The purpose of the current study is to find exact travelling wave solutions of the Rosenau equation. By the use of the extended auxiliary equation method, various exact solutions are obtained in terms of Jacobi elliptic functions and exponential functions. Moreover, several solitary and periodic wave solutions are given as special cases. When the parameters take some values, some graphical illustrations are shown in order to understand the behaviour of these new solutions. Furthermore, we compare our solutions with some familiar solutions, which can be considered as special cases.
The current manuscript investigates the exact solutions of the modified Benjamin-Bona-Mahony (BBM) equation. Due to its efficiency and simplicity, the modified auxiliary equation method is adopted to solve the problem under consideration. As a result, a variety of the exact wave solutions of the modified BBM equation are obtained. Furthermore, the findings of the current study remain strong since Jacobi function solutions generate hyperbolic function solutions and trigonometric function solutions, as liming cases of interest. Some of the obtained solutions are illustrated graphically using appropriate values for the parameters.
<abstract><p>This paper presents a novel concept of $ G $-Hardy-Rogers functional operators on metric spaces endowed with a graph. It investigates sufficient circumstances under which such a mapping becomes a Picard operator. As applications of the principal idea discussed herein, a few important corresponding fixed point results in ordered metric spaces and cyclic operators are pointed out and analyzed. For upcoming research papers in this field, comparative graphical illustrations are created to highlight the pre-eminence of proposed notions with respect to the existing ones.</p></abstract>
We prove the existence of positive radial solutions to the problemwhere ∆pu = div(|∇u| p−2 ∇u), N > p > 1, Ω = {x ∈ R N : |x| > r 0 > 0}, f : (0, ∞) → R is p-superlinear at ∞ with possible singularity at 0, and λ is a small positive parameter. A nonexistence result is also established when f has semipositone structure at 0.
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