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