The clue to voltage is in it's proper name: Potential Difference.
Voltage isn't a real thing, it's just the difference between two things.
Imagine, if you will, a tank of water. Say it's 1m on each side, and you fill it to a depth of 10cm. You lift the tank up off the ground to a height of 1m. So the top of the water is 1.1m above the ground.
Now you make a hole in the bottom of the tank. The water flows out of the hole.
Until you had made that hole (created a circuit) the difference in heights still existed.
Now imagine the same scenario, but you instead fill the tank right to the top. The top of the water is now 2m above the ground. You have a greater potential difference between the top of the water and the ground.
Make the same hole, and the water comes out with more force due to the increased weight of the water in the tank - the greater potential difference causing greater current to flow.
First, please don't try "supply fight" shenanigans in practice if you don't know what to expect.
Assuming both sources are exactly 10V, the current capacity wouldn't matter. Just use Kirchoff's circuit law as usual: 10V on one source, -10V on the other, so 0V on the resistive load. V=RI=0, I=0, no current.
Current capacity would matter if:
1) There was some voltage difference between the sources;
2) This voltage difference causes enough current to flow through the circuit so as to overload the current capacity of one or both sources.
In practice, there are probably no two sources that produce the exact same voltage, so condition 1 will always be true. However, condition 2 rarely will be satisfied: you need a large I, so you need a large V/R. If both sources are rated as 10V, they probably shouldn't differ much, so V will be small. And R will be at least the sum of the output impedances of the voltage sources.
If you actually want to explore how the current limitations would affect the outcome, you should consider what happens when 1 and 2 are true. I'd suggest assuming that one voltage source is +10V/10A and the other is, say, -9V/100A, and R is sufficiently tiny (say 0.05 ohms).
Best Answer
This cannot be analysed because you have an anomaly where the two induced voltages (net effect zero volts) is placed across a voltage source of 10 volts. This will cause infinite current to flow and makes it non-analysable.
Even if the two induced voltages were additive or anything that resulted in a net voltage other than 10V it produces infinite current.
With a voltage source (as shown) across a 100 ohm resistor, this part of the circuit can be taken in isolation and the current will be 0.2 amps.