In this paper, we study the local existence of weak solutions for parabolic problem involving the fractional p-Laplacian. Our technique is based on the Galerkin method combined with the theory of Young measures. In addition, an example is given to illustrate the main results.

Given S a semigroup. We study two Pexider-type functional equations
f(xy) + g(xy) = f(x) + f(y) + g(x)g(y),   x,y ∈ S,
and
∫_Sf(xyt)dμ(t) + ∫_Sg(xyt)dμ(t) = f(x) + f(y) + g(x)g(y),   x,y ∈ S,
for unknown functions f and g mapping S into ℂ, where μ is a linear combination of Dirac measures (δ_{z_i}})_i∈I for some fixed elements (z_i)_i∈I contained in S such that ∫_Sdμ(t) = 1.
The main goal of this paper is to solve the above two functional equations and examine whether or not they are equivalent to the systems of equations
f(xy) = f(x) + f(y),  g(xy) = g(x)g(y),   x,y ∈ S,
and
∫_Sf(xyt)dμ(t) = f(x) + f(y),  ∫_Sg(xyt)dμ(t) = g(x)g(y),   x,y ∈ S,
respectively.

Let S be a semigroup, let (H, +) be a uniquely 2-divisible, abelian group and let ϕ, ψ be two endomorphisms of S that need not be involutive. In this paper, we express the solutions f : S→H of the following quadratic functional equation
f(xϕ(y)) + f(ψ(y)x) = 2f(x) + 2f(y),   x,y ∈ S,
in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation. Some applications of this result are presented.

In this paper we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations
-div[A(x,u,∇u)ω₁ + B(x,u,∇u)ν₁] + ????(x,u,∇u)ν₂ + |u|^p-2uω₂ - Σ_i,j=1ⁿD_j(a_ij(x)D_iu(x)) = f₀(x) - Σ_j=1ⁿD_jf_j(x)   in Ω,
u(x) = 0   on ∂Ω,
in the setting of the weighted Sobolev spaces.

Let S be a semigroup, ℍ be the skew field of quaternions, and ψ: S→S be an anti-endomorphism. We determine the general solution of the functional equation
g(xy) - g(xψ(y)) = 2g(x)g(y),    x,y ∈ S,
where g: S→ℂ is the unknown function. And when S = M is a monoid, we solve the functional equation
g(xy) + g(xψ(y)) = 2g(x)g(y),   x,y ∈ M,
where g: M→ℍ is the unknown function.

The sequence spaces ru`∞(O, ∇q), ru`p(O, ∇q), ruc(O, ∇q), ruc0(O, ∇q), rumφ(O, ∇q, p), runφ(O, ∇q, p), rumφ(O, ∇q), runφ(O, ∇q) are defined by the Orlicz function in this article. We examine all of its characteristics, including symmetry, solidity, and completeness. A few geometric properties on convexity on the space rumφ(O, ∇q, p) are also examined in this article.

In this paper, the almost everywhere convergence of Cesàro means of Walsh–Kaczmarz–Fourier series in a varying parameter setting is investigated. In particular, we define subsequence ℕ_{α_n,q} of natural numbers and prove that the maximal operator
sup_{n∈ℕ_{α_n,q}}|σ_n^α_nf|
is of strong type (H¹,L¹), where H¹ is a Hardy space.

In this study, we conduct a literature review on normed linear spaces whose strengths are between rotundity and uniform rotundity. In this discourse, we also explore inter-relationships and juxtapositions between the subjects under consideration. There has been some discussion on the extent to which the geometry of the factor spaces has an impact on the geometry of the product spaces, as well as the degree to which the quotient spaces and subspaces inherit the geometry of the space itself. A comprehensive review has been conducted on the applications of most of these rotundities to some fields within the realm of approximation theory. In addition, some open problems are enumerated in the paper.

In this paper, by Schauder's fixed point theorem and the Banach contraction principle, we consider the existence, uniqueness, and stability of convex solutions of a nonhomogeneous iterative functional differential equation. Finally, some examples were considered by our results.

In this work we focus on a dynamical system with jumps, where the intensity of the jumps depends on the system’s state. By verifying the assumptions of the theorem from [4], we show that our model satisfies the central limit theorem.