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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.

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.

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.