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1
Content available Existence of positive continuous weak solutions for some semilinear elliptic eigenvalue problems
Zeddini Noureddine, Sari Rehab Saeed
Opuscula Mathematica
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2022
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Vol. 42, no. 3
489--519
EN
Let D be a bounded C1,1-domain in Rd, d ≥ 2. The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions K(D) that was defined by N. Zeddini for d = 2 and by H. Mâagli and M. Zribi for d ≥ 3 and adapted to study some nonlinear elliptic problems in D. The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants λ and μ to the following system Δu = λf(x, u, v), Δv = μg(x, u, v) in D, u = ϕ1 and v = ϕ2 on ∂D, where ϕ1 and ϕ2 are nontrivial nonnegative continuous functions on ∂D. The functions f and g are nonnegative and belong to a class of functions containing in particular all functions of the type f(x, u, v) = p(x)uαh1(v) and g(x, u, v) = q(x)h2(u)vβ with α ≥ 1, β ≥ 1, h1, h2 are continuous on [0,∞) and p, q are nonnegative functions in K(D).
2
Content available Solvability of a quadratic integral equation of Fredholm type via a modified argument
Dal İlyas, Temizer Ӧmer Faruk
Journal of Mathematics and Applications
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2020
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Vol. 43
47--66
EN
This article concerns with the existence of solutions of thea quadratic integral equation of Fredholm type with a modified argument, [wzór], where p, k are functions and F is an operator satisfying the given conditions. Using the properties of the Hölder spaces and the classical Schauder fixed point theorem, we obtain the existence of solutions of the equation under certain assumptions. Also, we present two concrete examples in which our result can be applied.
3
Content available On the existence of continuous positive monotonic solutions of a self-reference quadratic integral equation
El-Sayed Ahmed M.A., Ebead Hanaa R.
Journal of Mathematics and Applications
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2020
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Vol. 43
67--80
EN
In this work we study the existence of positive monotonic solutions of a self-reference quadratic integral equation in the class of continuous real valued functions. The continuous dependence of the uniquesolution will be proved. Some examples will be given.
4
Content available remote The solvability of functional quadratic volterra-urysohn integral equations on the half line
Metwali M. M. A.
Fasciculi Mathematici
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2018
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Nr 61
109--123
EN
We study the solvability of general quadratic Volterra integral equations in the space of Lebesgue integrable functions on the half line. Using the conjunction of the technique of measures of weak noncompactness with modified Schauder fixed point principle we show that the integral equation, under certain conditions, has at least one solution. Moreover, that result generalizes several ones obtained earlier in many research papers and monographs.
5
Content available Positive solutions with specific asymptotic behavior for a polyharmonic problem on Rn
Dhifli A.
Opuscula Mathematica
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2015
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Vol. 35, no. 1
5--19
EN
This paper is concerned with positive solutions of the semilinear polyharmonic equation [formula] on Rn, where m and n are positive integers with n > 2m, α ∈ e (—1,1). The coefncient a is assumed to satisfy[formula], where Λ ∈ (2m,∞) and [formula]is positive with [formula], one also assumes that [formula]. We prove the existence of a positive solution u such that [formula], with [formula] and a function L, given explicitly in terms of L and satisfying the same condition as infinity. (Given positive functions ∫ and g on Rn, ∫≈ g means that [formula]for some constant c > 1.)
6
Content available Bounded, asymptotically stable, and L1 solutions of caputo fractional differential equations
Islam M.N.
Opuscula Mathematica
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2015
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Vol. 35, no. 2
181--190
EN
The existence of bounded solutions, asymptotically stable solutions, and L1 solu­tions of a Caputo fractional differential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular and hence the standard techniques that are normally applied on Volterra integral equations do not apply here. This hurdle is overcomed using a resolvent equation and then applying some known properties of the resolvent. In the analysis Schauder's fixed point theorem and Liapunov's method have been employed. The existence of bounded solutions are obtained employing Schauder's theorem, and then it is shown that these solutions are asymptotically stable by a definition found in [C. Avramescu, C. Vladimirescu, On the existence of asymptotically stable solution of certain integral equations, Nonlinear Anal. 66 (2007), 472-483]. Finally, the L1 properties of solutions are obtained using Liapunov's method
7
Content available On the existence of positive continuous solutions for some polyharmonic elliptic systems on the half space
Zine El Abidine Z.
Opuscula Mathematica
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2012
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Vol. 32, no. 1
91-113
EN
We study the existence of positive continuous solutions of the nonlinear polyharmonic system (-Δ)mu + λqg(v) = 0, (-Δ)mv + μpf(u) = 0 in the half space [formula] where m ≥1 and n>2m.The nonlinear term is required to satisfy some conditions related to the Kato class [formula]. Our arguments are based on potential theory tools associated to (-Δ)m and properties of functions belonging to [formula].
8
Content available Existence of solutions for a four-point boundary value problem of a nonlinear fractional differential equation
Dou X., Li Y., Liu P.
Opuscula Mathematica
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2011
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Vol. 31, no. 3
359-372
EN
In this paper, we discuss a four-point boundary value problem for a nonlinear differential equation of fractional order. The differential operator is the Riemann-Liouville derivative and the inhomogeneous term depends on the fractional derivative of lower order. We obtain the existence of at least one solution for the problem by using the Schauder fixed-point theorem. Our analysis relies on the reduction of the problem considered to the equivalent Fredholm integral equation.
9
Content available remote On a mixed type integral equation and fractional-order functional differential equations
El-Sayed A. M. A., Sherif N., Ibrahim I. A.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
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2005
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Vol. 45, [Z] 2
237--247
EN
In this paper we study the existence of solution of a nonlinear integral equation of (mixed type) Volterra-Fredholm type. As an application we prove the existence of solution of an initial value problem of fractional order in the space of Lebesgue integrable functions on the interval [0,1].
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