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1
Content available Uniqueness for a class p-Laplacian problems when a parameter is large
Alreshidi B., Hai D. D.
Opuscula Mathematica
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2024
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Vol. 44, no. 1
5--17
EN
We prove uniqueness of positive solutions for the problem−Δpu = λf(u) in Ω, u = 0 on ∂Ω, where 1 < p < 2 and p is close to 2, Ω is bounded domain in Rn with smooth boundary ∂Ω, f : [0,∞) → [0,∞) with f(z) ∼ zβ at ∞ for some β ∈ (0, 1), and λ is a large parameter. The monotonicity assumption on f is not required even for u large.
2
Content available Study of fractional semipositone problems on RN
Biswas Nirjan
Opuscula Mathematica
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2024
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Vol. 44, no. 4
445--470
EN
Let s ∈ (0, 1) and N > 2s. In this paper, we consider the following class of nonlocal semipositone problems: (−Δ)su = g(x)ƒa(u) in RN, u > 0 in RN, where the weight g ∈ L1(RN) ∩ L∞(RN) is positive, a > 0 is a parameter, and ƒa ∈ C(R) is strictly negative on (−∞, 0]. For ƒa having subcritical growth and weaker Ambrosetti–Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution ua, provided a is near zero. To obtain the positivity of ua, we establish a Brezis–Kato type uniform estimate of (ua) in Lr(RN) for every r ∈ [formula].
3
Content available remote On a first-order differential system with initial and nonlocal boundary conditions
Ngoc Le Thi Phuong, Long Nguyen Thanh
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
277--296
EN
This paper is devoted to the existence of solutions and the multiplicity of positive solutions of an initial-boundary value problem for a nonlinear first-order differential system with nonlocal conditions. The main tool is the fixed-point theorem in which we construct the novel representation of the associated Green’s functions with useful properties and define a cone in the Banach space suitably. Some examples are also given to demonstrate the validity of the main results.
4
Existence of positive solutions of a new class of nonlocal p(x)-Kirchhoff parabolic systems via sub-super-solutions concept
Zediri Sounia, Guefaifia Rafik, Boulaaras Salah
Journal of Applied Analysis
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2020
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Vol. 26, nr 1
49--58
EN
Motivated by the idea which has been introduced by Boulaaras and Guefaifia [S. Boulaaras and R. Guefaifia, Existence of positiveweak solutions for a class of Kirchhoff elliptic systems with multiple parameters, Math. Methods Appl. Sci. 41 (2018), no. 13, 5203-5210] and by Afrouzi and Shakeri [G. A. Afrouzi, S. Shakeri and N. T. Chung, Existence of positive solutions for variable exponent elliptic systems with multiple parameters, Afr. Mat. 26 (2015), no. 1-2, 159-168] combined with some properties of Kirchhoff-type operators, we prove the existence of positive solutions for a new class of nonlocal p(x)-Kirchhoff parabolic systems by using the sub- and super-solutions concept.
5
Content available On a Robin (p, q)-equation with a logistic reaction
Papageorgiou Nikolaos S., Vetro Calogero, Vetro Francesca
Opuscula Mathematica
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2019
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Vol. 39, no. 2
227--245
EN
We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a p-Laplacian and of a q-Laplacian ((p,q)-equation) plus an indefinite potential term and a parametric reaction ol logistic type (superdiffusive case). We prove a bilurcation-type result describing the changes in the set ol positive solutions as the parameter λ > 0 varies. Also, we show that lor every admissible parameter λ > 0, the problem admits a smallest positive solution. Keywords: positive solutions, superdiffusive reaction, local minimizers, maximum principle, min
6
Content available Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunov-type inequalities
Chidouh A., Torres D. F. M.
Opuscula Mathematica
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2018
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Vol. 38, no. 1
31--40
EN
We prove existence of positive solutions to a boundary value problem depending on discrete fractional operators. Then, corresponding discrete fractional Lyapunov-type inequalities are obtained.
7
Content available Positive solutions of boundary value problems with nonlinear nonlocal boundary conditions
Padhi S., Pati S., Hota D. K.
Opuscula Mathematica
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2016
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Vol. 36, no. 1
69--79
EN
We consider the existence of at least three positive solutions of a nonlinear first order problem with a nonlinear nonlocal boundary condition given by [formula] where r : [0,1] → [0, ∞) is continuous; the nonlocal points satisfy [formula] the nonlinear function ƒi and [formula] are continuous mappings from [0,1] x [0,∞) → [0,∞) for i = 1,2,... ,m and j = 1, 2,. .. , n respectively, and λ > 0 is a positive parameter.
8
Content available Existence and boundary behavior of positive solutions for a Sturm-Liouville problem
Masmoudi S., Zermani S.
Opuscula Mathematica
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2016
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Vol. 36, no. 5
613--629
EN
In this paper, we discuss existence, uniqueness and boundary behavior of a positive solution to the following nonlinear Sturm-Liouville problem [formula] where [formula], A is a positive differentiable function on (0,1) and a is a positive measurable function in (0,1) satisfying some appropriate assumptions related to the Karamata class. Our main result is obtained by means of fixed point methods combined with Karamata regular variation theory.
9
Content available Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities
Jaroś J., Takaśi K.
Opuscula Mathematica
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2015
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Vol. 35, no. 1
47--69
EN
We consider n-dimensional cyclic systems of second order differential equations [formula] (*) under the assumption that the positive constants α and β satisfy α1...αn > β1...βn and pi(t) and qi(t) are regularly varying functions, and analyze positive strongly increasing so­lutions of system (*) in the framework of regular variation. We show that the situation for the existence of regularly varying solutions of positive indices for (*) can be characterized completely, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of growth precisely. We give examples demonstrating that the main results for (*) can be applied to some classes of partial differential equations with radial symmetry to acquire accurate information about the existence and the asymptotic behavior of their radial positive strongly increasing solutions.
10
Positive solutions for discrete boundary value problems with p-Laplacian
Ji D., Ge W.
Journal of Applied Analysis
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2010
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Vol. 16, nr 1
107-119
EN
In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for the following two-point discrete boundary value problem with p-Laplacian: Δ(∅p(Δu(k - 1))) + e(k)f(u(k)) = 0 , k∈ N(1,T) , u(0) - B0 (Δu(0)) = 0 , u(T + 1) + B1 (Δu(T)) = 0. The main tool is the monotone iterative technique.
11
Content available remote On a class of rational difference equations
Atasever N., Yalcinkaya I.
Fasciculi Mathematici
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2010
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Nr 43
13-18
EN
In this paper we study the behavior of the positive solutions of the following nonlinear difference equation ...[wzór], n = 0, 1, 2, ... where the initial values ...[wzór] and k = 0, 1, 2, . . ..
12
Content available remote On the solutions of a class of difference equations systems
Elabbasy E., El-Metwally H., Elsayed E.
Demonstratio Mathematica
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2008
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Vol. 41, nr 1
109-122
EN
In this paper we discuses the periodic solutions of particular cases of the following general system of difference equations [...]. The results obtained are new and contained as special cases some other results.
13
Content available Further properties of the rational recursive sequence [formula]
Andruch-Sobiło A., Migda M.
Opuscula Mathematica
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2006
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Vol. 26, no. 3
387-394
EN
In this paper we consider the difference equation [formula], n=0, 1... with positive parameters a and c, negative parameter b and nonnegative initial conditions. We investigate the asymptotic behavior of solutions of equation (E).
14
Content available On the number of positive solutions to a class of integral equations
Wang L., Yu W., Zhang L.
Control and Cybernetics
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2003
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Vol. 32, no 2
383-395
EN
By using the complete discrimination system for polynomials, we study the number of positive solutions in C[0,1] to the integral equation phi(x) = integral[...] k(x,y)phi^n(y)dy, where k(x,y) = phi1(x)phi1(y)+phi2(x)phi2(y),[phi]i(x) > 0,[phi]i(y) > 0,0 < x,y < 1,i = 1,2, are continuous functions on [0,1], n is a positive integer. We prove the following results: when n = 1, either there does not exist, or there exist infinitely many positive solutions in C[0,1]; when n [is greater than or equal] 2, there exist at least 1, at most n + 1 positive solutions in C[0,1]. Necessary and sufficient conditions are derived for the cases: 1) n = 1, there exist positive solutions; 2) n [is greater than or equal to] 2, there exist exactly m (m belongs to {1,2,..., n + 1}) positive solutions. Our results generalize the ones existing in the literature, and their usefulness is shown by examples.
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