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In this paper, a class of fractional integrodifferential equations of mixed type with time-varying generating operators and nonlocal conditions is considered. Using a contraction mapping principle and Krasnoselskii's fixed point theorem via Gronwall's inequailty, the existence and uniqueness of mild solution are given. The existence of optimal pairs of systems governed by fractional integrodifferential equations of mixed type with time-varying generating operators and nonlocal conditions is also presented.

Abstract. In this paper, we study the existence and uniqueness of solutions to the nonlocal problems for the fractional differential equation in Banach spaces. New sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculus and fixed point method under some suitable conditions. Two examples are given to illustrate the results.

Situated between the North China Craton to the east and the Tarim Craton to the west, the northern Alxa area in westernmost Inner Mongolia in China occupies a key location for interpreting the late-stage tectonic evolution of the southern Central Asian Orogenic Belt. New LA-ICP-MS zircon U–Pb dating results reveal 282.2 ± 3.9 Ma gabbros and 216.3 ± 3.2 Ma granites from the Yagan metamorphic core complex in northern Alxa, NW China. The gabbros are characterized by low contents of Si, Na, K, Ti and P and high contents of Mg, Ca, Al and Fe. These gabbros have arc geochemical signatures with relative enrichments in large ion lithophile elements and depletions in high field strength elements, as well as negative εNd(t) (−0.91 to −0.54) and positive εHf(t) (2.59 to 6.37) values. These features indicate that a depleted mantle magma source metasomatized by subduction fluids/melts and contaminated by crustal materials was involved in the processes of magma migration and emplacement. The granites show high-K calc-alkaline and metaluminous to weakly peraluminous affinities, similar to A-type granites. They have positive εNd(t) (1.55 to 1.99) and εHf(t) (5.03 to 7.64) values. These features suggest that the granites were derived from the mixing of mantle and crustal sources and formed in a postcollisional tectonic setting. Considering previous studies, we infer that the final closure of the Palaeo-Asian Ocean in the central part of the southern Central Asian Orogenic Belt occurred in late Permian to Early–Middle Triassic times.

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In this paper, zircon U–Pb geochronology, major and trace elements, and Sr–Nd isotope geochemistry of the Baiyanghe dolerites in northern West Junggar of NW China are presented. The U–Pb dating of zircons from the dolerites yielded ages of 272.2±4 Ma and 276.7±6.2 Ma, which indicate the emplacement times. The dolerites are characterized by minor variations in SiO2(46.89 to 49.07 wt%), high contents of Al2O3(13.60 to 13.92 wt%) and total Fe2O3(11.14 to 11.70 wt%), and low contents of MgO (2.67 to 3.64 wt%) and total alkalis (Na2O+K2O, 5.1 to 5.97 wt%, K2O/Na2O = 0.37–0.94), which indicate affinities to metaluminous tholeiite basalt. The REE pattern ((La/Sm)N= 2.25–2.34, (La/Yb)N= 7.42–8.36), V–Ti/1000 and 50*Zr–Ti/50–Sm discrimination diagrams show that these rocks are OIB-type. The high contents of Zr and Ti indicate a within-plate tectonic setting, and samples plot in the ‘plume source’ field shown on the Dy/Yb(N)versus Ce/Yb(N)diagram. The positive εNd(t) values (+7.09 to +7.48), high initial87Sr/86Sr ratios (0.70442 to 0.70682) and depletions of Nb and Ta elements in the samples can be explained by the involvement of subducted sediments. In summary, it is possible that the Baiyanghe dolerites were derived from an OIB-like mantle source and associated with a mantle plume tectonic setting. Therefore, our samples provide the youngest evidence for the existence of a mantle plume, which may provide new insights into the Late Palaeozoic tectonic setting of West Junggar.

This paper deals with complete controllability of systems governed by linear and semilinear conformable differential equations. By establishing conformable Gram criterion and rank criterion, we give sufficient and necessary conditions to examine that a linear conformable system is null completely controllable. Further, we apply Krasnoselskii’s fixed point theorem to derive a completely controllability result for a semilinear conformable system. Finally, three numerical examples are given to illustrate our theoretical results.