appl.

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“…The existence of solutions of the Neumann and periodic boundary value problems of semilinear differential equations has been extensively studied by many authors via the following Mawhin continuation theorem (see [19][20][21][22][23] and references therein).…”

Section: N -1}mentioning

confidence: 99%

Discrete Neumann boundary value problem for a nonlinear equation with singular ϕ-Laplacian

2018

Adv Differ Equ

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Let I ⊂ R be an open interval with 0 ∈ I, and let g ∈ C 1 (I, (0, +∞)). Let N ∈ N be an integer with N ≥ 4, [2, N-1] Z := {2, 3,. .. , N-1}. We are concerned with the existence of solutions for the discrete Neumann problem ⎧ ⎨ ⎩ ∇(k n-1 v k √ 1-(v k) 2) = nk n-1 [-g (ψ-1 (v k)) √ 1-(v k) 2 + g(ψ-1 (v k))H(ψ-1 (v k), k)], k ∈ [2, N-1] Z , v 1 = 0 = v N-1 which is a discrete analogue of the Neumann problem about the rotationally symmetric spacelike graphs with a prescribed mean curvature function in some Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, where ψ(s) := s 0 dt g(t) , ψ-1 is the inverse function of ψ, and H : R × [2, N-1] Z → R is continuous with respect to the first variable. The proofs of the main results are based upon the Brouwer degree theory.

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“…18 in the way that the nonhomogeneous term h contains the

(ν - 1)

th‐order difference

▵_{ν - 2}^{ν - 1} x

. More importantly, the boundary conditions of () imply resonance which means that the linear operator

L x = ▵_{ν - 2}^{ν} x

is noninvertible and the problem cannot be converted to a fixed point problem 20,26 . To deal with this difficulty, we apply the coincidence degree theory for semilinear operators 27,28 …”

Section: Introductionmentioning

confidence: 99%

Existence of solutions for a class of fractional difference equations at resonance

et al. 2021

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We study a class of nonlinear fractional difference equations with nonlocal boundary conditions at resonance. The system is inspired by the three‐point boundary value problem for differential equations that have been extensively studied. It is also an extension to a fractional difference equation arising from real‐world applications. Converting the problem to an equivalent system corresponding to the integral operator and Green's function for differential equations, we are able to apply the coincidence degree theory for semilinear operators to obtain sufficient conditions for the existence of solutions. In addition, we prove a new property of the Gamma function and construct a family of examples to illustrate the applications of the results.

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Second-order ordinary differential systems with nonlocal Neumann conditions at resonance

Cited by 8 publications

References 8 publications

A topological degree approach to a nonlocal Neumann problem for a system at resonance

A topological degree approach to a nonlocal Neumann problem for a system at resonance

Discrete Neumann boundary value problem for a nonlinear equation with singular ϕ-Laplacian

Existence of solutions for a class of fractional difference equations at resonance

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