Nonlocal differential equations with concave coefficients of convolution type

Nonlocal differential equations with (p,q)$ {(p,q)}$ growth

“…Remark As we have mentioned in our recent work (for example, [27, Remark 2.7]), the invocation of Lemma 2.4 requires that $$T$$ map all of $$\overline{\widehat{V}_{\rho _2}}$$ into $$\mathcal {K}$$. However, the method we used in the proof of Theorem 2.3 is justified (see, for example, Granas and Dugundji [40, Theorem 2.5] or Infante [42]).…”

“…In a series of recent papers Goodrich [27–32] together with Lizama [35] has greatly expanded the types of nonlocal problems from the model cases (1.4)–(1.5). More precisely, by using a specially tailored order cone, namely, $$$$\begin{equation*} \mathcal {K}:=\Big \lbrace u\in \mathcal {C}{\left([0,1]\right)}\ : \ u\geqslant 0\text{ and }(a*u)(1)\geqslant C_0\Vert u\Vert _{\infty }\Big \rbrace ,\nonumber \end{equation*}$$$$for a positive constant $$C_0$$ to be defined in Section 2, the far more general problem (1.1) can be considered.…”

“…In a series of recent papers Goodrich [27][28][29][30][31][32] together with Lizama [35] has greatly expanded the types of nonlocal problems from the model cases (1.4)-(1.5). More precisely, by using a specially tailored order cone, namely,…”