Abstract Owing to the non-ideal properties of the Green’s function associated with the boundary value problem of Caputo-Hadamard fractional differential equations, a complete theoretical framework for studying such solutions has remained elusive. To address this challenge, this paper introduces novel substitutions and conducts a meticulous analysis of the Green’s function’s properties. This approach enables us to successfully derive the maximum and minimum iterative solutions for a singular p -Laplacian Caputo-Hadamard fractional differential equation. Subsequently, by employing the theory of mixed monotone operators, we establish the existence and uniqueness of positive solutions. Furthermore, we demonstrate that the unique solution u λ ⋆ ${u}_{\lambda }^{\star }$ is parameter-dependent, exhibiting the asymptotic behavior lim λ → ∞ ‖ u λ ⋆ ‖ = 0 ${\mathrm{lim}}_{\lambda \to \infty }{\Vert}{u}_{\lambda }^{\star }{\Vert}=0$ and lim λ → 0 + ‖ u λ ⋆ ‖ = ∞ ${\mathrm{lim}}_{\lambda \to {0}^{+}}{\Vert}{u}_{\lambda }^{\star }{\Vert}=\infty $ . The validity of our main theoretical findings is confirmed through a numerical example, with graphical illustrations depicting the solution’s variation with the parameter.
Abstract We establish sufficient conditions for the existence and multiplicity of solutions to third-order ordinary differential equations involving all lower-order derivatives, subject to Riemann–Stieltjes integral boundary conditions. The analysis relies on the method of upper and lower solutions combined with the Leray–Schauder topological degree theory. Finally, examples are provided to illustrate the main results.
Abstract We study strong lacunary harmonic summability of fractional difference sequences in q -rung orthopair fuzzy normed spaces. The fractional difference is defined on its convergence domain, with an explicit sufficient condition and a truncation estimate for bounded Banach-space-valued sequences. In the canonical norm model, the summability condition is equivalent, on bounded transformed sequences, to strong summability of the norms. We prove linearity, uniqueness and subsequence results for proper admissible ideals, and establish a transfer theorem for distinct lacunary partitions whose harmonic symmetric difference is negligible. In complete spaces, a Cauchy criterion is obtained under bounded normalized row mass; this mass bound holds automatically when β = 1. Counterexamples explain the failure of a fixed-centre zero-limit condition and the obstruction caused by unbounded row mass. The general summability results concern the transformed sequence, whereas the domain and truncation estimates use the fractional coefficients explicitly.
Abstract In this work, we focus on the existence and asymptotic behavior of radial solutions for the k -Hessian equation with the nonlinear gradient terms S k σ D 2 u = λ b ( | x | ) φ ( − u ) | ∇ u | m , i n Ω , u = 0 , o n ∂ Ω , $$\begin{cases}_{k}\left(\sigma \left({D}^{2}u\right)\right)=\lambda b\left(\vert x\vert \right)\varphi \left(-u\right)\vert \nabla u{\vert }^{m}, \mathrm{i}\mathrm{n} {\Omega},\hfill \\ u=0, \mathrm{o}\mathrm{n} \partial {\Omega},\hfill \end{cases}$$ where λ > 0, 0 ≤ m < k and |∇ u | denotes the gradient of u . The main methods of proof rely on the eigenvalue theory established by Guo and Lakshmikantham, as well as Guo’s fixed point theorem, combined with techniques such that Jensen inequality.
Consider L(H) $L\left(\mathcal{H}\right)$ , the algebra of all bounded linear operators acting on a complex Hilbert space H $\mathcal{H}$ . This paper shows that if A,B,C∈L(H) $A,B,C\in L\left(\mathcal{H}\right)$ where A and B are positive, and the operator matrixAC*CB $$\left[\begin{matrix}A& {C}^{{\ast}}\\ C& B\end{matrix}\right]$$
