In this article, by the use of the lower and upper solutions method, we prove the existence of a positive solution for a Riemann–Liouville fractional boundary value problem. Furthermore, the uniqueness of the positive solution is given. To demonstrate the serviceability of the main results, some examples are presented.

In this paper, we introduce a noncommutative extension of the Gross Laplacian, called quantum Gross Laplacian, acting on some analytical operators. For this purpose, we use a characterization theorem between this class of operators and their symbols. Applying the quantum Gross Laplacian to the particular case where the operator is the multiplication one, we establishes a relation between the classical and the quantum Gross Laplacians.Keywords Space of entire functions with growth condition · Symbols and kernels of operators · Convolution product · Classical and quantum Gross Laplacian Mathematics Subject Classification (2000) 60H40 · 60H15 · 46F25 · 46G20 · 81S25

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In the present paper, we are concerned to prove under some hypothesis the existence of fixed points of the operator L defined on C(I) by Lu(t) = ?w0 G(t,s)h(s) f(u(s))ds, t ? I, ? ? {1,?}, where the functions f ? C([0,?); [0,?)), h ? C(I; [0,?)), G ? C(I x I) and (I = [0,1]; if ? = 1, I = [0,?), if ? = 1. By using Guo Krasnoselskii fixed point theorem, we establish the existence of at least one fixed point of the operator L.

In this work, we study the existence and the multiplicity of non-negative solutions for the following problem, is a bounded smooth domain, λ, p, q are positive real numbers, s ∈ (0, 1), a, b are continuous functions, and L is a nonlocal operator defined later by (1.1). We establish the existence and we give a multiplicity of solutions by constrained minimization of the Euler-Lagrange functional corresponding to the problem (P λ ), on suitable subsets of Nehari manifold and using the fibering maps. Precisely, we show the existence of λ 0 > 0, such that for all λ ∈ (0, λ 0 ), problem (P λ ) has at least two non-negative solutions.Keywords Non-local operator • Fractional Laplacian • Multiple solutions • Sign-changing weight functions • Nehari manifold • Fibering maps B A. Ghanmi

In this paper, an iterative method is applied to solve some p-Laplacian boundary value problem involving Riemann-Liouville fractional derivative operator. More precisely, we establish the existence of two positive solutions. Moreover, we prove that these solutions are one maximal and the other is minimal. An example is presented to illustrate our main result. Finally, a numerical method to solve this problem is given.

The present work is aimed at studying the effect of fractional order and thermal relaxation time on an unbounded fiber-reinforced medium. In the context of generalized thermoelasticity theory, the fractional time derivative and the thermal relaxation times are employed to study the thermophysical quantities. The techniques of Fourier and Laplace transformations are used to present the problem exact solutions in the transformed domain by the eigenvalue approach. The inversions of the Fourier-Laplace transforms hold analytical and numerically. The numerical outcomes for the fiber-reinforced material are presented and graphically depicted. A comparison of the results for different theories under the fractional time derivative is presented. The properties of the fiber-reinforced material with the fractional derivative act to reduce the magnitudes of the variables considered, which can be significant in some practical applications and can be easily considered and accurately evaluated.