The present paper has enormous applications including fields of crystal growth, material processing, spacecraft, underground spread of chemical contaminants, petroleum reservoirs, waste dispersal and fertilizer migration in saturated soil, alloy solidification, and many more. The importance of double-diffusive convection has been recognized in various engineering applications, and it has thoroughly been investigated theoretically. In the presence of a constant heat source/sink on both layers, the double component-magneto-Marangoni-convection in a composite layer is examined. Due to heat, this combined layer is enclosed by adiabatic boundaries and exposed to basic temperature gradients. For the system of ordinary differential equations, the thermal surface-tension-driven (Marangoni) number, which also happens to be the Eigenvalue, is solved in closed form. For basic temperature gradients, the eigenvalues, or thermal Marangoni numbers are determined in closed form for lower rigid and higher free boundaries with surface tension, depending on both temperature and concentration. The effect of several parameters on the eigenvalue against depth ratio was studied. It is observed that the inverted parabolic temperature gradient on double component magneto-Marangoni-convection in a combined layer is the most stable of the three temperature gradients and for larger values of depth ratios, all physical parameters are nominal for the porous region dominant combined system.

The paper is concerned with asymptotic and stability behaviors of fixed solutions of the second order nonlinear delay difference equation of the form ?2(xn +pnxn-k -qnxn-l)+ f(xn) = 0, n = 0,1,2, · ··,. Examples are provided to illustrate the results.

The present work addresses the nonlocal problems for a controllability of functional differential equation in Banach spaces. Sufficient conditions are derived by using Schaefer’s fixed point and semigroup theory.

Definiti onA graph G consists of a pair (V, E), where V is a non-empty fin ite set whose elements are called vertices(points) and E is a set of unordered pair of distinct elements of E are called edges (line or lin k) of thegraph G. Example (A graph wi th 4 vertices and 6 edges)V 1 , V 2 , V 3 , V 4 are vertices e v e 2 , e 3 , e 4 , e 5 , e 6 are edges. Definiti onThe degree of the vertex v in a graph G is the number of the edges incident with V. The degree of the vertex v is denoted by deg(v) or d (v).The minimu m degree and the maximu m degree of a graph of G are usually denoted by special symbol & (G) and  (G) respectively. Example deg (V 1 ) = 3 deg (V 4 ) = 4