arXiv subjects

Dhruba R. Adhikari

Publications and source records attributed to Dhruba R. Adhikari.

3 recordsLinked to original sources

arXiv2023

Existence Results for Multivalued Compact Perturbations of ${m}$-Accretive Operators

Let $X$ be a real Banach space with its dual $X^*$ and $G$ be a nonempty, bounded and open subset of $X$ with $0\in G$. Let $T: X\supset D(T)\to 2^{X}$ be an $m$-accretive operator with $0\in D(T)$ and $0\in T(0)$, and let $C$ be a compact operator from $X$ into $X$ with $D(T)\subset D(C)$. We prove that $f\in \overline{R(T)}+\overline{R(C)}$ if $C$ is multivalued and $f\in \overline{R(T+C)}$ if $C$ is single-valued, provided $Tx+Cx+\varepsilon x\not\ni f$ for all $x\in D(T)\cap \partial G$ and $\varepsilon >0.$ The surjectivity of $T+C$ is proved if $T$ is expansive and $T+C$ is weakly coercive. Analogous results are given if $T$ has compact resolvents and $C$ is continuous and bounded. Various results by Kartsatos, and Kartsatos and Liu are improved, and a result by Morales is generalized.

math.FA↗

arXiv2022

Solvability of Inclusions Involving Perturbations of Positively Homogeneous Maximal Monotone Operators

Let $X$ be a real reflexive Banach space and $X^*$ be its dual space. Let $G_1$ and $G_2$ be open subsets of $X$ such that $\bar G_2\subset G_1$, $0\in G_2$, and $G_1$ is bounded. Let $L: X\supset D(L)\to X^*$ be a densely defined linear maximal monotone operator, $A:X\supset D(A)\to 2^{X^*}$ be a maximal monotone and positively homogeneous operator of degree $γ>0$, $C:X\supset D(C)\to X^*$ be a bounded demicontinuous operator of type $(S_+)$ w.r.t. $D(L)$, and $T:\bar G_1\to 2^{X^*}$ be a compact and upper-semicontinuous operator whose values are closed and convex sets in $X^*$. We first take $L=0$ and establish the existence of nonzero solutions of $Ax+ Cx+ Tx\ni 0$ in the set $G_1\setminus G_2.$ Secondly, we assume that $A$ is bounded and establish the existence of nonzero solutions of $Lx+Ax+Cx\ni 0$ in $G_1\setminus G_2.$ We remove the restrictions $γ\in (0, 1]$ for $Ax+ Cx+ Tx\ni 0$ and $γ= 1$ for $Lx+Ax+Cx\ni 0$ from such existing results in the literature. We also present applications to elliptic and parabolic partial differential equations in general divergence form satisfying Dirichlet boundary conditions.

math.FA↗

arXiv2022

Continuity of the Yosida Approximants Corresponding to General Duality Mappings

Let $X$ be a real locally uniformly convex Banach space and $X^*$ be the dual space of $X$. Let $φ:\mathbf R_+\to \mathbf R_+$ be a strictly increasing and continuous function such that $φ(0) = 0$, $φ(r) \to \infty$ as $r\to\infty$, and let $J_φ$ be the duality mapping corresponding to $φ$. We will prove that for every $R>0$ and every $x_0\in X$ there exists a nondecreasing function $ψ= ψ(R, x_0) :\mathbf R_+\to \mathbf R_+$ such that $ψ(0) = 0$, $ψ(r)>0$ for $r>0$, and $\langle x^*- x_0^*, x-x_0\rangle \ge ψ(\|x-x_0\|) \|x-x_0\|$ for all $x$ satisfying $\|x-x_0\|\le R$ and all $x^*\in J_φx$ and $x_0^*\in J_φx_0.$ This result extends the previous results of Prüss and Kartsatos who studied the normalized duality mapping $J$ (with $φ(r)=r$) for uniformly convex and locally uniformly Banach spaces, respectively. As an application of the above result, we give a concise proof of the continuity of the Yosida approximants $A_λ^φ$ and resolvents $J_λ^φ$ of a maximal monotone operator $A:X\supset X\to 2^{X^*}$ on $(0, \infty) \times X$ for an arbitrary $φ$ when $X$ is reflexive and both $X$ and $X^*$ are locally uniformly convex. In addition, we discuss pseudomonotone homotopy of the Yosida approximants $A_λ^φ$ with reference to the Browder degree.

math.FA↗