Let (X, [[sigma].sub.b]) be a b-metric-like space with parameter s [greater than or equal to] 1, and let f, g : X [right arrow] X and [alpha] : X x X [right arrow] [0, [infinity]) be given mappings and

arbitrary constant p such that p [greater than or equal to] 2.

(b) By the first equation of (7), we easily obtain the fact that [r.sub.1]u(1-u/K)-[rho][u.sup.[alpha]]v [less than or equal to] [r.sub.1]u(1-u/K) in [0, +[infinity])x[OMEGA]; the first result follows easily from the simple comparison argument for parabolic problems, and thus there exists T [member of] (0, +[infinity]) such that u(t, x) [less than or equal to] K + [epsilon] in [T, +[infinity]) x [OMEGA] for an

arbitrary constant [epsilon] > 0.

where [c.sub.0] is an

arbitrary constant. From (4), we have

For

arbitrary constant values, [[alpha].sub.2] = [[alpha].sub.1] = l, [[alpha].sub.7] = -1, [[alpha].sub.6] = 1, [[alpha].sub.10] = 0, [[alpha].sub.11] = 1, and [[alpha].sub.8] = 5.

Proof If f is a constant function, say f(x) = c for all x [member of] G, then using the functional equation (10) we have g(x) = c(2c- 1) for any

arbitrary constant c [member of] C.

Here, we note, for rigid body displacement field, that [u.sub.r] = a + b x r, where a and b are

arbitrary constant vectors, and, for constant electric potential, [[phi].sub.r] [tau]([u.sub.r], [[phi].sub.r]) = 0.

Here we propose [f.sub.2](T) = T + [epsilon][T.sup.2], where e is an

arbitrary constant. For this model, manipulation of (13)-(15) yields

with l [right arrow] -3p[mu]/(4[[pi].sup.2][p.sup.3] - 2[rho]), as [alpha] = [e.sup.[pi]i[tau]] [right arrow] 0, where [[??].sub.0] is an

arbitrary constant.

After integration we get [[lambda].sub.2]g' = [c.sup.2][phi](g') + [c.sub.1], where [c.sub.1] is an

arbitrary constant. Suppose that [lim.sub.x[right arrow][infinity]]([partial derivative]u/[partial derivative]x)(x, 0) = A.

Optical band gap is denoted as Eg for the crystal and 'A' is the

arbitrary constant. Plank's constant is denoted by 'h' and frequency of incident photons is denoted by 'Y'.

where m is an

arbitrary constant. These values of (eq.) and (eq.) are define the famous Schwarzschild spacetime.

Later, inhomogeneous ODE (17) is solved uniformly in all three cases using method of

arbitrary constant variation [5].