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In this work, we give a qualitative study on the existence of positive solutions for local and nonlocal elliptic integro-differential quasilinear systems, by using the concept of super- and subsolutions combined with Schauder’s fixed point theorem.

In this paper, we prove that under weak assumptions on the reaction terms and diffusion coefficients, a positive solution exists for a one-dimensional case and a positive radial solution to a multidimensional case of a nonlocal elliptic problem. Additionally, we establish the uniqueness of the solution, with the fixed point theorem being the primary tool employed. Our results are new and generalize several existing results.

Our goal in this paper is to present well-posed results for nonclassical diffusion systems which have applications in population dynamics. First, we establish the existence and uniqueness of a mild solution to the initial value problem. The asymptotic behavior of the mild solution is also considered when the parameter tends to zero. Second, we obtain a local well-posedness result for nonclassical diffusion systems with a nonlocal time condition. The main idea to obtain the above theoretical results is to use Banach’s theorem and some techniques in Fourier series analysis. Some numerical tests are also presented to illustrate the theory.

We demonstrate that the presence of summands with involution in the argument in an ordinary differential equation can significantly affect the well-posedness of the Cauchy and other problems. Furthermore, we show that the above effects can influence the well-posedness of classical boundary value problems for partial differential equations, particularly in the case of parabolic and pseudoparabolic equations.

In this article, the existence of solutions for fully nonlinear Kirchhoff-type problem − M ( ∫ ( Φ ( | ∇ u | ) + Φ ( | u | ) ) d x ) [ d i v ( a ( | ∇ u | ) ∇ u ) + a ( | u | ) u ] = λ ∑ i = 1 k ( t q i ( x ) − 1 − t r i ( x ) − 1 ) is proved via variational method. Finally, some new problems are introduced.

In this article, we consider the following problem { ( − Δ ) s u = α u + − β u − + f ( u ) + h in Ω u = 0 on ℝ n \\ Ω , , where Ω ⊂ ℝ n is a bounded domain with Lipschitz boundary, n > 2s, 0 < s < 1, (α, β) ∈ ℝ², ƒ : ℝ → ℝ is a bounded and continuous function and h ∈ L²(Ω). We prove the existence results in two cases: first, the nonresonance case where (α, β) is not an element of the Fučik spectrum. Second, the resonance case where (α, β) is an element of the Fučik spectrum. Our existence results follows as an application of the saddle point theorem. It extends some results, well known for Laplace operator, to the nonlocal operator.

In this paper, we are concerned with the following nonlocal problem: − ( a − ϵ ∫ R 3 K ( x ) | ∇ u | 2 d x ) div ( K ( x ) ∇ u ) = λ K ( x ) f ( x ) | u | q − 2 u + K ( x ) | u | 4 u , x ∈ R 3 , where a , λ > 0 , 1 < q < 2 , K ( x ) = exp ( | x | α / 4 ) with α ≥ 2 , ϵ > 0 is small enough, and f ( x ) ≥ 0 satisfies some integrability condition. By using the Ekeland variational principle and the concentration compactness principle, we establish the existence of two positive solutions for the problem and prove that at least one of them is a positive ground state solution.

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0

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