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In this paper, we study the optimization problem (P) of minimizing a convex function over a constraint set with nonconvex constraint functions. We do this by given new characterizations of Robinson's constraint qualification, which reduces to the combination of generalized Slater's condition and generalized sharpened nondegeneracy condition for nonconvex programming problems with nearly convex feasible sets at a reference point. Next, using a version of the strong CHIP, we present a constraint qualification which is necessary for optimality of the problem (P). Finally, using new characterizations of Robinson's constraint qualification, we give necessary and sufficient conditions for optimality of the problem (P). [ABSTRACT FROM AUTHOR]

In this paper, we proved two new Riemann–Liouville fractional Hermite–Hadamard type inequalities for harmonically convex functions using the left and right fractional integrals independently. Also, we have two new Riemann–Liouville fractional trapezoidal type identities for differentiable functions. Using these identities, we obtained some new trapezoidal type inequalities for harmonically convex functions. Our results generalize the results given by İşcan (Hacet J Math Stat 46(6):935–942, 2014). [ABSTRACT FROM AUTHOR]

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In this paper, we study the partial differential equation Here $$\Delta _\alpha =-(-\Delta )^{\alpha /2}$$ , $$0<\alpha <2$$ , is the fractional Laplacian, $$k,h:[0,\infty )\rightarrow [0,\infty )$$ are continuous functions and $$\varphi :\mathbb {R}\rightarrow [0,\infty )$$ is a convex differentiable function. If $$0\le u_0\in C_b(\mathbb {R}^d)\cap L^1(\mathbb {R}^d)$$ we prove that (1) has a non-negative classical global solution. Imposing some restrictions on the parameters we prove that the mass $$M(t)=\int _{\mathbb {R}^d}u(t,x)\mathrm{d}x$$ , $$t>0$$ , of the system u does not vanish in finite time, moreover we see that $$\lim _{t\rightarrow \infty }M(t)>0$$ , under the restriction $$\int _0^\infty h(s)\mathrm{d}s<\infty $$ . A comparison result is also obtained for non-negative solutions, and as an application we get a better condition when $$\varphi $$ is a power function. [ABSTRACT FROM AUTHOR]