Fractional Differential Equations

Introduction to Potential Theory. By L. L. Helms. Pp. ix, 282. £7. 1970. (Wiley.)

By means of the recent $$\psi $$
 -Hilfer fractional derivative and of the Banach fixed-point theorem, we investigate stabilities of Ulam–Hyers, Ulam–Hyers–Rassias and semi-Ulam–Hyers–Rassias on closed intervals [a, b] and $$[a,\infty )$$
 for a particular class of fractional integro-differential equations.

Ulam–Hyers–Rassias Stability for a Class of Fractional Integro-Differential Equations

By applying the properties of the unique classical solution to the singular boundary value problem on half line −p″(s)=g(p(s)),p(s)>0,s∈(0,∞),p(0)=0,lims→∞p′(s)=b⩾0, and constructing the new comparison functions, they show the existence and the optimal global estimates of solutions to singular nonlinear Dirichlet problems − Δ u=k(x)g(u),u>0,x∈Ω,u| ∂Ω =0 , where Ω is a bounded domain with smooth boundary in RN;g(s) is nonincreasing and positive in (0, ∞), ∫ 1 ∞ g(t) dt and lim s→0 + g(s)=+∞;k∈C α (Ω) is positive in Ω , and may be singular or zero on the boundary.

Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems

We use recent results by Diaz and Rakotoson concerning very weak solutions to linear boundary value problems in order to improve previous work on existence and properties of weak positive solutions to a model example of semilinear singular elliptic problem.

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On Very Weak Positive Solutions to Some Semilinear Elliptic Problems With Simultaneous Singular Nonlinear and Spatial Dependence Terms

Existence and Estimates of Solutions for Singular Nonlinear Elliptic Problems

Abstract. We establish an existence result of positive solutions to the following boundary value problem: where is a bounded -domain in ℝn, and are nonnegative functions in , , satisfying some appropriate assumptions related to Karamata regular variation theory. We give estimates on such solutions where appear the combined effects of singular and sublinear terms in the nonlinearity.

Combined effects in nonlinear singular elliptic problems in a bounded domain

In this paper, we study the asymptotic behavior of the unique positive classical solution to the following semilinear boundary value problem Δ u + a ( x ) u α = 0 , x ∈ Ω , u > 0  in  Ω , u | ∂ Ω = 0 . Here Ω is a bounded C 1 , 1 domain, α 1 and the function a is in C l o c γ ( Ω ) , 0 γ 1 such that there exists c > 0 satisfying for each x ∈ Ω , 1 c ≤ a ( x ) δ ( x ) λ exp ( − ∫ δ ( x ) η z ( t ) t d t ) ≤ c , where λ ≤ 2 , η > d = d i a m ( Ω ) , δ ( x ) = d i s t ( x , ∂ Ω ) and z is a continuous function on [ 0 , η ] with z ( 0 ) = 0 .

Asymptotic behavior of positive solutions of a semilinear Dirichlet problem

We take up the existence and the exact asymptotic behavior near the boundary ∂ Ω of the unique classical solution to a singular Dirichlet problem − Δ u = a ( x ) g ( u ) , x ∈ Ω , u > 0  in  Ω , u ∣ ∂ Ω = 0 , where Ω is a C 1 , 1 -bounded domain in R n , n ≥ 2 and the function a is singular near the boundary and belongs to the Kato class K ( Ω ) .

Habib Mâagli

Papers

Existence and Estimates of Solutions for Singular Nonlinear Elliptic Problems

Combined effects in nonlinear singular elliptic problems in a bounded domain

Asymptotic behavior of positive solutions of a semilinear Dirichlet problem

Exact asymptotic behavior near the boundary to the solution for singular nonlinear Dirichlet problems

Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem