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We consider the following quasilinear Neumann boundary-value problem of the type $$ \begin {cases} -\displaystyle \sum _{i=1}^{N}\frac {\partial }{\partial x_{i}}a_{i}\Big (x,\frac {\partial u}{\partial x_{i}}\Big ) + b(x)|u|^{p_{0}(x)-2}u = f(x,u)+ g(x,u) &\text {in} \ \Omega , \\ \quad \dfrac {\partial u}{\partial \gamma } = 0 &\text {on} \ \partial \Omega . \end {cases} $$ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev...

We prove the existence of solutions to nonlinear parabolic problems of the following type: $$\left\{\begin{array}{cc}{\displaystyle \frac{\partial b\left(u\right)}{\partial t}}+A\left(u\right)=f+\mathrm{div}\left(\Theta (x;t;u)\right)\hfill & \text{in}Q,\hfill \\ u(x;t)=0\hfill & \text{on}\partial \Omega \times [0;T],\hfill \\ b\left(u\right)(t=0)=b\left(u_0\right)\hfill & \text{on}\Omega ,\hfill \end{array}\right.$$ where $b:ℝ\to ℝ$ is a strictly increasing function of class ${𝒞}^1$, the term $$A\left(u\right)=-\mathrm{div}\left(a\right(x,t,u,\nabla u\left)\right)$$ is an operator of Leray-Lions type which satisfies the classical Leray-Lions assumptions of Musielak type, $\Theta :\Omega \times [0;T]\times ℝ\to ℝ$ is a Carathéodory, noncoercive function which satisfies the following condition: ${sup}_{\left|s\right|\le k}\left|\Theta (·,·,s)\right|\in E_{\psi }\left(Q\right)$ for all $k>0$, where $\psi$ is the Musielak complementary function of $\Theta$, and the second term $f$ belongs to $L^1\left(Q\right)$.

In this paper we study the global existence of positive integrable solution for the nonlinear integral inclusion of fractional order $$x\left(t\right)\in p\left(t\right)+I^{\alpha }{F}_1(t,I^{\beta }{f}_2(t,x\left(\varphi \left(t\right)\right))),t\in (0,1).$$ As an application the global existence of the solution for the initial-value problem of the arbitrary (fractional) orders differential inclusion $$\frac{dx\left(t\right)}{dt}\in p\left(t\right)+I_1(t,D^{(}t)\left)\right),\text{a.e.}tgt0$$ will be studied.

The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let $\mathscr{H}$ be a Hilbert space and let $A$ be a positive bounded operator on $\mathscr{H}$. The semi-inner product ${\langle h\mid k\rangle }_A:=\langle Ah\mid k\rangle$, $h,k\in \mathscr{H}$, induces a semi-norm ${\parallel ·\parallel }_A$. This makes $\mathscr{H}$ into a semi-Hilbertian space. An operator $T\in {ℬ}_A\left(\mathscr{H}\right)$ is said to be $(n,m)$-$A$-normal if $[T^n,{\left(T^{{♯}_A}\right)}^m]:=T^n{\left(T^{{♯}_A}\right)}^m-{\left(T^{{♯}_A}\right)}^m{T}^n=0$ for some positive integers $n$ and $m$.