Theaim of this research is to establish existence and uniqueness results for the Riemann–Liouville fractional integral equation of order $\alpha$$\varkappa \left(t\right)=f\left(t\right)+{\displaystyle \frac{\lambda }{\Gamma \left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}g\left(s,\varkappa \left(s\right)\right)ds,t\in [0,1],$ by developing common fixed point theorems for generalized contractions involving control functions of two variables in the framework of complex valued suprametric spaces. The proposed results extend and generalize several existing findings in the literature, and some illustrative examples are provided to demonstrate the novelty and applicability of the main theorem.

In the present work, we consider the space-time fractional Whitham-Broer-Kaup (FWBK) equation and then find its analytical solutions under the framework of the Riccati-Bernoulli sub-ordinary differential equation method along with the Bäcklund transformation. The derived solutions are described in terms of the hyperbolic rational and trigonometric functions and resolve unique wave features. A unique feature of this work is the investigation of the impact of the fractional order on wave steepness in $ 2D $, $ 3D $, and contour plots. Also, it should be pointed that the found relationship shows that the change of the fractional order parameter really describes the traveling periodic waves, for example, Stokes waves, with potential importance for the analysis of wave propagation and stability in the nonlinear fractional structures. This work carries theoretical developments of fractional calculus and ideas for applications in fluid dynamics and wave mechanics.

We relate the exponential integrability of the conjugate function $${\tilde{f}}$$f~ to the size of the gap in the essential range off. Our main result complements a related theorem of Zygmund.

We propose in this paper a class of generalized nonlinear mixed variational inequality problems on Banach spaces, which is composed of three elliptic generalized nonlinear mixed variational inequality problems. We demonstrate that, under appropriate assumptions, the ideas of the Kakutani-Ky Fan fixed point theorem and Minty techniques, the existence of the solution set for generalized nonlinear mixed variational inequality problem is nonempty, weakly compact, and unique. Furthermore, we provide a stability analysis by perturbing the duality mappings and addressing the optimal control problem for the solution set of the system of generalised nonlinear mixed variational inequality problems.

This text deals with exploring random solutions for generalized nonlinear variational inequality problems. Using the Fan-KKM theorem and Aumann's measurable selection theorem, we are able to prove the existence and uniqueness of random solution sets under the conditions of monotonicity and convexity. Additionally, we use Minty's lemma to demonstrate the compactness and convexity of the random solution sets.

In this paper, we consider two extensions of Stevic-Sharma operator and find estimations for the essential norm of them from QK(p; q) and H1 into weighted Bloch spaces.